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abstract: 'We present here a new model and algorithm which performs an efficient Natural gradient descent for Multilayer Perceptrons. Natural gradient descent was originally proposed from a point of view of information geometry, and it performs the steepest descent updates on manifolds in a Riemannian space. In particular, we extend an approach taken by the “Whitened neural networks” model. We make the whitening process not only in feed-forward direction as in the original model, but also in the back-propagation phase. Its efficacy is shown by an application of this “Bidirectional whitened neural networks” model to a handwritten character recognition data (MNIST data).'
author:
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Yuki Fujimoto$^{\ast}$ and Toru Ohira$^{\ast\ast}$\
\
\
bibliography:
- 'bwnn.bib'
title: A Neural Network model with Bidirectional Whitening
---
Introduction
============
Interests for developing and efficient learning algorithm for multilayer neural networks have grown rapidly due to recent upheaval of the deep learning and other machine learnings. Natural gradient descent(NGD) is considered as one of the strong methods. It was proposed from a point of view of information geometry[@amari1998natural], where neural networks are considered as manifolds in a Riemannian space with a measure given by the Fisher information matrix (FIM). Then, the learning process can be interpreted as an optimization problem of a function in a Riemannian space. The idea of applying the NGD to multilayer neural networks was initiated by Amari. Recently, it has regained interests from machine learning researchers[@martens2014new; @pascanu2013revisiting].
However, difficulty exists for using the NGD: the computational costs of estimating the FIM and obtaining its inverse is high. Much attention and research efforts have gone into solving this difficulty[@NIPS2015_5953; @martens2015optimizing; @park2000adaptive; @ioffe2015batch; @NIPS2016_6114].
In this paper, we will focus on one of such approaches, and extend the work of [@NIPS2015_5953]. In their approach “Whitened neural networks” model was proposed. There, a neural network architecture, whose FIM is closer to the identity matrix with less computational demands, is explored. Extra neurons and connections are added to achieve this whitening approximation. In particular, they have used this scheme for the forward direction of inputs to neurons and achieved lower computational costs.
Our main proposal in this paper is to further push the approximation of the FIM being closer to the identity by implementing the whitening process also in the back-propagation phase. This model, which we term as the “bidirectional whitened neural networks” model, will be described in the following. Its efficacy is also shown through its application to a handwritten character recognition data (MNIST data).
Multilayer Perceptron and Natural Gradient Descent
==================================================
We present here a brief review of the Multilayer Perceptron and the Natural Gradient Descent, which we focus in this paper. The first level of approximation for the FIM is also discussed.
Multilayer Perceptron {#sec: multilayer perceptron}
---------------------
Multilayer Perceptron is a model of neural networks which has feed-forward structure with no recurrent loops. They have multiple layers called input, hidden, and output, and neurons have all to all connections between successive layers. Let us consider a $N$ layer Perceptron, and set the values of the input as $\bm{z}^{(0)} = \bm{x}$, the hidden layer values as $\bm{z}^{(i)} = \bm{h}^{(i)}$, ($1 \leq i \leq N - 1$), and the output of the entire network as $\bm{z}^{(N)} = f(\bm{x}; \bm{w})$.
This $f(\bm{x}; \bm{w})$ can be viewed as a function of $\bm{x}$ by fixing the parameters $\bm{w}$, and thus called as a “multilayer Perceptron function”. The rules of computing the value of the $i$ layer from the $i - 1$ in the network is given as follows ($1 \leq i \leq N$). $$\begin{aligned}
\label{eq: forward propagation1}
\bm{a}^{(i)} &=& W^{(i)}\bm{z}^{(i-1)} + \bm{b}^{(i)} \\
\label{eq: forward propagation2}
&=& \bar{W}^{(i)}\bar{\bm{z}}^{(i-1)} \\
\label{eq: forward propagation3}
\bm{z}^{(i)} &=& \bm{\phi}^{(i)}(\bm{a}^{(i)})\end{aligned}$$ Here, $\bm{\phi}^{(i)}(\cdot)$is an activation function applied to each element of $\bm{a}$. Typically, the sigmoid function or $ReLU$ function are used for this activation function. Also, is a shortened notation by setting $\bar{W}^{(i)} \equiv (\bm{b}^{(i)}, W^{(i)}), \bar{\bm{z}}^{(i)} \equiv (1, \bm{z}^{(i)^{T}})^{T}$.
Hence, the multilayer Perceptron function (MPF) is defined by setting
$\{(W^{(i)}, \bm{b}^{(i)})\}$. It is often convenient to denote these parameters by $\bm{w}$, defined by $$\begin{aligned}
\label{eq: vectolization}
\bm{w} \equiv (\mathrm{vec}(\bar{W}^{(1)})^{T}, \ldots, \mathrm{vec}(\bar{W}^{(N)})^{T})^{T}
$$ where $\mathrm{vec}(A)$ means a compound vector of column vectors of a matrix $A$
The learning process of multilayer Perceptrons is an optimization problem set by the following statistical inference. The training data of input and output pairs is given as $D \equiv \{(\bm{x}_{k}, \bm{y}_{k})\}_{k=1}^{K}$. We assume this data set is generated by the same joint distribution $Q(X, Y)$ independently. In order to estimate this input output probabilistic relations, a statistical model $\{p(\bm{x}, \bm{y}; \bm{w})\}_{\bm{w} \in \Theta}$ is considered using the MPF. Here $p(\bm{x}, \bm{y}; \bm{w})$ is a joint probability density function and $\Theta \subset \mathbb{R}^{M}$ is a set of parameters. The problem is to find the parameter $\bm{w}$ which makes $p(\bm{x}, \bm{y}; \bm{w})$ as a best estimate of $Q(X, Y)$. The maximum likelihood method is employed to obtain such $\bm{w^*}$. $$\begin{aligned}
\bm{w}^{\ast} &\equiv& \operatorname*{arg\, max}_{\bm{w} \in \Theta} \prod_{k=1}^{K} p(\bm{x}_{k}, \bm{y}_{k}; \bm{w})\end{aligned}$$ It is known that this estimation is the same as the following minimization problem. $$\begin{aligned}
\label{eq: MLP Learning1}
\bm{w}^{\ast} &\equiv& \operatorname*{arg\, min}_{\bm{w} \in \Theta} \sum_{k=1}^{K} -\log p(\bm{x}_{k}, \bm{y}_{k}; \bm{w}) \\
\label{eq: MLP Learning3}
&=& \operatorname*{arg\, min}_{\bm{w} \in \Theta} M(\bm{w})\end{aligned}$$ Here, we have set the target function to minimize as $M(\bm{w})$. Research on efficient algorithms for this optimization problem is the central issue in the following.
Natural Gradient Method {#sec: natural gradient}
-----------------------
Natural Gradient Method is a steepest descent method in a Riemannian space. It is proposed from the information geometry where statistical models are manifolds in a Riemannian space with a metric of the Fisher Information Matrices[@amari2007methods]. Thus, we can view the learning by the multilayer Perceptrons as an optimization problem in a Riemannian space as presented in \[sec: multilayer perceptron\].
Let us start by defining the Fisher information matrix and the Natural Gradient Descent.
[**Definition: Fisher Information Matrix**]{}
We set $l(\bm{x} ; \bm{w}) \equiv \log p(\bm{x};\bm{w})$. For $\bm{w} \in \Theta$, a square matrix $G(\bm{w}) = (g_{ij} (\bm{w}))$ is defined as follows. $$\begin{aligned}
\label{eq:fisher information matrix}
G(\bm{w}) \equiv E \left[\nabla l(X; \bm{w}) \nabla l(X; \bm{w})^{T}\right]
$$ can be expressed by each elements as, $$\begin{aligned}
g_{ij}(\bm{w}) = E \left[ \dfrac{\partial l}{\partial w_{i}} (X ; \bm{w}) \dfrac{\partial l}{\partial w_{j}} (X ; \bm{w}) \right]
= \int \dfrac{\partial l}{\partial w_{i}} (\bm{x} ; \bm{w}) \dfrac{\partial l}{\partial w_{j}} (\bm{x};\bm{w}) p(\bm{x}; \bm{w})d\bm{x}
$$ We call this matrix $G$ the Fisher information matrix (FIM).
[**[Definition: Natural Gradient Descent]{}**]{}
We call the following gradient method as the Natural Gradient Descent (NGD). $$\begin{aligned}
\label{eq: natural gradient descent}
\bm{w}(t + 1) \!=\! \bm{w}(t) \!-\! \eta(t) G^{-1}(\bm{w}(t)) \nabla M(\bm{w}(t))\end{aligned}$$ Here $\eta(t)$ is a rate of the learning.
Then, $-G^{-1}(\bm{w}(t)) \nabla M(\bm{w}(t))$ is the direction of the maximal decrease of the target function $M$ given a fixed step size. We note that this NGD reduces to the ordinary gradient descent, when $G$ is the identity matrix.
Approximation of the Fisher Information Matrix {#sec: approximation of FIM}
----------------------------------------------
As discussed in the previous section, the FIM and its inverse play important roles in the calculation in the NGD. We, thus, present a preliminary approximation of the FIM in order to lessen the computational burdens[@martens2015optimizing].
Let us first compute the FIM for the multilayer Perceptrons. The probability density function associated with the multilayer Perceproton function (MPF) is given as follows. $$\begin{aligned}
p(\bm{x}, \bm{y}; \bm{w}) &=& p(\bm{y} | f(\bm{x}; \bm{w})) p(\bm{x})\end{aligned}$$ Also, the gradient vector are written concisely as in , $$\begin{aligned}
\dfrac{\partial l}{\partial \bm{w}} \equiv \left(\mathrm{vec}\left(\dfrac{\partial l}{\partial \bar{W}^{(1)}}\right)^{T}, \ldots, \mathrm{vec}\left(\dfrac{\partial l}{\partial \bar{W}^{(N)}}\right)^{T}\right)^{T}\end{aligned}$$ Then, the FIM for the MLP is given as follows. $$\begin{aligned}
\label{eq: neural fisher information matrix}
G(\bm{w}) &=& \begin{pmatrix}
G_{1, 1} & G_{1, 2} & \cdots & G_{1, N} \\
G_{2, 1} & G_{2, 2} & \cdots & G_{2, N} \\
\vdots & \vdots & \ddots & \vdots \\
G_{N, 1} & G_{N, 2} & \cdots & G_{N, N} \\
\end{pmatrix} \\
\label{eq: neural fisher information matrix2}
G_{i, j} &\equiv& E\left[ \mathrm{vec}\left(\dfrac{\partial l}{\partial \bar{W}^{(i)}}\right) \mathrm{vec}\left(\dfrac{\partial l}{\partial \bar{W}^{(j)}}\right)^{T} \right]\end{aligned}$$ Hence, the FIM for the MLP is composed of the block matrices $G_{i, j}$.
If we further set $\delta^{(i)}_{j} = \dfrac{\partial l}{\partial a^{(i)}_{j}}$, the following is obtained. $$\begin{aligned}
\label{eq: loss gradient}
\dfrac{\partial l}{\partial \bar{W}^{(i)}} = \bm{\delta}^{(i)}\bar{\bm{z}}^{(i-1)^{T}}\end{aligned}$$ By putting into , the $G_{i, j}$ can now be expressed as $$\begin{aligned}
G_{i, j}
&=& E\left[
\bar{\bm{z}}^{(i-1)} \bar{\bm{z}}^{(j-1)^{T}} \otimes \bm{\delta}^{(i)}\bm{\delta}^{(j)^{T}}
\right]\end{aligned}$$ (Here, $\otimes$ is the Kronecker product. )
For the efficient computation, it is essential to approximate this FIM. The preliminary approximation consists of two steps.
The first step approximation of $G_{i, j}$ is given as $\tilde{G}_{i, j}$ which is defined as follows. $$\begin{aligned}
G_{i, j}
\label{eq: K-FAC approximation}
&\approx& E\left[\bar{\bm{z}}^{(i-1)}\bar{\bm{z}}^{(j-1)^{T}}\right] \otimes
E\left[\bm{\delta}^{(i)}\bm{\delta}^{(j)^{T}}\right] \\
&\equiv& \bar{Z}_{i-1, j-1} \otimes D_{i, j} \nonumber \\
&\equiv& \tilde{G}_{i, j} \nonumber\end{aligned}$$ This approximation means that we are inter-changing the expectation of the Kronecker products with the Kronecker products of the expectations. The matrix $\tilde{G}$, whose elements are given by replacing $G_{i, j}$ of with $\tilde{G}_{i, j}$, is the first step approximation of the FIM. We note that the FIM is decomposed into two parts by this approximation: $\bar{\bm{z}}^{(i-1)}$ (the feed-forward phase part) and $\bm{\delta}^{(i)}$ (the back-propagtaing phase part).
We perform the second step approximation on $\tilde{G}$ to obtain $\breve{G}$. $$\begin{aligned}
\label{eq: block diagonalization}
\breve{G} &\equiv& {\mathop{\mathrm{diag}}\nolimits}(\tilde{G}_{1, 1}, \tilde{G}_{2, 2}, \ldots, \tilde{G}_{N, N})\end{aligned}$$ Here, ${\mathop{\mathrm{diag}}\nolimits}(\cdots)$ denotes a block diagonal matrix, whose non-zero diagonals are given by the elements. In other words, $\breve{G}$ is obtained from $\tilde{G}$ by setting non-diagonal elements as the zero matrix, $$\begin{aligned}
\tilde{G}_{i, j} = O \quad (i \neq j)\end{aligned}$$ This approximation allows us to compute the FIM layer by layer independently.
Whitened Neural Networks {#sec: whitening neural networks}
========================
In this section, we present algorithms which aim to perform Natural Gradient Descent efficiently with the approximated FIM, $\breve{G}$.
Natural Gradient Descent by Whitening {#sec: activation whitening}
-------------------------------------
Let us first describe Whitened Neural Networks[@NIPS2015_5953]. The main idea of this method is to perform the NGD by reconfiguring the network and parameters, so that the FIM becomes closer to the identity matrix. When the FIM is the identity matrix, the NGD is the same as the ordinary gradient descent, thus can be implemented simply with less computational costs.
### Whitened Neural Network
The architecture of the Whitened Neural Networks (WNN) is obtained by changing through into the following form. $$\begin{aligned}
\label{eq: whitening forward prop1}
\bm{z}^{\dagger^{(i-1)}} &=& U^{(i-1)}(\bm{z}^{(i-1)} - \bm{c}^{(i-1)}) \\
\label{eq: whitening forward prop2}
\bm{a}^{(i)} &=& W^{\dagger^{(i)}} \bm{z}^{\dagger^{(i-1)}} + \bm{b}^{\dagger^{(i)}} \\
\label{eq: whitening forward prop3}
\bm{z}^{(i)} &=& \bm{\phi}^{(i)} \left( \bm{a}^{(i)} \right)\end{aligned}$$ Here $\{(U^{(i-1)}, \bm{c}^{(i-1)})\}$ are the new parameters introduced as “Whitening” parameters. $\{(W^{\dagger^{(i)}}, \bm{b}^{\dagger^{(i)}})\}$ are the new model parameters associated with this new architecture. These are the ones which we want to estimate and update using gradient descent methods as in the normal multilayer Perceptrons.
We present in the Figure \[fig:Whitening Neural Networks\] the new architecture defined by , , . It shows the $i-1$th layer to the $i$th layer. We note the gray layer in the Figure \[fig:Whitening Neural Networks\] is the new inserted layer for the purpose of “whitening”. This change of network configuration is the essence of WNN.
![Architecture of Whitened Neural Networks[]{data-label="fig:Whitening Neural Networks"}](wnn.eps){width="10cm"}
From , the approximated FIM $\tilde{G}_{i, i}$ in the WNN, then, is expressed as the following. $$\begin{aligned}
\label{eq: fisher approx}
\tilde{G}_{i, i} = E\left[\bar{\bm{z}}^{\dagger^{(i-1)}}\bar{\bm{z}}^{\dagger^{(i-1)^{T}}}\right] \otimes
E\left[\bm{\delta}^{(i)}\bm{\delta}^{(i)^{T}}\right]\end{aligned}$$ The essential idea of the whitening is to make $\breve{G}$ closer to the identity by defining the whitening parameters $\{(U^{(i-1)}, \bm{c}^{(i-1)})\}$ as $$\begin{aligned}
\label{eq: whitening condition}
E\left[\bar{\bm{z}}^{\dagger^{(i-1)}}\bar{\bm{z}}^{\dagger^{(i-1)^{T}}}\right] = I\end{aligned}$$ for each $i$ and performs the gradient descent. (Our idea, which will be described later in \[sec: gradient whitening\], is to further extend the whitening to the latter factor $E\left[\bm{\delta}^{(i)}\bm{\delta}^{(i)^{T}}\right]$ in )
### Updating of the Whitening Parameters
We calculate here explicitly $\{(U^{(i-1)}, \bm{c}^{(i-1)})\}$, which satisfies the condition .
As $\bar{\bm{z}}^{\dagger^{(i-1)}} = (1, \bm{z}^{\dagger^{(i-1)^{T}}})^{T}$, can be decomposed into $$\begin{aligned}
\label{eq: whitening condition2}
\begin{pmatrix}
1 & E\left[\bm{z}^{\dagger^{(i-1)^{T}}} \right] \\
E\left[\bm{z}^{\dagger^{(i-1)}}\right] & E\left[\bm{z}^{\dagger^{(i-1)}}\bm{z}^{\dagger^{(i-1)^{T}}}\right]
\end{pmatrix}
= I\end{aligned}$$ Thus, $$\begin{aligned}
\label{eq: whitening condition3}
E\left[\bm{z}^{\dagger^{(i-1)}}\right] &=& \bm{0} \\
\label{eq: whitening condition4}
E\left[\bm{z}^{\dagger^{(i-1)}}\bm{z}^{\dagger^{(i-1)^{T}}}\right] &=& I\end{aligned}$$ are required to satisfy this condition.
Let us look at these conditions. can be satisfied by $$\begin{aligned}
\label{eq: update c}
\bm{c}^{(i-1)} &\leftarrow& E\left[\bm{z}^{(i-1)}\right]\end{aligned}$$ Also, for , we first set the matrix $\check{Z}_{i-1, i-1}$ by the following $$\begin{aligned}
\check{Z}_{i-1, i-1}
\equiv E\left[(\bm{z}^{(i-1)} \! - \! \bm{c}^{(i-1)})(\bm{z}^{(i-1)} \! - \! \bm{c}^{(i-1)})^{T}\right]\end{aligned}$$ Then, becomes $$\begin{aligned}
\label{eq: whitening condition6}
E\left[ \bm{z}^{\dagger^{(i-1)}} \! \bm{z}^{\dagger^{(i-1)^{T}}} \right]
= U^{(i-1)} \check{Z}_{i-1, i-1} U^{(i-1)^{T}}
= I\end{aligned}$$ Because $\check{Z}_{i-1, i-1}$ is a symmetric matrix, there exists a orthogonal matrix $P$, which makes it diagonal. $$\begin{aligned}
\check{Z}_{i-1, i-1} = P \Lambda P^{T}\end{aligned}$$ Here $\Lambda$ is the diagonalized matrix. Then, if we set $$\begin{aligned}
\label{eq: update U}
U^{(i-1)} &\leftarrow& (\Lambda + \varepsilon I)^{-\frac{1}{2}} \cdot P^{T}\end{aligned}$$ the condition is approximately satisfied. (Here, $\varepsilon$ is a small positive constant to avoid division by zero. )
By this process, called the whitening process, according to and , we update the whitening parameters satisfying . We note that, in this updating, the calculation of $\bm{z}^{(i-1)}$ in feed-forward phase is essential.
### Updating of the model parameters
We now turn our attention to the updating of the model parameters $\{(W^{\dagger^{(i)}}, \bm{b}^{\dagger^{(i)}})\}$. We need to pay attention so that the inclusion of the whitening process and the associated layer does not change the value of the multilayer Perceptron function (MPF) itself. In concrete, we need to do the following. Let us assume the whitening parameters $\{(U^{(i-1)}, \bm{c}^{(i-1)})\}$ are updated to $\{(U^{(i-1)}_{new}, \bm{c}^{(i-1)}_{new})\}$. We want to keep the value of unchanged by this updating. This places a constrains in the way we update the model parameters $\{(W^{\dagger^{(i)}}_{new}, \bm{b}^{\dagger^{(i)}}_{new})\}$. Namely, for any value of $\bm{z}^{(i-1)}$, the following must be satisfied. $$\begin{aligned}
{W^{\dagger^{(i)}}U^{(i-1)}(\bm{z}^{(i-1)} - \bm{c}^{(i-1)}) + \bm{b}^{\dagger^{(i)}}
} = W^{\dagger^{(i)}}_{new}U^{(i-1)}_{new}(\bm{z}^{(i-1)} - \bm{c}^{(i-1)}_{new}) + \bm{b}^{\dagger^{(i)}}_{new}\end{aligned}$$ We can obtain the following by solving these equations. $$\begin{aligned}
\label{eq: model param update1}
W^{\dagger^{(i)}}_{new} &=& W^{\dagger^{(i)}}U^{(i-1)}U^{(i-1)^{-1}}_{new}\\
\label{eq: model param update2}
\bm{b}^{\dagger^{(i)}}_{new} &=& \bm{b}^{\dagger^{(i)}} - W^{\dagger^{(i)}}U^{(i-1)}\bm{c}^{(i-1)} + W^{\dagger^{(i)}}_{new}U^{(i-1)}_{new}\bm{c}^{(i-1)}_{new}\end{aligned}$$ By putting together and , we can set $\{(W^{(i)}, \bm{b}^{(i)})\}$ as $$\begin{aligned}
W^{(i)} &=& W^{\dagger^{(i)}}U^{(i-1)} \\
\bm{b}^{(i)} &=& \bm{b}^{\dagger^{(i)}} - W^{\dagger^{(i)}}U^{(i-1)}\bm{c}^{(i-1)}\end{aligned}$$ Using these $\{(W^{(i)}, \bm{b}^{(i)})\}$, we can re-write and as $$\begin{aligned}
\label{eq: update W}
W^{\dagger^{(i)}} &\leftarrow& W^{(i)}U^{(i-1)^{-1}}_{new}\\
\label{eq: update b}
\bm{b}^{\dagger^{(i)}} &\leftarrow& \bm{b}^{(i)} + W^{(i)}\bm{c}^{(i-1)}_{new}\end{aligned}$$ Thus, we can keep MPF the same by updating whitening parameters first as in and and then update model parameters with and .
As we change model parameters, the values of $E[\bm{z}^{(i-1)}], \check{Z}_{i-1, i-1}$ changes, which in turn requires the update of the whitening parameters to keep the FIM close to the identity matrix. However, it is computationally expensive to update both set of parameters at every iterations. In particular, the update of the whitening parameters for a layer of $M$ neurons takes computation of the order of $O(M^{3})$. Thus, in actual implementations, the update of the whitening parameters are performed at certain fixed time intervals[@NIPS2015_5953], though this makes a gradual digression from the NGD for that time interval between the successive updating of the whitening parameters.
The method and algorithm described above is called “Projected Natural Gradient Descent”(PRONG)[@NIPS2015_5953], which is outlined in Algorithm \[alg: PRONG\].
training set $D$, initial parameter $\bm{w}(0)$ updating period of whitening parameters $\tau$ $\bullet$ $U^{(i)} \leftarrow I; \bm{c}^{(i)} \leftarrow \bm{0}; t \leftarrow 0$ $\bullet$ Computation of standard parameters $\{(W^{(i)}, \bm{b}^{(i)})\}$. $\bullet$ Estimations of $E[\bm{z}^{(i-1)}], \check{Z}_{i-1, i-1}$. $\bullet$ Updating of the Whitening parameters $\{(U^{(i-1)}, \bm{c}^{(i-1)})\}$. $\bullet$ Updating of the model parameters $\{(W^{\dagger^{(i)}}, \bm{b}^{\dagger^{(i)}})\}$. $\bullet$ Updating of $\{(W^{\dagger^{(i)}}, \bm{b}^{\dagger^{(i)}})\}$ by the ordinary gradient descent. $\bullet$ $t \leftarrow t + 1$
Extension of Whitening {#sec: gradient whitening}
----------------------
Here, we describe our proposal of the new extended whitening algorithms based on \[sec: activation whitening\].
In the whitening method described above, in order to keep the approximated FIM, $\breve{G}$, closer to the identity matrix, updating of the whitening parameters $\{(U^{(i)}, \bm{c}^{(i)})\}$ are performed. This makes the first factor $E\left[\bar{\bm{z}}^{\dagger^{(i-1)}}\bar{\bm{z}}^{\dagger^{(i-1)^{T}}}\right]$ in $$\begin{aligned}
\tilde{G}_{i, i} &=& E\left[\bar{\bm{z}}^{\dagger^{(i-1)}}\bar{\bm{z}}^{\dagger^{(i-1)^{T}}}\right] \otimes
E\left[\bm{\delta}^{(i)}\bm{\delta}^{(i)^{T}}\right]\end{aligned}$$ closer to the identity matrix.
The main idea of our method is to make the second factor $E\left[\bm{\delta}^{(i)}\bm{\delta}^{(i)^{T}}\right]$ toward the identity as well, so that $\tilde{G}_{i, i}$ is even better approximated by the identity matrix. This turns out that we implement whitening process not only in the feed-forward phase but also in the back-propagating phase.
### Bidirectional Whitened Neural Networks
In order to perform the back-whitening, we modify the forward-whitening process described by , and into the following. $$\begin{aligned}
\label{eq: biwhitening forward prop1}
\bm{z}^{\dagger^{(i-1)}} &=& U^{(i-1)}(\bm{z}^{(i-1)} - \bm{c}^{(i-1)}) \\
\label{eq: biwhitening forward prop2}
\bm{a}^{\dagger^{(i)}} &=& W^{\dagger^{(i)}} \bm{z}^{\dagger^{(i-1)}} + \bm{b}^{\dagger^{(i)}} \\
\label{eq: biwhitening forward prop3}
\bm{a}^{(i)} &=& R^{(i)^{T}} \bm{a}^{\dagger^{(i)}} \\
\label{eq: biwhitening forward prop4}
\bm{z}^{(i)} &=& \bm{\phi}^{(i)} \left( \bm{a}^{(i)} \right)\end{aligned}$$ Here, $\{R^{(i)^{T}}\}$ is a newly introduced parameter, called the back-whitening parameter.
We show, as in Figure \[fig:Whitening Neural Networks\], the architecture of this extended method defined by , , , in the Figure \[fig:Gradient\_Whitening\_Neural\_Networks\]. The dark gray part in the Figure \[fig:Gradient\_Whitening\_Neural\_Networks\] is the newly introduced layer to accommodate the back-whitening parameter $\{R^{(i)^{T}}\}$.
As mentioned above, this proposed method performs whitening process both in feed-forward and back-propagating phase. Thus, we call this new architecture as the Bidirectional Whitened Neural Networks (BWNN).
![Architecture of the Bidirectional Whitened Neural Networks[]{data-label="fig:Gradient_Whitening_Neural_Networks"}](bwnn.eps){width="10cm"}
We introduce a new parameter $\bm{\delta}^{\dagger^{(i)}}$ in place of $\bm{\delta}^{(i)}$ as in the following. $$\begin{aligned}
\label{eq: transformed delta}
\bm{\delta}^{\dagger^{(i)}} \equiv \dfrac{\partial l}{\partial \bm{a}^{\dagger^{(i)}}}\end{aligned}$$ Then, the approximation of $\tilde{G}_{i, i}$ is then expressed as $$\begin{aligned}
\tilde{G}_{i, i} = E\left[ \bar{\bm{z}}^{\dagger^{(i-1)}}\bar{\bm{z}}^{\dagger^{(i-1)^{T}}} \right] \otimes
E\left[ \bm{\delta}^{\dagger^{(i)}}\bm{\delta}^{\dagger^{(i)^{T}}} \right]\end{aligned}$$
In analogy with Section \[sec: activation whitening\], we will fix the back-whitening parameter $\{R^{(i)^{T}}\}$ so that $$\begin{aligned}
\label{eq: gradient whitening condition}
E\left[\bm{\delta}^{\dagger^{(i)}}\bm{\delta}^{\dagger^{(i)^{T}}}\right] = I\end{aligned}$$
### Updating of the back-whitening parameter
Let us explicitly find $\{R^{(i)^{T}}\}$ to satisfy . From , we have $$\begin{aligned}
\label{eq: delta relation}
\delta_{j}^{\dagger^{(i)}} = \sum_{k} \dfrac{\partial l}{\partial a_{k}^{(i)}} \dfrac{\partial a_{k}^{(i)}}{\partial a_{j}^{\dagger^{(i)}}} = \sum_{k} \delta_{k}^{(i)} r_{kj}^{(i)^{T}}\end{aligned}$$ Thus, $\bm{\delta}^{\dagger^{(i)}}$ is a linear transformation of $\bm{\delta}^{(i)}$, which can be written as $$\begin{aligned}
\label{eq: delta relation2}
\bm{\delta}^{\dagger^{(i)}} &=& R^{(i)}\bm{\delta}^{(i)}\end{aligned}$$ By inserting into , we obtain $$\begin{aligned}
E\left[\bm{\delta}^{\dagger^{(i)}}\bm{\delta}^{\dagger^{(i)^{T}}}\right]
= R^{(i)}D_{i, i}R^{(i)^{T}} = I\end{aligned}$$ Hence, in analogy with , $R^{(i)}$ which satisfies is given by the following $$\begin{aligned}
\label{eq: update R}
R^{(i)} \leftarrow (\Lambda + \varepsilon I)^{-\frac{1}{2}} \cdot P^{T}\end{aligned}$$ Here, $\Lambda, P$ are the diagonalized and the orthogonal matrices associated with $D_{i, i}$, and $\varepsilon$ is the small positive parameter to avoid a division by zero.
Altogether, as in the case of the forward-whitening parameters, is satisfied by updating of the back-whitening parameters according to , which, in turn, depends on the calculation of $\bm{\delta}^{(i)}$ in the back-propagating phase.
### Updating of the model parameters
As in the feed-forward phase, we update the model parameters $\{(W^{\dagger^{(i)}}, \bm{b}^{\dagger^{(i)}})\}$ so that the values of the multilayer Perceptron function are kept the same when the back-whitening parameters are updated.
In order to achieve this, the model parameters $\{(W^{\dagger^{(i)}}, \bm{b}^{\dagger^{(i)}})\}$ need to be updated as follows, given the back-whitening parameters are updated from $R^{(i)}$ to $R_{new}^{(i)}$. $$\begin{aligned}
W^{\dagger^{(i)}} &\leftarrow& (R^{(i)^{T}}_{new})^{-1}R^{(i)^{T}}W^{\dagger^{(i)}} \\
\bm{b}^{\dagger^{(i)}} &\leftarrow& (R^{(i)^{T}}_{new})^{-1}R^{(i)^{T}}\bm{b}^{\dagger^{(i)}}\end{aligned}$$
We will call the above algorithm as “Bidirectional Projected Natural Gradient Descent”(BPRONG) because it performs whitening both in feed-forward and back-propagaing phase. Its outline is shown in Algorithm \[alg: BiPRONG\]. Also, as in the forward-whitening, we can perform the back-whitening update in a fixed intervals. They can both be done at the same time, or independently. In the following section, we will employ the latter method for a numerical application.
training set $D$,initial parameter $\bm{w}(0)$ parameters for forward-whitening $\tau_{1}, c_{1}$,parameters for back-whitening $\tau_{2}, c_{2}$ $\bullet$ $U^{(i)} \leftarrow I; \bm{c}^{(i)} \leftarrow \bm{0}; R^{(i)^{T}} \leftarrow I; t \leftarrow 0$ $\bullet$ forward-whitening (cf. Algorithm \[alg: PRONG\]). $\bullet$ Estimation of $D_{i, i}$. $\bullet$ Computation of the back-whitening parameters $\{R^{(i)^{T}}_{new}\}$. $\bullet$ Updating the model parameters $\{(W^{\dagger^{(i)}}, \bm{b}^{\dagger^{(i)}})\}$. $\bullet$ Updating the back-whitening parameters $\{R^{(i)^{T}}\}$. $\bullet$ Updating of $\{(W^{\dagger^{(i)}}, \bm{b}^{\dagger^{(i)}})\}$ by the ordinary gradient descent. $\bullet$ $t \leftarrow t + 1$
Numerical Experiment {#sec: numerical experiment}
====================
In order to see the efficacy of our proposed method BPRONG in \[sec: gradient whitening\], we have applied it to a problem of hand-written character (digits) recognition using the MNIST data set (http://yann.lecun.com/exdb/mnist/) and compared against three other methods: ordinary Stochastic Gradient Descent(SGD), Batch
Normalization(BN)[@ioffe2015batch], and PRONG. The network architecture is common to all the compared methods with 5 layers of 784-100-100-100-10 neurons from input to output. Also, common learning rate of $0. 01$ is taken and the mini-batch size is $100$. The training data contains $60000$ sets and the test data has $10000$. We call updates of $600$ as $1$ epoch, and plot, at each epoch, the training loss with the training set, and the validation loss with the test data sets.
We observe the advantage of BPRONG with respect to the iteration numbers both in the training and the validation losses as shown in Figures \[fig: train\_loss\] and \[fig: validation\_loss\]. With respect to the actual computation times, BPRONG is faster than PRONG, and about the same speed as the BN (Figures \[fig: train\_loss\_time\] and \[fig: validation\_loss\_time\]). This is due to the fact that eigenvalue decomposition associated with the whitening is computationally costly to offset the advantage over BN with respect to iteration numbers.
Altogether, our proposed method, BPRONG, has shown its potential. If we can find methods to speed up the whitening process, BPRONG can show its effectiveness further.
![Training loss as a function of the iteration numbers[]{data-label="fig: train_loss"}](train_loss.eps){width="10cm"}
![Validation loss as a function of the iteration numbers[]{data-label="fig: validation_loss"}](validation_loss.eps){width="10cm"}
![Training loss as a function of the computational time[]{data-label="fig: train_loss_time"}](train_loss_time.eps){width="10cm"}
![Validation loss as a function of the computational time[]{data-label="fig: validation_loss_time"}](validation_loss_time.eps){width="10cm"}
Discussion {#sec: discussion}
==========
We presented here an extended model of the previously proposed Whitened Neural Networks[@NIPS2015_5953] as a method to realize the Natural Gradient Descent. Our extension, which we call Bidirectional Whitened Neural Networks, aims to make the Fisher Information Matrix closer to the identity matrix. It has shown its potential as an efficient method thorough a numerical application to a hand-written digits recognition problem.
We note two points as topics to be investigated further. First, the proposed model should be tested for larger and deeper network architectures for a check of its efficacy and stability. It may require further modifications for improvements on these aspects, particularly by exploring matrix decomposition methods. Secondly, we want to find more dynamical way for whitening process. In other words, we would like to keep the Fisher Information Matrix constantly closer to the identity by continuous whitenings. Though it is computationally more expensive, we may build on some previous studies, such as adaptive calculations of transforming matrices[@cardoso1996equivariant].
|
---
abstract: 'Recently, the discovery of a Venus-mass planet orbiting a brown-dwarf host in a binary system was reported from the analysis of the microlensing event OGLE-2013-BLG-0723. We reanalyze the event considering the possibility of other interpretations. From this, we find a new solution where the lens is composed of 2 bodies in contrast to the 3-body solution of the previous analysis. The new solution better explains the observed light curve than the previous solution with $\Delta\chi^2\sim 202$, suggesting that the new solution is a correct model for the event. From the estimation of the physical parameters based on the new interpretation, we find that the lens system is composed of two low-mass stars with $\sim 0.2\ M_\odot$ and $\sim 0.1\ M_\odot$ and located at a distance $\sim 3$ kpc. The fact that the physical parameters correspond to those of the most common lens population located at a distance with a large lensing probability further supports the likelihood of the new interpretation. Considering that two dramatically different solutions can approximately explain the observed light curve, the event suggests the need of carefully testing all possible lens-system geometries.'
author:
- 'Cheongho Han$^{1}$, David P. Bennett$^{2,3}$, Andrzej Udalski$^{4}$, and Youn Kil Jung$^{1}$'
title: 'A New Non-Planetary Interpretation of the Microlensing Event OGLE-2013-BLG-0723'
---
Introduction
============
Recently, the microlensing discovery of a Venus-mass planet (OGLE-2013-BLG-0723LBb) orbiting a brown dwarf in a binary system was reported by @Udalski2015. The discovery of the planet was of special scientific interest in many aspects. First, the planet itself is the lowest-mass planet among those discovered by using the microlensing method. Second, the host of the planet is a substellar-mass brown dwarf suggesting the possibility that an ice-rock planets can be formed in the outer parts of the accretion disk around a brown dwarf. Third, the planet belongs to a binary system where the planet orbits the lower-mass component of the binary. Finally, the planet/host mass ratio indicates that the system may be viewed either as a scaled-down version of a planet plus a star or as a scaled-up version of a moon plus a planet orbiting a star, suggesting that the formation processes of companions within accretion disks around stars, brown dwarfs, and planets are similar.
The light curve of the microlensing event OGLE-2013-BLG-0723 is complex and composed of multiple anomalous features. See the light curve in Figure \[fig:one\]. In chronological order, there exists a short-term anomaly occurred at ${\rm HJD}'={\rm HJD}-2450000\sim 6424$ followed by a bump at ${\rm HJD}'\sim 6463$ and two sharp spikes occurred at ${\rm HJD}'\sim 6472$ and 6492. According to the interpretation of @Udalski2015, the main anomalous features (the bump and the two spikes) of the light curve were produced by a wide binary where the projected separation between the binary components is greater than the angular Einstein radius $\theta_{\rm E}$ of the lens system and the short-term anomaly was produced by a planetary-mass object accompanied to the lower-mass component of the binary. See the geometry of the lens system presented in Figure 2 of @Udalski2015.
For lensing light curves produced by binary objects, it is known that there can be a pair of degenerate solutions where one solution has a binary separation normalized to the Einstein radius, $s$, is greater than unity, $s>1$ (wide binary), and the other solution has a separation smaller than unity, $s<1$ (close binary) [@Griest1998; @Dominik1999]. According to the interpretation of @Udalski2015, the main features of the light curve were explained by a wide-binary solution. They did find a close-binary solution[^1], but the solution was rejected because it resulted in unphysical lens parameters.
In this work, we present another interpretation of the lensing event OGLE-2013-BLG-0723 based on a new solution of lensing parameters found from the reanalysis of the event. According to this interpretation, all features of the lensing light curve including the short-term anomaly can be explained by a close-binary model, that was not found in the previous analysis, without the need to introduce an additional planet.
Reanalysis
==========
Microlensing light curves produced by binary objects are described by many parameters. For the simplest case of a rectilinear lens-source relative motion, one needs 7 basic parameters. Among these parameters, 3 parameters describe the relative lens-source motion, including the time of the closest source approach to a reference position of the lens, $t_0$, the separation between the source and the reference position at $t_0$, $u_0$ (normalized to the angular Einstein radius $\theta_{\rm E}$), and the angle between the source trajectory and the binary axis, $\alpha$. In our modeling, we use the barycenter of the binary as a reference position. The Einstein time scale $t_{\rm E}$, which is defined as the time required for the source to cross the Einstein radius, is needed to characterize the time scale of an event. Another two parameters of the projected binary separation, $s$ (also normalized to $\theta_{\rm E}$), and the mass ratio between the binary components, $q$, characterize the binary lens components. The last parameter is the normalized source radius $\rho=\theta_*/\theta_{\rm E}$, where $\theta_*$ is the angular source radius. This parameter is needed to describe the caustic-crossing features that are affected by finite-source effects. See Figure 6 of @Jung2015 for the graphical presentation of the binary lensing parameters.
Modeling based on the basic parameters is often not enough to precisely describe lensing light curves and additional parameters are needed to consider higher-order effects. In order to consider parallax effects, which are caused by the positional change of the observer induced by the orbital motion of the Earth around the Sun [@Gould1992], one needs two additional parameters $\pi_{{\rm E},N}$ and $\pi_{{\rm E},E}$. These parameters represent the two components of the lens parallax vector ${\mbox{\boldmath $\pi$}}_{\rm E}$ projected onto the sky along the north and east equatorial coordinates, respectively. The magnitude of the parallax vector corresponds to $\pi_{\rm E}=\pi_{\rm rel}/\theta_{\rm E}$, where $\pi_{\rm rel}={\rm AU}(D_{\rm L}^{-1}-D_{\rm S}^{-1})$ is the relative lens-source parallax, $D_{\rm L}$ and $D_{\rm S}$ are the distances to the lens and source, respectively. The direction of ${\mbox{\boldmath $\pi$}}_{\rm E}$ is that of the relative lens-source motion in the frame of the Earth at a reference time of the event.
[lrrrrrr]{} $\chi^2$ & 5536.9 & 4292.3 & 4211.1 & 3936.2 & 3925.1 & 3930.1\
$t_0$ (HJD’) & $6485.057\pm 0.028$ & $ 6485.055\pm 0.038$ & $ 6485.285\pm 0.048$ & $ 6485.976\pm 0.054$ & $ 6486.004\pm 0.056$ & $ 6486.007\pm 0.069$\
$u_0$ & $0.0133 \pm 0.001$ & $ 0.019 \pm 0.002$ & $-0.013 \pm 0.001$ & $ 0.032 \pm 0.002$ & $ 0.030 \pm 0.001$ & $-0.028 \pm 0.001$\
$t_{\rm E}$ (days) & $48.60 \pm 0.07 $ & $ 55.24 \pm 0.22 $ & $ 47.97 \pm 0.22 $ & $ 51.93 \pm 0.42 $ & $ 50.65 \pm 0.52 $ & $ 51.33 \pm 0.48 $\
$s$ & $0.668 \pm 0.001$ & $ 0.670 \pm 0.001$ & $ 0.665 \pm 0.001$ & $ 0.672 \pm 0.003$ & $ 0.672 \pm 0.003$ & $ 0.666 \pm 0.003$\
$q$ & $0.585 \pm 0.003$ & $ 0.411 \pm 0.005$ & $ 0.562 \pm 0.009$ & $ 0.529 \pm 0.007$ & $ 0.562 \pm 0.011$ & $ 0.560 \pm 0.011$\
$\alpha$ (rad) & $4.892 \pm 0.002$ & $-4.939 \pm 0.005$ & $ 4.879 \pm 0.004$ & $-4.991 \pm 0.005$ & $-4.975 \pm 0.007$ & $ 4.963 \pm 0.005$\
$\rho$ ($10^{-3}$) & $1.04 \pm 0.01 $ & $ 1.08 \pm 0.02 $ & $ 1.06 \pm 0.02 $ & $ 1.19 \pm 0.02 $ & $ 1.19 \pm 0.02 $ & $ 1.16 \pm 0.02 $\
$\pi_{{\rm E},N}$ & - & $-1.14 \pm 0.04 $ & $ 0.70 \pm 0.04 $ & - & $ 0.05 \pm 0.04 $ & $ 0.10 \pm 0.07 $\
$\pi_{{\rm E},E}$ & - & $ 0.04 \pm 0.05 $ & $ 1.15 \pm 0.09 $ & - & $ 0.26 \pm 0.07 $ & $ 0.30 \pm 0.08 $\
$ds/dt$ (${\rm yr}^{-1}$) & - & - & - & $ 0.06 \pm 0.04 $ & $ 0.08 \pm 0.04 $ & $-0.01 \pm 0.03 $\
$d\alpha/dt$ (${\rm yr}^{-1}$) & - & - & - & $-0.46 \pm 0.03 $ & $-0.53 \pm 0.03 $ & $ 0.44 \pm 0.03 $
[lr]{} $\chi^2$ & 4126.8\
$t_0$ (HJD’) & 6484.526 $\pm$ 0.037\
$u_0$ & -0.079 $\pm$ 0.002\
$t_{\rm E}$ (days) & 68.48 $\pm$ 0.01\
$s_1$ & 5.07 $\pm$ 0.02\
$q_1$ & 3.11 $\pm$ 0.02\
$\alpha$ (rad) & -1.195 $\pm$ 0.003\
$s_2$ & 0.97 $\pm$ 0.02\
$q_2$ $(10^{-5})$ & 6.61 $\pm$ 0.01\
$\psi_0$ (rad) & -4.936 $\pm$ 0.005\
$\rho$ $(10^{-3})$ & 1.40 $\pm$ 0.02\
$\pi_{{\rm E},N}$ & -0.05 $\pm$ 0.01\
$\pi_{{\rm E},E}$ & 1.35 $\pm$ 0.02\
$ds_2/dt$ $({\rm yr}^{-1})$ & 0.81 $\pm$ 0.02\
$d\psi/dt$ $({\rm yr}^{-1})$ & -0.50 $\pm$ 0.02
Another higher-order effect that is often needed to consider in binary-lens modeling is the orbital motion of the lens [@Park2013]. To first-order approximation, the lens-orbital effect is described by two parameters $ds/dt$ and $d\alpha/dt$, which represent the change rates of the binary separation and the source trajectory angle, respectively [@Albrow2000].
Ideally, a solution of lensing parameters can be searched for by comparing an observed light curve with all possible model light curves resulting from the combination of lensing parameters. However, thorough coverage of the vast parameter space is limited by computing power and thus the grid spacing of each parameter cannot be arbitrarily small to completely cover the parameter space. As a result, solutions of lensing parameters, especially ones located in a very localized region in the parameter space, can be occasionally missed.[^2]
The reanalysis of this event was initiated by one of us (DPB) using the initial condition grid search method of @Bennett2010 with the data set used in the original paper [@Udalski2015]. To limit the parameter space to be searched, the initial search was conducted using only a stellar binary model without microlensing parallax or orbital motion. This search yielded a candidate solution with a caustic crossing feature at ${\rm HJD}'\approx 6414.5$ where there is a gap in the data. This is relatively close to the observed light curve bump at ${\rm HJD}'\approx 6424$, suggesting that a model including parallax and/or orbital motion might naturally explain the light curve. Several similar models were explored, and a model including microlensing parallax with the early light curve peak at ${\rm HJD}'\approx 6424$ was found. This solution was sent to the lead authors of the original paper, while models also including orbital motion were investigated.
Considering the possibility that there may exist missed solutions, especially in the close binary regime, we conduct reanalysis of the lensing event OGLE-2013-BLG-0723. In this analysis, we narrow down the grid spacing of parameters in order to minimize the possibility of missing localized solutions. We also consider higher-order effects in the initial solution search to avoid the possibility that local solutions are missed due to the neglect of higher-order effects. The grid search is conducted in the space of the parameters $(s,q,\alpha)$ because lensing light curves can change dramatically with the small changes of these parameters. To search for other parameters, for which lensing light curves vary smoothly with the change of the parameters, a downhill approach is used. For the downhill approach, we use a Markov Chain Monte Carlo (MCMC) method.
It is needed to consider finite-source effects in order to describe the caustic-crossing parts of the light curve. We incorporate the effects by using a numerical ray-shooting method. In this method, uniform rays are shot from the image plane, bent by the lens equation, and then collected in the source plane. Then, finite magnifications are computed as the ratio of the ray number density on the source plane to the density on the image plane. Since precise computations of finite magnifications require a large number of rays, this numerical method demands large amount of computing power. For efficient production of theoretical light curves, we apply the “map-making” method [@Dong2006], where one can produce many light curves resulting from different source trajectories based on a single ray-shooting map for a given set of the binary parameters $s$ and $q$. In computing finite-source magnifications, we consider the surface-brightness variation of the source star by modeling the surface-brightness profile as $S_\lambda \propto 1-\Gamma_\lambda (3\cos\psi/2),$ where $\Gamma_\lambda$ is the linear limb-darkening coefficient $\psi$ is the angle between the line of sight toward the source center and the normal to the surface. The limb-darkening coefficient $\Gamma_I=0.36$ is adopted from @Claret2000 based on the source type. For the detailed procedure of determining the source type, see section 4.
We note that it is difficult to consider lens-orbital effects in the initial search for solutions based on the map-making method. This is because the binary separation and orientation vary in time during events and thus a single map cannot be used to produce multiple light curves. We, therefore, consider orbital effects after a preliminary solution is found from the initial search.
For direct comparison of models, we use the same data sets as those used in @Udalski2015. These data sets are composed of 4067 $I$-band and 19 $V$-band data acquired by the Optical Gravitational Lensing Experiment (OGLE) group and 62 $I$-band data obtained by the Wise group. We also use the same error-bar normalization. Since $\chi^2$ per degree of freedom is normalized to unity, $\Delta\chi^2=n^2$ corresponds to the statistical importance of $n\sigma$ level.
New Interpretation
==================
From the grid search, we find a new candidate close-binary solution that was not found in the previous analysis. It turns out that the new solution was missed because the grid spacing was not small enough to locate the isolated solution in the parameter space. Once the approximate region of the solution in the parameter space is located, we gradually refine the solution first by allowing grid parameters to vary and then by considering higher-order effects.
Based on the newly found local minima, we test 5 different models. In the “standard” binary-lens model, we model the light curve based on the 7 basic lensing parameters. In the “parallax” and “orbital” models, we separately consider the lens parallax and orbital effects, respectively. In the “orbital + parallax” model, we consider both lens-orbital motion and parallax effects. In order to check the well-known “ecliptic degeneracy” in the determination of the lens parallax [@Skowron2011], we test two models with $u_0>0$ and $u_0<0$ for models considering parallax effects.
In Table \[table:one\], we present the best-fit parameters of the individual tested models along with their $\chi^2$ values. In Table \[table:two\], we also present the parameters of the best-fit 3-body solution of @Udalski2015 for comparison. We note that notations of the 3-body lensing parameters are different from those of the 2-body parameters due to the addition of one more lens component. For example, $(s_1,q_1)$ and $(s_2,q_2)$ denote the separations and mass ratios between $M_1-M_2$ and $M_1-M_3$ pairs, respectively, and $M_1$, $M_2$, and $M_3$ denote the lens components according to the order of heavier mass. The notation $\psi_0$ denotes the angle between the binary axis and the planet-host axis at $t_0$ and $ds_2/dt$ and $d\psi/dt$ are the change rates of $s_2$ and $\psi$, respectively.
We find that higher-order effects are important for the precise description of the event. It is found that the standard model with $\chi^2=5536.9$ cannot explain the short-term anomaly. Furthermore, it leaves noticeable residuals in the regions of the light curve around the other anomalous features. Consideration of the higher-order effects significantly improves the fit. We find that inclusion of parallax effects improves the fit by $\Delta\chi^2=1326$. The improvement by considering lens-orbital effects is $\Delta\chi^2=1601$. When both higher-order effects are simultaneously considered, the improvement is $\Delta\chi^2=1612$. Considering that (1) $\chi^2$ improvement by the orbital effect is significantly greater than the improvement by the parallax effect and (2) the additional improvement from the orbital model to the orbital + parallax model is minor ($\Delta\chi^2=11$), we judge that the lens orbital motion is the dominant higher-order effect. With the inclusion of the higher-order effects, the lensing light curve fits all anomalous features.
In Figure \[fig:one\], we present the light curve of the close-binary model obtained from the reanalysis. For comparison, we also present the triple-lens model light curve that is constructed based on the best-fit parameters presented in @Udalski2015. Figure \[fig:two\] shows the lens-system geometry of the close binary model where the source trajectory with respect to the caustics are presented. According to the new close-binary model, the short-term anomaly was produced by the passage of the source over the tip of the 3-cusp peripheral caustic, and the two caustic-crossing spikes were produced by the source passage over the 4-cusp central caustic. The bump between the short-term anomaly and the caustic-crossing spikes was produced when the source passed over the narrow strip of the strong anomaly region formed along the line connecting the sharp cusps of the central and peripheral caustics. In Figure \[fig:three\], we also present the distribution of the lensing parameters of the best-fit solution.
We find that both the previous 3-body and the new 2-body solutions almost equally well explain the short-term anomaly. In Figure \[fig:four\], we present the enlargement of the anomaly region over which we plot both model light curves. The lower panel shows the $\chi^2$ difference between the two models. Positive $\Delta\chi^2$ value means that the binary model provides a better fit and vice versa. One finds that $\chi^2$ difference for nearly all data points are $\lesssim 1$, implying that the anomaly is well explained by both solutions. We note that despite the almost the same goodness of the fits, the two models are greatly different; the binary solution predicts a huge 1.5 – 2.0 mag brightening, while the triple model predicts a mild variation. If there existed a few points at the peak of the anomaly, the two models could have been clearly distinguished. Unfortunately, this part of the light curve was not covered by data.
Although it is difficult to resolve the degeneracy between the two solutions based on the short anomaly, we find that the degeneracy can be resolved from the overall light curve. One can see the goodness of the 2-body fit over the 3-body fit from the comparison of $\chi^2$ values of the fits presented in Tables \[table:one\] and \[table:two\] and the cumulative function of $\Delta\chi^2$ between the two models as a function of time presented in Figure \[fig:five\]. It is found that the 2-body solution with $\chi^2=3925.1$ provides a better fit over the 3-body solution with $\chi^2=4126.8$, i.e. $\Delta\chi^2=201.7$. From the cumulative distribution, it is found that the 2-body solution better explains the observed light curve in the regions between the caustic crossings and the declining part of the light curve after the caustic crossings. We find that the $\chi^2$ difference between the two models is $\Delta\chi^2=202$. We note that $\chi^2\sim 50$ improvement is achieved during $6510 \lesssim {\rm HJD}'\lesssim 6540$, when the phase of the Moon was close to full and thus data show somewhat larger scatter. We note, however, that the fit improvement is not attributed to the effect of elevated sky background or other noise because such an effect was accounted for in the error bar estimation during photometry procedure. Furthermore, the sharp increases in the cumulative $\chi^2$ plot in other regions, especially around the caustic-crossing regions, demonstrate the better fit of the new solution. @Udalski2015 mentioned a systematic trend in the baseline magnitude of the source star caused by a nearby bright star. In Appendix, we show this trend. We also describe the procedure to correct the systematic trend. We note that the improvement of the fit by the new solution is not attributed to this systematic trend because we use data set where the trend was corrected. Considering that not only the new solution provides a better fit but also the 2-body model is simpler than the 3-body model, the 2-body solution is likely to be the correct solution according to Occam’s razor.
We consider two tests that may further support our new interpretation. The first test is checking the brightnesses of the source and blend predicted by the two models from high-resolution observations. We find that this method would not work because the estimated source and blend fluxes of the binary solution $(F_s,F_b)=(0.151\pm 0.001,0.170\pm 0.001)$ are not much different from $(F_s,F_b)=(0.191\pm 0.001,0.132\pm 0.001)$ of the triple-lens solution. The second test is measuring the relative lens-source proper motion. The detailed procedure of computing the proper motion is described in the next section. We find that the heliocentric proper motion estimated from the binary model is $\mu_\odot = 6.4\pm 0.5\ {\rm mas\ yr}^{-1}$, while the expected value for the triple-lens model is $\mu_\odot = 14.5\pm 1.3\ {\rm mas\ yr}^{-1}$. The difference between the estimated proper motions is considerable, implying that future observations with high-resolution instrument will be able to distinguish the two solutions.
In the aspect that there exist multiple interpretations for an isolated short-term anomaly located away from the main light curve, OGLE-2013-BLG-0723 is similar to the case of MACHO-97-BLG-41. For MACHO-97-BLG-41, there existed two interpretations: one interpretation by a 3-body circumbinary planetary system [@Bennett1999] and the other by a 2-body orbiting binary system [@Albrow2000; @Jung2013]. Considering that two solutions with dramatically different lens systems can explain observed light curves, the two events suggest the need of carefully testing all possible lens-system geometries.
New Estimation of Physical Parameters
=====================================
For the new solution, both finite-source and parallax effects are detected and thus we are able to determine the angular Einstein radius $\theta_{\rm E}$ and the lens parallax $\pi_{\rm E}$. With these values, the mass and distance to the lens are uniquely determined by $$M_{\rm tot}={\theta_{\rm E} \over \kappa \pi_{\rm E}};\qquad
D_{\rm L}={{\rm AU} \over \pi_{\rm E}\theta_{\rm E}+\pi_{\rm S}},$$ where $\kappa=4G/(c^2{\rm AU})$ and $\pi_{\rm S}={\rm AU}/D_{\rm S}$ is the parallax of the source star [@Gould2000]. The lens parallax is determined from modeling. The angular Einstein radius is estimated from the relation $\theta_{\rm E}=\theta_*/\rho$, where the normalized source radius $\rho$ is measured from modeling and the angular source radius $\theta_*$ is estimated from the source type that is determined based on the de-reddened color and brightness. For the calibration of the color and brightness of the source star, we use the centroid of bulge giant clump as a reference [@Yoo2004]. In Figure \[fig:six\], we present the position of the source with respect to the centroid of the giant clump in the color-magnitude diagram. The estimated de-reddened color and brightness of the source star are $(V-I,I)_0=(0.52\pm 0.05,17.3\pm 0.01)$, indicating that the source is an F-type main-sequence star. We then convert $V-I$ color into $V-K$ color using the color-color relation of @Bessell1988 and obtain the angular source radius $\theta_*$ using the relation between $V-K$ and $\theta_*$ presented in @Kervella2004. The estimated source radius is $\theta_*=0.90 \pm 0.06\ \mu{\rm as}$. We note that the source radius is slightly smaller than the value estimated in @Udalski2015 because of the slight color difference, which is caused by the difference in the ratios of the source to blended light between the two models. From the angular source radius, the estimated the angular Einstein radius is $\theta_{\rm E}=0.75 \pm 0.05$ mas.
[lrr]{} Primary mass & $0.22 \pm 0.06\ M_\odot$ & $0.19 \pm 0.04\ M_\odot$\
Companion mass & $0.13 \pm 0.04\ M_\odot$ & $0.10 \pm 0.02\ M_\odot$\
Distance to the lens & $3.11 \pm 0.57$ kpc & $2.74 \pm 0.45$ kpc\
Projected separation & $1.57 \pm 0.29$ AU & $1.38 \pm 0.23$ AU\
Geocentric lens-source proper motion & $5.42 \pm 0.39$ ${\rm mas\ yr}^{-1}$ & $5.39 \pm 0.39$ ${\rm mas\ yr}^{-1}$\
Heliocentric lens-source proper motion (north) & $0.91 \pm 0.07$ ${\rm mas\ yr}^{-1}$ & $1.58 \pm 0.11$ ${\rm mas\ yr}^{-1}$\
Heliocentric lens-source proper motion (east) & $6.43 \pm 0.47$ ${\rm mas\ yr}^{-1}$ & $6.46 \pm 0.46$ ${\rm mas\ yr}^{-1}$\
Ratio of the projected kinetic to potential energy & 0.04 & 0.02
In Table \[table:three\], we list the determined physical parameters. Although the $u_0>0$ solution is slightly preferred over the $u_0<0$ solution, it is found that the $\chi^2$ difference between the two models is merely $\Delta\chi^2=5.0$. Since such a level of $\Delta\chi^2$ can be ascribed to systematics in data, we present the physical parameters corresponding to both solutions. We note, however, that the estimated lens parallax values from the two models are similar each other and thus the resulting physical parameters are also similar. Once the distance to the lens is known, the projected separation between the binary components and the geocentric lens-source proper motion are estimated by $$a_\perp=s D_{\rm L}\theta_{\rm E}$$ and $$\mu_\oplus={\theta_{\rm E}\over t_{\rm E}},$$ respectively. We note that the geocentric reference frame is set with respect to the Earth position at $t_0$ (${\rm HJD}\sim 2456486$), which approximately corresponds to the time of the source star’s closest approach to the center of mass of the binary lens. The heliocentric proper motion is estimated from the geocentric proper motion by $${\mbox{\boldmath $\mu$}}_\odot = \mu_\oplus {{\mbox{\boldmath $\pi$}}_{\rm E}\over \pi_{\rm E}} +
{{\bf v}_{\oplus,\perp} \over {\rm AU}} \pi_{\rm rel},$$ where ${\bf v}_{\oplus,\perp}=(-2.1,26.3)\ {\rm km\ s}^{-1}$ is the velocity of the Earth projected on the sky at the reference time, i.e. $t_0$. We also present ratio of the projected kinetic to potential energy [@Dong2009] that is computed by $$\left({{\rm KE}\over{\rm PE}}\right)_\perp =
{ (r_\perp/{\rm AU})^2 \over 8\pi^2(M_{\rm tot}/M_\odot)}
\left[
\left( {1\over s}{ds\over dt}\right)^2 +
\left( {d\alpha\over dt}\right)^2
\right].$$ To be a bound system, the ratio of the binary lens should follow $({\rm KE}/{\rm PE})_\perp \leq {\rm KE}/{\rm PE} \leq 1$. For both the $u_0>0$ and $u_0<0$ solutions, this condition is satisfied.
The characteristics of the lens determined in the new analysis is greatly different from those determined in the previous analysis. The newly estimated masses of the binary components are $M_1\sim 0.2\ M_\odot$ and $M_2\sim 0.1\ M_\odot$, indicating that they are low-mass stars, which are the most common population of Galactic lenses. On the other hand, the previous analysis characterized the lens system as a 3-body system where a terrestrial planet is orbiting sub-stellar brown-dwarf host accompanied by a low-mass companion. The distance to the lens estimated in the new analysis is $D_{\rm L}\sim 3$ kpc, which corresponds to roughly a halfway distance between the observer and the source and thus the lensing probability is maximized. By contrast, the previous analysis estimated a very close lens distance of $D_{\rm L}\lesssim 500$ for which the lensing probability is low. Therefore, the likelihood of the new interpretation is further supported by the fact that the physical parameters of the lens system correspond to the most probable values predicted by the mass function and distribution of Galactic matter.
Summary
=======
We presented a new interpretation of the microlensing event OGLE-2013-BLG-0723 that had been previously interpreted as a 3-body lensing event produced by a Venus-mass planet orbiting a brown-dwarf host in a binary system. The new solution, where the lens is composed of 2 bodies, can explain all anomalous features in the lensing light curve without the need to introduce an additional planetary companion. The fact that the new solution better explains the observed light curve than the previous solution with a simpler model implies that the new model is preferred, a conclusion that is also supported by Occam’s razor. In addition, the fact that the physical parameters estimated from the new solution correspond to those of the most common lens population located at a distance of a large lensing probability further supports the likelihood of the new interpretation. Considering that two dramatically different solutions can explain the observed light curve, the event suggests the need of carefully testing all possible lens-system geometries. In particular, care must be taken to ensure that close binary solutions with caustic motion due to orbital motion are considered. OGLE-2013-BLG-0723 is the second close stellar binary microlensing event that was originally misinterpreted as a binary star system with a planet.
Work by C. Han was supported by Creative Research Initiative Program (2009-0081561) of National Research Foundation of Korea. D.P.B. was supported by grants NASA-NNX13AF64G and NNX15AJ76G. The OGLE project has received funding from the National Science Centre, Poland, grant MAESTRO 2014/14/A/ST9/00121 to AU.
Correction of Baseline Variation
================================
The light curve of OGLE-2013-BLG-0723 shows a systematic decline in the baseline. See the upper panel of Figure \[fig:seven\]. A similar long term linear trend (of opposite sign) was seen in OGLE-2013-BLG-0341 and was eventually traced to a nearby bright star that was gradually moving toward (in that case) the source star, so that more of its flux was being “captured” in the tapered aperture used to estimate the source flux. We searched for such a moving bright star by examining the difference of two images, from 2004 and 2012. We indeed find a dipole from a bright star roughly 1.5" away, which is the characteristic signature of such moving stars. Having identified the cause of this trend, we conduct a linear fit for it and remove it. In the middle and lower panels of Figure \[fig:seven\], we present the baseline of the source star before and after the correction, respectively.
[99]{}
Albrow, M. D., Beaulieu, J.-P., Caldwell, J. A. R., et al. 2000, , 534, 894 Bennett, D.P. 2010, , 716, 1408 Bennett, D. P., Rhie, S. H., Becker, A. C., et al. 1999, Natur, 402, 57 Bessell, M. S., & Brett, J. M. 1988, , 100, 1134 Claret, A. 2000, , 363, 1081 Dominik, M. 1999, , 341, 943 Dong, S., Gould, A., Udalski, A., et al. 2009, , 695, 970 Dong, Subo, DePoy, D. L., Gaudi, B. S., et al. 2006, , 642, 842 Gould, A. 1992, , 392, 442 Gould, A. 2000, , 542, 785 Griest, K., & Safizadeh, N. 1998, , 500, 37 Jung, Y. K., Han, C., Gould, A., & Maoz, D. 2013, , 768, L7 Jung, Y. K.,Udalski, A., Sumi, T., et al. 2015, , 798, 123 Kervella, P., Bersier, D., Mourard, D., et al. 2004, , 428, 587 Park, H., Udalski, A., Han, C., et al. 2013, , 778, 134 Skowron, J., Udalski, A., Gould, A., et al. 2011, , 738, 87 Udalski, A., Jung, Y. K., Han, C., et al. 2015, , 812, 47 Yoo, J., DePoy, D. L., Gal-Yam, A., et al. 2004, , 603, 139
[^1]: This solution is different from that one presented in this work.
[^2]: An adaptive mesh refinement approach, which locally adds more grids where they are needed, can help to minimize the probability of missing solutions, but such a code has not yet been developed in microlensing analysis due to the difficulty in providing a prior condition for denser grid regions.
|
---
abstract: |
[Abstract.]{}
For $0\leq \alpha<1$ and prime number $p$ let $L(\alpha,p)$ be the sum of the first $[\alpha p]$ values of Legendre symbol modulo $p$. We study positivity of $L(\alpha,p)$ and prove that for $|\alpha-\frac13|<2\cdot 10^{-6}$ and for rational $\alpha\leq \frac12$ with denominators in the set $\{1,2,3,4,5,6,8,12\}$ the inequality $L(\alpha,p)\geq 0$ holds for majority of primes.
address: 'Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia '
author:
- 'A.B. Kalmynin'
title: Long nonnegative sums of Legendre symbols
---
Introduction
============
Let $p$ be an odd prime number and $\chi_p(n)={\left(\frac{n}{p}\right)}$ be the Legendre symbol modulo $p$. It is well-known that the sum
$$\sum_{n\leq p/2}{\left(\frac{n}{p}\right)}$$
is always nonnegative. In other words, there are at least as many quadratic residues as nonresidues modulo $p$ below $p/2$. More precisely, Dirichlet proved the following formula for this sum
Let $p$ be an odd prime number. Then the equality
$$\sum_{n\leq p/2}{\left(\frac{n}{p}\right)}=\begin{cases}
\left(2-{\left(\frac{2}{p}\right)}\right)h(-p)\text{ if }p\equiv 3 \pmod 4\\
0 \text{ if }p\equiv 1 \pmod 4
\end{cases}$$
holds, where $h(-p)$ is the class number of the number field $\mathbb Q(\sqrt{-p}).$
Other cases of connection between character sums of this type and class numbers of quadratic fields are given in [@suw]. For example, the sum still will be nonnegative if we replace $p/2$ by $p/3$ or $p/4$. This leads us to the general question about nonnegativity of the sum of length $\alpha p$ for any real number $\alpha$. Let us define $$L(\alpha,p)=\sum_{n\leq \alpha p}{\left(\frac{n}{p}\right)}.$$ Numerical evidence suggests that for any $\alpha\leq \frac{1}{2}$ most primes satisfy the inequality
$$\label{main}
L(\alpha,p)\geq 0.$$
For example, among first $1000$ primes there are $896$ with $L(\frac25,p)\geq 0$, $917$ with $L(\frac38,p)\geq 0$, $884$ with $L(\frac{1}{12},p)\geq 0$, $812$ with $L(\frac{1}{2\pi},p)\geq 0$ and $937$ with $L(\frac{1}{e},p)\geq 0$. For $10000$ these numbers are $8915, 9122, 8799, 8019$ and $9340$, respectively, and we get $89041, 91036, 87868, 79784$ and $93260$ for the first $100000$ prime numbers. As we can see, for all our choices of parameter $\alpha$ the proportion of $p$ with nonnegative $L(\alpha,p)$ seems to be even more than $75\%$. Based on that, let us formulate our main conjecture:
For all $0\leq \alpha\leq \frac{1}{2}$ the lower asymptotic density of primes that satisfy \[main\] is at least $\frac{1}{2}$. In other words, the inequality \[main\] holds for majority of prime numbers: $$\label{goal}
\liminf_{x\to +\infty}\frac{1}{\pi(x)}\#\{p\leq x: \sum_{n\leq \alpha p}{\left(\frac{n}{p}\right)}\geq 0\}\geq \frac{1}{2}.$$
We were able to prove Conjecture 1 for rational $\alpha$ with small denominator and also for all real $\alpha$ inside small neighbourhood of the point $\frac{1}{3}$. So, our two main results are as follows:
Conjecture 1 holds for all rational $\alpha\leq \frac{1}{2}$ with denominators in the set $\{1,2,3,4,5,6,8,12\}$.
Conjecture 1 holds for all real $\alpha$ satisfying $\frac{1}{3}-2\cdot10^{-6}\leq \alpha\leq \frac{1}{3}+2\cdot10^{-6}$.
To prove these two theorems, we are going to reduce the initial problem to the study of certain fixed random variable, using various methods of Fourier analysis, probability theory and distribution of primes in arithmetic progressions.
Fourier expansion
==================
In this section we are going to prove the following simple result:
For any $\alpha$ and all large enough prime numbers $p$ the equality
$$\label{four}
L(\alpha,p)=\tau\left({\left(\frac{\cdot}{p}\right)}\right)\sum_{m\in \mathbb Z, m\neq 0}\frac{1-e^{-2\pi i\alpha m}}{2\pi i m}{\left(\frac{m}{p}\right)}$$
holds, where $\tau\left({\left(\frac{\cdot}{p}\right)}\right)=\sum\limits_{n=0}^p e^{2\pi in/p}{\left(\frac{n}{p}\right)}$ is quadratic Gauss sum.
Theorem is trivial for $\alpha\in \mathbb Z$, so it is enough to assume that $\alpha$ is not an integer. Also, $L(\alpha,p)$ is a periodic function of $\alpha$ with period 1 and so is the right-hand side of our formula. Thus, we can also assume that $0<\alpha<1.$ As periodic characteristic function $\chi_{[0,\alpha]}(\{x\})$ of the interval $[0,\alpha]$ is smooth everywhere except for the discontinuity points, by Dini’s critertion for $x \not\equiv 0, \alpha \pmod 1$ we have
$$\chi_{[0,\alpha]}(\{x\})=\sum_{m \in \mathbb Z}\mathcal F\chi_{[0,\alpha]}(m)e^{2\pi i mx},$$
where
$$\mathcal F\chi_{[0,\alpha]}(m)=\int_0^\alpha e^{-2\pi i mx}dx=\begin{cases}
\alpha \text{ if } m=0\\
\frac{1-e^{-2\pi i \alpha m}}{2\pi i m} \text{ if } m\neq 0.
\end{cases}$$
Therefore, due to the fact that ${\left(\frac{0}{p}\right)}=0$ and $\alpha p$ is not an integer for large enough $p$, the equality
$$L(\alpha,p)=\sum_{n=0}^p \chi_{[0,\alpha]}(n/p){\left(\frac{n}{p}\right)}=\sum_{n=0}^p \sum_{m\in \mathbb Z} \mathcal F\chi_{[0,\alpha]}(m)e^{2\pi nm/p}{\left(\frac{n}{p}\right)}$$
holds for $p$ large enough. Changing the order of summation and using the fact that the sum of Legendre symbols over a complete system of residues is equal to 0, we get
$$L(\alpha,p)=\sum_{m\neq 0}\frac{e^{-2\pi i\alpha m}-1}{2\pi i m}\sum_{n=0}^p e^{2\pi i nm/p}{\left(\frac{n}{p}\right)}.$$
From multiplicativity of Legendre symbol, we easily obtain
$$\sum_{n=0}^p e^{2\pi inm/p}{\left(\frac{n}{p}\right)}=\tau\left({\left(\frac{\cdot}{p}\right)}\right){\left(\frac{m}{p}\right)},$$
from which we get the desired result.
Probabilistic reduction
========================
Here we show that, roughly speaking, in our formula for $L(\alpha,p)$ one can replace all the Legendre symbols by the random multiplicative function. It turns out that it is possible to reduce certain properties of linear combinations of Legendre symbols to properties of random completely multiplicative functions $f$ satisfying $f(n)=\pm 1$ for all $n \in \mathbb N$.
Let us give a few definitions. The main object of our study is the random prime number:
For $\varepsilon=\pm 1$ and $x\geq 5$ by $\mathfrak p_x^\varepsilon$ we denote the random variable which is uniformly distributed among primes $\leq x$ that are congruent to $\varepsilon$ modulo 4.
Next, we need to define the random multiplicative function:
Let $X_2,X_3,X_5,X_7,X_{11}\ldots$ be the sequence of independent identically distributed random variables which are distributed according to the Rademacher distribution and indexed by prime numbers. In other words, for any prime $p$ we have $$\mathbb P(X_p\leq x)=
\begin{cases}
0 \text{ if }x<-1\\
1/2 \text{ if }-1\leq x<1\\
1 \text{ if }x\geq 1
\end{cases}$$
For any natural number $n$ we define $X_n$ by the formula
$$X_n=\prod_{p} X_p^{\nu_p(n)},$$
where $\nu_p(n)$ is the largest $k$ with $p^k \mid n$. Note that the product contains finite number of terms and also that $X_{ab}=X_aX_b$ for all $a$ and $b$.
Using constructed random variables, we define certain random series.
Let $\{a_n\}$ be the sequence of complex numbers, $x\geq 5$ and $\varepsilon=\pm 1$. We define
$$L(a,x,\varepsilon)=\sum_{n=1}^{+\infty}\frac{a_n}{n}{\left(\frac{n}{\mathfrak p_x^\varepsilon}\right)}$$
and
$$L(a)=\sum_{n=1}^{+\infty}\frac{a_nX_n}{n}.$$
In this section we will show that $L(a)$ is often a rather good model for $L(a,x,\varepsilon)$. But first of all, we need to show that $L(a)$ is well-defined.
If the sequence $\{a_n\}$ is bounded, then the series defining $L(a)$ converges almost surely.
Assume that $|a_n|\leq C$ for all $n$. Consider the fourth moment of the weighted sum of $X_n$:
$$\mathbb E(|a_1X_1+\ldots+a_nX_n|^4)=\sum_{p,q,r,s\leq n}a_pa_q\overline{a}_r\overline{a}_s\mathbb EX_{pqrs}.$$
Using the boundedness of $a_n$ and noticing that $\mathbb EX_n=0$ unless $n$ is a square, in which case $\mathbb EX_n=1$, we get
$$\mathbb E(|a_1X_1+\ldots+a_nX_n|^4)\leq C^4 \sum_{\substack{pqrs=m^2 \\ p,q,r,s\leq n}} 1.$$
Now, if $pqrs=m^2$ and $p,q,r,s\leq n$ then $m\leq n^2$, so
$$\mathbb E(|a_1X_1+\ldots+a_nX_n|^4)\leq C^4\sum_{m\leq n^2}\tau_4(m^2).$$
As $\tau_4(m^2)\ll m^\varepsilon$ for any $\varepsilon>0$ we obtain
$$\mathbb E(|a_1X_1+\ldots+a_nX_n|^4)\ll n^{2+\varepsilon}.$$
Choosing $\varepsilon=1/6$ we get by Markov’s inequality
$$\mathbb P(|a_1X_1+\ldots+a_nX_n|\geq n^{5/6})\ll \frac{n^{2+1/6}}{n^{20/6}}=n^{-7/6}.$$
Hence, by Borel-Cantelli lemma we have $a_1X_1+\ldots+a_nX_n=O(n^{5/6})$ almost surely. Using partial summation, we obtain the convergence of
$$\sum_{n\geq 1}\frac{a_nX_n}{n}.$$
Now we are going to use some results on primes in arthmetic progressions to prove the following theorem
Let $\{a_n\}$ be the sequence of real numbers such that the inequality
$$\max_N \sum_{n\leq N}a_n{\left(\frac{n}{p}\right)}\ll \sqrt{p}\ln p$$
holds for all but at most $o(\pi(x))$ primes $p\leq x$ as $x\to +\infty$ (i.e. abovementioned inequality is true for almost all primes). Then for $\varepsilon=\pm 1$ random variables $L(a,x,\varepsilon)$ converge to $L(a)$ in distribution as $x \to +\infty$.
The proof will be divided into several parts in which we will treat different chunks of our series differently. We will formulate and use several lemmas concerning the distribution of prime numbers and the method of moments inside the proof. First of all, we split the sum in the definition of $L(a,x,\varepsilon)$ into three sums as follows:
$$L(a,x,\varepsilon)=\sum_{n\leq\ln^3 x}\frac{a_n}{n}{\left(\frac{n}{\mathfrak p_x^\varepsilon}\right)}+\sum_{\ln^3 x<n\leq \sqrt{x}\ln^2 x}\frac{a_n}{n}{\left(\frac{n}{\mathfrak p_x^\varepsilon}\right)}+\sum_{n>\sqrt{x}\ln^2 x}\frac{a_n}{n}{\left(\frac{n}{\mathfrak p_x^\varepsilon}\right)}=$$
$$A(a,x,\varepsilon)+B(a,x,\varepsilon)+C(a,x,\varepsilon).$$
We will prove that as $x\to +\infty$ the variables $B(a,x,\varepsilon)$ and $C(a,x,\varepsilon)$ both converge to 0 in probability and then show that $A(a,x,\varepsilon)$ converges to $L(a)$ in distribution via the method of moments.
Let us prove that the infinite sum $C(a,x,\varepsilon)$ is usually small. This easily follows from the conditions of Theorem 5 because with probability $1-o(1)$ we have $\sum\limits_{n\leq N}{\left(\frac{n}{p}\right)}\ll \sqrt{p}\ln p$ so for almost all realizations $p$ of $\mathfrak p_x^\varepsilon$ we get
$$\sum_{n>\sqrt{x}\ln^2 x}\frac{a_n}{n}{\left(\frac{n}{p}\right)}=\int\limits_{\sqrt{x}\ln^2 x}\frac{d\left(\sum_{n\leq t}a_n{\left(\frac{n}{p}\right)}\right)}{t}=-\frac{\sum_{n\leq \sqrt{x}\ln^2 x}a_n{\left(\frac{n}{p}\right)}}{\sqrt{x}\ln^2 x}+$$ $$\int\limits_{\sqrt{x}\ln^2 x}\frac{\sum_{n\leq t}a_n{\left(\frac{n}{p}\right)}}{t^2}dt=O\left(\frac{\sqrt{p}\ln p}{\sqrt{x}\ln^2 x}\right)=O\left(\frac{1}{\ln x}\right),$$
as $p\leq x$. Therefore, $C(a,x,\varepsilon)$ converges to 0 in probability as $x\to +\infty$. To handle $B(a,x,\varepsilon)$, let us estimate the expectation of $B(a,x,\varepsilon)^2$. Notice the following property of ${\left(\frac{n}{p}\right)}$:
For any nonzero integer $a$ there is a Dirichlet character $\chi_a$ of modulus at most $4|a|$ such that for any odd integer $n$ we have ${\left(\frac{a}{n}\right)}=\chi_a(n)$ and $\chi_a$ is nonprincipal if and only if $a$ is not a square.
For any $a$ there are $b$ and $c$ such that $a=bc^2$, $b$ is a fundamental discriminant and $2c$ is an integer. Now, for any odd integer $n$ we have
$${\left(\frac{a}{n}\right)}={\left(\frac{b}{n}\right)}{\left(\frac{c^2}{n}\right)}={\left(\frac{b}{n}\right)}\chi_{0,2c}(n).$$
As $b$ is a fundamental discriminant, ${\left(\frac{b}{n}\right)}$ is a Dirichlet character to the modulus $|b|$ (see [@IK], p. 53). Therefore ${\left(\frac{a}{n}\right)}={\left(\frac{b}{n}\right)}\chi_{0,2c}(n)=\chi_a(n)$, where $\chi_a$ has modulus at most $2|b|c\leq 4|b|c^2=4a$, which completes the proof.
Now, the expectation of $B(a,x,\varepsilon)^2$ equals
$$\frac{1}{\pi(x,4,\varepsilon)}\sum_{\substack{p\leq x \\ p\equiv \varepsilon\pmod 4}} \left|\sum_{\ln^3 x<n\leq \sqrt{x}\ln^2 x}\frac{a_n}{n}{\left(\frac{n}{p}\right)}\right|^2$$
Due to nonnegativity of summands, we can sum over all odd numbers instead of primes and get
$$\mathbb EB(a,x,\varepsilon)^2\leq \frac{1}{\pi(x,4,\varepsilon)}\sum_{\substack{d\leq x \\ 2\nmid d}}\left|\sum_{\ln^3 x<n\leq \sqrt{x}\ln^2 x} \frac{a_n}{n}{\left(\frac{n}{d}\right)}\right|^2=$$
$$=\frac{1}{\pi(x,4,\varepsilon)}\sum_{\substack{d\leq x \\ 2\nmid d}} \sum_{\ln^3 x<n,k<\sqrt{x}\ln^2 x}\frac{a_na_k}{nk}{\left(\frac{nk}{d}\right)}=\frac{1}{\pi(x,4,\varepsilon)}\sum_{\substack{d \leq x \\ 2\nmid d}}\sum_{\ln^6 x<m<x\ln^4 x}\frac{b_m}{m}{\left(\frac{m}{d}\right)},$$
where $b_m=\sum\limits_{\ln^3 x<n,k<\sqrt{x}\ln^2 x}a_na_k$. As $|a_n|=O(1)$ for all $n$, we have $b_m=O(\tau(m))$. Changing the order of summation and using Lemma 2 we deduce the inequality
$$\mathbb EB(a,x,\varepsilon)^2\leq \frac{1}{\pi(x,4,\varepsilon)}\sum_{\ln^6 x<m<x\ln^4 x}\frac{|b_m|}{m}\left|\sum_{d\leq x, 2\nmid d} \chi_m(d)\right|$$
Now, if $m$ is a square, then inner sum is trivially estimated by $x$, while if it isn’t a square, the Pólya-Vinogradov inequality gives us the bound $O(\sqrt{m}\ln m)$. Also, $\pi(x,4,\varepsilon)\sim \frac{x}{2\ln x}$ therefore
$$\mathbb EB(a,x,\varepsilon)^2\ll \frac{\ln x}{x}\left(x\sum_{\ln^3 x<l<\sqrt{x}\ln^2 x} \frac{\tau(l^2)}{l^2}+\sum_{\ln^6 x<m<x\ln^4 x}\frac{\tau(m)\ln m}{\sqrt{m}}\right).$$
As for any $\delta>0$ the inequality $\tau(m)\ll m^\delta$ holds, we obtain
$$\sum_{\ln^3 x<l<\sqrt{x}\ln^2 x}\frac{\tau(l^2)}{l^2}\ll \frac{1}{\ln^2 x}$$
and
$$\sum_{\ln^6 x<m<x\ln^4 x}\frac{\tau(m)\ln m}{\sqrt{m}}\ll x^{2/3}.$$
These bounds result in the estimate
$$\mathbb EB(a,x,\varepsilon)^2 \ll \frac{1}{\ln x},$$
which implies that $B(a,x,\varepsilon)$ converges to $0$ in probability as $x\to +\infty$.
So, we are left with the shortest part of our sum. To prove that $A(a,x,\varepsilon)$ converges to $L(a)$ in distribution, we are going to use the moment method in the following form:
Let $\xi_n$ be a sequence of real random variables such that for some random variable $\xi$ and all natural numbers $k$ the identity $$\lim_{n\to +\infty}\mathbb E\xi_n^k=\mathbb E\xi^k$$
holds. Assume that $\xi$ satisfies the Carleman’s criterion
$$\sum_{n\geq 1}(\mathbb E\xi^{2n})^{-1/2n}=+\infty.$$
Then the convergence in distribution
$$\xi_n \xrightarrow{d} \xi$$
takes place.
See [@Akh], p. 85.
In moment computation we will use the following classical result on primes in arithmetic progressions
Let $A>0$ be a fixed real number. Then there is a positive constant $c_A$ such that for any real $x>0$ and nonprincipal Dirichlet character $\chi$ modulo $q\leq (\ln x)^A$ the estimate
$$\pi(x;\chi)=\sum_{p\leq x}\chi(p)=O\left(xe^{-c_A\sqrt{\ln x}}\right)$$
holds.
See [@K], p. 138.
As an immediate consequence, we obtain the following result on expectations of Legendre symbols:
For any positive integer $k$ there is a positive constant $c_k$ such that for all $0<n\leq (\ln x)^k$ and $\varepsilon=\pm 1$ we have
$$\mathbb E{\left(\frac{n}{\mathfrak{p}_x^\varepsilon}\right)}=\square(n)+O(\exp(-c_k\sqrt{\ln x})),$$
where $\square(n)=1$ if $n$ is a square and $0$ otherwise.
Indeed, this expectation can be rewritten as
$$\mathbb E{\left(\frac{n}{\mathfrak{p}_x^\varepsilon}\right)}=\frac{1}{\pi(x,4,\varepsilon)}\sum_{\substack{p\leq x \\ p\equiv \varepsilon \pmod 4}}{\left(\frac{n}{p}\right)}=\frac{1}{2\pi(x,4,\varepsilon)}\sum_{p\leq x}\left({\left(\frac{n}{p}\right)}+\varepsilon{\left(\frac{-4n}{p}\right)}\right).$$
Applying Lemma 2 and Lemma 4, we get the desired result.
Using Corollary 1, one can easlily deduce the formula for $k-$th moment of $A(a,x,\varepsilon)$. Note first that
$$A(a,x,\varepsilon)^k=\sum_{n_i\leq \ln^3 x}\frac{a_{n_1}a_{n_2}\ldots a_{n_k}}{n_1\ldots n_k}{\left(\frac{n_1\ldots n_k}{\mathfrak{p}_x^\varepsilon}\right)}.$$
Rearranging the summands according to the product of variables, we get
$$A(a,x,\varepsilon)^k=\sum_{n\leq \ln^{3n}x}\frac{\tau_k(n;a,x)}{n}{\left(\frac{n}{\mathfrak{p}_x^\varepsilon}\right)},$$
where $\tau_k(n;a,x)=\sum\limits_{n_1\ldots n_k=n, n_i\leq \ln^3 x} a_{n_1}\ldots a_{n_k}$. Also, one can easily show that $|\tau_k(n;a,x)|\ll \tau_k(n)$ and $\tau_k(n;a,x)\to \tau_k(n;a)=\tau_k(n;a,\infty)$. Using the linearity of expectation and Corollary 1, we deduce that
$$\mathbb EA(a,x,\varepsilon)^k=\sum_{n\leq \ln^k x}\frac{\tau_k(n^2;a,x)}{n^2}+O(\exp(-0.5c_k\sqrt{\ln x})).$$
Thus, by dominated convergence theorem we obtain
$$\lim_{x\to +\infty}\mathbb EA(a,x,\varepsilon)^k=\sum_{n}\frac{\tau_k(n^2;a)}{n^2}.$$
Now, using dyadic subdivision, one can prove that $k-$th moment of $L(a)$ exists and equals
$$\mathbb EL(a)^k=\sum_{n_1,\ldots,n_k}\frac{a_{n_1}\ldots a_{n_k}\mathbb EX_{n_1\ldots n_k}}{n_1\ldots n_k}=\sum_n \frac{\tau_k(n^2;a)}{n^2},$$
because the expectation in this sum is only nonzero when $n_1\ldots n_k$ is a square. We need to prove that $L(a)$ satisfies the Carleman’s condition. To do so, notice that $|\tau_k(n^2;a)|\leq C^k \tau_k(n^2)$. Therefore,
$$EL(a)^{2k}\leq C^{2k}\sum_n \frac{\tau_{2k}(n^2)}{n^2}.$$
As $\tau_{2k}(n^2)$ is multiplicative, we get
$$EL(a)^{2k}\leq C^{2k}\prod_p\left(\frac{1}{2}\left(\frac{1}{(1-p^{-1})^{2k}}+\frac{1}{(1+p^{-1})^{2k}}\right)\right).$$
Let us split this product in two parts. For primes $p\leq k^2$ we use the inequality
$$\frac{1}{2}\left(\frac{1}{(1-p^{-1})^{2k}}+\frac{1}{(1+p^{-1})^{2k}}\right)\leq \frac{1}{(1-p^{-1})^{2k}},$$
from which we deduce by Mertens’ third theorem that
$$\prod_{p\leq k^2} \left(\frac{1}{2}\left(\frac{1}{(1-p^{-1})^{2k}}+\frac{1}{(1+p^{-1})^{2k}}\right)\right)\leq \prod_{p\leq k^2}\frac{1}{(1-p^{-1})^{2k}}\leq B^{2k}(\ln k)^{2k}$$
for some absolute constant $B$. Further, for $p>k^2$ the relation $$\frac{1}{2}\left(\frac{1}{(1-p^{-1})^{2k}}+\frac{1}{(1+p^{-1})^{2k}}\right)=\frac{1+p^{-2}{2k\choose 2}+p^{-4}{2k\choose 4}+\ldots+p^{-2k}{2k\choose 2k}}{(1-p^{-2})^{2k}}$$ holds.
Using the inequality ${2k\choose m}\leq (2k)^m$ for all $m$, we obtain for $k>2$ the estimate
$$\frac{1}{2}\left(\frac{1}{(1-p^{-1})^{2k}}+\frac{1}{(1+p^{-1})^{2k}}\right)\leq \frac{1+8p^{-2}k^2}{(1-p^{-2})^{2k}}$$
Thus for all $k>2$ we get $$\prod_{p>k^2} \left(\frac{1}{2}\left(\frac{1}{(1-p^{-1})^{2k}}+\frac{1}{(1+p^{-1})^{2k}}\right)\right)\leq \prod_{p>k^2}\frac{1+8p^{-2}k^2}{(1-p^{-2})^{2k}}$$ $$\leq \left(\frac{\pi^2}{6}\right)^{2k}\exp\left(\sum_{p>k^2}\frac{8k^2}{p^2}\right)\ll\left(\frac{\pi^2}{6}\right)^{2k}.$$
Combining these bounds we obtain for some fixed $B$ and $C$
$$\mathbb EL(a)^{2k}\ll \left(\frac{\pi^2}{6}BC\ln k\right)^{2k}.$$
This upper bound implies that
$$\sum_{k\geq 1}(\mathbb EL(a)^{2k})^{-1/2k}\gg \sum_{k\geq 3}\frac{1}{\ln k}=+\infty.$$
Therefore $L(a)$ satisfies the Carleman’s condition and this completes the proof.
Theorem 5 can be used in very different settings, but we are going to use the following corollary:
If $a_n=\sum_m \lambda_m e^{2\pi i n\alpha_m}$ for some finite sequence of complex $\lambda_m$ and real $\alpha_m$ and $a_n$ is real for all $n$, then $L(a,x,\varepsilon)$ converges to $L(a)$ in distribution.
Obviously, $a_n$ is bounded. Therefore, it is enough to check that for any real $\alpha$ the estimate
$$\sum_{n\leq N}e^{2\pi i \alpha n}{\left(\frac{n}{p}\right)}\ll \sqrt{p}\ln p$$
holds for almost all primes $p$. To show that this is indeed the case, note that if $\alpha$ is rational and nonzero then $||\alpha p||\gg 1$ for $p$ large enough and if $\alpha$ is irrational then the sequence $\alpha p_n$ is uniformly distributed modulo 1, so for almost all $p$ the inequality $||\alpha p||\geq \frac{1}{\ln p}$ holds. Using the Fourier expansion for the Legendre symbol, we get for almost all $p$
$$\sum_{n\leq N}e^{2\pi i \alpha n}{\left(\frac{n}{p}\right)}=\frac{1}{\tau\left({\left(\frac{\cdot}{p}\right)}\right)}\sum_{a=1}^{p-1}\sum_{n\leq N}\exp(2\pi in(\alpha-a/p))\ll \frac{1}{\sqrt{p}}\sum_{a=1}^{p-1}||\alpha-a/p||^{-1}\ll$$
$$\ll \frac{1}{\sqrt{p}}\sum_{a=-p/2}^{p/2}\frac{p}{|a+1/\ln p|}\ll\sqrt{p}\ln p,$$
which completes the proof for $\alpha\neq 0$. If $\alpha=0$ then the desired bound is just the Pólya-Vinogradov inequality.
Corollary 2 has a very useful implication on our main problem. Namely, from Section 2 we deduce that we need to study the distribution of $L(a^{\pm},x,\pm1)$ with $a^{\pm}_n$ being an exponential polynomial of variable $n\alpha.$ We are also going to use the following well-known formula for the Gauss sum:
Let $p$ be an odd prime number. Then we have
$$\tau\left({\left(\frac{\cdot}{p}\right)}\right)=\begin{cases}
\sqrt{p}\text{ if }p\equiv 1\pmod 4\\
i\sqrt{p}\text{ if }p\equiv 3\pmod 4.
\end{cases}$$
See [@IK], p. 49.
From this we get the following:
Let $\alpha$ be a real number, $a^{+}_n(\alpha)=\sin 2\pi n\alpha$ and $a^{-}_n(\alpha)=1-\cos2\pi n\alpha$. Let $c(\alpha)$ be a lower density among primes of primes $p$, satisfying $L(\alpha,p)\geq 0$, $c^{+}(\alpha)$ and $c^{-}(\alpha)$ be the probabilities of positivity of random variables $L(a^{+}(\alpha))$ and $L(a^{-}(\alpha))$. Then we have
$$c(\alpha)\geq \frac{c^{+}(\alpha)+c^{-}(\alpha)}{2}.$$
In particular, if $c^{+}(\alpha)+c^{-}(\alpha)>1$, then Conjecture 1 is true for $\alpha.$
First, it is enough to only consider large enough odd prime numbers, because we can ignore any finite number of primes. Now, if $p\equiv 1 \pmod 4$ then we have by Theorem 4 and
$$L(\alpha,p)=\tau\left({\left(\frac{\cdot}{p}\right)}\right)\sum_{m\in \mathbb Z, m\neq 0}\frac{1-e^{-2\pi i \alpha m}}{2\pi i m}{\left(\frac{m}{p}\right)}=$$ $$=\sqrt{p}\sum_{m> 0}\left(\frac{1-e^{-2\pi i \alpha m}}{2\pi i m}-\frac{1-e^{2\pi i \alpha m}}{2\pi i m}\right){\left(\frac{m}{p}\right)}=\frac{\sqrt{p}}{\pi}\sum_{m>0}\frac{\sin 2\pi m\alpha}{m}{\left(\frac{m}{p}\right)},$$
because ${\left(\frac{-m}{p}\right)}={\left(\frac{-1}{p}\right)}{\left(\frac{m}{p}\right)}={\left(\frac{m}{p}\right)}.$ Therefore, the proportion of primes $p\leq x$ with $p\equiv 1 \pmod p$ with $L(\alpha,p)\geq 0$ among all primes $p\leq x$ is
$$\frac{\pi(x;4,1)}{\pi(x)}\mathbb P(L(a^{+},x,1)\geq 0)\sim \frac{\mathbb P(L(a^{+},x,1)\geq 0)}{2}$$
as $x\to +\infty$. Now, as $L(a^{+},x,1)$ converges to $L(a^{+})$ in distribution and the subset $\mathbb R_{>0} \subset \mathbb R$ is open, we have
$$\liminf_{x\to +\infty}\mathbb P(L(a^{+},x,1)\geq 0)\geq \liminf_{x\to +\infty}\mathbb P(L(a^{+},x,1)>0)\geq \mathbb P(L(a^{+})>0)=c^{+}(\alpha).$$
If $p\equiv 3 \pmod 4$, then we have similarly
$$L(\alpha,p)=\tau\left({\left(\frac{\cdot}{p}\right)}\right)\sum_{m\in \mathbb Z, m\neq 0}\frac{1-e^{-2\pi i \alpha m}}{2\pi i m}{\left(\frac{m}{p}\right)}=$$ $$=i\sqrt{p}\sum_{m> 0}\left(\frac{1-e^{-2\pi i \alpha m}}{2\pi i m}+\frac{1-e^{2\pi i \alpha m}}{2\pi i m}\right){\left(\frac{m}{p}\right)}=\frac{\sqrt{p}}{\pi}\sum_{m>0}\frac{1-\cos 2\pi m\alpha}{m}{\left(\frac{m}{p}\right)}.$$
Hence, the proportion of primes $p\leq x$ with $p\equiv 3 \pmod p$ with $L(\alpha,p)\geq 0$ among all primes $p\leq x$ is
$$\frac{\pi(x;4,3)}{\pi(x)}\mathbb P(L(a^{-},x,-1)\geq 0)\sim \frac{\mathbb P(L(a^{-},x,-1)\geq 0)}{2}.$$
And we also have
$$\liminf_{x\to +\infty}\mathbb P(L(a^{-},x,1)\geq 0)\geq \liminf_{x\to +\infty}\mathbb P(L(a^{-},x,1)>0)\geq \mathbb P(L(a^{-})>0)=c^{-}(\alpha).$$
Therefore, we obtain
$$\liminf_{x\to +\infty}\frac{\#\{p\leq x: L(\alpha,p)\geq 0\}}{\pi(x)}\geq \frac{c^{+}(\alpha)+c^{-}(\alpha)}{2},$$
as needed.
Rational alpha with small denominators
======================================
In this section we are going to prove Theorem 2. In other words, we are going to prove Conjecture 1 for the following values of $\alpha$:
$$\alpha=0,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{6},\frac{1}{8},\frac{3}{8},\frac{1}{12},\frac{5}{12},\frac{1}{5},\frac{2}{5}.$$
All the denominators of these numbers, except for $5$, are precisely the numbers $n$ such that the group $\left(\mathbb Z/n\mathbb Z\right)^*$ has exponent $1$ or $2$, i.e. such that $a^2\equiv 1\pmod n$ for all $a$ coprime to $n$. The reason for such a choice of denominators is that for such $n$ all the Dirichlet characters modulo $n$ are real-valued and so we don’t need to consider any complex Euler products. On the other hand, in the case of $\alpha=\frac{1}{5}$ or $\frac{2}{5}$ we need to compute arguments of certain complex-valued Euler products, which is going to make things more complicated. Probably, the list of $n$ such that Conjecture 1 follows from conditions $n\alpha\in \mathbb Z$ and $\alpha<\frac{1}{2}$ can be expanded in a way similar to what is presented in this section, but we don’t know if it works for all rational $\alpha$.
For a natural number $m$ the function $\chi_{0,m}(\cdot)$ will be the principal character modulo $m$. If $\chi$ is a Dirichlet character and $\beta$ is not an integer then we are going to define $\chi(\beta)$ to be equal $0$. So, for example, $\chi_{0,2}\left(\frac{n}{2}\right)$ is a 4-periodic function with values $0,1,0,0,0,1,\ldots$
Let us start with a few simple cases.
If $\alpha=0$ then our conjecture is trivial, as we always have $L(0,p)=0$, so $c(0)=1$.
If $\alpha=\frac{1}{2}$, then
$$a^{+}_n(\alpha)=\sin \pi n=0$$
for all $n$ and
$$a^{-}_n(\alpha)=1-\cos\pi n=\begin{cases}
2 \text{ if }n\text{ is odd}\\
0 \text{ otherwise}
\end{cases}$$
From these formulas we get
$$L(a^{+})=0$$
for all possible realizations of $(X_n)$. Therefore, for all primes $p\equiv 1\pmod 4$ we have $L(1/2,p)=0$. Indeed, in the previous section we learned that for $p \equiv 1 \pmod 4$ the quantity $\frac{\pi}{\sqrt{p}}L(\alpha,p)$ is a particular realization of random variable $L(a^{+})$, while for $p\equiv 3 \pmod 4$ it is a realization of random variable $L(a^{-})$. From our formulas we also get
$$L(a^{-})=2\sum_{n>0}\frac{\chi_{0,2}(n)X_n}{n}=2\prod_{p}\left(1-\frac{\chi_{0,2}(p)X_p}{p}\right)^{-1}\geq 0,$$
due to multiplicativity of $\chi_{0,2}(n)X_n$. Therefore, for all primes $p$ we have $L(1/2,p)\geq 0$, so $c(1/2)=1$.
Now, for $\alpha=\frac{1}{3}$ one can easily check that
$$\label{aplus3}
a_n^{+}(1/3)=\sin\frac{2\pi n}{3}=\frac{\sqrt{3}}{2}{\left(\frac{n}{3}\right)}$$
and
$$\label{amin3}
a_n^{-}(1/3)=1-\cos\frac{2\pi n}{3}=\frac{3}{2}\chi_{0,3}(n).$$
and we again get nonnegativity of all realizations from the Euler products for corresponding series. Hence in this case we again have $c(\alpha)=c(1/3)=1$.
The same is true for $\alpha=\frac{1}{4}$, because in that case we get
$$a_n^{+}(1/4)=\sin\frac{\pi n}{2}={\left(\frac{-4}{n}\right)}=\chi_4(n)$$
and
$$a_n^{-}(1/4)=1-\cos\frac{\pi n}{2}=\chi_{0,2}(n)+2\chi_{0,2}\left(\frac{n}{2}\right),$$
therefore $c(1/4)=1$, because
$$L(a^{+})=\prod_p \left(1-\frac{\chi_4(p)X_p}{p}\right)^{-1}\geq 0$$
and
$$L(a^{-})=\prod_p \left(1-\frac{\chi_{2,0}(p)X_p}{p}\right)^{-1}+X_2\prod_p \left(1-\frac{\chi_{0,2}(p)X_p}{p}\right)^{-1}=$$ $$(1+X_2)\prod_{p>2} \left(1-\frac{X_p}{p}\right)^{-1}\geq 0.$$ The factor $X_2$ comes from the fact that for any $\chi$ and any $d$ we have
$$\sum_{n}\frac{\chi(n/d)X_n}{n}=\sum_{n}\frac{\chi(n)X_{nd}}{nd}=\frac{X_d}{d}\sum_n \frac{\chi(n)X_n}{n}.$$
For $\alpha=\frac{1}{6}$ we once again obtain $c(1/6)=1$, but for a more complicated reasons.
Here we have
$$a_n^{+}(\alpha)=\sin\frac{\pi n}{3}.$$
The values of $a_n^{+}$ are $\frac{\sqrt{3}}{2},-\frac{\sqrt{3}}{2},0,-\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2},0,\ldots$ (the sequence is 6-periodic) so we get
$$a_n^{+}(\alpha)=\frac{\sqrt{3}}{2}(\chi_6(n)+{\left(\frac{n/2}{3}\right)}).$$
Here $\chi_6(n)=\pm 1$ if $n\equiv \pm 1\pmod 6$ and 0 otherwise. Note also that $\chi_6(n)=\chi_{0,2}(n){\left(\frac{n}{3}\right)}.$ Therefore,
$$L(a^{+})=\frac{\sqrt{3}}{2}\left(\sum_n\frac{\chi_{0,2}(n){\left(\frac{n}{3}\right)}X_n}{n}+\sum_n\frac{{\left(\frac{n/2}{3}\right)}X_n}{n}\right)=$$
$$\frac{\sqrt{3}}{2}(1+X_2)\prod_{p}\left(1-\frac{{\left(\frac{p}{3}\right)}X_p}{p}\right)^{-1}\geq 0,$$
so we again have $L(\alpha,p)\geq 0$ for all $p\equiv 1 \pmod 4$.
Values of $a_n^{-}$ are $\frac{1}{2},\frac{3}{2},2,\frac{3}{2},\frac12,0\ldots$ and this function can be expanded as follows:
$$a_n^{-}=1-\cos\frac{\pi n}{3}=2\chi_{0,2}\left(\frac{n}{3}\right)+\frac{1}{2}\chi_{0,3}(n)+\chi_{0,3}\left(\frac{n}{2}\right).$$
Nonnegativity is not obvious from this formula, so let us work with Euler products
$$L(a^{-})=\sum_{n}\frac{a_n^{-}}{n}=\frac{2X_3}{3}\sum_n \frac{\chi_{0,2}(n)X_n}{n}+\frac{1+X_2}{2} \sum_{n}\frac{\chi_{0,3}(n)X_n}{n}=$$ $$=\left(\frac{2X_3}{3}\left(1-\frac{X_2}{2}\right)+\frac{1+X_2}{2}\left(1-\frac{X_3}{3}\right)\right)\sum_n \frac{X_n}{n}=\frac{1+X_2+X_3-X_2X_3}{2}\sum_n \frac{X_n}{n}\geq 0.$$ This calculation shows that $c(1/6)=1.$
Now we are going to consider a different set of values $\alpha$ for which we don’t have $c(\alpha)=1$.
If $\alpha=\frac18$ then the first $8$ values of $a_n^{+}(1/8)=\sin\frac{\pi n}{4}$ are equal to $$\frac{\sqrt{2}}{2},1,\frac{\sqrt{2}}{2},0,-\frac{\sqrt{2}}{2},-1,-\frac{\sqrt{2}}{2},0,$$ which is equal to $\frac{\sqrt{2}}{2}{\left(\frac{-2}{n}\right)}+\chi_4\left(\frac{n}{2}\right)$ thus we obtain
$$L(a^{+})=\frac{\sqrt{2}}{2}\sum_n\frac{{\left(\frac{-2}{n}\right)}X_n}{n}+\frac{X_2}{2}\sum_n \frac{\chi_4(n)X_n}{n}$$
and hence $c^{+}(1/8)>\frac12$, because for $X_2=+1$ we have $L(a^{+})\geq 0$ and for $X_2=-1$ our random variable is positive with positive probability, because
$$\mathbb E(L(a^{+})\mid X_2=-1)=\frac{\sqrt{2}-1}{2}\sum_{2\nmid n}\frac{1}{n^2}=\frac{(\sqrt{2}-1)\pi^2}{18}>0.$$
On the other hand, the first $8$ values of $a_n^{-}(1/8)$ are $$1-\frac{\sqrt{2}}{2},1,1+\frac{\sqrt{2}}{2},2,1+\frac{\sqrt{2}}{2},1,1-\frac{\sqrt{2}}{2},0$$
and one can check that we have
$$a_n^{-}(1/8)=\chi_{0,2}(n)+\chi_{0,2}(n/2)+2\chi_{0,2}(n/4)-\frac{\sqrt{2}}{2}{\left(\frac{2}{n}\right)}.$$
Therefore
$$L(a^{-})=\left(\frac32+\frac{X_2}{2}\right)\sum_n \frac{\chi_{0,2}(n)X_n}{2}-\frac{\sqrt{2}}{2}\sum_n \frac{{\left(\frac{2}{n}\right)}X_n}{n}\geq$$ $$\geq \sum_n \frac{\chi_{0,2}(n)X_n}{n}-\frac{\sqrt{2}}{2}\sum_n \frac{{\left(\frac{2}{n}\right)}X_n}{n}=F-\frac{\sqrt{2}}{2}G.$$
Now, let us notice, that the distribution of $(\lambda(n)X_n)$, where $\lambda(n)$ is the Liouville’s function, coincides with the distribution of $(X_n)$, as $-X_p$ are also independent and have Rademacher’s distribution. On the other hand, if we replace $X_n$ by $\lambda(n)X_n$, $F$ and $G$ will transform into
$$\sum_n \frac{\lambda(n)\chi_{0,2}(n)X_n}{n}=\prod_{p>2}\left(1+\frac{X_p}{p}\right)^{-1}=\prod_{p>2}\left(1-\frac{1}{p^2}\right)^{-1}\left(1-\frac{X_p}{p}\right)=\frac{\pi^2}{9F}$$
and (by the same argument) $\frac{\pi^2}{9G}$, respectively. Also, $\ln F-\ln G$ has continuous distribution (because characteristic function goes to $0$ at infinity), so $\mathbb P(L(a^{-})>0)\geq \frac12$, because $F-\frac{\sqrt{2}}{2}G\geq 0$ holds either for $X_n$ or for $\lambda(n)X_n$, so we get $c(1/8)>\frac12$.
As for the second case of denominator $8$, i.e. $\alpha=\frac38$, using $a_{3n}^{\pm}(1/8)=a_{n}^{\pm}(3/8)$ we get
$$L(a^{+}(3/8))=\frac{\sqrt{2}}{2}\sum_n\frac{{\left(\frac{-2}{n}\right)}X_n}{n}-\frac{X_2}{2}\sum_n \frac{\chi_4(n)X_n}{n},$$
so $c^{+}(3/8)=c^{+}(1/8)>\frac12$. Also, in the formula for $L(a^{-})$ we will get $+G$ instead of $-G$, so that
$$L(a^{-})\geq F+\frac{\sqrt{2}}{2}G\geq 0.$$
Hence we get $c^{-}(3/8)=1$ and $c(3/8)\geq \frac34$.
The last two cases without any complex Dirichlet characters are $\alpha=\frac{1}{12}$ and $\alpha=\frac{5}{12}$.
Let us start with $\alpha=\frac{1}{12}$. For the $a_n^{+}=\sin\frac{\pi n}{6}$ we get the first twelve values
$$\frac12,\frac{\sqrt{3}}{2},1,\frac{\sqrt{3}}{2},\frac12,0,-\frac12,-\frac{\sqrt{3}}{2},-1,-\frac{\sqrt{3}}{2},-\frac12,0$$
and find the formula
$$a_n^{+}(1/12)=\frac12 \chi_4(n)\chi_{0,3}(n)+\frac{\sqrt{3}}{2}{\left(\frac{n/2}{3}\right)}\chi_{0,2}(n/2)+\chi_4(n/3)+\frac{\sqrt{3}}{2}{\left(\frac{n/4}{3}\right)}$$
and get
$$L(a^{+})=\frac{1+X_3}{2}\sum_n \frac{\chi_4(n)X_n}{n}+\frac{\sqrt{3}(X_2+1)}{4}\sum_n \frac{\chi_3(n)X_n}{n},$$
thus, $L(1/12,p)\geq 0$ for all $p\equiv 1 \pmod 4$.
For cosines we compute
$$1-\cos\frac{\pi n}{6}=1-\frac{\sqrt{3}}{2},\frac12,1,\frac32,1+\frac{\sqrt{3}}{2},2,1+\frac{\sqrt{3}}{2},\frac32, 1,\frac12,1-\frac{\sqrt{3}}{2},0,$$
so that
$$a_n^{-}=-\frac{\sqrt{3}}{2}{\left(\frac{12}{n}\right)}+\chi_{0,2}(n)+\frac12\chi_{0,6}(n/2)+\frac32 \chi_{0,3}(n/4)+2\chi_{0,2}(n/6).$$
Observe that $a_n^{-}$ is always nonnegative, so that we have $\mathbb EL(a^{-})>0$ and $c^{-}(1/12)>0$, which results in $c(1/12)>\frac12.$
When $\alpha=\frac{5}{12}$ we apply the formula $a_n^{\pm}(5/12)=a_{5n}(1/12)$, and for $a^{+}$ this results in
$$L(a^{+})=\frac{1+X_3}{2}\sum_n \frac{\chi_4(n)X_n}{n}-\frac{\sqrt{3}(X_2+1)}{4}\sum_n \frac{\chi_3(n)X_n}{n},$$
so that $L(a^{+})>0$ for almost all $X_n$ with $X_2=-1$ and $X_3=1$, therefore $c^{+}(5/12)\geq\frac14$. On the other hand, we have ${\left(\frac{12}{5}\right)}=-1$, so
$$a_n^{-}=\frac{\sqrt{3}}{2}{\left(\frac{12}{n}\right)}+\chi_{0,2}(n)+\frac12\chi_{0,6}(n/2)+\frac32 \chi_{0,3}(n/4)+2\chi_{0,2}(n/6),$$
which gives
$$L(a^{-})=\frac14(1+(1-X_2)(1-X_3))\sum_n \frac{X_n}{n}+\frac{\sqrt{3}}{2}\sum_n \frac{{\left(\frac{12}{n}\right)}X_n}{n},$$
so that $c^{+}(5/12)=1$ and $c(5/12)\geq \frac{1+1/4}{2}=\frac{5}{8}$.
Now, for the last case, where we have to deal with some complex characters. If $\alpha=\frac15$, then the function $a_n^{-}=1-\cos\frac{2\pi n}{5}$ takes values
$$\frac{5-\sqrt{5}}{4},\frac{5+\sqrt{5}}{4},\frac{5+\sqrt{5}}{4},\frac{5-\sqrt{5}}{4},0$$
and we get
$$L(a^{-})=\sum_n \frac{\left(\frac52\chi_{0,5}(n)-\frac{\sqrt{5}}{2}{\left(\frac{n}{5}\right)}\right)X_n}{n}=\frac52 H-\frac{\sqrt{5}}{2}T,$$
where
$$H=\sum_n \frac{\chi_{0,5}(n)X_n}{n}=\prod_{p\neq 5}\left(1-\frac{X_p}{p}\right)^{-1}$$
and
$$T=\sum_n \frac{{\left(\frac{n}{5}\right)}X_n}{n}=\prod_{p\neq 5}\left(1-\frac{{\left(\frac{n}{p}\right)}X_p}{p}\right)^{-1}.$$
The map $X_n\mapsto \lambda(n)X_n$ transforms $H$ into $\frac{4\pi^2}{25H}$ and $T$ into $\frac{4\pi^2}{25T}$, so $c^{-}(1/5)\geq \frac12.$
Expressions for $\sin\frac{2\pi n}{5}$ involve nested radicals, which results in formula
$$\sin\frac{2\pi n}{5}=\frac{A-iB}{2}\kappa(n)+\frac{A+iB}{2}\overline{\kappa(n)},$$
where $\kappa$ is a Dirichlet character modulo $5$ with $\kappa(2)=i$ and
$$A=\sqrt{\frac{5+\sqrt{5}}{8}}, B=\sqrt{\frac{5-\sqrt{5}}{8}}.$$
Therefore, the inequality $L(a^{+})\geq 0$ is equivalent to
$$\mathrm{Re}\,\left((A-iB)\sum_{n}\frac{\kappa(n)X_n}{n}\right)\geq 0$$
Now, $A+iB=\sqrt{A^2+B^2}e^{i\varphi}$ with $\varphi=\arctan\left(\frac{B}{A}\right)\approx 0.553$ and our inequality takes the form
$$\cos\left(-\varphi+\mathrm{arg}\,\left(\sum_n \frac{\kappa(n)X_n}{n}\right)\right)\geq 0.$$
The sum inside cosine can be expanded into the Euler product:
$$\sum_n \frac{\kappa(n)X_n}{n}=\prod_p\left(1-\frac{\kappa(p)X_p}{p}\right)^{-1}.$$
Obviously, if $\kappa(p)$ is real, then $p$ gives no contribution to the sum, and if $\kappa(p)$ is not real (which is equivalent to $p\equiv \pm 2\pmod 5$) then
$$\left(1-\frac{\kappa(p)X_p}{p}\right)^{-1}=\frac{\exp(\kappa(p)X_p\arctan{\frac{1}{p}})}{\sqrt{1+p^{-2}}}.$$
From this we deduce the formula
$$\sum_n \frac{\kappa(n)X_n}{n}=Me^{i\xi},$$
where $M$ is an almost surely positive real random variable and
$$\xi=\sum_{p\equiv \pm 2\pmod 5}\frac{\kappa(p)}{i}X_p\arctan{\frac{1}{p}},$$
which is also real. Therefore, our desired inequality takes form
$$\cos(\xi-\varphi)\geq 0.$$
Notice that if it is not true, then we should have $|\xi-\varphi|>\frac{\pi}{2}$, in which case $|\xi|>\frac{\pi}{2}-\varphi.$
By Chebyshev inequality, probability of this event is at most
$$\frac{\mathbb E\xi^2}{\left(\frac{\pi}{2}-\varphi\right)^2}\leq \frac{0.3536}{1.077}<\frac13,$$
because
$$\mathbb E\xi^2=\sum_{p\equiv \pm 2\pmod 5}\arctan^2{\frac{1}{p}}\approx 0.35355.$$
From this we obtain $c^{+}(1/5)\geq \frac23$ and so $c(1/5)\geq \frac{7}{12}>\frac12$, as needed.
If $\alpha=\frac25$, then the proof is easier, because in this case we have
$$a_n^{-}=1-\cos\frac{4\pi n}{5}=\frac52\chi_{0,5}(n)+\frac{\sqrt{5}}{2}{\left(\frac{n}{5}\right)},$$
thus
$$L(a^{-})=\frac52 H+\frac{\sqrt{5}}{2}T\geq 0,$$
so for all $p\equiv 3 \pmod 4$ we have $L(\frac25,p)\geq 0$. On the other hand, the set of $p\equiv 1 \pmod 4$ with $L(\frac25,p)>0$ has a positive lower density, because we have
$$\mathbb EL(a^{+})=\sum_n \frac{\sin\frac{4\pi n^2}{5}}{n^2}=\sqrt{\frac{5-\sqrt{5}}{8}}\sum_{n>0}\frac{{\left(\frac{n}{5}\right)}}{n^2}>0.$$
From this we get $c(2/5)>\frac12$, which concludes our proof.
Values of alpha close to one-third
==================================
In the previous section we proved Conjecture 1 for several special values of $\alpha$, using expansion with Dirichlet characters and certain symmetry considerations. Here we are going to use a different approach to the main conjecture. One can notice that the all random variables that we constructed are linear combinations of a fixed sequence of random variables $X_n$ with coefficients that are smooth functions of a parameter $\alpha$. Using this observation, we are going to prove Theorem 3.
Let us define certain class of random variables that appears naturally when we study $L(a^{\pm})$.
Let $\sigma^2>0$. We say that random variable lies in the class $L(\sigma^2)$, if there is an infinite sequence of real numbers $a_1,a_2,\ldots$ with
$$\sum_{i}a_i^2\leq \sigma^2.$$
and a sequence $\kappa_1,\kappa_2,\ldots$ of independent Rademacher random variables such that
$$\eta=\sum_i a_i\kappa_i.$$
The next lemma will allow us to control the tail of distribution from the class $L(\sigma^2)$.
Let $\eta$ be in $L(\sigma^2)$. Then for any $T>0$ we have
$$\mathbb P(\eta\geq T)\leq \exp\left(-\frac{T^2}{2\sigma^2}\right).$$
Consider the moment generating function of $\eta$:
$$F(t)=\mathbb Ee^{t\eta}.$$
It is easy to see that for any positive $t$ the inequality
$$\mathbb P(\eta\geq T)\leq e^{-tT}F(t)$$
is true. Let us show that $F(t)\leq \exp(t^2\sigma^2/2)$. Indeed, $\eta$ lies in $L(\sigma^2)$, so for some $a_i$ and $\kappa_i$ we have
$$F(t)=\mathbb Ee^{t\sum_i a_i\kappa_i}=\prod_i \mathbb Ee^{ta_i\kappa_i}=\prod_i \left(\frac{e^{-a_i t}+e^{a_it}}{2}\right).$$ The expectation of exponent equals the product of expectations due to independence of $\kappa_i$.
Note now that for any real $t$ the inequality
$$\frac{e^{-t}+e^t}{2}=1+\frac{t^2}{2}+\frac{t^4}{4!}+\ldots\leq 1+\frac{t^2}{2}+\frac{\left(\frac{t^2}{2}\right)^2}{2!}+\ldots=e^{t^2/2}$$
holds. The upper bound is true due to inequality $(2n)!\geq 2^n n!=(2n)!!$. Therefore, for any real $t$ we have
$$F(t)\leq \prod_i e^{a_i^2t^2/2}\leq e^{\sigma^2t^2/2},$$
from this we get
$$\mathbb P(\eta\geq T)\leq e^{-tT}e^{\sigma^2t^2/2}.$$
Choosing $t=T/\sigma^2$, we prove the desired inequality.
From this lemma we immediately deduce the bound for probability of negativity of random variables that are close to $\exp(\eta)$ for some $\eta\in L(\sigma^2)$.
Assume that real random variables $X$ and $Y$ satisfy $X=e^{\eta}$ for some $\eta \in L(\sigma^2)$ and
$$\mathbb E(X-Y)^2\leq D.$$ Then for any $0<u<1$ we have
$$\mathbb P(Y\leq 0)\leq \exp\left(-\frac{\ln^2 u}{8\sigma^2}\right)+\frac{D}{u}.$$
Indeed, if $Y\leq 0$ then either $(X-Y)^2\geq u$ or $X^2 \leq u$. Probability of the first event is at most $\frac{D}{u}$ due to Markov’s inequality. On the other hand, the second event implies that $\eta\leq-\frac{\ln u}{2}$, therefore $-\eta\geq\frac{\ln u}{2}$. Notice now that if $\eta\in L(\sigma^2)$ then the same is true for $-\eta$, as one can choose $-a_i$ instead of $a_i$ in the defining formula. Thus, Lemma 6 implies that the probability of second event is at most $\exp\left(-\frac{\ln^2 u}{8\sigma^2}\right)$. This proves the desired estimate.
Let us show now, that random variables $L(a^{\pm}(1/3))$ are proportional to exponents of certain random variables from the class $L(\sigma^2).$
For $\sigma^2=0.395$ there are $\eta_1$ and $\eta_2$ from $L(\sigma^2)$ such that
$$L(a^{+}(1/3))=\frac{\pi}{\sqrt{3}}e^{\eta_1}$$
and
$$L(a^{-}(1/3))=\frac{\pi}{3}e^{\eta_2}.$$
Due to the formulas (\[aplus3\]) and (\[amin3\]) we have
$$L(a^{+}(1/3))=\frac{3}{2}\sum_{n>0}\frac{X_n\chi_{0,3}(n)}{n}=\frac32\prod_{p\neq 3}\left(1-\frac{X_p}{p}\right)^{-1}$$
and
$$L(a^{-}(1/3))=\frac{\sqrt{3}}{2}\sum_{n>0}\frac{{\left(\frac{n}{3}\right)}X_n}{n}=\frac{\sqrt{3}}{2}\prod_{p\neq 3}\left(1-\frac{X_p{\left(\frac{p}{3}\right)})}{p}\right)^{-1}$$
Let us observe that for any $\varepsilon=\pm 1$ the identities
$$\left(1-\frac{\varepsilon}{p}\right)\left(1+\frac{\varepsilon}{p}\right)=1-\frac{1}{p^2}$$
and
$$\left(1-\frac{\varepsilon}{p}\right)\left(1+\frac{\varepsilon}{p}\right)^{-1}=\left(\frac{p-1}{p+1}\right)^{\varepsilon}$$ hold. Multiplying this two equalities and taking a square root, we deduce
$$1-\frac{\varepsilon}{p}=\left(\frac{p-1}{p+1}\right)^{\varepsilon/2}\left(1-\frac{1}{p^2}\right)^{1/2}.$$
If we now choose $\varepsilon$ to be $X_p$ and $X_p{\left(\frac{p}{3}\right)}$, we obtain the relations
$$L(a^{-}(1/3))=A_1e^{\eta_1}$$ and $$L(a^{+}(1/3))=A_2e^{\eta_2},$$
where
$$A_1=\frac32 \prod_{p\neq 3}\left(1-\frac{1}{p^2}\right)^{1/2}=\frac32 \sqrt{\frac{\pi^2}{6}(1-1/9)}=\frac32 \sqrt{\frac{4\pi^2}{27}}=\frac{\pi}{\sqrt{3}},$$
$$A_2=\frac{A_1}{\sqrt{3}}=\frac{\pi}{3},$$
$$\eta_1=\sum_{p\neq 3}\frac{1}{2}\ln\left(\frac{p-1}{p+1}\right)X_p$$
and
$$\eta_2=\sum_{p\neq 3}\frac{1}{2}\ln\left(\frac{p-1}{p+1}\right){\left(\frac{p}{3}\right)}X_p.$$
Direct computation shows that the inequality
$$\sum_{p\neq 3}\frac{1}{4}\ln^2\left(\frac{p-1}{p+1}\right)<0.395,$$
is true, which concludes the proof of lemma.
Now we are going to show that if $\alpha$ is close to $\frac13$ then random variables $L(a^{\pm}(\alpha))$ and $L(a^{\pm}(1/3))$ are in some sense close to each other. But first let us prove two auxiliary propostitions:
Let $\tau(n)=\sum\limits_{d\mid n}1$ be the divisor function. Then for any complex $s$ with $\mathrm{Re}\,s>1$ the equality
$$\sum_{n=1}^{+\infty}\frac{\tau(n^2)}{n^s}=\frac{\zeta(s)^3}{\zeta(2s)}$$
holds.
Due to multiplicativity of $\tau(n^2),$ we have
$$\sum_{n=1}^{+\infty}\frac{\tau(n^2)}{n^s}=\prod_p \left(1+3p^{-s}+5p^{-2s}+\ldots\right)=\prod_p\left(\frac{1-p^{-2s}}{(1-p^{-s})^3}\right),$$
which gives the desired equality.
Using this lemma, we will prove the following bound for $L^2-$distance between two random variables of certain type.
Let $f:\mathbb R\to \mathbb R$ — be a function with Lipschitz constant $L$ such that $|f(x)|\leq C$ for any real $x$. Then for any pair of real $\alpha$ and $\beta$ we have
$$\mathbb E(L(f(\alpha))-L(f(\beta)))^2\leq 92|\alpha-\beta|^{2/3}L^{2/3}C^{4/3},$$
where for any real $\gamma$ the sequence $f_n(\gamma)$ is defined by formula $f_n(\gamma)=f(n\gamma)$ and $L(f(\gamma))$ then given by Definition 3 for $a_n=f_n(\gamma).$
By definition of $L(f(\alpha))-L(f(\beta))$, we have
$$\mathbb E(L(f(\alpha))-L(f(\beta)))^2=\sum_{n,m>0}\frac{(f(n\alpha)-f(n\beta))(f(m\alpha)-f(m\beta))}{nm}\mathbb E(X_nX_m).$$
Let us notice that $\mathbb E(X_nX_m)=1$ if $nm=d^2$ for some integer $d$ and $0$ otherwise. Splitting the resulting sum according to range of $d$ into $d>A$ and $d\leq A$ parts, we get
$$\mathbb E(L(f(\alpha))-L(f(\beta)))^2=\sum_{nm=d^2, d\leq A}\frac{(f(n\alpha)-f(n\beta))(f(m\alpha)-f(m\beta))}{nm}+$$ $$+\sum_{nm=d^2, d>A}\frac{(f(n\alpha)-f(n\beta))(f(m\alpha)-f(m\beta))}{nm}.$$
Every summand of the first sum is at most $L^2|\alpha-\beta|^2$ due to the Lipschitz property, while in the second sum every summand is bounded by $\frac{4C^2}{d^2}$. Furthermore, any $d$ corresponds to exactly $\tau(d^2)$ summands. From this we obtain the inequality
$$\mathbb E(L(f(\alpha))-L(f(\beta)))^2\leq |\alpha-\beta|^2L^2\sum_{d\leq A}\tau(d^2)+4C^2\sum_{d>A}\frac{\tau(d^2)}{d^2}.$$
To get a more convenient estimate, we multiply every summand of the first sum by $(A/d)^{4/3}$, every summand of second sum by $(d/A)^{2/3}$. Obviously, both sums will not decrease. Thus,
$$\sum_{d\leq A}\tau(d^2)\leq A^{4/3}\sum_{d\leq A}\frac{\tau(n^2)}{n^{4/3}}=A^{4/3}s_1(A),$$
due to Lemma 8. Similarly,
$$\sum_{d>A}\frac{\tau(d^2)}{d^2}\leq A^{-2/3}\sum_{d>A} \frac{\tau(d^2)}{d^{4/3}}=A^{-2/3}s_2(A).$$ Observing that $s_1(A)+s_2(A)=\frac{\zeta(4/3)^3}{\zeta(8/3)}$ by lemma 8, we get for all positive $A$ $$\mathbb E(L(f(\alpha))-L(f(\beta)))^2\leq |\alpha-\beta|^2L^2A^{4/3}s_1(A)+4C^2A^{-2/3}s_2(A)\leq$$ $$\leq\frac{\zeta(4/3)^3}{\zeta(8/3)}\max(|\alpha-\beta|^2L^2A^{4/3},4C^2A^{-2/3})$$ Choosing $A=2CL^{-1}|\alpha-\beta|^{-1}$ and using relation $\frac{\zeta(4/3)^3}{\zeta(8/3)}2^{4/3}<92,$ we obtain the desired estimate.
Applying Lemma 9 to the functions $f(x)=\sin(2\pi x)$ and $f(x)=\cos(2\pi x)$ with constants $C=1$ and $L=2\pi$, we obtain the inequality
$$\label{fund}
\mathbb E(L(a^{\pm}(\alpha))-L(a^{\pm}(\beta)))^2\leq 313.3|\alpha-\beta|^{2/3}.$$
Now we have enough instruments to prove Theorem 2.
Due to Lemma 5 and inequality \[fund\], the estimates
$$\mathbb E(e^{\eta_1}-\frac{\sqrt{3}}{\pi}L(a^{-}(\alpha)))^2=\frac{3}{\pi^2}\mathbb E(L(a^{-}(1/3))-L(a^{-}(\alpha)))^2\leq \frac{926.9}{\pi^2}|\alpha-1/3|^{2/3}\leq 94|\alpha-1/3|^{2/3}$$
and $$\mathbb E(e^{\eta_2}-\frac{3}{\pi}L(a^{+}(\alpha)))^2=\frac{9}{\pi^2}\mathbb E(L(a^{+}(1/3))-L(a^{+}(\alpha)))^2\leq \frac{2780.7}{\pi^2}|\alpha-1/3|^{2/3}\leq 282|\alpha-1/3|^{2/3}$$ hold.
From this for $|\alpha-1/3|<2\cdot10^{-6}$ we deduce, using Corollary 4, that the probability of negativity of $L(a^{-}(\alpha))$ satisfies for any $0<u<1$ the inequality
$$\mathbb P(L(a^{-}(\alpha))\leq0)\leq \exp\left(-\frac{\ln^2 u}{3.16}\right)+\frac{0.015}{u}$$
and similarly
$$\mathbb P(L(a^{+}(\alpha)\leq0))\leq \exp\left(-\frac{\ln^2 u}{3.16}\right)+\frac{0.0447}{u}.$$
Computation shows that the minimal values of resulting expressions are attained at the points $u_{-}\approx 0.0756$ and $u_{+}\approx 0.12957$. Values of these expressions in corresponding points are less than or equal to $0.32$ and $0.612$ respectively. From this we obtain the inequality
$$c(\alpha)\geq 1-\frac{0.32+0.612}{2}=0.534$$ in the desired range. Therefore, if $\alpha$ lies inside $2\cdot10^{-6}$-neighbourhood of $1/3$, then the sum $L(\alpha,p)$ is nonnegative for at least 53.4% of all primes, which concludes the proof.
Conclusion and open problems
============================
In this paper we were able to reduce Conjecture 1 to the study of one fixed random variable and partially prove the conjecture. Although the progress we made simplifies the original problem, there are still a lot of questions one may ask even beyond the Conjecture 1. For example, is it true, that the $\liminf$ in \[goal\] can be replaced by $\lim$? If so, what are the properties of this limit as a function of $\alpha$? Also, from section 4 we see that sometimes probability of events $L(a^{-})=0$ or $L(a^{+})=0$ is nonzero: for example, if $X_2=X_3=-1$ then $L(a^{+}(1/6))=L(a^{-}(1/6))=0$. Is it possible to describe all $\alpha$ with $\mathbb P(L(a^{+}(\alpha))=0)+\mathbb P(L(a^{-}(\alpha))=0)>0$ and is this set finite? On the other hand, one might also ask about a computational perspectives of our results. As we use Siegel-Walfisz theorem in the proof of Theorem 5, the proof provides no method to estimate the rate of convergence of resulting distribution. It is probably possible to overcome this obstacle with the help of Page theorems but we don’t know if this can be useful for numerical verification of inequality $c(\alpha)>\frac12.$
[99]{} Akhiezer, N. I.<<The Classical Moment Problem and Some Related Questions in Analysis>>, Oliver & Boyd, 1965 Iwaniec H., Kowalski E., <<Analytic Number Theory>>, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. Karatsuba A.A., <<Basic Analytic Number Theory>>, Springer-Verlag Berlin Heidelberg, 1993 Schinzel A. Urbanowicz J. Van Wamelen P. <<Class Numbers and Short Sums of Kronecker Symbols>>, J. Number Theory, 78 (1999), 62-84.
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author:
- |
Neil D. Lambert$^{a}$ and Ivo Sachs$^{b}$\
$^{a}$Laboratoire de Physique Th[é]{}orique CNRS-ENS[^1]\
24 rue Lhomond, F-75231 Paris Cedex 05, France\
$^{b}$Theoretische Physik\
Ludwig-Maximilians Universität\
Theresienstrasse 27, 80333 Munich, Germany
title: '**NON-ABELIAN FIELD THEORY OF STABLE NON-BPS BRANES**'
---
1[[11]{}]{}
Introduction
============
Non-BPS branes in type II string theory [@sen1; @Bergman] have recently attracted considerable attention. On one hand they are of interest because the are stable, non-perturbative states in string theory without preserving any supersymmetry. These states are therefore important to gain a deeper understanding of the non-perturbative spectrum of string theory. On the other hand, BPS D-branes have played a key role in the recent success of describing non-perturbative properties of supersymmetric Yang-Mills theory in terms of string theory [@Witten1]. The hope is now that non-BPS states could play a similarly important role in the analysis of non-perturbative properties of non-supersymmetric Yang-Mills theory. Previous work in this direction has focused on D-branes in non-supersymmetric type 0 theory [@Tseytlin1]. A successful generalisation of the AdS/CFT correspondence [@malda] to these non-supersymmetric backgrounds of a stack of D-branes has however been hampered by the tachyonic instability in the closed string sector of that theory[^2] [@Tseytlin2].
In Sen’s non-BPS branes of type II theory such instabilities in the closed string sector are absent. Instead one has to deal with a tachyon in the open string sector. However, it turns out that these instabilities can be cured by considering type II theory on an appropriate orbifold. In doing so stable non-BPS branes have been obtained [@sen1; @Bergman]. While the possible application of these objects for a string-theoretic description of non-supersymmetric, non-abelian field theory has been suggested [@sen1] we are not aware of any concrete progress in this direction.
In this paper we initiate a systematic analysis of the low energy dynamics of a stack of non-BPS branes in type II theory. In particular we construct the low energy effective field theory on a stack of non-BPS 3-branes. Before proceeding it is perhaps necessary to clarify some subtleties that arise in the present situation. It has been argued that generically there is a force between two non-BPS branes generated at loop one loop in open string theory [@sen2]. If so, a stack of non-BPS branes is unstable. In such a situation it is not clear that the effective field theory is appropriate. Indeed one motivation for our work is to address this question in a concrete example. In this paper we shall determine the low energy interactions of the light fields from tree-level string theory[^3]. The effective action is then simply defined to be a local quantum field theory of these light modes which reproduces the correct amplitudes. It turns out that, as in the supersymmetric case, many of the couplings such as the gauge couplings can be inferred solely from geometrical considerations and T-duality. In addition, the Yukawa couplings are restricted by general properties of the string S-matrix such as winding number conservation. In this way one can fix all couplings apart from the potential for the scalars $\chi_i$ originating in the tachyonic sector. These in turn will be fixed below by an explicit calculation of the four-tachyon amplitude in the presence of non-BPS branes. The potential obtained in this way shows some interesting features. The four scalar interaction ($[\chi_i,\chi_j]^2$) always present in supersymmetric theories arises also in the potential for $\chi_i$, but with the opposite sign! The corresponding instability is however cured by another $\chi^4$-term that comes with exactly the right coefficient to transform this instability into a flat direction of the potential. Note however, that contrary to supersymmetric theories, these new flat directions do not lie in the commuting Cartan subalgebra. As a consequence, unlike in the supersymmetric case, at a generic point in this branch of the moduli space all scalars corresponding to the separations of the branes acquire masses. This in turn leads to a condensation of the non-BPS branes. Nevertheless, the gauge symmetry is spontaneously broken.
We then discuss how quantum corrections affect these flat directions within the field theory approximation. In vacua where $\chi_i\es 0$ the one-loop correction to the potential for the scalars $\phi_I$ (corresponding to separating the non-BPS branes) leads to a repulsive force, unless the orbifold is tuned in such a way that the masses of the scalars from the tachyonic sector vanish. This is in agreement with a one-loop open string calculation [@sen2] and is a result of a Bose-Fermi degeneracy. However we will see that this degeneracy does not persist at the level of the interacting field theory and therefore we suspect that the effective potential will not vanish at two loops at the critical radii. We also consider the one loop correction to the effective potential for the scalars $\chi_i$, which removes all flat directions in the corresponding branch when $m_i=0$. The new minimum prefers a non-vanishing, non-abelian expectation value of $\chi_i$. This, in turn induces a mass term for other scalars $\phi_I$ resulting in an attractive force between the branes. This raises the possibility that a stack of non-BPS branes can condense even when the orbifold is not tuned precisely to criticality. This is encouraging in view of a possible existence of a gravitational background dual to the field theory on the branes. Indeed, a necessary condition for a successful description of non-Abelian YM-theory in terms of these non-BPS branes is, of course, that a stack of such branes is stable, i.e. does not fly apart. Our result for the field theory action of these branes provides sufficient information to further verify this condition at higher order. We leave this lengthy but, in principle straightforward computation for future work.
The rest of this paper is organised as follows. In section two we review some basic features of non-BPS 3-branes in type IIA string theory. In section three we determine the low energy field theory on parallel branes. Lastly in section four we calculate the one-loop corrections to the effective potentials for the scalar fields.
Non-BPS 3-branes
================
In this section we review the basic features of stable non-BPS branes in type II string theory. We will consider a non-BPS 3-brane in type IIA string theory for the sake of clarity. However, the generalisation to other branes is clear. We use a “mostly plus” metric and the conventions that indices $m,n=0,1,2,...,9$ run over all of ten-dimensional space-time, $\mu,\nu =0,1,2,3$ run over the 3-brane world volume, $i,j = 6,7,8,9$ and $I,J = 4,5$ label the transverse directions. Group indices are labeled by $a,b$ and we choose a hermitian basis with $\Tr (t^at^b) =
\delta^{ab}$. We will suppress all spinor indices.
As explained in [@sen1] the excitations of a single non-BPS D-brane are carried by two types of open strings with CP factor $I$ and $\sigma_{1}$ respectively. The CP $I$-sector is precisely the same as for a BPS D-brane and gives rise to massless $N=4$ vector multiplet $A_\mu,\phi^I,\chi^i,\lambda$ on the brane. Here $\lambda$ is a ten-dimensional spinor which, as a consequence of the GSO projection, is chiral (in a ten-dimensional sense). If there are $N$ branes then the gauge group is $U(N)$. The CP $\sigma_1$ sector has GSO-projection $(-1)^{F}\es -1$ and hence contains a tachyonic state in its spectrum. The lightest modes are therefore a (real) tachyon $\tau$ with $m^2=-\frac{1}{2\al'}$ from the NS ground state and a massless a ten-dimensional fermion $\psi$ from the R ground state. The two fermions $\lambda$ and $\psi$ have opposite ten-dimensional chirality. All fields are in the adjoint of $U(N)$.
The tachyonic instability of the non-BPS brane can be removed by compactifying the directions $x^i\cong x^i + R_i$ and introducing an orbifold $T^{4}/{\cal I}_{4}(-1)^{F_L}$, where ${\cal I}_4: x^{i}\mapsto
-x^{i}$ and $F_L$ is the left-moving spacetime fermion number. The effect of this orbifold in the $I$-sector is simply to remove the scalars $\phi^i$ and project onto six-dimensional chiral fermions $\lambda_+$, $\Gamma^{6789}\lambda_+=\lambda_+$. This leaves an $N=2$ vector multiplet in four dimensions. In the $\sigma_1$-sector the ${\cal I}_{4}$ also projects on to six-dimensional chiral fermions $\psi_-$, $\Gamma^{6789}\psi_-=\psi_-$. In addition ${\cal I}_4$ reverses the sign of the tachyon winding modes and the orbifold keeps only those components which are odd under ${\cal I}_4$. Thus after the orbifold the lightest field in the NS sector with CP-factor $\sigma_{1}$ originates in the ground state of strings with winding number one around the coordinate $x_{i}$. As there are four compact directions we will have eight different scalar fields $\tau_i^\pm$ associated with these modes, where $\tau_i^\pm$ is the component of the tachyon with winding number $\pm1$ around $x^i$. In particular only the combinations \_i = [\_i\^+ - \_i\^-]{} , survive. Their mass is given by m\_i\^2 = ([R\_i’]{})\^[2]{}-[12’]{} , where $R_i$ is the radius of the compact direction $x^i$. Thus, so as long as $R_i\geq R_{c}\es \sqrt{\frac{\al'}{2}}$, these lowest mass states are not tachyons and hence a non-BPS brane is stable. On the other hand we can tune the radii so that the $m_i^2$ are small compared to the string scale. Therefore these states are relevant for the low energy dynamics and should be included in the field action of the brane.
To summarise, after the orbifold, the light fields on the non-BPS branes consist of an $N=2$ vector multiplet $(A_\mu,\phi^I,\lambda_+)$ with gauge group $U(N)$ coupled to a massless six-dimensional fermion $\psi_-$ and four massive scalars $\chi_i$ all in the adjoint representation. Note that at the critical radius we obtain the field content of an $N=4$ super-Yang-Mills gauge theory. However we will shortly see that the interactions of these fields are never supersymmetric.
Non-Abelian action for non-BPS 3-branes
=======================================
In [@Senw1] Sen proposed a world volume action for a single non-BPS p-brane of the form S=T\_[p]{}d\^[4]{}\[\_[i]{} ()\^[2]{}-V()\]+ …, where $V(\chi)\es 1 + m^{2}\chi^{2}+O(\chi^{3})$ and the ellipses denote fermionic terms. We now want to find the non-Abelian action describing a stack of non-BPS 3-branes of type IIA string theory on the orbifold $T^{4}/{\cal I}_{4}(-1)^{F_L}$. For this we first recall that the fields originating in the CP-factor $I$ are the same as for a BPS-brane. Therefore the action for these fields is the same as for the BPS case, i.e. an $N=2$, $U(N)$ gauge theory. For the fields originating in the $\sigma_{1}$-sector some parts of the action (namely those up to $O(\chi^{2})$) can, in fact, be obtained from geometric considerations and T-duality alone as we shall now demonstrate.
Consider for example a pair of non-BPS D0-branes, separated by a distance $\Delta X_{1}$, and a $\sigma_{1}$-string starting at one $D0$ wrapping around the orbifold and ending on the other $D0$. The mass of such a string is given by \[m1\] M\^[2]{}\_i=()\^[2]{}+[N-’]{} , where $N$ is the oscillator number of the string state. Now, from (Fig. 1) $(\Delta X)^{2}\es (R_i)^{2}+(\Delta
X_{i})^{2}$. This mass must then be reproduced within the world volume theory. A similar argument applies to the mass of $\psi_-$ which is non-zero if the $\sigma_1$ string is stretched (with no $-\ha$ in [(\[m1\])]{}). Thus, after rescaling the fields to the canonical dimensions in field theory we conclude that the effective action has the form \[wa2\] S&=&[Tr]{}\^[4]{}x ,where ${\cal L}_{\sigma_1}$ denotes additional terms involving fields from the $\sigma_1$-sector and $D_\mu = \partial_\mu + i\gy[A_\mu,\ ]$ is the gauge covariant derivative. For simplicity we continue to label the fermions in terms of ten dimensional spinors and therefore the $\Gamma$-matrices are also ten-dimensional. In this notation $\lambda_+$ and $\psi_-$ are constrained to have opposite ten-dimensional chiralities and the subscript $\pm$ refers to their chirality under $\Gamma^{012345}$. This action reproduces the correct masses for the scalars $\phi_{I}$ in the CP factor $I$ and the tachyons $\chi_{i}$ in the CP factor $\sigma_{1}$. The covariant derivative arises because the $\sigma_1$-sector fields are in the adjoint. This also follows via T-duality of the couplings $[\phi_{I},\chi_{i}][\phi^{I},\chi^{i}]$ and $\bar\psi_-\Gamma^I[\phi^I,\psi_-]$ in [(\[wa2\])]{}. Alternatively, one can deduce the $([\phi_I,\chi_i])^2$ and $\bar\psi_-[\phi_I,\psi_-]$ couplings from T-duality and fact that $\chi$ and $\psi$ are in the adjoint.
Next we determine the remaining terms ${\cal L}_{\sigma_1}$ that appear in [(\[wa2\])]{}. Let us first discuss the fermion-tachyon coupling. In addition to the terms |\_-\^I\[\_[I]{},\_-\] in [(\[wa2\])]{} one might expect, in view of the similarity of [(\[wa2\])]{} with $N=4$ Yang-Mills, further Yukawa couplings involving the tachyons $\chi_i$. If both fermions originate in the same sector then this coupling is excluded as it involves a trace over an odd number of $\sigma_{1}$’s. On the other hand this does not exclude coupling of the form |\_+\^k\[\_[k]{},\_-\] . Indeed precisely such a coupling occurs in an $N=4$ super-Yang-Mills action. However, this coupling is excluded in the field theory limit because, due to winding number conservation, one of the fermions would have to have a non-zero winding number and therefore a mass of $O(\al')$. The absence of these terms is a first indication that [(\[wa2\])]{} is never supersymmetric in contrast to the effective action for BPS D3-branes.
The only remaining undetermined term in the effective action is the potential for the scalars originating in the tachyonic NS sector V(\_i)=-m\_i\^2\_i\^2-[L]{}\_[\_1]{} . Since have already determined the mass term in $V(\chi_i)$ let us concentrate on the terms $O(\chi^{3})$ and $O(\chi^{4})$. To determine the exact form of these terms one can analyse disk three-and four point functions respectively. In fact the three-point function must vanish, as can be seen in a number ways. One way is to note that the trace over an odd number of $\sigma_{1}$ vanishes. In addition winding number conservation implies that processes involving an odd number of $\chi_i$ fields must vanish. This leads to a ${\bf Z}_2^4$ symmetry of the effective theory generated by $\chi_i \leftrightarrow -\chi_i$. Thus we are left with the four-point function $S_{D}(k_{1},\cdots,k_{4})$. The relevant graphs for this are given in Fig. 2. The rest of this section is devoted to calculating the corresponding amplitude and deducing the form of the four-tachyon term in the effective action.
If we denote by $k$ the “field theory” momentum (i.e. momentum tangent to the brane) then the full ten-dimensional momentum $K$ of the i-th tachyon winding mode is Kk + . Here $\vec{w}_i$ is the winding vector of the tachyon around the compact directions $x^i$. The lowest mass states $\tau_i^\pm$ left after the orbifold have $\vec{w_6} = (\pm R_6,0,0,0)$, $\vec{w_7}=(0,\pm R_7,0,0)$, $\vec{w_8}=(0,0,\pm R_8,0)$ and $\vec{w_9}=(0,0,0,\pm R_9)$. To fix the gauge we need to fix three of the bosonic coordinates and two of the fermionic coordinates in this correlator. The fixed (-1 picture) and the integrated (0-picture) vertex operators are V\_\^[(0)]{}=i2’gc(x)(K(x))e\^[iKX(x)]{} V\_\^[(-1)]{}=igc(x) e\^[-(x)]{}e\^[iKX(x)]{} , respectively. The corresponding amplitude is found to be S\_[D]{}(k\_[1]{},,k\_[4]{})&=& 4’g\^2(2)\^4 \_[\_i]{}\^[4]{} (\_[i=1]{}\^[4]{}k\_[i]{})\_[f]{}\[t\^[a\_[1]{}]{},,t\^[a\_[4]{}]{}\]\
&&K\_[3]{}K\_[4]{} \_[-]{}\^x\_[4]{} \_[i<j]{}|x\_[ij]{}|\^[2’ K\_[i]{}K\_[j]{}]{}+(K\_[2]{}K\_[3]{}), where $\de_{\sum\vec{w}_i}$ ensures the winding number conservation. In terms of the ten-dimensional Mandelstam variables S=-(K\_[1]{}+K\_[2]{})\^[2]{}T=-(K\_[1]{}+K\_[3]{})\^[2]{}U=-(K\_[1]{}+K\_[4]{})\^[2]{}, with S+T+U-2/’, the amplitude takes the form \[amp\] S\_[D]{}(k\_[1]{},,k\_[4]{})&=&2g\^2(2)\^4 \_[p=1]{}\^[4]{}\_ \^[4]{}(\_[i=1]{}\^[4]{}k\_[i]{})\
&&\
&&[.+(1+’T) (t\^[a\_[2]{}]{}t\^[a\_[3]{}]{}t\^[a\_[4]{}]{}t\^[a\_[1]{}]{}+t\^[a\_[3]{}]{}t\^[a\_[2]{}]{}t\^[a\_[1]{}]{}t\^[a\_[4]{}]{}) B(-’S,-’U).]{}\
&&[.+(1+’S) (t\^[a\_[3]{}]{}t\^[a\_[2]{}]{}t\^[a\_[4]{}]{}t\^[a\_[1]{}]{}+t\^[a\_[2]{}]{}t\^[a\_[3]{}]{}t\^[a\_[1]{}]{}t\^[a\_[4]{}]{}) B(-’T,-’U)\]]{},where $B(a,b)$ is the Euler beta function. The amplitude [(\[amp\])]{} has no pole at $\al'S\es -\ha$. In the bosonic string this pole is related to the tachyonic intermediary state. Its absence is the consistent with the absence of a three tachyon coupling, as explained above. After orbifolding the full four-tachyon amplitude is the given by the sum over all diagrams in figure 2. Winding number conservation implies that the external states must be of the form $\tau_i^+,\tau_i^-,\tau_j^+,\tau_j^-$. There are two cases to consider: either $i\ne j$ or $i=j$.
Let us first consider diagrams of Fig. 2 with $i\ne j$. Here we find that there are four separate graphs which contribute to $\chi_i,\chi_i,\chi_j,\chi_j$ scattering. It is helpful here to introduce the Mandelstam variables for the field theory momenta s = -(k\_1+k\_2)\^2 ,t= -(k\_1+k\_3)\^2 ,u = -(k\_1+k\_4)\^2 . For the four graphs of interest we find that S = -1/’ + s - (m\_i\^2+m\_j\^2) , T = -1/’ + t - (m\_i\^2+m\_j\^2) , U = u . We may now expand the amplitude [(\[amp\])]{} to lowest order in $\al'$, noting that $s,t$ and $u$ are all $O(1)$. In this way we find \[ij\] S\^[ij]{}\_D(k\_1,,k\_4) &=& g\^2(2)\^4 \^[4]{}(\_[i=1]{}\^[4]{}k\_[i]{}) ((-+ {t\^[a\_2]{}, t\^[a\_4]{}}{t\^[a\_1]{}, t\^[a\_3]{}} .\
&&.+ ([t-s2u]{})(\[t\^[a\_2]{}, t\^[a\_3]{}\]\[t\^[a\_1]{}, t\^[a\_4]{}\]) ) . The pole corresponds to an exchange of a gauge boson in the u-channel. Note that, for $i\ne j$, the incoming states can only annihilate to a zero winding number state such as a gauge boson in the u-channel.
Let us now consider diagrams in Fig. 2 where $i=j$. Winding number conservation now implies that there are six graphs that must be summed over. These graphs come in three pairs with S = s , T=t ,U = -2/’ + u - 4m\_i\^2 , and similarly for the other two pairs. If we now sum of all these contributions we find \[ii\] S\^[ii]{}\_D(k\_1,,k\_4) &=&g\^2(2)\^4 \^[4]{}(\_[i=1]{}\^[4]{}k\_[i]{}) (([t-su]{}) (\[t\^[a\_2]{}, t\^[a\_3]{}\]\[t\^[a\_1]{}, t\^[a\_4]{}\]).\
&&.+([s-ut]{}) (\[t\^[a\_1]{}, t\^[a\_3]{}\]\[t\^[a\_2]{}, t\^[a\_4]{}\]).\
&&.+([t-us]{}) (\[t\^[a\_1]{}, t\^[a\_2]{}\]\[t\^[a\_3]{}, t\^[a\_4]{}\]) ) . Thus here there are only pole contributions corresponding to the exchange of a gauge boson. Note that for $i=j$ it is possible for the incoming states to annihilate into a zero winding state in any of the three channels. Comparing [(\[ij\])]{} and [(\[ii\])]{} with the tree-level field theory amplitude we find after a lengthy but straightforward computation that the string tree-level amplitude is reproduced by the following potential for the scalars $\chi$ originating in the tachyonic sector \[potential\] V() = (\_i m\_i\^2\^2\_i + \_[ij]{}(\[\_i,\_j\])\^2 + \^2\_[ij]{}(\_i)\^2(\_j)\^2) . This is the main result of this paper. Before discussing its implications a comment about the uniqueness of the effective potential [(\[potential\])]{} is in order. As is well known the effective action for tachyons is ambiguous [@Banks] due to the possibility of replacing $m^2T$ by $\nabla^2 T$. In the present situation this ambiguity can in principle arise for the $\chi^4$ term and should be kept in mind for the cases when $m_i\neq 0$.
Let us briefly describe the classical vacuum moduli space of the theory. Of course without supersymmetry we do not expect any of these vacuum moduli to persist in the quantum theory and hence we must interpret the term moduli space loosely. We note that the complete potential for all the fields can be written as \[fullpot\] . Clearly if $m_i^2<0$ for some $i$, say $i=1$, then the potential is not bounded from below because we may take only $\chi_1$ to be non-vanishing and thereby make the potential as negative as we wish. On the other hand if $m_i^2\ge0$ for all $i$ then since we have chosen a hermitian basis for the Lie algebra it is not hard to see that all the terms appearing in [(\[fullpot\])]{} are positive. Thus the potential is bounded below by zero.
For generic values of the orbifold radii the potential is minimized by $\chi_i=0$ and $[\phi_4,\phi_5]=0$. In this case the moduli space of vacua are given by the Cartan subalgebra of $U(N)$. This corresponds to a Coulomb branch with $N$ massless $U(1)$ gauge fields.
If $m_i^2=0$ then there are additional branches of the vacuum moduli space that can arise where $\phi_I=0$. We are then left with the requirement $\{\chi_i,\chi_j\}=0$ for $i\ne j$, $m^2_i=m^2_j=0$ (and $\chi_i=0$ if $m_i^2>0$), i.e. this vacuum is parameterised by anti-commuting scalar vevs. Generically in these branches the gauge group is completely higgsed leaving only the massless fields from the $U(1)$ corresponding to the overall translation invariance. Note that here the scalars $\phi_I$ become massive so that the non BPS-branes are bound together. In addition the ${\bf Z}_2^4$ symmetry of the theory is spontaneously broken.
One-Loop Effective Potential
============================
In this section we compute within the the field theory approximation the one-loop corrections to the tree level potentials obtained in the previous section. For simplicity we take all the radii to be equal and assume that there are only two non-BPS branes. It has been shown using open string theory [@sen2] that there is a one-loop repulsive force between the two non-BPS branes which vanishes at the critical radius $R\es R_{c}$. We will reproduce this force in the field theory approximation where it corresponds to the lifting of the Coulomb branch by a one-loop effective potential for the scalars $\phi_I$, when the $\sigma_1$-scalars $\chi_i$ are set to zero. We will also discuss the lifting of the Higgs branch discussed above when $m=0$ and $\phi_I=0$. As we shall see, the one-loop quantum corrections remove the $\chi_i=0$ vacuum, indicating that the branes can condense.
Let us first describe the lifting of the Coulomb branch resulting in a force between the branes when $R_i>R_{c}$ and the scalars from the $\sigma_{1}$-sector are massive. A quick way to compute the potential is to first consider the massless limit. In this case the field content becomes the same as that of $N=4$ super-Yang-Mills. In addition the interaction terms which break supersymmetry do not contribute to the $\phi_I$ effective potential at one loop. Therefore the effective potential is the same as for an $N=4$ super Yang-Mills theory and hence vanishes on account of the Bose-Fermi degeneracy. Then, because the mass term only appears in the fluctuation determinant of the tachyon[^4], the effective potential at non-zero $m$ is simply given by V\_[eff]{}(X\_[1]{})=2, where $\M_m\es -\ha g_{ab}\de_{ij}(-\nabla^{2} +m_i^{2})$ is the fluctuation operator for the scalars in the $\sigma_1$-sector. We use the $\zeta$-function definition of the regularised functional determinants. The one-loop effective potential is then found to be $(\phi_{I}^{a}\to\de^{a3}\de_{I5}F+\phi_{I}^{a})$ 2\^2 V\_[eff]{}(F)&=&-(m\^2+4\^2 F\^2)\^2\
&&+(4\^2 F\^2)\^2-m\^4 .\
Of physical interest to us is the force between to branes &=&-F\^3{ () .\
&&.-()} , where $\Lambda$ is the UV cut-off which should be set to the characteristic string scale ($\al'$) in the present situation [@Tseytlin3]. Here we take the one-loop mass of $m_\phi$ to vanish at[^5] $\mu\es\Lambda$. From [(\[m1\])]{} 4\^2 F\^[2]{}=()\^2()\^[2]{} , so that &=& -A,where r=A= c=1-(m\^2’). Note that in the field theory approximation we have $X^{2}<<\al'$ and $m^{2}\al'<<1$ always. Nevertheless we can distinguish between $r<<1$ and $r>>1$. In the first case the force between two non-BPS branes is approximated by -Acr, whereas in the second case the leading behaviour is Ar(). In particular the force is repulsive in both limits. This is in agreement with the string theory result [@sen2]. Of course the effective potential is bounded from below however the minimum is outside the validity of our one-loop and field theory approximations. Nevertheless one does expect a stable minimum to arise in the field theory.
Finally let us turn to the effective potential for the $\sigma_1$-scalars $\chi_i$. Due to the absence of the corresponding Yukawa couplings in [(\[potential\])]{} only scalar and vector loops contribute to the effective potential. For simplicity, rather than a general background we consider the ansatz \_6=\_1,\_7= \_2\_8=\_9=0 , where $\sigma_1,\sigma_2$ and $\sigma_3$ are Pauli matrices. The corresponding one-loop effective potential is given by 16\^2V(v\_1,v\_2)&=&-6(v\_1\^2+v\_2\^2)\^2\
&&-8 v\_1\^4-8 v\_2\^4.The minimum of this potential is at $v_1^2\es v_2^2\es O(\Lambda^2)$. Therefore it favours a non-abelian expectation value for $\chi$. This, in turn, induces a mass term for $\phi_I$ and hence an attraction between the non-BPS branes at tree-level in $\phi_I$. The loop correction in the $\phi$-channel in a non-vanishing $\chi$-background will be repulsive but this is should be of sub-leading order in $\gy$.
Of course, this result is to be taken with a grain of salt as the predicted expectation value of $\chi_i$ is beyond the range of validity of both the one-loop approximation in field theory and furthermore the field theory approximation altogether ($\Lambda\es(\al')^{-\ha}$). A more reliable result should come from a two-loop computation in the field theory or, better still, a string loop computation in a $\chi$-background. We leave this challenge for future work and highlight here the possibility of brane condensation due to the existence of a non-trivial, non-abelian minima in the potential for $\chi_i$.
[**Acknowledgments**]{}
We would like to thank Peter West for discussions. I.S. would like to thank Kings College London, where this project was started, for hospitality. We are also indebted to the referee for pointing out a mistake in our review of non-BPS branes. N.D.L. was supported by the EU grant ERBFMRX-CT96-0012 and I.S. was supported in parts by Swiss Government TMR Grant, BBW Nr. 970557.
[99]{} A. Sen, [*Stable Non-BPS States in String Theory*]{}, JHEP [**98**]{} (1998) 007, hep-th/9803194; [*Stable Non-BPS Bound States of BPS D-branes*]{}, JHEP [**08**]{} (1998) 010, hep-th/9805019; [*BPS D-branes on Non-supersymmetric Cycles*]{}, JHEP [**12**]{} (1998) 021, hep-th/9812031; [*Non-BPS States and Branes in String Theory*]{}, hep-th/9904207.
O. Bergman and M. R. Gaberdiel, [*Stable non-BPS D-particles*]{}, Phys. Lett. [**B441**]{} (1998) 133, hep-th/9806155; [*Non-BPS States in Heterotic - Type IIA Duality*]{}, JHEP [**03**]{} (1999) 013, hep-th/9901014. A. Hanany and E. Witten, [*Type IIB Superstrings, BPS Monopoles, And Three-Dimensional Gauge Dynamics*]{}, Nucl. Phys. [**B492** ]{} (1997) 152, hep-th/9611230; E. Witten, [*Solutions Of Four-Dimensional Field Theories Via M Theory*]{}, Nucl. Phys. [**B500**]{} (1997) 3, hep-th/9703166; A. Brandhuber, J. Sonnenschein, S. Theisen and S. Yankielowicz, [*Brane Configurations and 4D Field Theory Dualities*]{}, Nucl. Phys. [**B502** ]{} (1997) 125, hep-th/9704044; P. S. Howe, N. D. Lambert and P. C. West, [*Classical M-Fivebrane Dynamics and Quantum N=2 Yang-Mills*]{}, Phys. Lett. [**B418**]{} (1998) 85, hep-th/9710034.
A. Polyakov, [*The Wall of the Cave*]{}, Int.J.Mod. Phys. [**A14** ]{} (1999) 645, hep-th/9809057; I.R. Klebanov and A.A. Tseytlin, [*D-Branes and Dual Gauge Theories in Type 0 String Theory*]{}, Nucl. Phys. [**B546** ]{} (1999) 155, hep-th/9811035; A. Armoni and B. Kol, [*Non-Supersymmetric Large N Gauge Theories from Type 0 Brane Configurations*]{}, JHEP [**07**]{} (1999) 011, hep-th/9906081. R. Blumenhagen, A. Font and D. Lust, [*Non-Supersymmetric Gauge Theories from D-Branes in Type 0 String Theory*]{}, Nucl. Phys. [**B560** ]{} (1999) 66, hep-th/990610; M. S. Costa, [*Intersecting D-branes and Black Holes in Type 0 String Theory*]{}, JHEP [**04**]{} (1999) 016, hep-th/9903128; I. Sachs, [*Electric Black Holes in Type 0 String Theory*]{}, JHEP [**11**]{} (1999) 011, hep-th/9907201.
J.M. Maldacena, [*The Large N Limit of Superconformal Field Theories and Supergravity*]{}, Adv. Theor. Math. Phys. [**2** ]{} (1998) 231, hep-th/9711200.
I.R. Klebanov and A.A. Tseytlin, [*Asymptotic Freedom and Infrared Behavior in the Type 0 String Approach to Gauge Theory*]{}, Nucl. Phys. [**B547** ]{} (1999) 143, hep-th/9812089; J. A. Minahan, [*Asymptotic Freedom and Confinement from Type 0 String Theory*]{}, JHEP [**04**]{} (1999) 007, hep-th/9902074 .
C. Angelantonj and A. Armoni, [*Non-Tachyonic Type 0B Orientifolds, Non-Supersymmetric Gauge Theories and Cosmological RG Flow*]{}, hep-th/9912257.
M.R. Gaberdiel, A. Sen, [*Non-supersymmetric D-Brane Configurations with Bose-Fermi Degenerate Open String Spectrum*]{}, JHEP [**11**]{} (1999) 008, hep-th/9908060.
I. Pesando, [*On the Effective Potential of the Dp- anti-Dp System in Type II Theories*]{}, Mod. Phys. Lett. [**A14** ]{} (1999) 1545, hep-th/9902181;
A. Sen, [*Supersymmetric World-Volume Action for Non-BPS D-branes*]{}, JHEP [**10**]{} (1999) 008, hep-th/9909062;
T. Banks, [*The Tachyon Potential in String Theory*]{}, Nucl. Phys. [**B361** ]{} (1991) 166; A.A. Tseytlin, [*On the Tachyonic Terms in the String Effective Action*]{}, Phys. Lett. [**B264** ]{} (1991) 311.
D.G.C. McKeon, I. Sachs and I.A. Shovkovy, [*SU(2) Yang-Mills Theory with Extended Supersymmetry in a Background Magnetic Field*]{}, Phys. Rev. [**D59** ]{} (1999) 105010, hep-th/9807059.
A.A. Tseytlin and K. Zarembo, [*Effective Potential in Non-Supersymmetric SU(N) X SU(N) Gauge Theory and Interactions of Type 0 D3-Branes*]{}, Phys. Lett. [**B457** ]{} (1999) 77, hep-th/9902095.
[^1]: UMR 8549, Unit[é]{} Mixte de Recherche du Centre National de la Recherche Scientifique et de l’ Ecole Normale Sup[é]{}rieure.
[^2]: For recent progress though see [@angel]
[^3]: See also [@Pesando] for a similar calculation of the tachyon potential for the $D\bar D$ system in type II theory.
[^4]: We choose an $N\es 2$ supersymmetric version of the $R_{\xi}$-gauge [@Gerry]
[^5]: for $\mu\neq\Lambda$ the effective mass will then be given by $m_\phi^2\es
\frac{\gy^2m^2}{\pi^2}\log(\mu/\Lambda)$.
|
---
abstract: 'This paper presents results of a work on crawling CEUR Workshop proceedings[^1] web site to a Linked Open Data (LOD) dataset in the framework of ESWC 2014 Semantic Publishing Challenge 2014[^2]. Our approach is based on using an extensible template-dependent crawler and DBpedia for linking extracted entities, such as the names of universities and countries.'
author:
- Maxim Kolchin
- Fedor Kozlov
bibliography:
- 'references.bib'
subtitle: 'A template-based information extraction from web sites with unstable markup'
title: 'Unstable markup:'
---
Introduction
============
The work that is presented in this paper aims to provide a solution for Task 1 of ESWC 2014 Semantic Publishing Challenge. The task is to crawl CEUR Workhop proceedings web siteand create a LOD dataset containing detailed information about workshops, proceedings volumes, papers and their authors and etc.
The source code and instructions to run the crawler are located at our Github repository[^3].
Challenges
----------
At first glance, the task looks a pretty straightforward, but there are several challenges that need to be solved:
- the web site has a quite unstable and in some cases invalid HTML markup because of absence of a standardised and strict template for creation of pages for proceedings volumes, so it makes it harder to crawl such pages, because usually crawlers are written for web sites with fixed markup;
- only a small percentage of proceedings volumes uses RDFa markup and microformats are used only for volumes starting from 559th one, so at the time of writing around 49% of volume pages don’t have any metadata that could help in crawling;
- according to the rules of the web site, proceedings should comply with some requirements regarding numbers of invited and regular papers, therefore there are joint proceedings of several workshops. Such workshop and proceedings should be represented in the dataset accordingly;
- the web site includes proceeding not only in English, but also in German. In addition it’s quite common practise for authors of papers written in English to use names of their universities or companies in a native language;
Our approach
============
We developed an extensible template-dependent crawler that uses sets of special predefined templates for each type of entity. The main aim of this templates is to cover entire variety of entity representations in HTML format. Some of templates used for extracting papers from workshops pages are:
- a template based on RDFa metadata,
- a template based on Microformats,
- and two templates specific to some similar HTML markups .
When HTML page parsing begins, the crawler consecutively runs predefined templates till one of the templates returns the valid data. Validation based on template’s structure. Template’s parsing process extracts data from HTML page using XPath Language and regular expressions. XPath Language is used for searching text data by elements and properties in HTML markup. Regular expressions is used for extracting entity tokens from plain text. When data is extracted template parsing process converts data into ontology instances and properties. The templates are completely independent from each other and the crawler uses a mapping of the templates to the types of contents where they are applied to such as the index page, a workshop page and a publication, which makes the crawler easily extensible. More about the extensibility in the next section.
The main advantage of our approach is a flexibility of different data representations in HTML markup with usage of the same code of the crawler and support of invalid HTML.
Architecture
------------
The parser is implemented in Python and based on Grab Spider framework[^4]. This framework allows to build asynchronous site crawlers. Crawler downloads all workshop’s pages and papers and then runs the parsing tasks. There is a collection of specific parsers for each entity. Each parser in collection process a part of some HTML page to build properties and entity relations.
The overall system architecture is shown in Fig. \[fig:overall\_sys\_arch\].
![The overall system architecture[]{data-label="fig:overall_sys_arch"}](overall_sys_arch){width="\textwidth"}
### Extensibility
The crawler provides two basic templates with the default implementations: a template for a single entity, in example a workshop page, and a template for a list of entities, in example the index page with the list of workshop names. And to add a new template, one of the basic ones should be extended by implementing the method responsible for the template matching. The rest of the work is done by the default implementations responsible for translating data to triples and writing them to disk.
Data representation
-------------------
To represent crawled data we use several different ontologies such as Semantic Web Conference Ontology (SWC)[^5], Semantic Web for Research Communities ontology (SWRC)[^6], The Bibliographic Ontology (BIBO)[^7], The Timeline Ontology (TIMELINE)[^8], Friend of a Friend (FOAF)[^9], Dublin Core (DC and DCTERMS) [^10] and DBpedia Ontology (DBPEDIA-OWL)[^11] and RDF Schema (RDFS)[^12].
A part of the data representation schema is shown on Fig. \[fig:schema\]. Representation of time and time intervals doesn’t use The Event Ontology (EVENT) because it assumes inclusion of blank nodes. Since RDFLib doesn’t work well with them we decided to use TIMELINE ontology instead. TIMELINE ontology provides timeline:atDate property for setting a date to an instance and *timeline:beginsAtDateTime* and *timeline:endsAtDateTime* properties for a time interval.
On CEUR Workshop proceedings web site some proceedings volumes has links to each other. These links usually relate a proceedings of a workshop to the previous its editions and we uses *rdfs:seeAlso* property to represent this relationships.
![Schema representing the crawled data[]{data-label="fig:schema"}](SemPub2014_Ontologies){width="\textwidth"}
Specific solutions
------------------
In most cases all problems are solved by an appropriate template, but there are some problems requiring specific solutions.
### Extraction of countries and affiliations
Identification of countries and affiliations in papers was done with external datasets. In case of extracting countries parser extracts the first page from the PDF document. Country-candidates are extracted using regular expressions with predefined templates. Parser sends request with a list of country-candidates to SPARQL-endpoint of DBpedia[@dbpedia-swj] resource to get list of unique country’s IRIs.
The country extraction query must support different naming conventions of country-candidates. Hence the following SPARQL-query is suggested.
SELECT DISTINCT ?country {
VALUES ?search { "The Netherlands" }
?country a dbpedia-owl:Country .
{ ?name_uri dbpedia-owl:wikiPageRedirects ?country ;
rdfs:label ?label .
}
UNION
{ ?country rdfs:label ?label }
FILTER( STR(?label) = ?search )
}
Creation of properties and relations for the current paper entity is based on received list.
### Identification of related workshops
As mentioned above, in most cases *skos:related* property is used to relate to a previous edition of the corresponding workshops. But sometimes it’s not correct. Especially in case of joint proceedings. To identify correct links we implemented the algorithm measuring similarity of two workshops based on its full name and acronym. In case of absence of an acronym we generate one from the full name’s upper case characters. For example for “Concept Extraction Challenge at Making Sense of Microposts 2013“ workshop the “CECMSM“ acronym is generated. String similarity measurement uses the basic Ratcliff-Obershelp algorithm[@ratcliff1988pattern]. This algorithm was selected because it is being provided by the Python Standard Library.
Conclusion
==========
Task 1 of Semantic Publishing Challenge 2014 is solved with developed parser based on Grab Spider framework. This parser uses SWC, SWRC, BIBO, TIMELINE ontologies, DBpedia datasets and the basic Ratcliff-Obershelp algorithm for string similarity measurement. Our approach based on templates of web site blocks, the schema representing extracted information and solutions for some specific problems. The main advantages of our approach are flexible representation of different data templates in HTML markup and support of invalid HTML.
Unsolved issues
---------------
In most cases our solution works well, but there are several “places“ where it doesn’t work well and therefore may not pass some tests completely:
- extraction of country and university candidates from papers works only for texts consisting only of US-ASCII characters because PDFMiner[^13] which we use to extract text from PDF files doesn’t work well with Unicode symbols;
- [papers written in PostScript or HTML are completely ignored;]{}
Future work
-----------
This work can be further extended by solving the known issues and implementing additional functionality in the crawler for deeper information extraction to make the dataset more useful for further analysis. To achieve it, the following particular tasks could be done:
- use external repositories such as DBLP[^14], Semantic Web Dog Food[^15] and other open datasets to extract, link and enrich the information about authors and editors,
- optimise extraction of authors’ affiliations from papers and particular connections between the authors and the affiliations,
- extraction of authors’ e-mail addresses from papers to improve aligning of ontology instances representing authors and editors.
### Acknowledgments. {#acknowledgments. .unnumbered}
This work has been partially financially supported by the Government of Russian Federation, Grant \#074-U01.
[^1]: CEUR Workshop proceedings web site, URL: <http://ceur-ws.org>
[^2]: ESWC 2014 Semantic Publishing Challenge, URL: <http://2014.eswc-conferences.org/semantic-publishing-challenge>
[^3]: The source code and instructions, URL: <https://github.com/ailabitmo/sempubchallenge2014-task1>
[^4]: Grab framework, URL: <http://grablib.org/>
[^5]: Semantic Web Conference Ontology, URL: <http://data.semanticweb.org/ns/swc/ontology>
[^6]: Semantic Web for Research Communities, URL: <http://ontoware.org/swrc/>
[^7]: The Bibliographic Ontology, URL: <http://purl.org/ontology/bibo/>
[^8]: The Timeline Ontology, URL: <http://purl.org/NET/c4dm/timeline.owl#>
[^9]: The Friend of a Friend (FOAF), URL: <http://www.foaf-project.org/>
[^10]: Dublin Core, URL: <http://purl.org/dc/elements/1.1/>
[^11]: DBpedia Ontology, URL: <http://dbpedia.org/ontology/>
[^12]: RDF Schema, URL: <http://www.w3.org/2000/01/rdf-schema#>
[^13]: PDFMiiner, URL: <http://www.unixuser.org/~euske/python/pdfminer/>
[^14]: DBLP, URL: <http://www.informatik.uni-trier.de/~ley/db/>
[^15]: Semantic Web Dog Food, URL: <http://data.semanticweb.org/>
|
---
abstract: 'We present the results of [[*SPITZER*]{}]{} mid-infrared spectroscopic observations of two highly-obscured massive X-ray binaries: [IGR J16318–4848]{} and [GX 301–2]{}. Our observations reveal for the first time the extremely rich mid-infrared environments of this type of source, including multiple continuum emission components (a hot component with $T$ $>$ 700 K and a warm component with $T$ $\sim$ 180 K) with apparent silicate absorption features, numerous recombination lines, many forbidden ionic lines of low ionization potentials, and pure rotational [$\rm H_{\rm 2}$]{} lines. This indicates that both sources have hot and warm circumstellar dust, ionized stellar winds, extended low-density ionized regions, and photo-dissociated regions. It appears difficult to attribute the total optical extinction of both sources to the hot and warm dust components, which suggests that there could be an otherwise observable colder dust component responsible for the most of the optical extinction and silicate absorption features. The observed mid-infrared spectra are similar to those from Luminous Blue Variables, indicating that the highly-obscured massive X-ray binaries may represent a previously unknown evolutionary phase of X-ray binaries with early-type optical companions. Our results highlight the importance and utility of mid-infrared spectroscopy to investigate highly-obscured X-ray binaries.'
author:
- 'Dae-Sik Moon, David L. Kaplan, William T. Reach, Fiona A. Harrison, Jeong-Eun Lee, Peter G. Martin'
title: 'The Rich Mid-Infrared Environments of Two Highly-Obscured X-ray Binaries: [[*SPITZER*]{}]{} Observations of [IGR J16318–4848]{} and [GX 301–2]{}'
---
Introduction {#sec_intro}
============
Recently, a large number of highly-obscured (e.g., [$N_{\rm H}$]{} $\ge$ 10$^{23}$ cm$^{-2}$) massive X-ray binaries have been discovered by the [[*INTEGRAL*]{}]{} hard X-ray ($\ge$ 15 keV) satellite [@wet03]. The prototypical case is [IGR J16318–4848]{} which shows variable high obscuration in the X-ray, sometimes reaching [$N_{\rm H}$]{} $\simeq$ 2 $\times$ 10$^{24}$ [@coet03; @walteret03]. Its bright optical and near-infrared (IR) counterpart is an early B-type supergiant star with numerous emission lines [@fc04]. Interestingly the obscuration toward [IGR J16318–4848]{}obtained in the optical and near-IR wavebands ([$A_{\rm V}$]{} $\sim$ 18) is almost two orders of magnitude smaller than that inferred from the X-rays, which suggests that the extreme obscuration seen in the X-ray is intrinsic only to the X-ray source. However, [$A_{\rm V}$]{} $\sim$ 18 is still greater than the interstellar obscuration, indicating the existence of substantial circumstellar material around the supergiant companion.
In order to fully understand the implications of the [[*INTEGRAL*]{}]{}discoveries — specifically if they imply existence of a separate class of highly-obscured X-ray binaries — we must investigate any similarities between the new [[*INTEGRAL*]{}]{} sources and previously known sources. As already suggested by @ret03, the X-ray pulsar [GX 301–2]{}appears similar to the new [[*INTEGRAL*]{}]{} highly-obscured massive X-ray binaries: it has variable X-ray obscuration of [$N_{\rm H}$]{} $\simeq$ 10$^{23}$–10$^{24}$ cm$^{-2}$, the compact source is a neutron star, and the optical companion is an early B-type supergiant [or hypergiant; @ket95], like [IGR J16318–4848]{}. Recently, @kmr06 [; hereafter Paper I], using optical, near-, and mid-IR ($\le$ 20 $\mu$m) spectral energy distributions (SEDs), have found that both sources have strong mid-IR excesses that they identify as continuum emission from hot dust. This is suggestive that they have very similar circumstellar material which may be related to their strong X-ray obscuration. In this [*Letter*]{}, we present the results of [[*SPITZER*]{}]{} mid-IR spectroscopic observations of [IGR J16318–4848]{} and [GX 301–2]{}, showing that both sources indeed have very similar rich mid-IR properties previously unknown for X-ray binaries.
Observations and Data Reduction {#sec_obs}
===============================
We observed [IGR J16318–4848]{} and [GX 301–2]{} with [*InfraRed Spectrogrpah*]{} (IRS) aboard [[*SPITZER*]{}]{} [@jhet04; @wet04] on 2005 September 8 and 2005 July 2, respectively. The duration of the total observations was 1.32 ([IGR J16318–4848]{}) and 0.82 ([GX 301–2]{}) hours with the all six IRS modules in operation in the standard staring mode, obtaining two nods of spectra separated by 1/3 of the slit length. For the basic data reduction, we used the co-added Basic Calibration Data (BCD) produced through the standard [[*SPITZER*]{}]{} Science Center data reduction pipeline (S13.2.0), and also the [[*SPITZER*]{}]{} IRS Custom Extraction (i.e., SPICE) software (v1.3-beta1). In the low-resolution mode where the slit lengths are large (i.e., 57 for 5.2–14.5 $\mu$m and 168 for 14.0–38.0 $\mu$m), we subtracted out background emission of one nod using the spectrum of the another nod, and combined the two background-subtracted spectra for the final spectrum. However, in the high-resolution mode, the small slit length (i.e., 113 for 9.9–19.6 $\mu$m and 223 for 18.7–37.2 $\mu$m), together with the high brightnesses of the sources, made it impossible to subtract out the background. Therefore, we just combined the spectra from the two nods for the final high-resolution spectrum without any background subtraction. In addition, we used IRSCLEAN (v1.5) to mask and clean rogue pixels from the co-added BCD images, and then interpolated over neighboring normal pixels.
Low-Resolution Spectrum: Continuum and Broad Features {#sec_lowres}
=====================================================
Some emission features, such as ionic forbidden lines, [$\rm H_{\rm 2}$]{} lines, and Polycyclic Aromatic Hydrocarbon (PAH) emission, present in the low-resolution spectra appear extended. Here we consider just the emission from point sources, and defer discussion of the extended emission to a separate paper. Figure \[fig:igrj\] presents the background-subtracted low-resolution spectrum of [IGR J16318–4848]{}, as well as the best-fit spectra of the continuum emission and the residual of the SED fit (see below). Note that the low-resolution spectrum is extracted for the continuum source (i.e., [IGR J16318–4848]{}) based on the standard point source extraction procedure of SPICE. Several features are prominent in Figure \[fig:igrj\], including complicated multi-component continuum emission, broad and strong absorption around 9.7 and 18 $\mu$m, and numerous emission lines. The spectrum of [GX 301–2]{} (Figure \[fig:gx\]) is similar, with strong continuum emission, silicate absorptions feature around 9.7 $\mu$m, and several emission lines.
As in Figures \[fig:igrj\] and \[fig:gx\], the silicate absorption features and numerous emission lines make it difficult to carry out any meaningful independent SED fits of the shorter wavelength (i.e., $\lambda$ $\le$ 8 $\mu$m) data of the IRS spectra. Therefore, instead, we first examined if the IRS SEDs at the shorter wavelengths are consistent with what have been found in Paper I using only the line-free regions. According to Paper I, the SEDs of [IGR J16318–4848]{} and [GX 301–2]{} at $\lambda$ $\le$ 8 $\mu$m have contributions from hot (i.e.,T $\simeq$ 1040 K for [IGR J16318–4848]{} and 720 K for [GX 301–2]{}) dust emission and stellar emission of a B0I star with temperature of 26000 K, while the contribution of free-free emission is negligible. Our IRS SEDs of the line-free regions in Figures \[fig:igrj\] and \[fig:gx\] at $\lambda$ $\le$ 8$\mu$m were fitted very well with those parameters, resulting in the reduced chi-square $\chi^2_\nu$ $\le$ 1. We used the extinction [$A_{\rm V}$]{} = 18.5 and 7.1 for [IGR J16318–4848]{} and [GX 301–2]{}, respectively, as in Paper I. However, in the longer wavelength (i.e., $\lambda$ $\ge$ 22 $\mu$m) range, we were unable to fit the IRS SEDs with the given hot dust and stellar parameters because the best fit resulted in unacceptably large chi-square values. We therefore performed SED fits with three components (i.e., hot dust, stellar, and warm dust component) using the entire line-free SEDs in the $\lambda$ $\le$ 8 $\mu$m and $\ge$ 22 $\mu$m range. Over the fits the parameters of the warm dust component were treated as free parameters, while those of the hot dust and stellar component were fixed to be the values in Paper I. As a result, we obtained the warm dust temperature of 190 K ([IGR J16318–4848]{}) and 170 K ([GX 301–2]{}) with $\chi^2_\nu$ $\simeq$ 1.2 ([IGR J16318–4848]{}) and 1.5 ([GX 301–2]{}). We used the extinction coefficients compiled in @m2000 excluding the 8–22 $\mu$m range in order to avoid the uncertainties of the coefficients associated with the silicate absorption features. We instead interpolated the extinction coefficients at $\lambda$ $\le$ 8 and $\ge$ 22 $\mu$m to calculate the silicate absorption-free extinction coefficients in the $\lambda$ = 8–22 $\mu$m range, and used them in the fits. The main panel of Figures \[fig:igrj\] and \[fig:gx\] presents the results of our best SED fit obtained by excluding the silicate absorption features, where we can confirm that the fit matches nicely the observed SED except the $\lambda$ = 8–22 $\mu$m range. In the bottom panel, on the other hand, we show the redisuals of the fit obtained by subtracting the fluxes computed by the best fit from the observed values. For this residual construction, in order to see if our best SED fit is consistent with the observed silicate absorption features, we combined the best-fit parameters obtained at $\lambda$ $\le$ 8 and $\ge$ 22 $\mu$m with the extinction coefficients in the $\lambda$ = 8–22 $\mu$m range [@m2000], and calculated the silicate absorption feature-associated fluxes expected by the best SED fit in the $\lambda$ = 8–22 $\mu$m range. The residuals in Figures \[fig:igrj\] and \[fig:gx\] show that overall our best SED fit agrees with the silicate absorption features reasonably well, although there appears to be some small discrepancies for [IGR J16318–4848]{}.
The masses of the dust components were estimated under the optically thin assumption [@het77]: $M_{\rm dust} =
F_{\lambda}d^2 / \kappa_{\lambda}B_{\lambda}(T_{\rm dust})$, where $d$ is the distance to the source, $\kappa_{\lambda}$ is the dust mass absorption coefficient, and $B_\lambda(T)$ is the Planck function at temperature $T$. Using the interstellar mass absorption coefficients of @draine03 at 8 and 25 $\mu$m, the hot and warm dust masses are estimated to be $M_{hot}$ $\simeq$ 4.5 $\times$ 10$^{-9}$ $d_5^2$ [$M_{\odot}$]{}and $M_{warm}$ $\simeq$ 3.1 $\times$ 10$^{-7}$ $d_5^2$ [$M_{\odot}$]{}for [IGR J16318–4848]{}; $M_{hot}$ $\simeq$ 5.7 $\times$ 10$^{-9}$ $d_5^2$ [$M_{\odot}$]{} and $M_{warm}$ $\simeq$ 1.4 $\times$ 10$^{-7}$ $d_5^2$ [$M_{\odot}$]{} for [GX 301–2]{}, where $d_5$ is the distance to the sources normalized by 5 kpc.
The silicate absorption features seen in Figures \[fig:igrj\] and \[fig:gx\] have a tight linear correlation with the optical extinction in the diffuse interstellar medium (ISM), although the correlation has been reported to break down in dense enviornment where the silicate feature does not show a monotonic increase with extinction at [$A_{\rm V}$]{} $\geq$ 12 mag [e.g., @cet07 and references therein]. The correlation in the diffuse ISM is [$A_{\rm V}$]{}/$\tau_{\rm 9.7}$ = 18.5 . If we adopt the above correlation given in the diffuse ISM, the intensity ratio of the observed spectrum to the unabsorbed spectrum in Figures \[fig:igrj\] and \[fig:gx\] at 9.7 $\mu$m correspond to [$A_{\rm V}$]{} $\sim$ 18.5 and $\sim$ 6 for [IGR J16318–4848]{} and [GX 301–2]{}, respectively, These are very similar to the values obtained in previous studies for the total optical extinctions of the both sources. This suggests that a substantial portion, if not all, of the silicate features seen in Figures \[fig:igrj\] and \[fig:gx\] are associated with the optical extinction of [IGR J16318–4848]{} and [GX 301–2]{}.
High-Resolution Spectrum: Line Emission {#sec_highres}
=======================================
Figures \[fig:igrjhigh\] and \[fig:gxhigh\] present high-resolution spectrum of [IGR J16318–4848]{} and [GX 301–2]{}, revealing numerous emission lines. (Note that the background emission is [*NOT*]{} subtracted.) The majority of them are lines, while the rest consists of metallic forbidden lines, pure rotational [$\rm H_{\rm 2}$]{} lines, and 11.3 $\mu$m PAH emission feature. Table \[tbl\_HI\] lists the intensities of some of the bright lines identified in both sources with high ($>$ 3) signal-to-noise ratio. The line intensities and the signal-to-noise ratio were estimated with Gaussian profile fits. The intensities of the lines are consistent with gas at $T$ $\simeq$ 10$^4$ K and with [$N_{\rm e}$]{} $\simeq$ 10$^4$ [$\rm cm^{\rm -3}$]{} in the Menzel Case B state, following @fc04, although the intensity ratios are not very sensitive to those parameters [@hs87].
Table \[tbl\_other\] lists the parameters of forbidden ionic lines and [$\rm H_{\rm 2}$]{} lines. For the forbidden lines, [\[\]]{}, [\[\]]{}, [\[\]]{}, and [\[\]]{} were detected in both sources, while [\[\]]{} and [\[\]]{} were only in [IGR J16318–4848]{}; [\[\]]{}is only in [GX 301–2]{}. The [$\rm H_{\rm 2}$]{} lines were detected in both sources. The intensity ratio of the [\[\]]{} lines of 18.7 and 33.5 $\mu$m is diagnostic of the number density in the temperature range of 5000–20000 K. The extinction-corrected [\[\]]{} line surface brightness ratio of 18.7/33.5 $\mu$m is 1.21 for [IGR J16318–4848]{} and 1.68 for [GX 301–2]{}: according to the calculation of @het84 and @aet99, the ratios correspond to the electron number density [$N_{\rm e}$]{} $\sim$ 1000 [$\rm cm^{\rm -3}$]{} for both sources. On the other hand, the intensity ratio of the [\[\]]{} line (12.8 $\mu$m) and the [\[\]]{} line (15.6 $\mu$m) is sensitive to the hardness of the radiation, or temperature of the environment. If we use the calculation of @ket96, the extinction-corrected intensity ratio of [\[\]]{}/[\[\]]{} is consistent with $\sim$ 4 $\times$ 10$^4$ K for [IGR J16318–4848]{} and $\sim$ 4.5 $\times$ 10$^4$ K for [GX 301–2]{}. The three [$\rm H_{\rm 2}$]{} lines detected in the both sources are pure rotational transition lines of [$\rm H_{\rm 2}$]{} (0,0) S(0,1,2) which are usually excited by collisions. Assuming that the bottom two lines follow the distribution of the local thermodynamic equilibrium and are optically thin, the surface brightness ratios are consistent with the excitation temperature $T_{\rm ex}$ $\sim$ 1000 K. The estimated [$\rm H_{\rm 2}$]{} column densities are $\sim$ 1 $\times$ 10$^{19}$ cm$^{-2}$ ([IGR J16318–4848]{}) and 5 $\times$ 10$^{18}$ cm$^{-2}$ ([GX 301–2]{}). This implies that the optical extinction associated with the warm molecular gas is very small, [$A_{\rm V}$]{} $\ll$ 1, based on the relation [$N_{\rm H}$]{} = 1.87 $\times$ 10$^{21}$ [$A_{\rm V}$]{} [@draine03].
Discussion and Conclusion {#sec_dis_sum}
=========================
Our [[*SPITZER*]{}]{} spectroscopic observations of [IGR J16318–4848]{} and [GX 301–2]{} have revealed for the first time the rich mid-IR environment of highly-obscured X-ray binaries. This includes two dust components with prominent silicate absorption, numerous recombination lines, many forbidden ionic lines, and pure rotational [$\rm H_{\rm 2}$]{} lines. Based on the observed spectra, we infer the following components for [IGR J16318–4848]{} and [GX 301–2]{}: (1) hot ($T$ $>$ 700 K) and warm ($T$ $\sim$ 180 K) circumstellar dust; (2) ionized stellar winds responsible for the lines; (3) extended low-density ionized regions for the forbidden lines; and (4) photo-dissociated regions associated with the PAH, [$\rm H_{\rm 2}$]{} and possibly the [\[\]]{} line emission. For the forbidden lines, all the detected lines have relatively low ionization potentials like in starburst galaxies where the radiation is relatively soft (compared with active galactic nuclei, for instance). This may indicate that the illumination of hard X-rays from the central compact X-ray source is not primarily responsible for the forbidden line emission. However, the inferred temperature for the radiation exciting the [\[\]]{} and [\[\]]{} lines is hotter than the stellar photospheres, so there can be some contribution from the compact object. Considering that [\[\]]{} and [\[\]]{} were detected only in [IGR J16318–4848]{} while [\[\]]{} was only in [GX 301–2]{}, the radiation field of [IGR J16318–4848]{} may be softer than [GX 301–2]{}, as demonstrated by the small temperature difference between the two sources (see § \[sec\_highres\]).
Perhaps the most natural explanation for the origin of the hot dust component relies on dust formation in the dense outflows from the early-type companions of [IGR J16318–4848]{} and [GX 301–2]{}, as is the case in most sgB\[e\] stars (i.e., B-type supergiants with forbidden emission lines). However, the origin of the warm circumstellar dust component is very uncertain, and B\[e\] stars seldom show evidence for this type of warm dust. If both the hot and warm dust components have spherical shell geometres around the central star, then the associated optical extinctions are: [$A_{\rm V}$]{} $\simeq$ 4 $\times$ 10$^{-4}$ $Q_{\rm abs}$ ($M_{\rm d}$/10$^{-6}$ [$M_{\odot}$]{}) ($T_{\rm d}$/100 K)$^4$ ($L_{\rm UV}$/10$^{39}$ ergs s$^{-1}$), where $M_{\rm d}$ and $T_{\rm d}$ are the mass and temperature of the dust components, $Q_{\rm abs}$ $<$ 1 is the dust absorption coefficient, and $L_{\rm UV}$ is the ultra-violet luminosity of the central star. This gives the optical extinctions [$A_{\rm V}$]{} $\ll$ 1 for both the hot and warm dust components of [IGR J16318–4848]{} and [GX 301–2]{}. (Here we use 1 $\times$ 10$^{39}$ ergs s$^{-1}$ for $L_{\rm UV}$ for both sources.) Therefore, under the assumption of spherical shell geometry, both the hot and warm dust components of [IGR J16318–4848]{} and [GX 301–2]{} contribute very little to the total optical extinction, suggesting that the hot and warm dust components are [*NOT*]{} strongly associated with the silicate absorption features. This also applies to the dust associated with the warm extended [$\rm H_{\rm 2}$]{} gas since the optical extinction from this component is tiny (i.e., Av $\ll$ 1; see § \[sec\_highres\]). What’s then the origin of the silicate absorption features? If it is due to the foreground ISM, we would expect to see the $\rm CO_2$ ice feature around 15 $\mu$m, especially for [IGR J16318–4848]{}, based on the intensity ratio between the silicate absorption feature and ice feature found in the ISM [@ket05]. The absence of the ice feature in our spectra supports the interpretation that the silicate absorption features are probably not associated with the foreground ISM. One possibility may be the existence of an undisclosed colder (e.g., $\ll$ 100 K) circumstellar dust component which is responsible for the silicate absorption features and most of the optical extinction. We need further longer wavelength observatons to confirm this possibility. Considering that the 9.7 $\mu$m silicate absorption feature represents the oxygen-rich material, the existence of the 9.7 micron silicate absorption feature may indicate that the origin of the potential colder dust component is related to the nucleosynthesis of the progenitors of [IGR J16318–4848]{} and [GX 301–2]{}(or their companions).
The optical/near-IR companion of [IGR J16318–4848]{} is a sgB\[e\] star. Such stars are known to have hot ($T$ $\sim$ 1000 K) circumstellar dust, and probably are evolving into Luminous Blue Variables (LBVs) or Wolf-Rayet stars. Our mid-IR spectra of [IGR J16318–4848]{} and [GX 301–2]{} are very similar to that of the LBV P Cygni which shows many lines and forbidden ionic lines [@let96b]. The difference is that while the mid-IR continuum of P Cygni is due to the free-free emission in stellar winds, the main mid-IR emission of [IGR J16318–4848]{} and [GX 301–2]{} is from multiple dust continua. (The SEDs of both sources observed here are inconsistent with the free-free emission, as mentioned in Paper I.) However, we note that some LBV stars have also been observed to have thermal mid-IR dust emission [@let96a]. Based on the fact that B\[e\] stars seldom show mid-IR forbidden line emission, the B-type supergiant (or hypergiant) companions of [IGR J16318–4848]{} and [GX 301–2]{} may be in the evolutionary track to LBVs, implying that the extremely high obscuration seen in some massive X-ray binaries may be a phenomenon associated with the evolutionary phase. This scenario is also consistent with the fact that the highly-obscured massive X-ray binary Cygnus X-3 has a Wolf-Rayet star companion, together with the circumstellar dust emission of $T$ $\sim$ 250 K [@ket02].
This work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA. D.-S.M thanks Elise Furlan for her help in IRS data analysis and Marten van Kerkwijk for comments. This research was partly supported by the Discovery Grant (327277) of Natural Science and Engineering Research Council of Canada to D.-S.M.
Alexander, T., Sturm, E., Lutz, D., Sternberg, A., Netzer, H., & Genzel, R. 1999, , 512, 204 Bodaghee, A., et al. 2007, A&A, accepted (astro-ph/0703043) Chiar, J. E., et al. 2007, , 666, L73 Courvoisier, T. J.-L., et al. 2003, IAU Circ., 8063, 3 Filliatre, P., & Chaty, S. , 616, 469 Draine, B. T. 2003, , 41, 241 Hildebrand, R. H., et al. 2004, , 154, 18 Houck, J. R., Shure, M. A., Gull, G. E., & Herter, T. 1984, , 287, L11 Houck, J. R., et al. 2004, , 154, 18 Hummer, D. G., & Storey, P. J. 1987, , 224, 801 Kaplan, D. L., Moon, D.-S., & Reach, W. T. 2006, , 649, L107 (Paper I) Kaper, L., et al. 1995, A&A, 300, 446 Koch-Miramond, L., Abraham, P., Fuchs, Y., Bonnet-Bidaud, J.-M., & Claret, A. 2002, A&A, 877, 396 Knez, C., et al. 2005, , 635, L145 Kunze, D., et al. 1996, A&A, 315, L101 Lamers, H. J. G. L. M., et al. 1996a, A&A, 315, L225 Lamers, H. J. G. L. M., et al. 1996b, A&A, 315, L229 Mathis, J. S. 2000, in Allen’s Astrophysical Quantities, ed. A. N. Cox (London: Athlone Press, Ltd.), 527 Revnivtsev, M. G., Sazonov, S. Yu., Gilfanov, M. R., & Sunyaev, R. A. 2003, Astron. Lett., 29, 587 Walter, R., et al. 2003, A&A, 411, L427 Werner, M. W., et al. 2004, , 154, 1 Winkler, C., et al. 2003, A&A, 411, L349
[crr||crr]{} 17–9(10.26) & 4.39 & 2.44 & 12–8(10.50) & 11.57 & 11.40\
16–9(10.80) & 5.60 & 2.20 & 9–7(11.31) & 25.43 & 13.50\
22–10(11.49) & 1.74 & 0.78 & 15–9(11.54) & 6.48 & 1.87\
20–10(12.16) & 2.84 & 1.35 & 7–6(12.37) & 72.02 & 42.12\
18–10(13.19) & 5.15 & 1.70 & 13–9(14.18) & 10.61 & 3.37\
23–11(14.30) & 2.00 & 0.93 & 22–11(14.71) & 5.02 & 0.87\
16–10(14.96) & 5.41 & 1.75 & 20–11(15.82) & 2.79 & 1.33\
10–8(16.21) & 14.66 & 7.32 & 15–10(16.41) & 3.83 & 1.83\
19–11(16.59) & 2.15 & 0.40 & 12–9(16.88) & 7.35 & 3.17\
14–10(18.62) & 4.04 & 1.38 & 16–11(20.92) & 2.69 & 1.14\
9–8(27.80) & 18.71 & 8.86 & & &\
[crr||crr]{} [\[\]]{}(10.68) & 10.25 & & [\[\]]{}(12.73) & 13.33 &\
[\[\]]{}(18.24) & 1.69 & & & &\
[\[\]]{}(12.81) & 21.44 & 23.41 & [\[\]]{}(15.56) & 2.60 & 15.67\
[\[\]]{}(17.94) & 22.54 & & [\[\]]{}(24.52) & 7.33 &\
[\[\]]{}(25.99) & 70.93 & & [\[\]]{}(35.35) & 15.46 &\
[\[\]]{}(22.93) & & 1.99 & & &\
[\[\]]{}(18.71) & 7.73 & 8.02 & [\[\]]{}(33.48) & 40.14 & 25.01\
[\[\]]{}(34.82) & 177.20 & 33.33 & & &\
[$\rm H_{\rm 2}$]{}(0,0)S(2)(12.28) & 2.88 & 1.44 & [$\rm H_{\rm 2}$]{}(0,0)S(1)(17.03) & 4.66 & 2.05\
[$\rm H_{\rm 2}$]{}(0,0)S(0)(28.22) & 10.08 & 4.24 & & &\
|
---
abstract: 'We have studied the effects of chirped femtosecond laser pulses on the formation of ultracold molecules in a Rb magneto-optical trap. We have found that application of chirped femtosecond pulses suppressed the formation of $^{85}\mathrm{Rb}_2$ and $^{87}\mathrm{Rb}_2$ $a^3\Sigma_u^+$ molecules in contrast to comparable non-chirped pulses, cw illumination, and background formation rates. Variation of the amount of chirp indicated that this suppression is coherent in nature, suggesting that coherent control is likely to be useful for manipulating the dynamics of ultracold quantum molecular gases.'
author:
- 'Benjamin L. Brown'
- 'Alexander J. Dicks'
- 'Ian A. Walmsley'
title: |
Coherent Control of Ultracold Molecule Dynamics in a\
Magneto-Optical Trap Using Chirped Femtosecond Laser Pulses
---
Achieving control of the dynamics of quantum systems has been a long-standing goal of physics and chemistry [@WARREN1993; @RABITZ2000]. Rapid advances in the manipulation of laser-matter interactions to obtain desired outcomes by means of tailored optical fields have been enabled by the development of ultrafast femtosecond optical sources and pulse-shaping techniques. In particular, recent successes include exciting *a priori* specified quantum states in atoms [@WEINACHT1999] and molecules [@BARDEEN1997], as well as in selectively cleaving chemical bonds in complex molecules [@ASSION1998]. Simultaneously, a very different thrust has been extending the regime of the ultracold ($T \leq
1\,\mathrm{mK}$) to simple molecular complexes [@BURNETT2002]. Robust samples of trapped ultracold molecules are expected to facilitate significant advances in molecular spectroscopy, collision studies, and perhaps quantum computation [@DEMILLE2002; @TESCH2002].
The difficulty in generalizing laser cooling techniques to molecules has stimulated exploration of alternative approaches to producing ultracold molecules. One approach, using magnetic Feshbach resonances, has led to the observation of molecular Bose-Einstein condensates [@JOCHIM2003; @GREINER2003; @ZWIERLEIN2003]. The study of other routes to ultracold molecule formation has also been a topic of intense activity [@MASNOUSEEUWS2001]. An extremely successful optical approach has been to photoassociate molecules from ultracold atoms [@THORSHEIM1987]. In this method, a sample of ultracold atoms is irradiated by a cw laser tuned to excite free atoms to weakly bound excited states. Stable ground-state molecules may then form by spontaneous emission, provided the Franck-Condon overlap factors are favorable. Ultracold molecules have been observed using this technique for a variety of homonuclear [@FIORETTI1998; @NIKOLOV1999; @GABBANINI2000; @FATEMI2002] and heteronuclear [@KERMAN2004; @MANCINI2004; @HAIMBERGER2004; @WANG2004A] alkali metal species.
![\[fig:potentials\] (Color online) Schematic of the chirped femtosecond pulse photoassociation experiment. The potential energy curves of $\mbox{Rb}_2$ involved in the design of the experiment are shown. (a) Ultracold atoms are excited over a range of internuclear separations by a chirped femtosecond pulse to bound states below the $5\mathrm{S}$+$5\mathrm{P}_{3/2}$ dissociation limit. The time-dependent frequency sweep ensures the unlikelihood of population cycling back to the ground state. (b) Excited molecules undergo spontaneous decay, some forming stable ground-state molecules. (c) These molecules are detected by resonance-enhanced two-photon ionization and time-of-flight ion-mass spectroscopy.](Fig1_Rb_chirp_PA_noRscale_vPRL2.eps)
Recent proposals have suggested controlling the interactions between ultracold atoms with tailored picosecond optical fields [@VALA2001A; @LUCKOENIG2004A; @LUCKOENIG2004; @KOCH2004]. The possibility of improved formation of ultracold molecules using femtosecond coherent control techniques is therefore enticing. When a pulse length is $< 100\,\mathrm{ps}$, the light-matter interaction is entirely coherent, and therefore the details of the temporal shape of the pulse electric field may effect the interaction significantly. This Letter describes the first experiments designed to enhance the molecule formation process in a magneto-optical trap (MOT) [@RAAB1987] with chirped femtosecond pulses. A schematic of the experiments is shown in Fig. \[fig:potentials\]. The broad bandwidth of femtosecond pulses permits excitation over a longer range of internuclear distances than picosecond pulses, thus addressing a larger number of atom pairs. Introducing a positively chirped molecular “$\pi$”-pulse allows wave packet dynamics to strongly suppress detrimental population cycling [@CAO1998]. This method also provides a route to optimal control of ultracold molecule formation [@BROWN2005]. Recent experiments using picosecond [@FATEMI2001] and chirped nanosecond [@WRIGHT2005] pulses for photoassociation (PA) provide encouragement for this approach. However, as we show in this Letter, the dynamics of molecule formation using femtosecond pulses appears to be significantly different from those predicted for picosecond pulses [@LUCKOENIG2004A; @LUCKOENIG2004]. Here we report a suppression of molecule formation rather than an enhancement; nevertheless, the mechanism responsible for this effect appears to be coherent.
Femtosecond (fs) chirped-pulse PA was undertaken in an Rb MOT because its trapping transition ($780.27\,\mathrm{nm}$) is near the peak of the gain profile for Ti:sapphire ($800\,\mathrm{nm}$). Each isotope of Rb could be trapped by adjustment of the MOT trapping and repumping laser frequencies. Typical traps contained $2 \times 10^{7}$ atoms in a spheroidal cloud of diameter $0.7\,\mathrm{mm}$, yielding a peak density on the order of $10^{10}\,\mbox{cm}^{-3}$. The temperature of the atoms in the MOT was measured to be $170\,\mu\mathrm{K}$ using the release and recapture method [@LETT1988].
Two different PA lasers were employed to study the effects of light on the molecule formation process in the MOT. The first was a cw free-running diode laser (Sanyo DL7140-201) with an output power of $56\,\mathrm{mW}$, temperature-tuned between $700$–$785\,\mathrm{nm}$. The second laser was a commercial femtosecond-pulse oscillator (Spectra-Physics Mai Tai) whose carrier wavelength was tunable between $750$–$850\,\mathrm{nm}$ with output power $\leq 800\,\mathrm{mW}$ over this range. This laser had a repetition rate of $80\,\mathrm{MHz}$ and produced Gaussian-envelope transform-limited pulses of temporal intensity full-width at half-maximum (FWHM) $<100\,\mathrm{fs}$, corresponding to a spectral bandwidth FWHM $> 150\,\mathrm{cm}^{-1}$ for the available carrier wavelength range. Its output spectrum was monitored with a spectrometer and its field profile was regularly characterized using Spectral Interferometry for Direct Electric Field Reconstruction (SPIDER) [@IACONIS1998].
Molecules formed in the MOT were detected via time-of-flight (TOF) ion-mass spectroscopy using a channel electron multiplier. A narrow-bandwidth ($< 3\,\mathrm{GHz}$) tunable pulsed dye laser ($9\,\mathrm{ns}$ pulse duration, $50\,\mathrm{Hz}$ repetition rate) ionized the MOT cloud with pulses of energy $500\,\mu\mathrm{J}$. The PA and ionization lasers were merged onto the same beampath with a dichroic beamsplitter and focused at the MOT cloud, ensuring that the sample that we ionized was the same as that affected by our PA lasers. We chose to detect ground triplet state $a^{3}\Sigma_{u}^{+}$ $\mbox{Rb}_{2}$ molecules, which have been observed to form spontaneously in the MOT through one or both of two channels: PA stimulated by the MOT lasers themselves, and three-body recombination [@GABBANINI2000; @CAIRES2005]. The ionization wavelength was tuned to $602.7\,\mathrm{nm}$, which ionizes ground triplet state molecules via a resonant two-photon excitation ($X^{2}\Sigma_{g} \leftarrow (2)^{3}\Pi_{g} \leftarrow
a^{3}\Sigma_{u}^{+}$). To ensure that the detected molecules were in their electronic ground state, the MOT and PA lasers were shut off for at least $1\,\mu\mathrm{s}$ before the arrival of the ionization laser pulse to allow the atoms and molecules in the MOT to decay to their ground states. For a $^{85}\mbox{Rb}$ MOT the background detection rate of $\mbox{Rb}^{+}_{2}$ ions was typically 2–3 ions per pulse, while for a $^{87}\mbox{Rb}$ MOT the rate was significantly lower (0.3–0.8 ions per pulse).
A typical experiment involved applying one of the PA lasers to the MOT, applying the ionizing laser, and accumulating ion counts for a set number of ionization laser shots $N_{\mathrm{shots}}$. A background scan with the PA laser blocked was then immediately acquired for another $N_{\mathrm{shots}}$ to determine the relative effect of the PA laser.
The cw PA laser was focused on the MOT with an intensity of around $10^3\,\mathrm{W}\,\mathrm{cm}^{-2}$. For an $^{87}\mbox{Rb}$ MOT, the molecular signal was increased for detunings of a few $\mbox{cm}^{-1}$ below the trapping transition, with significant enhancement (by a factor of 1.5–2) at frequencies corresponding to resonant transitions to excited molecular vibrational states. For $^{85}\mbox{Rb}$, the molecular signal was unaffected at resonant transition frequencies, and generally reduced below the background level by 50% at non-resonant frequencies. These results confirm those of Ref. [@GABBANINI2000], where the quenching phenomenon was interpreted as a coupling of molecules that form spontaneously in the $^{85}\mbox{Rb}$ MOT to excited dissociative states.
In order to promote efficient photoassociative excitations near the D2 line by the fs pulses, care was taken to select a pulse spectrum with power concentrated in the region just below the $5\mathrm{S}$+$5\mathrm{P}_{3/2}$ dissociation limit ($780\,\mathrm{nm}$). The carrier wavelength of the fs laser was tuned to $783\,\mathrm{nm}$. A spectral filter was used to ensure that the broad spectrum of the fs pulses did not contain significant power on the blue side of the D2 line (which would lead to direct excitation of atoms to repulsive states lying above the $5\mathrm{S}$+$5\mathrm{P}_{3/2}$ dissociation limit). The fs pulse power was attenuated to $\leq 0.3\,\mathrm{nJ}$ per pulse, corresponding to focused peak intensities on the order of $10^7\,\mathrm{W}\,\mathrm{cm}^{-2}$. Since this intensity is many orders of magnitude above the Rb D2 atomic transition saturation intensity ($6\,\mathrm{mW}\,\mathrm{cm}^{-2}$), the light-matter interaction is beyond the perturbative regime.
![\[fig:NDscan\] $^{85}\mathrm{Rb}^{+}_{2}$ counts per ionization pulse vs. peak intensity of the applied fs pulses (filled circles). Each data point represents the accumulated result of an identical number of ionization laser shots. An exponential decay fit of the data is shown (solid line).](Fig2_NDscan_results_plus_fit_v2.eps)
Application of the spectrally-filtered fs-pulse PA laser to $^{85}\mbox{Rb}$ and $^{87}\mbox{Rb}$ MOTs generally caused a strong reduction in the molecular ion signal as compared to background. In the case of a $^{85}\mbox{Rb}$ MOT, the quenching observed due to the fs PA laser was stronger than that induced by the cw PA laser. In the case of a $^{87}\mbox{Rb}$ MOT, contrary to the augmentative effect observed when the cw PA laser was applied, $\sim 50\%$ fewer molecular ions with respect to background were observed upon application of the fs laser under similar experimental conditions. $^{85}\mathrm{Rb}_2^+$ ionization spectra for 600–610$\,\mathrm{nm}$ recorded for the cases of no PA laser, cw PA laser, and fs PA laser had similar structure. Variation of the average power of the fs PA laser revealed evidence of an exponential-decay dependence of molecular ion yield on pulse power (see Fig. \[fig:NDscan\]).
We further explored the effects of fs pulses on the molecular formation rate by varying the spectral chirp of the pulses. The $100\,\mathrm{fs}$ FWHM pulses were chirped by passing the beam 13 times through a $5.0\,\mathrm{in.}$-length plane-parallel dispersive glass block. The chirped pulses had an estimated temporal intensity FWHM of $5.8\,\mathrm{ps}$. To test whether the quenching effect involved resonant excitation of ground state atoms by the fs laser, we also tuned its carrier wavelength to $850\,\mathrm{nm}$ (as far to the red as experimentally feasible) and removed the spectral filter. A spectrometer was used to confirm that the power spectrum of the chirped pulses did not differ substantially from that of non-chirped pulses.
 (a) Schematic of the experimental setup. The fs PA beam was split by a 50:50 beamsplitter; one beam was chirped by the dispersive glass block ($\phi''$), and the other (reference) beam was power-balanced using neutral density filters (ND). A chopper wheel synchronized with the detection apparatus allowed only one of the two beams to pass at a given instant. The two beams were merged at a second 50:50 beamsplitter and focused to the MOT. The dye laser was merged onto the same beampath with a dichroic beamsplitter (DC). (b) Reference/chirp beam detection timing. Alternating $80\,\mathrm{MHz}$ trains of reference (b-i) and chirped (b-ii) pulses were applied to the MOT. One dye laser ionization pulse (b-iii) fired for each train of pulses. Not shown here is the extinction of each pulse train for $1\,\mu\mathrm{s}$ prior to the firing of the dye pulse.](Fig3_MTPaper_MTsetup_vPRL2.eps)
Due to run-to-run drift of the MOT conditions and the significant time required to re-align the setup when removing and re-inserting the glass block into the PA laser beampath, we implemented a “real-time” differential measurement to compare the effects of chirped and non-chirped fs pulses. A schematic diagram of the experimental setup appears in Fig. \[fig:setup\]. A broadband 50:50 beamsplitter was used to split the fs laser beam into two identical components. The dispersive glass block was inserted into the path of one of these beams (hereafter the “chirped beam”). The second, transform-limited, beam (the “reference beam”) was attenuated with absorptive neutral density filters so that each beam had the same average power, to $\pm1\%$ accuracy. An optical chopper wheel was aligned such that at any given instant one beam was blocked and the other was allowed to pass. Downstream from the chopper, the two beams were merged onto the same beampath using a second 50:50 broadband beamsplitter. From this point forward, the two beams were aligned and focused to the MOT position as usual.
The experiment timing was set up so that for a single rotation of the chopper wheel, one chirped pulse experiment and one reference pulse experiment occurred in sequence (see Fig. \[fig:setup\]). The TOF data acquisition was performed in “toggle” mode: TOF traces acquired for chirped pulse cases were added to the TOF histogram and traces acquired for reference pulse cases were subtracted from the histogram, yielding a net difference TOF histogram. A baseline differential measurement for the case with the dispersive glass block removed in Fig. \[fig:setup\] revealed a difference between two identical beams of fewer than $0.04\,\mbox{Rb}_{2}^{+}$ ions detected per ionization pulse.
![\[fig:toggledata\] Difference (“chirped case” $-$ “reference case”) TOF histogram comparing the effects of chirped and transform-limited reference pulses. The vertical dashed lines indicate the approximate arrival times of $^{85}\mathrm{Rb}$ atomic and molecular ions at the channeltron. There is no difference in the atomic ion signal between the two cases. A significant negative feature appears at the molecular ion arrival time, indicating that more $\mbox{Rb}_{2}^{+}$ ions were detected after applying the reference pulses than the chirped pulses.](Fig4_Toggle_data_v3.eps)
The results of an experiment comparing the effects of a pulse train of $5.8\,\mathrm{ps}$ pulses to the effects of a pulse train of $100\,\mathrm{fs}$ pulses are shown in Fig. \[fig:toggledata\]. The differential TOF trace reveals no discernable difference in the number of atomic ions, and a sharp negative feature in the vicinity of the arrival time of $^{85}\mbox{Rb}_{2}^{+}$ ions. The magnitude of this feature exceeded the baseline experiment $\mbox{Rb}_{2}^{+}$ detection rate by 400%. The negative feature in Fig. \[fig:toggledata\] signifies that appreciably more $\mbox{Rb}_{2}^{+}$ ions were accumulated after application of the transform-limited reference pulses. Thus, the chirped pulses quenched the $\mbox{Rb}_{2}^{+}$ ion signal more than the reference pulses. Since the chirped pulses had a peak intensity $\sim70$ times smaller than the reference pulses, this quenching effect was not solely dependent on the peak intensity, but also on the phase of the applied optical field. The quenching tended to increase with increasing chirp magnitude, although this trend was not monotonic as some chirp values were significantly more effective than others. These results provide evidence that the quenching effect is a coherent phenomenon in which wave packet dynamics play some role.
The exponential decay dependence of molecular formation rate on applied fs pulse peak intensity is consistent with a single-photon excitation loss from either the detected $a^3\Sigma_u^+$ or the excited $(1)^3\Sigma_g^+\big/0_g^-$ states. There are several coherent excitation pathways for $850\,\mathrm{nm}$ light that could be responsible for a reduction in the number of detected $a^3\Sigma_u^+$ molecules. One is MOT-laser PA followed by excitation of excited molecules in the $(1)^3\Sigma_g^+\big/0_g^-$ states to bound states in potentials that dissociate to 5P+5P ($(6)^3\Sigma_u^+$, $(7)^3\Sigma_u^+$, and $(4)^3\Pi_u$)—all of which are “dark” to our detection scheme—or to dissociative states lying above the 5S+6P dissociation limit. Another possibility is that a wave packet is excited from the ground state $a^3\Sigma_u^+$ onto the $C_3R^{-3}$ potential of the $(1)^3\Sigma_g^+$ state. This wave packet may have enhanced Franck-Condon overlaps with either continuum states lying above the 5S+5S dissociation limit or undetected $a^3\Sigma_u^+$ bound states, thus resulting in a decrease in the number of detected $a^3\Sigma_u^+$ molecules [@KOCH2005A].
In summary, we observed that application of fs pulses designed to photoassociate ultracold atoms results in a decrease, rather than an increase, in the formation of stable ultracold $a^3\Sigma_u^+$ molecules. Related work has recently been undertaken which provides evidence that the shape of the pulse spectrum has an effect on the quenching of the molecular formation rate [@SALZMANN2005]. We have shown here that phase-shaping of the applied fs pulses can be used to control the quenching rate. This result provides evidence that the quenching is a coherent process, although we are unable to distinguish the specific excitation pathway stimulated by the fs pulses. Further experimental and theoretical study of fs pulses for optimal control of photoassociative ultracold molecule formation is warranted, particularly for the creation of stable singlet molecules whose deeper $X^1\Sigma_g^+$ potential may be better suited to the broad inherent bandwidth of fs pulses.
We thank Françoise Masnou-Seeuws, Christiane Koch, Thorsten Köhler, and Nick Bigelow for helpful discussions. This work was supported by the National Science Foundation, grant no.PHY9877023. AJD acknowledges support from an EPSRC DTA studentship.
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abstract: 'Let $\left\{ X_{n}\right\} $ be an integer valued Markov Chain with finite state space. Let $S_{n}=\sum_{k=0}^{n}X_{k}$ and let $L_{n}\left(x\right)$ be the number of times $S_{k}$ hits $x\in\ZZ$ up to step $n.$ Define the normalized local time process $l_{n}\left(t,x\right)$ by $$l_{n}\left(t,x\right)=\frac{L_{\left\lfloor nt\right\rfloor }\left(\left\lfloor \sqrt{n}x\right\rfloor \right)}{\sqrt{n}},\,\, x\in\RR.$$ The subject of this paper is to prove a functional, weak invariance principle for the normalized sequence $l{}_{n}\left(t,x\right),$i.e. we prove under the assumption of strong aperiodicity of the Markov Chain that the normalized local times converge in distribution to the local time of the Brownian Motion.'
author:
- Michael Bromberg and Zemer Kosloff
title: Weak invariance principle for the local times of partial sums of Markov Chains
---
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\[sub:General-Definitions\]Introduction.
========================================
Let $S\subseteq\ZZ$ be a finite set, $P:S\times S\rightarrow[0,1]$ an irreducible, aperiodic stochastic matrix. Let $\mu$ be some distribution on $S$ and let $\left\{ X_{i}\right\} _{i=0}^{\infty}$ be a Markov Chain generated by $P$ and the initial distribution $\mu$, i.e. $X_{i},\, i=0,1,2,...$ are random variables defined on a probability space $\left(X,\BB,P^{\mu}\right)$, taking values in $S$, such that the equality
$$\PP^{\mu}\left(X_{i_{1}}=s_{1},\, X_{i_{2}}=s_{2},...,\, X_{i_{n}}=s_{n}\right)=\sum_{s_{0}\in S}\mu_{s_{0}}\cdot P^{i_{1}}\left(s_{0},s_{1}\right)\cdot P^{i_{2}-i_{1}}\left(s_{1},s_{2}\right)...\cdot P^{i_{n}-i_{n-1}}\left(s_{n-1},s_{n}\right),$$ holds with $\mu_{s}=\mu\left\{ s\right\} .$ Under these assumptions, $P$ has a unique $P$-invariant probability distribution vector $\nu$, in the sense that the equality $\nu P=\nu$ holds.
We may assume that for every initial distribution $\mu$, $X=S^{\NN},$ $\BB$ is the Borel $\sigma$-field generated by cylinders of the type $\left[s_{0}s_{1}...s_{n}\right]=\left\{ \omega\in S^{\NN}:\,\omega_{i}=s_{i},\, i=0,1,...,n\right\} $ and $X_{i}\left(\omega\right)=\omega_{i}.$
Set $$S_{n}=\sum\limits _{k=0}^{n}X_{k},\ W_{n}(t)=\frac{1}{\sqrt{n}}S_{\left\lfloor nt\right\rfloor },t\in[0,1].$$
Let $D_{[s,t]}$, $D_{+}$ and $D$ denote the spaces of functions defined on $\left[s,t\right]$, $\left[0,\infty\right)$ and $\RR$ respectively, that are continuous on the right with left limits at each point (Cadlag functions). $D_{\left[s,t\right]}$, $D_{+}$ and $D$ endowed with the Skorokhod $J_{1}$ topology and its suitable generalization to $\left[0,\infty\right)$ and $\RR$, constitute Polish (complete and separable metrizable) spaces (see [@Bil]).
Under the assumption that $E^{\nu}\left(X_{0}\right)=0$, the weak invariance principle for the sequence $W_{n}$ holds for any initial distribution $\mu,$ i.e. the process $W_{n}\left(t\right)$ regarded as a sequence of random variables defined on $(X,\BB,P^{\mu})$ and taking values in $D_{+}$ converges in distribution to the Brownian motion, henceforth denoted by $W_{\sigma}\left(t\right),$ where $$\sigma^{2}=Var\left(W_{\sigma}\left(1\right)\right)=\lim\limits _{n\rightarrow\infty}\frac{E^{\nu}\left(S_{n}^{2}\right)}{n}$$ is the asymptotic variance of the process $S_{n}$ with respect to the initial distribution $\nu.$ Now, $\sigma^{2}=0$ if and only if $X$ is a coboundary, meaning that there exists a square integrable process $Y=\left\{ Y_{n}\right\} $ such that (see [@R-E Lemma 6] or [@HeH II.3 Thm.A]) $$X_{n}=Y_{n}-Y_{n+1}.$$
In this case the functional limit of $W_{n}$ is degenerate, so we restrict our attention to the case when $\sigma^{2}>0$. The invariance principle may be proved by techniques used in this article, or otherwise, see [@Bil].
Let $L_{n}(x)=\#\left\{ k\leq n:S_{k}=x\right\} =\sum_{k=0}^{n}\One_{\left\{ x\right\} }\left(S_{k}\right)$ denote the number of arrivals of the process $\left\{ S_{k}\right\} _{k\in\NN}$ at the point $x$ after $n$ steps, and let $l_{n}(t,x)=\frac{1}{\sqrt{n}}L_{\left[nt\right]}\left(\left[\sqrt{n}x\right]\right)$, $\left(t,x\right)\in\left[0,\infty\right]\times\RR$. $l_{n}\left(t,x\right)$ is the amount of time multiplied by $\sqrt{n}$ that the process $W_{n}\left(\cdot\right)$ spends up to the time instant $t$ at the point $\frac{\left[\sqrt{n}x\right]}{\sqrt{n}}$ .
Let $l\left(t,x\right)$ be the local time of the Brownian motion $W_{\sigma}\left(\cdot\right)$ at time $t$, i.e. $l\left(\cdot,\cdot\right)$ is a random function satisfying the equality $$\int\limits _{A}l\left(t,x\right)dx=\int_{0}^{t}\One_{A}\left(W_{\sigma}\left(t\right)\right)dt$$ for every Borel set $A\in\BB_{\RR}$ and $t\in\left[0,\infty\right]$, $\PP$ - a.s where $\PP$ is the Wiener measure on $C(\RR)$, the space of continuous functions on $\RR$ (see [@MP]).
Note, that for every $t\geq0$, the sequence $l_{n}\left(t,x\right)$ defines a $D$ valued process on $X$. Since, $l(t,\cdot)$ is a.s. continuous on $\RR$, we may also regard it as a $D$ valued random variable.
In this paper, we prove that under the assumption that the Markov chain is strongly aperiodic (see definition \[def:Aperiodicity\]) , the weak invariance principle for $W_{n}\left(t\right)$ implies an invariance principle for the local times, i.e. the convergence in distribution of $W_{n}$ to $W_{\sigma}$ implies that $l_{n}\left(t,x\right)$ converges to $l\left(t,x\right)$ (here, convergence is in the sense that there exists a probability space where $l_{n}\left(t,x\right)$ converges to $l\left(t,x\right)$ uniformly on compact sets). The case when $X_{i}$’s are i.i.d’s was treated by Borodin [@Bor], under the assumption that for every $t\notin2\pi\ZZ$, $Ee^{itX_{1}}\neq1$, an assumption which we refer to as non-arithmeticity. We also show that the assumption of strong aperiodicity may be exchanged for the weaker assumption of non-arithmeticity in the i.i.d case, and for a certain class of Markov chains (see the remark following Definition \[def:Aperiodicity\]).
### Conventions and Notations: {#conventions-and-notations .unnumbered}
- $\PP^{x}$ denotes the measure $\PP^{\mu}$, where $\mu$ assigns mass $1$ to the point $x\in S$.
- $E^{\mu}$ denotes integrals on $S$ with respect to the measure $\PP^{\mu}$ and by $E^{x}$ integrals with respect to the measure $\PP^{x}$.
- For two random variables $X$ and $Y$, $X\deq Y$ means that $X$ and $Y$ are equally distributed. Also $X_{n}\overset{d}{\Rightarrow}X$ means that $X_{n}$ converges in distribution to $X$.
- $\mathbb{C}^{S}$ denotes the space of complex valued functions on $S$, which is isomorphic to $\mathbb{C}^{n}$ with $n=\left|S\right|$.
- $\mathcal{L_{S}}$ denotes the space of linear operators from $\mathbb{C}^{S}$ into itself, and we denote by $\left\Vert \cdot\right\Vert $ the operator norm on this space.
- Throughout the paper we assume that $P$ is aperiodic and irreducible, $\nu$ always denotes the stationary distribution with respect to $P$.
Statement of the Main Theorem.
==============================
Before stating the main theorem we introduce the notion of strong aperiodicity (see [@GH]):
*\[def:Aperiodicity\]A function $f:S\times S\rightarrow\mathbb{Z}$ is said to be aperiodic if the only solutions for*
*$$e^{itf\left(x,y\right)}=\lambda\frac{\varphi\left(x\right)}{\varphi\left(y\right)},\quad\forall x,y\in S,\; such\: that\:\mathbb{P}^{x}\left(y\right)>0$$*
*with $t\in\RR$, $\left|\lambda\right|=1$, $\varphi\in\mathbb{C}^{S}$, $\left|\varphi\right|\equiv1$ are $t\in2\pi\ZZ,$ $\varphi\equiv const$. $f$ is is said to be periodic if it is not aperiodic.*
*Since, we are only interested in the case $f\left(x,y\right)=y$, which gives $f\left(X_{n-1},X_{n}\right)=X_{n}$, we say that the Markov Chain $\left\{ X_{n}\right\} $ is strongly aperiodic iff the function $f\left(x,y\right)=y$ is aperiodic.*
The assumption of strong aperiodicity may be dropped in case $\left\{ X_{i}\right\} $ is a sequence of i.i.d’s and may be dropped in the Markov case provided that the underlying Markov shift is almost onto (see section \[sec:The-Periodic-Case\]). Note that strong aperiodicity of the Markov Chain implies aperiodicity of the stochastic matrix $P$, while the reverse implication is generally false - a simple random walk where $X_{n}$ equals $\pm1$ with equal probability may serve as a counter example. Here for $t=\pi,\lambda=-1$ and $\varphi\equiv1$, $$e^{i\pi X_{n}}=-1,$$ hence $X_{n}$ is not strongly aperiodic. However $$P=\left(\begin{array}{cc}
\frac{1}{2} & \frac{1}{2}\\
\frac{1}{2} & \frac{1}{2}
\end{array}\right)$$ is aperiodic. For discussion of condition \[def:Aperiodicity\] in a more general setting, see [@AD].
\[thm: Borodin for MC\]Let $\left\{ X_{n}\right\} $ be an irreducible, strongly aperiodic, finite state Markov chain. Assume that $E^{\nu}\left(X_{0}\right)=0$and $$W_{n}\overset{d}{\Rightarrow}W_{\sigma},\ \sigma>0.$$ Then there exists a probability space $\left(X',\BB',\PP'\right)$ and processes $W'_{n},\, W'_{\sigma}:X\rightarrow D_{+}$ such that:
1. *$W_{n}\overset{d}{=}W'_{n};\, W_{\sigma}\overset{d}{=}W'_{\sigma}$.*
2. *With probability one $W'_{n}$ converges to $W'_{\sigma}$ uniformly on compact subsets of $\left[0,\infty\right)$.*
3. *For every $\epsilon,T>0$ the processes $l'_{n}\left(t,x\right)$ and $l'\left(t,x\right)$ defined with respect to $W'_{n}$ and $W'_{\sigma}$ satisfy the relationship: $$\lim_{n\rightarrow\infty}\PP'\left(\sup_{t\in\left[0,T\right],\, x\in\RR}\left|l'_{n}\left(t,x\right)-l'\left(t,x\right)\right|>\epsilon\right)=0.\label{eq: Uniform convergence on compacts for local time}$$*
We start by fixing the time variable at $1$, and for convenience denote: $t_{n}\left(x\right)\equiv l_{n}\left(1,x\right)$ and $t\left(x\right)\equiv l\left(1,x\right).$ The main step on the road to proving the previous theorem is to prove:
\[thm: Main Theorem\]Let $\left\{ X_{n}\right\} $ be a finite state Markov chain, $E^{\nu}\left(X_{0}\right)=0$ and $$W_{n}\overset{d}{\Rightarrow}W_{\sigma},\ \sigma>0,$$ then $$t_{n}\overset{d}{\Rightarrow}t_{\sigma}.$$
In order to prove convergence in distribution of $\left\{ t_{n}(x)\right\} $, we have to establish relative compactness of the family $\Pi=\left\{ t_{n}\left(x\right)\right\} _{n\in\NN},$ which by definition means that every sequence of elements from $\Pi$ has a subsequence that converges in distribution. This is done in section \[sub: proof of the main theorem\]. Then to complete the proof, in section \[sub:Identifying the Only Possible Limit Point\] we show that $t_{\sigma}\left(x\right)$ is the only possible distributional limit point.
### Organization of the paper: {#organization-of-the-paper .unnumbered}
We begin with the proof under the assumption that $X_{n}$ is strongly aperiodic. Section \[sub:The Characteristic Function Operator\] contains estimates of the characteristic functions of the relevant processes, which are later used, in section \[sub:Probabilistic Estimates\] to carry out the calculations needed for the proof of Theorem \[thm: Main Theorem\]. In sections \[sub: proof of the main theorem\] and \[sec:Borodin’s-Theorem.\] we prove Theorems \[thm: Main Theorem\] and \[thm: Borodin for MC\] respectively under the assumption of strong aperiodicity.
In Section \[sec:The-Periodic-Case\] we show the modifications needed to extend these results to the periodic case.
This work was motivated by Aaronson’s work on *random walks in random sceneries* with independent jump random variables. In section \[sec:Applications-to-complexity\] we explain the model of RWRS and why Theorem \[thm: Borodin for MC\] implies Aaronson’s result for RWRS with the distribution of a strongly aperiodic Markov Chain.
\[sub:The Characteristic Function Operator\]The Characteristic Function Operator $Q(t)$
=======================================================================================
From the irreducibility of $P$ it follows that there is a unique probability distribution $\nu$ on $S$, such that $\nu P=\nu$. This is equivalent to $1$ being a simple eigenvalue of $P$, with a right-hand eigenspace of vectors in $\mathbb{C}^{S}$ having equal entries. Moreover, from aperiodicity of $P$ it follows that all other eigenvalues of $P$ are of modulus strictly less than $1.$ The projection to the eigenspace belonging to $1$ is given by $\left\langle v,\cdot\right\rangle \One=E^{\nu}\left(\cdot\right)\One$ (here, $\One$ is a vector in $\mathbb{C}^{S}$ with all entries equal to $1$). In what follows, we omit the $\One$, so that depending on the context $E^{\nu}\left(f\right)$ may denote both a scalar, or a vector in $\mathbb{C}^{S}$ with all entries equal to $E^{\nu}\left(f\right)$. Hence, we can write
$$Pf=E^{\nu}\left(f\right)+Nf,$$ where $N$ is an operator with spectral radius $\rho\left(N\right)$ strictly less then $1$. It follows that for every $\rho\left(N\right)<\eta<1$, and $n$ large enough, we have $$\left\Vert N^{n}\right\Vert \leq\eta^{n}.$$
Let $Q\left(t\right):\mathbb{C}\rightarrow\mathcal{L_{S}}$ be an operator valued function defined by $$Q\left(t\right)f\left(x\right):=\int\limits _{S}e^{ity}f\left(y\right)\PP^{x}\left(dy\right)=E^{x}\left(e^{itX_{0}}f\right)=\sum_{y\in S}P\left(x,y\right)e^{ity}f\left(y\right).$$ Note that $Q\left(0\right)=P$.
Since, $S$ is a finite state space, it is easy to see that the function $Q\left(t\right)$ is holomorphic and its derivatives are given by
$$\left(\frac{d^{n}}{dt^{n}}Q\left(t\right)f\right)\left(x\right)=i^{n}\int y^{n}e^{ity}f\left(y\right)\,\PP^{x}\left(dy\right)=i^{n}\sum_{y\in S}y^{n}P\left(x,y\right)e^{ity}f\left(y\right).$$
Let $$\varphi_{n}^{\mu}\left(t\right)=E^{\mu}\left(e^{itS_{n}}\right),\ \varphi_{n}^{\mu}\left(t_{1},t_{2}\right)=E^{\mu}\left(e^{it_{1}S_{n}+it_{2}X_{n}}\right)$$ denote the characteristic functions of the processes $S_{n}$ and $\left(S_{n},\, X_{n}\right)$ respectively, under the initial distribution $\mu.$ Our primary interest in $Q\left(t\right)$ is due to the fact that $\varphi_{n}^{\mu}\left(t\right)=E^{\mu}\left(Q\left(t\right)^{n}\One\right),$ and $\varphi_{n}^{\mu}\left(t_{1},t_{2}\right)=E^{\mu}\left(Q\left(t_{1}\right)^{n}e^{it_{2}s}\right).$
Expansion of the largest eigenvalue.
------------------------------------
It may be shown using standard perturbation techniques (see [@Ka] or [@HeH Chapter 3]) that the largest eigenvalue of $Q\left(t\right)$ and its eigenspaces are analytic functions of $t$ in a neighborhood of $0.$ More precisely, we have that in a neighborhood $I$ of $0$, $Q$$\left(t\right)$ has a simple eigenvalue $\lambda\left(t\right)$ which is an analytic perturbation of $\lambda\left(0\right)$, and there is a positive gap between $\lambda\left(t\right)$ and all other eigenvalues of $Q\left(t\right)$, i.e. $$\inf\limits _{t\in I}\min\limits _{\tilde{\lambda}\left(t\right)}\left|\lambda\left(t\right)-\tilde{\lambda}\left(t\right)\right|\geq const>0,$$ where the maximum is taken over all remaining eigenvalues of $Q\left(t\right)$. The projection function to the corresponding eigenspaces $\Pi\left(t\right)$ is also analytic in $t$, and we may choose the eigenfunctions $v\left(t\right)$ to be analytic perturbations of $v\left(0\right)=\One.$
Continuity of $Q\left(t\right)$ immediately gives us that in a neighborhood $I$ of $0$
$$Q\left(t\right)=\lambda\left(t\right)\cdot\Pi\left(t\right)+N\left(t\right),\label{eq:Q(t) structure}$$
where $\sup_{t\in I}\left\Vert N\left(t\right)\right\Vert =1-\eta<1$.
We proceed by evaluating the functions $\lambda\left(t\right)$ and $\Pi\left(t\right)$ for small $t$. By the structure of $P=Q\left(0\right)$ discussed in the beginning of this section, we have
$$\begin{aligned}
\lambda\left(0\right) & = & 1,\nonumber \\
v\left(0\right) & = & 1,\label{eq: Q for small t}\\
\Pi(0)(\cdot) & = & \left\langle \nu,\cdot\right\rangle v\left(0\right)=E^{\nu}(\cdot).\nonumber \end{aligned}$$
\[lem: Expansion of Main Eigenvalue Lemma\]If $E^{\nu}\left(X_{0}\right)=0$, then there exists a real nonnegative constant $\sigma^{2}$ such that, $$\lambda\left(t\right)=1-\frac{\sigma^{2}}{2}t^{2}+\O\left(\left|t\right|^{3}\right),$$ and $$\Pi\left(t\right)\left(\cdot\right)=E^{\nu}\left(\cdot\right)+\O\left(\left|t\right|\right).$$
The statement concerning $\Pi\left(t\right)$ immediately follows from Taylor’s expansion and formula , while the expansion of the main eigenvalue is carried out in [@HeH Lemma IV.4’].
It may be shown [@HeH Lemma IV.3, pp 24-25] that $\sigma^{2}$ actually equals the asymptotic variance ${\displaystyle \lim_{n\to\infty}\frac{E^{\nu}\left(S_{n}^{2}\right)}{n}}$ and therefore, $\sigma^{2}=0$ corresponds to the case of a degenerate limit for $W_{n}$. Since this case is excluded in the main theorem, in what follows, we assume that the results of the previous lemma hold with $\sigma^{2}>0$.
The connection between strong aperiodicity and the spectrum of $Q(t)$.
----------------------------------------------------------------------
The next lemma connects the notion of strong aperiodicity with the spectrum of the characteristic function operator:
\[lem:Aperiodicity\] An aperiodic and irreducible Markov chain $\left\{ X_{n}\right\} $ is strongly aperiodic iff $\rho\left(Q\left(t\right)\right)<1$ for all real $t\neq2\pi\ZZ$ ($\rho\left(Q\left(t\right)\right)$ is the spectral radius of $Q\left(t\right)$).
Note that since $Q\left(t\right)$ is a finite dimensional operator of norm less than or equal to $1$, $\rho\left(Q\left(t\right)\right)<1$ is equivalent to demanding that $Q\left(t\right)$ has no eigenvalues of modulus $1$.
If $\left\{ X_{n}\right\} $ is not strongly aperiodic, we have $$e^{ity}=\lambda\frac{\varphi\left(x\right)}{\varphi\left(y\right)},\quad\forall x,y\in S\: such\: that\: P^{x}\left(y\right)>0$$ where $\left|\lambda\right|=1,$$\left|\varphi\right|\equiv1,$ $t\notin2\pi\mbox{\ensuremath{\ZZ}}$ or $t\in2\pi\mathbb{Z}$ and $\varphi\neq const$. In the first case notice that, $$\left(Q\left(t\right)\varphi\right)\left(x\right)=\int\limits _{S}e^{ity}\varphi\left(y\right)\PP^{x}\left(dy\right)=\lambda\varphi\left(x\right)\label{eq: eigenvalue of Q(t)}$$ whence $\rho\left(Q\left(t\right)\right)=1$ for $t\notin2\pi\mathbb{Z}.$
In the second case ( $t\in2\pi\mathbb{Z})$, since $y\in\mathbb{Z}$ we have $$\left(Q(t)\varphi\right)(x)=\left(Q(0)\varphi\right)(x)=\int_{S}\varphi(y)P^{x}(dy)=\lambda\varphi(x).$$ This is impossible for non constant $\varphi$ since we have a spectral gap property for $t=0$ (see the beginning of the section). This proves one direction of the iff statement.
To prove the opposite direction observe that if equation holds with $\left|\lambda\right|=1$ we have
$$\left|\varphi\left(x\right)\right|=\left|\left(Q\left(t\right)\varphi\right)\left(x\right)\right|\leq\int\limits _{S}\left|\varphi\left(y\right)\right|\PP^{x}\left(dy\right)=\left(P\left|\varphi\right|\right)\left(x\right).$$ Since, $1$ is a simple eigenvalue of $P$, we must have $\left|\varphi\right|\equiv const$ and therefore, we may assume that $\left|\varphi\right|\equiv1$. Hence since $$\int\limits _{S}e^{ity}\varphi\left(y\right)\PP^{x}\left(dy\right)=\lambda\varphi\left(x\right),$$ and $\left|\lambda\varphi(x)\right|=\left|\varphi(y)\right|=1$, it follows from Proposition \[Trivial proposition\] in the Appendix that $$e^{ity}\varphi\left(y\right)=\lambda\varphi\left(x\right),\quad\forall x,y\in S,\: with\:\PP^{x}\left(y\right)>0$$ and the claim follows.
\[sub:Probabilistic Estimates\]Estimates.
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In this section we derive the main probability estimates needed to prove the main theorem. When lemma \[lem: Expansion of Main Eigenvalue Lemma\] is used, it is always assumed to hold with $\xi>0$ (see the remark that follows the lemma).
\[lem:Non-Arit. Lemma\]If $\{X_{n}\}$ is strongly aperiodic, then for every $\delta>0,$ there are $N\in\NN,$ $0<r_{\delta}<1,$ $c>0$ such that for every $n>N,$ $t\in\left[-\pi,\pi\right]\setminus\left(-\delta,\delta\right),$ $$\left\Vert Q\left(t\right)^{n}\right\Vert \leq cr_{\delta}^{n}.$$
Fix $\delta>0$ and $t\in[-\pi,\pi]\backslash(-\delta,\delta)$. By strong aperiodicity (Lemma \[lem:Aperiodicity\]), $r_{t}:=r\left(Q\left(t\right)\right)<1.$
Fix, $r_{t}<\tilde{r}_{t}<1$ and choose $n_{0}$ such that $$r_{t}=\inf_{n\in\NN}\left\Vert Q(t)^{n}\right\Vert ^{\frac{1}{n}}\leq\left\Vert Q(t)^{n_{0}}\right\Vert ^{\frac{1}{n_{0}}}<\tilde{r}_{t}.$$ By continuity of $Q\left(\cdot\right)$ at $t$, there exists $\eta=\eta_{t}>0$, such that if $\tau\in(t-\eta,t+\eta)$, $$\left\Vert Q\left(\tau\right)^{n_{0}}\right\Vert <\tilde{r}_{t}^{n_{0}}.\label{eq: spectral bound on a neighborhood}$$ For $n\in\mathbb{N},$ write $n=n_{0}m+k$ where $k<n_{0}$. It follows from that for every $\tau\in\left(t-\eta,t+\eta\right)$, $$\left\Vert Q\left(\tau\right)^{n}\right\Vert \leq\left\Vert Q\left(\tau\right)^{n_{0}}\right\Vert ^{m}\cdot\left\Vert Q\left(\tau\right)^{k}\right\Vert \leq\tilde{r_{t}}^{n_{0}m}\cdot\left\Vert Q\left(\tau\right)^{k}\right\Vert \leq\tilde{r}_{t}^{n}\cdot\frac{\left\Vert Q\left(\tau\right)\right\Vert ^{k}}{\tilde{r}_{t}^{k}}.$$ Setting $c_{t}=\sup\frac{\left\Vert Q\left(x\right)\right\Vert ^{k}}{\tilde{r}_{t}^{k}}$where the supremum is taken over all $x\in\left(t-\eta,t+\eta\right),\, k\leq n_{0}$ we obtain $$\left\Vert Q\left(\tau\right)^{n}\right\Vert \leq c_{t}\tilde{r}_{t}^{n}.$$
Hence, for every $t\in\left[-\pi,\pi\right]\setminus\left(-\delta,\delta\right),$ we can pick positive numbers $n_{t},$ $\eta_{t},c_{t}$ and $r_{t}<1$ such that for every $n>n_{t}$ and $\tau\in\left(t-\eta_{t},t+\eta_{t}\right)$, $$\left\Vert Q\left(\tau\right)^{n}\right\Vert \leq c_{t}r_{t}^{n}.$$ Since, $\left[-\pi,\pi\right]\setminus\left(-\delta,\delta\right)$ is compact there exists a finite sequence $\left\{ t_{i}\right\} _{i=1,..l}$ such that $$\left[-\pi,\pi\right]\setminus\left(-\delta,\delta\right)\subset\bigcup\limits _{i=1}^{l}\left(t_{i}-\eta_{t_{i}},t_{i}+\eta_{t_{i}}\right).$$
The result follows by setting $N=\max\left\{ n_{t_{i}}\right\} ,$ $r_{\delta}=\max\left\{ r_{t_{i}}\right\} ,$ $c=\max\left\{ c_{t_{i}}\right\} $.
\[lem:LLT\]If $\left\{ X_{n}\right\} $ is strongly aperiodic and $E^{\nu}\left(X_{0}\right)=0,$ there exists a constant $C$ such that for any initial distribution $\mu,$ $n\in\NN,$ $x\in\ZZ,$ $$P^{\mu}\left(S_{n}=x\right)\leq\frac{C}{\sqrt{n}}.$$
In view of equation and lemma \[lem: Expansion of Main Eigenvalue Lemma\], there exist $\delta,\, a,\,\eta>0$ such that for $t\in(-\delta,\delta)$, $$Q\left(t\right)=\lambda\left(t\right)\Pi\left(t\right)+N\left(t\right)$$
where $$\left|\lambda\left(t\right)\right|\leq1-at^{2},\quad\left\Vert N\left(t\right)^{n}\right\Vert \leq\left(1-\eta\right)^{n}.$$
By the inversion formula for Fourier transform,
$$\begin{aligned}
\sqrt{n}\PP^{\mu}\left(S_{n}=x\right) & = & \frac{\sqrt{n}}{2\pi}\int\limits _{-\pi}^{\pi}E^{\mu}\left(Q\left(t\right)^{n}\One\right)e^{-itx}dt\\
& \leq & \left|\frac{\sqrt{n}}{2\pi}\int\limits _{-\delta}^{\delta}E^{\mu}\left(Q\left(t\right)^{n}\One\right)e^{-itx}dt\right|+\frac{\sqrt{n}}{2\pi}\int\limits _{R_{\delta}}\left\Vert Q\left(t\right)^{n}\right\Vert dt\label{eq:LLT Bound 1}\end{aligned}$$
By lemma \[lem:Non-Arit. Lemma\], there are $N\in\NN,$ $0<r_{\delta}<1,$ $c>0$ such that for any $n>N$, $$\frac{\sqrt{n}}{2\pi}\int\limits _{R_{\delta}}\left\Vert Q\left(t\right)^{n}\right\Vert dt\leq c\sqrt{n}r_{\delta}^{n}.$$ Therefore, the second term in is uniformly bounded in $n$ . We proceed with estimating the first term in . $$\begin{aligned}
\left|\frac{\sqrt{n}}{2\pi}\int\limits _{-\delta}^{\delta}E^{\mu}\left(Q\left(t\right)^{n}\One\right)e^{-itx}dt\right| & \leq & \frac{\sqrt{n}}{2\pi}\int\limits _{-\delta}^{\delta}\left|\lambda\left(t\right)\right|^{n}dt+\frac{\sqrt{n}}{2\pi}\int\limits _{-\delta}^{\delta}\left\Vert N\left(t\right)^{n}\right\Vert dt\\
& \leq & \frac{\sqrt{n}}{2\pi}\int\limits _{-\delta}^{\delta}\left(1-at^{2}\right)^{n}dt+\frac{2\delta\sqrt{n}}{2\pi}\left(1-\eta\right)^{n}.\label{eq: LLT Bound 2}\end{aligned}$$
It is obvious that the second term in tends to $0$. To see that the first term is bounded, a change of variables $t=\frac{x}{\sqrt{n}}$ gives us,
$$\begin{aligned}
\frac{\sqrt{n}}{2\pi}\int\limits _{-\delta}^{\delta}\left(1-at^{2}\right)^{n}dt & = & \frac{1}{2\pi}\int\limits _{-\delta\sqrt{n}}^{\delta\sqrt{n}}\left(1-\frac{ax^{2}}{n}\right)^{n}dx\\
& \leq & \frac{1}{2\pi}\int\limits _{-\delta\sqrt{n}}^{\delta\sqrt{n}}e^{-ax^{2}}dx.\end{aligned}$$
Since, the last integral is uniformly bounded in $n$, this completes the proof.
(Estimation of the Potential Kernel, see [@JP])\[pro:Potential Kernel Estimate\] If $\left\{ X_{n}\right\} $ is strongly aperiodic and $E^{\nu}\left(X_{0}\right)=0$, then there exists $C>0$ such that for every $x,y\in\mathbb{Z}$ and $s\in S$, and initial distribution $\mu,$ $$\sum_{n=0}^{\infty}\left|\PP^{\mu}\left(S_{n}=x,\, X_{n}=s\right)-\PP^{\mu}\left(S_{n}=y,\, X_{n}=s\right)\right|\leq C\left|x-y\right|.$$
We denote by $\gamma_{t}\left(\cdot\right):S\rightarrow\mathbb{T}$ the function $e^{it\left(\cdot\right)}$ from the state space $S$ to the circle $\mathbb{T}=\left\{ z\in\mathbb{C}:\left|z\right|=1\right\} $.
For simplicity of notation we assume that $y=0$. The generalization to the case when $y\neq0$ is straightforward. Note that since $x\in\mathbb{Z}$, its enough to prove that the term converges for every $x$ and is $O(|x|)$ as $|x|\to\infty$.
By lemma \[lem: Expansion of Main Eigenvalue Lemma\] and equation we may fix $\delta>0$ and positive constants $C_{1},\, C_{2},C_{3}\,\eta$ such that for $t\in\left(-\delta,\delta\right),$
$$\begin{aligned}
\left|\lambda\left(t\right)\right| & \leq & 1-C_{1}t^{2}\\
\left|\Im\lambda\left(t\right)\right| & \leq & C_{2}\left|t^{3}\right|\\
\left\Vert N\left(t\right)\right\Vert & \leq & 1-\eta\\
\Pi\left(t\right)\left(\cdot\right) & = & E^{\nu}\left(\cdot\right)+\epsilon\left(t\right)\left(\cdot\right)\end{aligned}$$
where $\left\Vert \epsilon\left(t\right)\right\Vert \leq C_{3}\cdot\left|t\right|$. Set $R_{\delta}=\left[-\pi,\pi\right]\setminus\left(-\delta,\delta\right).$
By the inversion formula for the Fourier Transform, $$\begin{aligned}
& & \sum_{n=0}^{\infty}\left|\PP^{\mu}\left(S_{n}=0,\, X_{n}=s\right)-\PP^{\mu}\left(S_{n}=x,\, X_{n}=s\right)\right|\\
& = & \sum_{n=0}^{\infty}\left|\Re\left\{ \frac{1}{\left(2\pi\right)^{2}}\int\limits _{-\pi}^{\pi}\intop\limits _{-\pi}^{\pi}E^{\mu}\left(Q\left(t_{1}\right)^{n}\gamma_{t_{2}}\right)\cdot\left(1-e^{-it_{1}x}\right)\cdot e^{-it_{2}s}\, dt_{1}dt_{2}\right\} \right|\label{eq:Fourier Transform Inverse Formula}\end{aligned}$$
By lemma \[lem:Non-Arit. Lemma\] there are positive constants $n_{0},$$c,$ $r_{\delta}<1$ such that for every $n\geq n_{0},$ $t\in R_{\delta},$ $$\left\Vert Q\left(t\right)^{n}\right\Vert \leq cr_{\delta}^{n}.$$ Thus, $$\begin{aligned}
& & \sum_{n=n_{0}}^{\infty}\frac{1}{(2\pi)^{2}}\intop_{-\pi}^{\pi}\int\limits _{R_{\delta}}\left|E^{\mu}\left(Q\left(t_{1}\right)^{n}\gamma_{t_{2}}\right)\cdot\left(1-e^{-it_{1}x}\right)\cdot e^{-it_{2}s}\right|\, dt_{1}dt_{2}\\
& \leq & \sum_{n=n_{0}}^{\infty}\frac{1}{\left(2\pi\right)^{2}}\int\limits _{-\pi}^{\pi}\intop\limits _{R_{\delta}}cr_{\delta}^{n}\cdot2\, dt_{1}dt_{2}<\infty\end{aligned}$$ Here $R_{\delta}=\left[-\pi,\pi\right]\backslash\left(-\delta,\delta\right)$. It follows that $$\sum\limits _{n=0}^{\infty}\left|\Re\left\{ \frac{1}{\left(2\pi\right)^{2}}\int\limits _{-\pi}^{\pi}\intop\limits _{R_{\delta}}E^{\mu}\left(Q\left(t_{1}\right)^{n}\gamma_{t_{2}}\right)\cdot\left(1-e^{-it_{1}x}\right)\cdot e^{-it_{2}s}\, dt_{1}dt_{2}\right\} \right|<\infty$$ which is enough for the statement of the lemma. Hence, it remains to estimate expression where the inner integration is carried over the set $\left(-\delta,\delta\right).$
We have,
$$\begin{aligned}
& \sum_{n=0}^{\infty}\left|\Re\left\{ \frac{1}{\left(2\pi\right)^{2}}\int\limits _{-\pi}^{\pi}\intop\limits _{-\delta}^{\delta}E^{\mu}\left(Q\left(t_{1}\right)^{n}\gamma_{t_{2}}\right)\cdot\left(1-e^{-it_{1}x}\right)\cdot e^{-it_{2}s}\, dt_{1}dt_{2}\right\} \right|\nonumber \\
& \leq\sum_{n=0}^{\infty}\left|\Re\left\{ \frac{1}{\left(2\pi\right)^{2}}\int\limits _{-\pi}^{\pi}\intop\limits _{-\delta}^{\delta}E^{\mu}\left(\lambda\left(t_{1}\right)^{n}\Pi\left(t_{1}\right)\gamma_{t_{2}}\right)\cdot\left(1-e^{-it_{1}x}\right)\cdot e^{-it_{2}s}\, dt_{1}dt_{2}\right\} \right|\label{eq:Estimate1}\\
& +\sum_{n=0}^{\infty}\left|\frac{1}{\left(2\pi\right)^{2}}\int\limits _{-\pi}^{\pi}\intop\limits _{-\delta}^{\delta}E^{\mu}\left(N\left(t_{1}\right)^{n}\gamma_{t_{2}}\right)\cdot\left(1-e^{-it_{1}x}\right)\cdot e^{-it_{2}s}\, dt_{1}dt_{2}\right|\nonumber \end{aligned}$$
Since, $\left\Vert N\left(t\right)\right\Vert \leq1-\eta<1$ and $\left|\gamma_{t}\left(\cdot\right)\right|\equiv1$, we have $$\sum_{n=0}^{\infty}\left|\int\limits _{-\pi}^{\pi}\intop\limits _{-\delta}^{\delta}\frac{1}{\left(2\pi\right)^{2}}E^{\mu}\left(N\left(t_{1}\right)^{n}\gamma_{t_{2}}\right)\cdot\left(1-e^{-it_{1}x}\right)\cdot e^{-it_{2}s}\, dt_{1}dt_{2}\right|\leq\sum_{n=0}^{\infty}C^{'}\cdot\left(1-\eta\right)^{n}\leq C^{'}\frac{1}{\eta}\label{eq:Result1}$$ where $C^{'}$ is some constant, which is again enough for our purposes.
We proceed with estimating :
$$\begin{aligned}
& & \left|\Re\left\{ \frac{1}{\left(2\pi\right)^{2}}\int\limits _{-\pi}^{\pi}\intop\limits _{-\delta}^{\delta}E^{\mu}\left(\lambda\left(t_{1}\right)^{n}\Pi\left(t_{1}\right)\gamma_{t_{2}}\right)\cdot\left(1-e^{-it_{1}x}\right)\cdot e^{-it_{2}s}\, dt_{1}dt_{2}\right\} \right|\label{eq:Estimate2}\\
& \leq & \left|\Re\left\{ \frac{1}{\left(2\pi\right)^{2}}\int\limits _{-\pi}^{\pi}\intop\limits _{-\delta}^{\delta}\lambda\left(t_{1}\right)^{n}E^{\nu}\left(\gamma_{t_{2}}\right)\cdot\left(1-e^{-it_{1}x}\right)\cdot e^{-it_{2}s}\, dt_{1}dt_{2}\right\} \right|\nonumber \\
& + & \left|\Re\left\{ \frac{1}{\left(2\pi\right)^{2}}\int\limits _{-\pi}^{\pi}\intop\limits _{-\delta}^{\delta}E^{\mu}\left(\lambda\left(t_{1}\right)^{n}\cdot\epsilon\left(t_{1}\right)\left(\gamma_{t_{2}}\right)\right)\cdot\left(1-e^{-it_{1}x}\right)\cdot e^{-it_{2}s}\, dt_{1}dt_{2}\right\} \right|.\nonumber \end{aligned}$$
Since, $E^{\nu}\left(\gamma_{t_{2}}\right)=E^{\nu}\left(e^{it_{2}X_{0}}\right),$ by using the inversion formula and the Fubini theorem we obtain, $$\begin{aligned}
& & \left|\Re\left\{ \frac{1}{\left(2\pi\right)^{2}}\int\limits _{-\pi}^{\pi}\intop\limits _{-\delta}^{\delta}\lambda\left(t_{1}\right)^{n}E^{\nu}\left(\gamma_{t_{2}}\right)\cdot\left(1-e^{-it_{1}x}\right)\cdot e^{-it_{2}s}\, dt_{1}dt_{2}\right\} \right|\nonumber \\
& = & \left|\Re\left\{ \frac{\nu_{s}}{2\pi}\intop\limits _{-\delta}^{\delta}\lambda\left(t\right)^{n}\left(1-e^{-itx}\right)\, dt\right\} \right|\leq\left|\Re\left\{ \frac{1}{2\pi}\intop\limits _{-\delta}^{\delta}\lambda\left(t\right)^{n}\cdot\left(1-e^{-itx}\right)\, dt\right\} \right|\nonumber \\
& \leq & \frac{1}{2\pi}\intop\limits _{-\delta}^{\delta}\left|\lambda(t)\right|^{n}\cdot\left(1-\cos tx\right)\, dt+\frac{1}{2\pi}\intop\limits _{-\delta}^{\delta}\left|\Im\left\{ \lambda(t)^{n}\right\} \right|\left|\sin tx\right|\, dt.\label{eq:Estimate4}\end{aligned}$$
By summing over $n$ the first term in expression and by our choice of $\delta,$
$$\begin{aligned}
\sum_{n=0}^{\infty}\frac{1}{2\pi}\intop\limits _{-\delta}^{\delta}\left|\lambda\left(t\right)\right|^{n}\cdot\left(1-\cos tx\right)\, dt & \leq & \frac{1}{2\pi}\intop\limits _{-\delta}^{\delta}\sum_{n=0}^{\infty}\left(1-C_{1}t^{2}\right)^{n}\left(1-\cos tx\right)\, dt\\
& = & \frac{1}{2\pi}\intop\limits _{-\delta}^{\delta}\frac{1}{C_{1}t^{2}}\left(1-\cos tx\right)\, dt\end{aligned}$$
Now, $$\begin{aligned}
\frac{1}{2\pi}\intop\limits _{-\frac{1}{x}}^{\frac{1}{x}}\frac{1}{C_{1}t^{2}}\left(1-\cos tx\right)\, dt & = & \O\left(\left|x\right|\right),\,\left(\left|x\right|\rightarrow\infty\right)\end{aligned}$$ and $$\frac{1}{2\pi}\intop\limits _{\frac{1}{x}}^{\delta}\frac{1}{C_{1}t^{2}}\left(1-\cos tx\right)\, dt\leq\frac{1}{\pi}\intop\limits _{\frac{1}{x}}^{\delta}\frac{1}{C_{1}t^{2}}\, dt=\O\left(\left|x\right|\right),\,\left(\left|x\right|\rightarrow\infty\right).$$
Hence, from the integrand being an even function, we have $$\sum_{n=0}^{\infty}\frac{1}{2\pi}\intop\limits _{-\delta}^{\delta}\left|\lambda\left(t\right)\right|^{n}\cdot\left(1-\cos tx\right)\, dt=\O\left(\left|x\right|\right),\,\left(\left|x\right|\rightarrow\infty\right)\label{eq:Result2}$$
To estimate the sum over $n$ of the second term in expression we note that,
$$\left|\Im\left\{ \lambda\left(t\right)^{n}\right\} \right|\leq\left|\Im\lambda\left(t\right)\right|\cdot n\left|\lambda\left(t\right)\right|^{n-1}\leq C_{2}t^{3}\cdot n\left(1-C_{1}t^{2}\right)^{n-1}$$ where the first inequality is easily seen to be true by induction on $n$. Therefore, $$\begin{aligned}
\sum_{n=0}^{\infty}\frac{1}{2\pi}\intop\limits _{-\delta}^{\delta}\left|\Im\left\{ \lambda\left(t\right)^{n}\right\} \right|\left|\sin tx\right|\, dt & \leq & \frac{1}{2\pi}\intop\limits _{-\delta}^{\delta}\frac{C_{2}\left|t\right|^{3}}{C_{1}t^{4}}\left|\sin tx\right|\, dt\nonumber \\
& = & \O\left(\left|x\right|\right),\,\left(\left|x\right|\rightarrow\infty\right)\label{eq:Result3}\end{aligned}$$
This completes the estimation of the sum over $n$ of the first term in expression and all is left is to estimate the sum over $n$ of the second term in expression :
$$\begin{aligned}
& & \sum_{n=0}^{\infty}\left|\Re\left\{ \frac{1}{\left(2\pi\right)^{2}}\int\limits _{-\pi}^{\pi}\intop\limits _{-\delta}^{\delta}E^{\mu}\left(\lambda\left(t_{1}\right)^{n}\epsilon_{t_{1}}\left(\gamma_{t_{2}}\right)\right)\cdot\left(1-e^{-it_{1}x}\right)\cdot e^{-it_{2}s}\, dt_{1}dt_{2}\right\} \right|\nonumber \\
& \leq & \frac{1}{\left(2\pi\right)^{2}}\int\limits _{-\pi}^{\pi}\intop\limits _{-\delta}^{\delta}\sum_{n=0}^{\infty}\left(1-C_{1}t_{1}^{2}\right)^{n}C_{3}\left|t_{1}\right|\cdot\left|1-e^{-it_{1}x}\right|\, dt_{1}dt_{2}\nonumber \\
& = & \frac{1}{\left(2\pi\right)^{2}}\int\limits _{-\pi}^{\pi}\intop\limits _{-\delta}^{\delta}\frac{C_{3}}{C_{1}\left|t_{1}\right|}\cdot\left|1-e^{-it_{1}x}\right|\, dt_{1}dt_{2}=\O\left(\left|x\right|\right),\,\left(\left|x\right|\rightarrow\infty\right)\label{eq:Result4}\end{aligned}$$
The lemma follows by combining results , , and .
\[sub: proof of the main theorem\]Proof of theorem \[thm: Main Theorem\].
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Throughout this section we assume that $\left\{ X_{n}\right\} $ satisfies the assumptions of Theorem \[thm: Main Theorem\].
A fourth moment inequality for $L_{n}(x)-L_{n}(y)$.
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\[cor: Fourth moments inequality\]There exists $C>0$ such that for every $x,y\in\ZZ$, $$E^{\mu}\left(\left(L_{n}(x)-L_{n}(y)\right)^{4}\right)\leq C\cdot n\cdot\left|x-y\right|^{2}.$$
Set $Z_{n}=\left(S_{n},X_{n}\right)$. Throughout the proof we use the fact that for any initial distribution $\mu$ on $S$, $x,y\in\ZZ$, $n,j\in\NN,(j\leq n$) and $s_{1},s_{2}\in S$, $$\PP^{\mu}\left(\left.Z_{n}=\left(x,s_{2}\right)\right|Z_{j}=\left(y,s_{1}\right)\right)=\PP^{s_{1}}\left(Z_{n-j}=\left(x-y,s_{2}\right)\right).\label{eq:FourthMoment3}$$
Let $\One_{z}\left(t\right)$ be the indicator function of the set $\left\{ z\right\} $ and set $$\psi\left(t\right)=\One_{x}\left(t\right)-\One_{y}\left(t\right).$$
We have
$$E^{\mu}\left(\left(L_{n}(x)-L_{n}(y)\right)^{4}\right)=\sum_{\bar{i}}E^{\mu}\left(\prod_{j=1}^{4}\psi(S_{i_{j}})\right)$$ where the summation is carried over all vectors $\bar{i}=\left(i_{1},i_{2},i_{3},i_{4}\right)$ with coordinates equal to integers between $0$ and $n$.
By symmetry, we may assume that $i_{1}\leq i_{2}\leq i_{3}\leq i_{4}$. For a fixed $\bar{i}$
$$E^{\mu}\left(\prod_{j=1}^{4}\psi(S_{i_{j}})\right)=\sum_{\xi}\left(-1\right)^{\sigma}E^{\mu}\left(\One_{\xi_{1}}\left(S_{i_{1}}\right)\psi\left(S_{i_{2}}\right)\One_{\xi_{2}}\left(S_{i_{3}}\right)\psi\left(S_{i_{4}}\right)\right)\label{eq: FourthMoment2}$$
where the sum is carried out over all vectors $\bar{\xi}=\left(\xi_{1},\xi_{2}\right)$ with coordinates equal to either $x$ or $y$ and $\mbox{ \ensuremath{\sigma}=\ensuremath{\sigma\left(\bar{\xi}\right)} }$ is the number of coordinates that equal $y$.
Fixing $\bar{i}$ and $\bar{\xi}$ , and using we obtain
$$\begin{aligned}
E^{\mu}\left(\One_{\xi_{1}}\left(S_{i_{1}}\right)\psi\left(S_{i_{2}}\right)\One_{\xi_{2}}\left(S_{i_{3}}\right)\psi\left(S_{i_{4}}\right)\right) & =\sum_{\bar{s}\in S^{4}} & \left[\PP^{\mu}\left(Z_{i_{1}}=\left(\xi_{1},s_{1}\right)\right)\cdot\right.\nonumber \\
& & \left.F\left(i_{1},i_{2},i_{3}\right)\cdot\left(\PP^{s_{3}}\left(S_{i_{4}}=x\right)-\PP^{s_{3}}\left(S_{i_{4}}=y\right)\right)\right]\label{eq: FourthMoment1}\end{aligned}$$
where
$$\begin{aligned}
F\left(i_{1},i_{2},i_{3}\right) & = & \left[\PP^{s_{1}}\left\{ Z_{i_{2}-i_{1}}=\left(x-\xi_{1},s_{2}\right)\right\} \PP^{s_{2}}\left\{ Z_{i_{3}-i_{2}}=\left(\xi_{2}-x,s_{3}\right)\right\} \right.\\
& & \left.-\PP^{s_{1}}\left\{ Z_{i_{2}-i_{1}}=\left(y-\xi_{1},s_{2}\right)\right\} \PP^{s_{2}}\left\{ Z_{i_{3}-i_{2}}=\left(\xi_{2}-y,s_{3}\right)\right\} \right].\end{aligned}$$
At this point we take absolute values in and sum over all vectors $\bar{i}$ obtaining $$\begin{aligned}
\left|\sum_{\bar{i}}E^{\mu}\left(\One_{\xi_{1}}\left(S_{i_{1}}\right)\psi\left(S_{i_{2}}\right)\One_{\xi_{2}}\left(S_{i_{3}}\right)\psi\left(S_{i_{4}}\right)\right)\right| & \leq\sum_{\bar{s}\in S^{4}} & \sum_{\bar{i}}\left|\PP^{\mu}\left(Z_{i_{1}}=\left(\xi_{1},s_{1}\right)\right)\cdot\right.\\
& & \left.F\left(i_{1},i_{2},i_{3}\right)\cdot\left(\PP^{s_{3}}\left(S_{i_{4}}=x\right)-\PP^{s_{3}}\left(S_{i_{4}}=y\right)\right)\right|\\
& \leq\sum_{\bar{s}\in S^{4}} & \sum_{0\leq i_{1}\leq i_{2}\leq i_{3}\leq n}\left[C\cdot\frac{1}{\sqrt{i_{1}+1}}\cdot\right.\\
& & \left.\left|F\left(i_{1},i_{2},i_{3}\right)\right|\cdot\left|x-y\right|\right]\end{aligned}$$ where $C$ is some constant. Here, Lemma \[lem:LLT\] and Lemma \[pro:Potential Kernel Estimate\] were used to obtain the last inequality and $i_{1}+1$ appears instead of $i_{1}$, so that the expression will be defined for $i_{1}=0$.
Note that, $$\begin{aligned}
|F(i_{1},i_{2},i_{3})| & \leq & \PP^{s_{1}}\left(Z_{i_{2}-i_{1}}=\left(x-\xi_{1},s_{2}\right)\right)\left|\PP^{s_{2}}\left\{ Z_{i_{3}-i_{2}}=\left(\xi_{2}-x,s_{3}\right)\right\} -\PP^{s_{2}}\left\{ Z_{i_{3}-i_{2}}=\left(\xi_{2}-y,s_{3}\right)\right\} \right|\\
& + & \PP^{s_{2}}\left\{ Z_{i_{3}-i_{2}}=\left(\xi_{2}-y,s_{3}\right)\right\} \left|\PP^{s_{1}}\left\{ Z_{i_{2}-i_{1}}=\left(x-\xi_{1},s_{2}\right)\right\} -\PP^{s_{1}}\left\{ Z_{i_{2}-i_{1}}=\left(y-\xi_{1},s_{2}\right)\right\} \right|\end{aligned}$$
Substituting $|F(i_{1},i_{2},i_{3})|$ by the above expression, using Lemma \[lem:LLT\] to deduce that sums of the type $$\sum\limits _{k=0}^{n}P^{\mu}\left(Z_{n}=\left(x,s\right)\right)$$ are bounded by constant times $\sqrt{n}$, and applying Lemma \[pro:Potential Kernel Estimate\] we conclude that $$\left|\sum_{\bar{i}}E\left(\One_{\xi_{1}}\left(S_{i_{1}}\right)\psi\left(S_{i_{2}}\right)\One_{\xi_{2}}\left(S_{i_{3}}\right)\psi\left(S_{i_{4}}\right)\right)\right|\leq C\cdot n\left|x-y\right|^{2}.$$
Since there is only a finite number of possible vectors $\xi$ in (there are exactly $4$) the claim follows.
By Lemma \[cor: Fourth moments inequality\], $$\begin{aligned}
E^{\mu}\left(\left(t_{n}(x)-t_{n}(y)\right)^{4}\right) & = & \frac{1}{n^{2}}E^{\mu}\left[\left(L_{n}\left(\left\lfloor \sqrt{n}x\right\rfloor \right)-L_{n}\left(\left\lfloor \sqrt{n}y\right\rfloor \right)\right)^{4}\right]\\
& \leq & C\left|x-y\right|^{2}.\end{aligned}$$
The claim now follows by Chebychev’s inequality.
Relative compactness of $t_{n}\left(x\right)$ in $D$.
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A sequence $\left\{ X_{n}\right\} $ of random variables taking values in a standard Borel Space $\left(X,\BB\right)$ is called tight if for every $\epsilon>0$ there exists a compact $K\subset X$ such that for every $n\in\NN$, $$P_{n}(K)>1-\epsilon,$$ where $P_{n}$ denotes the distribution of $X_{n}$ . By Prokhorov’s Theorem relative compactness of $t_{n}(x)$ in $D$ is equivalent to tightness. Therefore we are interested in characterizing tightness in $D$.
For $x\left(t\right)$ in $D_{\left[-m,m\right]},$$T\subseteq\left[-m,m\right]$ set $$\omega_{x}\left(T\right)=\sup_{s,t\in T}\left|x(s)-x(t)\right|$$ and
$$\omega_{x}(\delta):=\sup_{\left|s-t\right|<\delta}\left|x\left(s\right)-x\left(t\right)\right|.$$
$\omega_{x}\left(\delta\right)$ is called the modulus of continuity of $x$. Due to the Arzela - Ascoli theorem it plays a central role in characterizing tightness in the space $C\left[-m,m\right]$ of continuous functions on $\left[-m,m\right]$, with a Borel $\sigma$-algebra generated by the topology of uniform convergence.
The function that plays in $D\left[-m,m\right]$ the role that the modulus of continuity plays in $C\left[-m,m\right]$ is defined by $$\omega_{x}'(\delta)=\inf_{\left\{ t_{i}\right\} }\max_{1\leq i\leq v}\omega(\left[t_{i},t_{i+1})\right),$$ where $\left\{ t_{i}\right\} $ denotes a $\delta$ sparse partition of $[-m,m]$, i.e. $\left\{ t_{i}\right\} $ is a partition $-m=t_{1}<t_{2}<...<t_{v+1}=m$ such that ${\displaystyle \min_{1\leq i\leq v}\left|t_{i+1}-t_{i}\right|>\delta}$. It is easy to check that if $\frac{1}{2}>\delta>0$, and $m\geq1,$ $$\omega'_{x}(\delta)\leq\omega_{x}\left(2\delta\right).$$ For details see [@Bil Sections 12 and 13]. The next theorem is a characterization of tightness in the space $D$.
[@Bil Lemma 3, p.173] (1)The sequence $t_{n}$ is tight in $D$ if and only if its restriction to $\left[-m,m\right]$ is tight in $D_{\left[-m,m\right]}$ for every $m\in\RR_{+}$.
\(2) The sequence $t_{n}$ is tight in $D_{\left[-m,m\right]}$ if and only if the following two conditions hold:
\(i) $\forall x\in\left[-m,m\right],\;\lim\limits _{a\rightarrow\infty}\limsup\limits _{n\rightarrow\infty}P\left[\left|t{}_{n}\left(x\right)\right|\geq a\right]=0.$
\(ii) $\forall\epsilon>0,\,\lim\limits _{\delta\rightarrow0}\limsup\limits _{n\rightarrow\infty}P\left[\omega_{t_{n}}^{'}\left(\delta\right)\geq\epsilon\right]=0.$
See [@Bil Thm. 13.2] and Corollary after wards. \[rem: Norm Boundness\]Conditions $\left(i\right)$ and $\left(ii\right)$ of the previous theorem imply that $$\lim\limits _{a\rightarrow\infty}\limsup\limits _{n\rightarrow\infty}P\left[\sup_{x\in[-m,m]}\left|t{}_{n}\left(x\right)\right|\geq a\right]=0.$$
\[pro:The-sequence is tight\]The sequence $\left\{ t_{n}(x)\right\} _{n=1}^{\infty}$ is tight.
We prove that condition $2(i)$ holds.
Fix $\epsilon>0,$ $x\in\RR.$ Since the Brownian Motion $W_{\sigma}(t)$ satisfies $$\lim_{M\to\infty}\PP\left(\sup_{t\in[0,1]}\left|W_{\sigma}(t)\right|>M\right)=0$$ and $W_{n}\left(t\right)$ converges in distribution to $W_{\sigma}\left(t\right)$ there are $M,n_{0}$ such that for all $n>n_{0},$
$$\PP^{\mu}\left(\sup_{t\in[0,1]}\left|W_{n}(t)\right|>M\right)<\epsilon.$$
By definition of $t_{n}\left(x\right),$ it follows that if $\left|x\right|>M,$ $n>n_{0},$ $$\PP^{\mu}\left(\left|t_{n}\left(x\right)\right|>0\right)<\epsilon.$$
Now, if $\left|x\right|\leq M$, by corollary \[cor:Tightness of Local Times\],
$$\begin{aligned}
\PP^{\mu}\left(\left|t_{n}\left(x\right)\right|>a\right) & \leq & \PP^{\mu}\left(\left|t_{n}\left(M+1\right)\right|>0\right)+\PP^{\mu}\left(\left|t_{n}\left(x\right)-t_{n}\left(M+1\right)\right|>a\right)\\
& \leq & \epsilon+\frac{4C\cdot\left(M+1\right)^{2}}{a^{4}}\end{aligned}$$
and the last expression can be made less then $2\epsilon$ for sufficiently large $a$.
To prove condition $2(ii)$ WLOG we may assume that $m\geq1$. Since $\omega'_{x}(\delta)\leq\omega_{x}\left(2\delta\right)$, it is sufficient to prove that the stronger condition $$\forall\epsilon>0.\ \lim\limits _{\delta\rightarrow0}\limsup\limits _{n\rightarrow\infty}\PP^{\mu}\left[\sup_{x,y\in[-m.m];\left|x-y\right|<\delta}\left|t_{n}(x)-t_{n}(y)\right|\geq\epsilon\right]=0\label{eq: sufficient condition for tightness}$$ holds.
Let $\epsilon>0$. By Corollary \[cor:Tightness of Local Times\] there exists $C>0$ such that for all $x,y:\ \frac{1}{\sqrt{n}}\leq|x-y|\leq1$ $$\PP^{\mu}\left(\left|t_{n}(x)-t_{n}(y)\right|>\epsilon\right)\leq\frac{C}{\epsilon^{4}}\left|x-y\right|^{2}.\label{eq:sufficient condition for tightness 2}$$ Let $\delta>0$ and $n>\delta^{-2}$, notice that $t_{n}(x)$ is constant on segments of the form $\left[\frac{j}{\sqrt{n}},\frac{j+1}{\sqrt{n}}\right)$, hence $$\PP^{\mu}\left(\sup_{x,y\in[-m.m];\left|x-y\right|<\delta}\left|t_{n}(x)-t_{n}(y)\right|\geq4\epsilon\right)\leq\sum\limits _{|k\delta|\leq m}\PP^{\mu}\left(\sup\limits _{k\delta\sqrt{n}\leq j\leq\left(k+1\right)\delta\sqrt{n}}\left|t_{n}\left(k\delta\right)-t_{n}\left(\frac{j}{\sqrt{n}}\right)\right|\geq\epsilon\right).$$ By [@Bil Theorem 10.2] it follows from that there exists $C_{2}>0$ such that $$\PP^{\mu}\left(\sup\limits _{k\delta\sqrt{n}\leq j\leq\left(k+1\right)\delta\sqrt{n}}\left|t_{n}\left(k\delta\right)-t_{n}\left(\frac{j}{\sqrt{n}}\right)\right|\geq\epsilon\right)\leq\frac{C_{2}}{\epsilon^{4}}\delta^{2}.\label{eq:Billingsley2}$$
Therefore, $$\PP^{\mu}\mbox{\ensuremath{\left(\sup_{x,y\in[-m.m];\left|x-y\right|<\delta}\left|t_{n}(x)-t_{n}(y)\right|\geq4\epsilon\right)\leq\frac{2C_{2}m}{\epsilon^{4}}\delta\xrightarrow[\delta\to0]{}}0. }$$
\[sub:Identifying the Only Possible Limit Point\]Identifying $t_{\sigma}\left(x\right)$ as the Only Possible Limit of a Subsequence of $\left\{ t_{n}\left(x\right)\right\} _{n\in\NN}$.
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\[pro:Identification ot the limit\]Assume that the sequence $\left\{ X_{n}\right\} $ satisfies the assumptions of Theorem \[thm: Main Theorem\]. Let $t_{n_{k}}\left(x\right)$ be some subsequence of $t_{n}\left(x\right)$ that converges in distribution to some limit $q\left(x\right).$ Then $q\left(x\right)\deq t_{\sigma}\left(x\right)$.
Let $q$ be a limit point of $\left\{ t_{n}(x)\right\} $. i.e. there exists $n_{k}\to\infty$ such that $t_{n_{k}}\overset{d}{\Rightarrow}q(x).$
By [@KS Section 2], for every $a,b\in\RR,$ the function on $D_{\left[0,1\right]}$ defined by $$f\rightarrow\int_{0}^{1}\One_{[a,b)}\left(f\right)dt$$ is continuous in the Skorokhod topology.
Therefore, since $W_{n}(t)\overset{d}{\Rightarrow}W_{\sigma}(t)$ , for every $a,b$ the random variables
$$\Lambda_{n}([a,b))=\int_{0}^{1}\One_{[a,b)}\left(W_{n}(t)\right)dt$$ converge in distribution to the occupation measure of the Brownian motion defined by $$\Lambda([a,b))=\int_{0}^{1}\One_{[a,b)}\left(W_{\sigma}(t)\right)dt.$$
The local time $t_{\sigma}$ of the Brownian motion is an almost surely continuous function of $x$ such that for fixed $a,b\in\RR$ the equality $$\int_{a}^{b}t_{\sigma}(x)dx=\int_{0}^{1}\One_{[a,b)}\left(W_{\sigma}(t)\right)dt\label{eq:B.motion local time and occupation}$$ holds almost surely (see [@MP]).
By straightforward calculations using definitions, we have $$\left|\int_{0}^{1}\One_{[a,b)}\left(W_{n}(t)\right)dt-\int_{a}^{b}t_{n}(x)dx\right|\leq\int\limits _{\frac{\left\lfloor \sqrt{n}a\right\rfloor }{\sqrt{n}}}^{\frac{\left\lfloor \sqrt{n}a\right\rfloor +1}{\sqrt{n}}}t_{n}\left(x\right)dx+\int\limits _{\frac{\left\lfloor \sqrt{n}b\right\rfloor }{\sqrt{n}}}^{\frac{\left\lfloor \sqrt{n}b\right\rfloor +1}{\sqrt{n}}}t_{n}\left(x\right)dx.\label{eq:Difference in densities}$$
Now, $$\begin{aligned}
\PP^{\mu}\left(\left|\int\limits _{\frac{\left\lfloor \sqrt{n}a\right\rfloor }{\sqrt{n}}}^{\frac{\left\lfloor \sqrt{n}a\right\rfloor +1}{\sqrt{n}}}t_{n}\left(x\right)dx\right|>\epsilon\right) & \leq & \PP^{\mu}\left(\sup_{x\in\left[a-1,b+1\right]}\left|t_{n}\left(x\right)\right|>M\right)\\
& + & \PP^{\mu}\left(\left|\int\limits _{\frac{\left\lfloor \sqrt{n}a\right\rfloor }{\sqrt{n}}}^{\frac{\left\lfloor \sqrt{n}a\right\rfloor +1}{\sqrt{n}}}t_{n}\left(x\right)dx\right|>\epsilon,\sup_{x\in\left[a-1,b+1\right]}\left|t_{n}\left(x\right)\right|\leq M\right).\end{aligned}$$
The second summand on the right side of the above inequality tends to $0$ since the integral is less than $\frac{M}{\sqrt{n}}$ .The first summand is arbitrarily close to $0$ for $M,n$ large enough, by Remark \[rem: Norm Boundness\]. Same reasoning applied to both summands of equation gives $$\left|\int_{0}^{1}\One_{[a,b)}\left(W_{n}(t)\right)dt-\int_{a}^{b}t_{n}(x)dx\right|\overset{d}{\Rightarrow}0.$$
It follows that the distributional limits of $\int_{0}^{1}\One_{[a,b)}\left(W_{n}(t)\right)dt$ and $\int_{a}^{b}t_{n}(x)dx$ coincide (see [@Bil Thm.3.1]). Therefore, for every $a,b\in\RR$, $$\int_{a}^{b}t_{n}(x)dx\overset{d}{\Rightarrow}\int_{a}^{b}t_{\sigma}(x)dx.$$ For every $a,b$ the function $$f\mapsto g_{a,b}(f)=\int_{a}^{b}f(x)dx$$ is a continuous function from $D$ to $\RR$. Therefore, since $t_{n_{k}}\overset{d}{\Rightarrow}q(x)$ , it follows that $$\int_{a}^{b}t_{n_{k}}(x)dx\overset{d}{\Rightarrow}\int_{a}^{b}q(x)dx.$$ Hence, $$\int_{a}^{b}q(x)dx\overset{d}{=}\int_{a}^{b}t_{\sigma}(x)dx.$$
Since the collection of functions $g_{a,b},$$a,b\in\RR$ generates $\mathcal{G}(D)$, it follows that
$$q\deq t_{\sigma}.$$
\[sec:Borodin’s-Theorem.\]Proof of Borodin’s Theorem (Theorem \[thm: Borodin for MC\]).
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We start by forming the mutual probability space on which, $W'_{n}$ and $W'_{\sigma}$ are defined.
Let $\left\{ t_{i}\right\} $ be a dense subset of $\left[0,\infty\right]$ and for each $n$ let us regard the infinite vector $$\xi_{n}:=\left(W_{n},\, l_{n}\left(t_{1},\cdot\right),\, l_{n}\left(t_{2},\cdot\right),\,...\right)$$ as a random variable taking values in the space $\Pi:=D_{+}\times D^{\NN}$
Since, a countable product of Polish Spaces is Polish, to prove convergence of the sequence $\left\{ \xi_{n}\right\} $ we need to establish tightness and uniquely identify the limit. By Tichonov’s theorem, tightness in each coordinate of the vectors $\xi_{n}$ separately (established in Theorem \[thm: Main Theorem\]) implies tightness for the whole sequence $\left\{ \xi_{n}\right\} $.
To identify the limit, generalizing the method in the proof of proposition \[pro:Identification ot the limit\] to $\Pi$, for each sequence $\alpha=\left\{ (a_{i},b_{i})\right\} _{i\in\NN}$ we define a continuous function $g_{\alpha}$ on $\Pi$ (taking values in $D_{+}\times\RR^{\NN}$) by $$\begin{aligned}
\left(f,h_{1},h_{2},...\right) & \overset{g_{\alpha}}{\rightarrow} & \left(f,\,\int\limits _{a_{1}}^{b_{1}}h_{1},\,\int\limits _{a_{2}}^{b_{2}}h_{2},...\right).\end{aligned}$$
The functions $\left\{ g_{\alpha}\right\} $ where $\alpha$ goes over all possible sequences of intervals in $\RR^{2}$ generate the Borel $\sigma$-field of $\Pi.$ Now, the arguments in the proof of proposition may be easily modified to show that $$g_{\alpha}\left(\xi_{n}\right)\overset{d}{\rightarrow}g_{\alpha}\left(\xi\right)$$ where $\xi:=\left(W'_{\sigma},\, l\left(t_{1},\cdot\right),\, l\left(t_{2},\cdot\right),\,...\right).$ This uniquely identifies the limit, establishing the convergence in distribution of the sequence $\xi_{n}$ to $\xi$. Now, by Skorokhod’s theorem (See [@Bil Theorem 6.7]), we may define processes $\xi'_{n}$ ,$\xi':\Omega'\rightarrow$ on some probability space $\left(\Omega',\BB',\PP\right)$ such that $\xi'_{n}$ and $\xi'$ have the same distribution as $\xi_{n}$ and $\xi$ respectively, and such that almost surely, $\xi'_{n}$ converges to $\xi'$. Note that for $n\in\NN$, by left continuity, for almost every $\omega\in\Omega'$, the vector $$\xi'_{n}=\left(W'_{\sigma},l'_{n}\left(t_{1},x\right),l'_{n}\left(t_{2},x\right),...\right)$$ uniquely determines a function $l'_{n}\left[t,x\right):\left[0,1\right]\times\left(-\infty,\infty\right)\rightarrow\RR,$ which coincides with the coordinates of $\xi'_{n}$ for $t\in\left\{ t_{i}\right\} $, and is Cadlag in each variable.
It is clear that the functions $l'_{n}\left(t,x\right)$ are a.s equal to the local time of $W'_{n}$. Similarly, the vector $\xi'$ uniquely, determines a function $l'\left(t,x\right)$ which a.s equals the Brownian local time of $W'_{\sigma},$ and is therefore continuous. The a.s. convergence of $\xi'_{n}$ to $\xi'$ implies a.s. convergence in the $J_{1}$ topology of $W'_{n}$ to $W'_{\sigma}$, thus proving *$\left({\rm 2}\right)$* of theorem \[thm: Borodin for MC\].
It remains to prove that $l'_{n},\, l',\, n=1,2,...$ satisfy the statement in *$\left({\rm 3}\right)$*. To see this fix $T>0.$ For $h>0,$ whose value is to be determined later, we extract from $\left\{ t_{i}\right\} $ a finite partition of $\left[0,T\right]$ with mesh less than $h$ and denote it by $$T:\,0=t_{0}<t_{1}<...<t_{l}=1.$$ Now for $i=1,..,l$, the process $l'_{n}\left(t_{i},\cdot\right)$, a.s converges to $l'\left(t_{i},\cdot\right)$ in the metric of the space $D$. Since, the limit is a continuous function, by properties of the $J_{1}$ topology, this convergence must be uniform on compact subsets of $\RR$. We may conclude that for every compact subset $K\subseteq\RR$ , every $\epsilon>0$, and almost all $\omega\in\Omega'$, there exists $N_{0}$, such that for every $n>N_{0}$, $$\sup_{x\in K,t\in T}\left|l'_{n}\left(t,x\right)-l'\left(t,x\right)\right|<\epsilon.\label{eq: Uniform Approx.}$$
Now, by monotonicity of the local time as a function of time, if $l'_{n}\left(t,x\right)-l'\left(t,x\right)\geq0$ then $$l'_{n}\left(t,x\right)-l'\left(t,x\right)\leq l'_{n}\left(t_{i+1},x\right)-l'\left(t_{i},x\right)$$ and if $l'_{n}\left(t,x\right)-l'\left(t,x\right)\leq0$ then $$l'\left(t,x\right)-l'_{n}\left(t,x\right)\leq l'\left(t_{i+1},x\right)-l'_{n}\left(t_{i},x\right)$$ where $t_{i},\, t_{i+1}$ are points in $T$ satisfying $$t_{i}\leq t\leq t_{i+1}.$$
Hence, $$\begin{aligned}
\PP\left(\sup_{t\in\left[0,1\right],x\in\RR}\left|l'_{n}\left(t,x\right)-l'\left(t,x\right)\right|>2\epsilon\right) & \leq & \PP\left(\sup_{t\in\left[0,1\right]}\left|W'_{n}\right|>M\right)+\PP\left(\sup_{t\in\left[0,1\right]}\left|W'_{\sigma}\right|>M\right)\label{eq:Loc.Time.1}\\
& & +\PP\left(\sup_{i,\, x\in\left[-M,M\right]}\left|l'_{n}\left(t_{i+1},x\right)-l'\left(t_{i},x\right)\right|>\epsilon\right)\label{eq:Loc.Time.2}\\
& & +\PP\left(\sup_{i,\, x\in\left[-M,M\right]}\left|l'_{n}\left(t_{i},x\right)-l'\left(t_{i+1},x\right)\right|>\epsilon\right)\label{eq:Loc.Time.3}\end{aligned}$$ where $i=1,...l.$
As in the proof of Proposition expression is small for sufficiently large $M$. Expressions and are handled similarly. We have $$\begin{aligned}
\PP\left(\sup_{i,\, x\in\left[-M,M\right]}\left|l'_{n}\left(t_{i+1},x\right)-l'\left(t_{i},x\right)\right|>\epsilon\right) & \leq & \PP\left(\sup_{t\in T,\, x\in\left[-M,M\right]}\left|l'_{n}\left(t_{i+1},x\right)-l'\left(t_{i+1},x\right)\right|>\frac{\epsilon}{2}\right)\\
& & +\PP\left(\sup_{i=0,1...l,\, x}\left|l'\left(t_{i+1},x\right)-l'\left(t_{i},x\right)\right|>\frac{\epsilon}{2}\right).\end{aligned}$$
The first expression on the right equals $0$ for all $n$ sufficiently large, by \[eq: Uniform Approx.\]. The second expression, by a.s. continuity of Brownian local time, can be made arbitrarily small by taking $h$ to be sufficiently small. This concludes the proof.
\[sec:The-Periodic-Case\]The periodic Case
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Let $X_{1},X_{2},....$ be an i.i.d. sequence in the domain of attraction of the Gaussian distribution. In this setting Borodin proved convergence of local times under the condition of what we call non-arithmeticity of the random walk $S_{n}$, that is $$E\left(e^{itX_{i}}\right)=1\ \Leftrightarrow\ t\in2\pi\ZZ.$$
Our notion of strong aperiodicity in the i.i.d case is equivalent to the stronger condition that
$$\left|E\left(e^{itX_{i}}\right)\right|=1\ \Leftrightarrow\ t\in2\pi\ZZ.$$
In this section we show that the methods of this paper are sufficient to replace the assumption of strong aperiodicity by a weaker non-arithmeticity in the i.i.d case and for Markov chains that are almost onto (see section \[sub:The-periodic-case\]).
We assume that $E\left(X_{1}\right)=0$ and that $$\Sigma:=\left\{ x|\PP\left(S_{n}=x\right)>0\,\, for\,\, some\,\, n\right\} =\ZZ.$$ Otherwise, since$\Sigma$ is a subgroup of $\ZZ$, in the recurrent case, we’ll need to relabel the state space.
Let $$p=\inf\left\{ k\in\NN:\ \PP\left(S_{k}=0\right)>0\right\}$$ be the period of the random walk. It follows that $$\left|E\left(e^{itX}\right)\right|=1\ \Leftrightarrow t\in\frac{2\pi}{p}\ZZ$$
In this case the random variable $X_{1}$ takes values in a proper coset of $\ZZ$. The periodic structure of $S_{n}$ is that for $n\in\NN$ and $k\in\left\{ 1,...,p-1\right\} $, $S_{np+k}$ takes values in one of the $p-1$ cosets of $\ZZ$ of the form $p\ZZ+j$, $j=0,...,p-1.$
In this case the potential kernel estimate( Lemma \[pro:Potential Kernel Estimate\]) should be corrected to state that for every $x,y\in\ZZ$, $$\sum_{n=0}^{\infty}\left|\sum_{k=0}^{p-1}\PP\left(S_{np+k}=x\right)-\PP\left(S_{np+k}=y\right)\right|\leq C\left|x-y\right|,\label{eq: JP periodic}$$ which is a special case of the corollary from lemma 7 in [@JP]. Lemma \[lem:LLT\] also holds in this case (see Lemma 2 in [@JP]). These two ingredients are enough to prove the fourth moment inequality, Lemma \[cor: Fourth moments inequality\]. To see how this is done notice that (here we keep the notation of Lemma \[cor: Fourth moments inequality\]) $$\begin{aligned}
E\left(\One_{\xi_{1}}\left(S_{i_{1}}\right)\psi\left(S_{i_{2}}\right)\One_{\xi_{2}}\left(S_{i_{3}}\right)\psi\left(S_{i_{4}}\right)\right) & = & \left[\PP\left(S_{i_{1}}=\xi_{1}\right)\cdot F\left(i_{1},i_{2},i_{3}\right)\cdot\right.\label{eq: sumation over xi}\\
& & \left.\left(\PP\left(S_{i_{4}-i_{3}}=x\right)-\PP\left(S_{i_{4}-i_{3}}=y\right)\right)\right]\nonumber \end{aligned}$$ where $$\begin{aligned}
F\left(i_{1},i_{2},i_{3}\right) & = & \left[\PP\left(S_{i_{2}-i_{1}}=x-\xi_{1}\right)\PP\left(S_{i_{3}-i_{2}}=\xi_{2}-x\right)\right.\label{eq: F for random walks}\\
& & \left.-\PP\left(S_{i_{2}-i_{1}}=y-\xi_{1}\right)\PP\left(S_{i_{3}-i_{2}}=\xi_{2}-y\right)\right]\\
& = & \PP\left(S_{i_{2}-i_{1}}=x-\xi_{1}\right)\left[\PP\left(S_{i_{3}-i_{2}}=\xi_{2}-x\right)-\PP\left(S_{i_{3}-i_{2}}=\xi_{2}-y\right)\right]\nonumber \\
& & +\PP\left(S_{i_{3}-i_{2}}=\xi_{2}-y\right)\left[\PP\left(S_{i_{2}-i_{1}}=x-\xi_{1}\right)-\PP\left(S_{i_{2}-i_{1}}=y-\xi_{1}\right)\right]\nonumber \end{aligned}$$ Therefore, since $$\begin{aligned}
E^{\mu}\left(L_{n}(x)-L_{n}(y)\right)^{4} & = & \sum_{\overrightarrow{\xi}}\sum_{\bar{i}}E^{\mu}\left(\One_{\xi_{1}}\left(S_{i_{1}}\right)\psi\left(S_{i_{2}}\right)\One_{\xi_{2}}\left(S_{i_{3}}\right)\psi\left(S_{i_{4}}\right)\right),\end{aligned}$$ from and it follows that we need to estimate two sums. The first is $$\begin{aligned}
\sum_{i_{1}=1}^{n}\left[\PP\left(S_{i_{1}}=\xi_{1}\right)\sum_{i_{2}=0}^{n-i_{1}}\PP\left(S_{i_{2}}=x-\xi_{1}\right)\sum_{i_{3}=0}^{n-i_{1}-i_{2}}\left\{ \PP\left(S_{i_{3}}=\xi_{2}-x\right)-\PP\left(S_{i_{3}}=\xi_{2}-y\right)\right\} \right.\label{eq: long ugly summation for periodic}\\
\left.\sum_{i_{4}=0}^{n-i_{1}-i_{2}-i_{3}}\left\{ \PP\left(S_{i_{4}}=x\right)-\PP\left(S_{i_{4}}=y\right)\right\} \right].\nonumber \end{aligned}$$ In order to bound this term we first notice that $$\sum_{i_{4}=0}^{n-i_{1}-i_{2}-i_{3}}\left\{ \PP\left(S_{i_{4}}=x\right)-\PP\left(S_{i_{4}}=y\right)\right\} =\sum_{l=0}^{\left\lfloor \left(n-i_{1}-i_{2}-i_{3}\right)/p\right\rfloor }\sum_{j=0}^{p-1}\left\{ \PP\left(S_{l\cdot p+j}=x\right)-\PP\left(S_{l\cdot p+j}=y\right)\right\} \pm p.$$ here $a=b\pm c$ means $\left|a-b\right|\leq c$. We do a similar rearrangement to the sum of $i_{3}$. It then follows that the term in is strictly smaller than $$\begin{aligned}
\sum_{i_{1}=1}^{n}\left[\PP\left(S_{i_{1}}=\xi_{1}\right)\sum_{i_{2}=0}^{n-i_{1}}\PP\left(S_{i_{2}}=x-\xi_{1}\right)\cdot\left(p+\sum_{l_{1}=0}^{\infty}\left|\sum_{j_{1}=0}^{p-1}\left\{ \PP\left(S_{i_{3}}=\xi_{2}-x\right)-\PP\left(S_{i_{3}}=\xi_{2}-y\right)\right\} \right|\right)\right.\\
\left.\cdot\left(p+\sum_{l=0}^{\infty}\left|\sum_{j=0}^{p-1}\left\{ \PP\left(S_{l\cdot p+j}=x\right)-\PP\left(S_{l\cdot p+j}=y\right)\right\} \right]\right)\right].\end{aligned}$$ The conclusion follows from a double application of the inequality and the bound in Lemma 2 of [@JP]. The second term is dealt with similarly and the rest of the proof of Borodin’s theorem remains the same. Thus we have:
Let $X_{1},X_{2},...,X_{n},..$ be an i.i.d sequence with $E\left(X_{1}\right)=0$ and $E\left(X_{1}^{2}\right)=1$ such that the random walk $S_{n}$ is recurrent. Then there exists a probability space $\left(X',\BB',\PP'\right)$ and processes $W'_{n},\, W'_{\sigma}:X\rightarrow D_{+}$ such that:
1. *$W_{n}\overset{d}{=}W'_{n};\, W\overset{d}{=}W'$.*
2. *With probability one $W'_{n}$ converges to $W'$ uniformly on compact subsets of $\left[0,\infty\right)$.*
3. *For every $\epsilon,T>0$ the processes $l'_{n}\left(t,x\right)$ and $l'\left(t,x\right)$ defined with respect to $W'_{n}$ and $W'_{\sigma}$ satisfy the relationship: $$\lim_{n\rightarrow\infty}\PP'\left(\sup_{t\in\left[0,T\right],\, x\in\RR}\left|l'_{n}\left(t,x\right)-l'\left(t,x\right)\right|>\epsilon\right)=0.$$*
\[sub:The-periodic-case\]The periodic case of finite state Markov chains
-------------------------------------------------------------------------
To drop the assumption of strong aperiodicity when $X_{1},..,X_{n},...$ is a Markov chain we use the dynamical setting introduced in section 1. Let $P:S\times S\to[0,1]$ be the transition matrix, $\mu$ be an initial distribution on $S$ and $P^{\mu}$ the probability measure on $S^{\mathbb{N}}$ generated by $\mu$ and $P$. We then look at the Markov shift $\left(S^{\mathbb{N}},\BB.P^{\mu},T\right)$ where $T$ is the shift and $\BB$ is the Borel $\sigma$-algebra generated by the cylinder sets $$\left[s_{0}s_{1}...s_{n}\right]=\left\{ \omega\in S^{\NN}:\,\omega_{i}=s_{i},\, i=0,1,...,n\right\} .$$
In this case $X_{n}=f\left(T^{n}\omega\right)=\omega_{n}$ and if the function $f$ is periodic there exists a solution to $$e^{itf\left(\omega\right)}=\lambda\frac{\varphi\left(\omega\right)}{\varphi\left(T\omega\right)}$$ where either $t\notin2\pi\ZZ$ or $\varphi$ is not constant. To make the same analysis as for random walks we need to exclude the case $\varphi\neq const$ (then summation over the Markov Chain will yield the same periodic structure as in the i.i.d case). A sufficient condition for that is that the system $\left(X,\BB,P_{\mu},T,\alpha\right)$ is almost onto with respect to the partition $\alpha=\left\{ \left[s_{0}\right]:\ s_{0}\in S\right\} $, meaning that for every $a,b\in\alpha$ there exist sets $a_{0},...,a_{n}$ such that $a_{0}=b$, $a_{n}=c$ and $Ta_{k}\cap Ta_{k+1}\neq\emptyset$ (for details see [@AD Section 3]).
\[sec:Applications-to-complexity\]Applications to complexity of random walks in random sceneries with a Markov chain base
=========================================================================================================================
A Random Walk in Random Scenery is a skew product probability preserving transformation which is defined as follows:
The *random scenery* is an invertible probability preserving transformation $\left(Y,\mathcal{C},\nu,S\right)$ and the *random walk in random scenery $\left(Y,\mathcal{C},\nu,S\right)$ with jump random variable $\xi$* (assumed $\ZZ$-valued) is the skew product $\left(Z,\BB(Z),m,T\right)$ defined by $$Z:=\Omega\times Y,\ m:=\mu_{\xi}\times\nu,\ T(w,y)=\left(Rw,S^{w_{0}}y\right)$$ where $$\left(\Omega,\BB\left(\Omega\right),\mu_{\xi},R\right)=\left(\ZZ^{\ZZ},\BB\left(\ZZ^{\ZZ}\right),\prod{\rm dist}\xi,{\rm shift}\right)$$ is the shift of the independent jump random variables.
Aaronson [@Aar], assuming $\xi$ is in the domain of attraction of an $\alpha-$stable law with $\alpha>0$, has studied the relative complexity and the relative entropy dimension of $T$ over the base. We refer the reader to [@Aar] for the definitions.
One can ask if Aaronson’s result [@Aar Theorem 3, Section 1] remain true if we change the base $\left(\Omega,\BB\left(\Omega\right),\mu_{\xi},R\right)$ to a strongly aperiodic, irreducible, finite state Markov Shift $\left(\Omega,\BB\left(\Omega\right),P^{\mu},R\right)$.
With Theorem \[thm: Borodin for MC\] at hand, Aaronson’s proof, which can even be simplified since the Brownian motion is a.s continuous, can be carried out verbatim.
Appendix
========
\[Trivial proposition\]Let $\left(X,\mathcal{B},\mu\right)$ be a probability space, $f:X\rightarrow\mathbb{S}^{1}$(the unit sphere) measurable. Then $$\left|\int f\left(x\right)\, d\mu\right|=1$$ implies $f=const$.\
Assume that $f$ is not constant. Then there are two disjoint intervals $I_{1},I_{2}\subseteq\mathbb{T}$ such that $\mu\left(f^{-1}\left(I_{1}\right)\right)>0$ and $\mu\left(f^{-1}\left(I_{1}\right)\right)>0$. It follows that $$\left|\int_{X}f\left(x\right)\, d\mu\right|\leq\left|\int_{f^{-1}(I_{1})}f(x)d\mu+\int_{f^{-1}(I_{2})}f(x)d\mu\right|+\mu\left(X\backslash\left(f^{-1}(I_{1})\cup f^{-1}(I_{2})\right)\right)<1.$$ This is true since if for $j\in\{1,2\}$ $$\left|\frac{1}{\mu\left(f^{-1}(I_{j})\right)}\int_{f^{-1}(I_{j})}f(x)d\mu\right|<1$$ then it is trivial, else $$\frac{1}{\mu\left(f^{-1}(I_{j})\right)}\int_{f^{-1}(I_{j})}f(x)d\mu\in I_{j},\ j=1,2$$ and the mean of two different values on the unit sphere is of modulus strictly less than $1$ .
[10]{} J.Aaronson, Relative Complexity of random walks in random sceneries. To appear in Annals of Probability.
J. Aaronson and M. Denker. Local limit theorems for partial sums of stationary sequences generated by GibbsMarkov maps. Stoch.Dyn. 1(2),193237 (2001).
Billingsley, P. Convergence of Probability Measures, second edition, Wiley, New York (1999).
Borodin, A.N. On the asymptotic behavior of local times of recurrent random walks with finite variance. Theory Probab. Appl. 26, pp. 758772 (1981).
K.L. Chung, (2nd ed.),Markov Chains with Stationary Transition Probabilities, Springer, Berlin (1967).
Y. Guivarc’h, J. Hardy, Theoremes limites pour une classe de chaines de Markov et applications aux diffeomorphismes d’Anosov. Ann. Inst. H. Poincare 24 , 73-98 (1988).
Hennion H., and Hervé, L. Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness. Lecture Notes in Mathematics, 1766. Springer-Verlag, Berlin (2001).
Jain N., Pruitt W. Asymptotic behavior of the local time of a recurrent random walk. Ann. Probab. Volume 12, Number 1, 64-85 (1984).
Kato, T. Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (1976).
Kesten, H. and Spitzer, F. A limit theorem related to a new class of self similar processes. Z. Wahrsch. Verw. Gebiete 50, no.1, pp. 5-25 (1979).
Mörters, P., Peres, Y.: Brownian Motion. Cambridge University Press, Cambridge (2009).
J. Rousseau-Egele, Un théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux, Ann. Probab. 11 (1983), pp. 772788.
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abstract: 'Bright fireballs or bolides are caused by meteoroids entering the Earth’s atmosphere at high speed. Some have a cometary origin, a few may have originated within the Venus-Earth-Mars region as a result of massive impacts in the remote past but a relevant fraction is likely the result of the break-up of asteroids. Disrupted asteroids produce clusters of fragments or asteroid families and meteoroid streams. Linking a bolide to a certain asteroid family may help to understand its origin and pre-impact dynamical evolution. On 2013 February 15, a superbolide was observed in the skies near Chelyabinsk, Russia. Such a meteor could be the result of the decay of an asteroid and here we explore this possibility applying a multistep approach. First, we use available data and Monte Carlo optimization (validated using 2008 TC$_{3}$ as template) to obtain a robust solution for the pre-impact orbit of the Chelyabinsk impactor ($a$ = 1.62 au, $e$ = 0.53, $i$ = 3$\fdg$82, $\Omega$ = 326$\fdg$41 and $\omega$ = 109$\fdg$44). Then, we use this most probable orbit and numerical analysis to single out candidates for membership in, what we call, the Chelyabinsk asteroid family. Finally, we perform $N$-body simulations to either confirm or reject any dynamical connection between candidates and impactor. We find reliable statistical evidence on the existence of the Chelyabinsk cluster. It appears to include multiple small asteroids and two relatively large members: 2007 BD$_{7}$ and 2011 EO$_{40}$. The most probable parent body for the Chelyabinsk superbolide is 2011 EO$_{40}$. The orbits of these objects are quite perturbed as they experience close encounters not only with the Earth–Moon system but also with Venus, Mars and Ceres. Under such conditions, the cluster cannot be older than about 20–40 kyr.'
date: 'Accepted 2013 July 30. Received 2013 July 9; in original form 2013 June 11'
title: 'The Chelyabinsk superbolide: a fragment of asteroid 2011 EO$_{40}$?'
---
celestial mechanics – minor planets, asteroids: general – minor planets, asteroids: individual: 2007 BD$_{7}$ – minor planets, asteroids: individual: 2008 TC$_{3}$ – minor planets, asteroids: individual: 2011 E0$_{40}$ – planets and satellites: individual: Earth.
Introduction
============
The decay of asteroids in the main belt region is one of the sources of small near-Earth asteroids or meteoroids. The shattered pieces resulting from the collisional, tidal or rotational break-up of a rubble-pile asteroid can spread along the entire orbit of the parent body on a time-scale of hundreds of years (Tóth, Vereš & Kornoš 2011). These meteoroid streams can cause meteor showers on the Earth when their paths intersect that of our home planet (e.g. Jopek & Williams 2013). Exceptionally bright meteors are popularly known as fireballs. More properly, relatively small impacting objects entering the Earth’s atmosphere at high speed and reaching an apparent magnitude of -14 or brighter are called bolides; if the magnitude is -17 or brighter they are known as superbolides (Ceplecha et al. 1999). Superbolides can produce very powerful ballistic shock waves as they move at hypersonic speeds and explosive shock waves when they fragment in the atmosphere or hit the ground to form an impact crater (e.g. Ens et al. 2012). They are also parents of meteorite showers as meteorite-dropping bolides, seeding the ground with fragments of extraterrestrial material (e.g. Foschini 2001). Although not capable of triggering global devastation, they are powerful enough to provoke a significant amount of local damage. These events are not exclusive of the Earth but have also been predicted (Dycus 1969; Adolfsson, Gustafson & Murray 1996; Christou & Beurle 1999; Bland & Smith 2000; Christou 2004, 2010; Domokos et al. 2007; Christou, Vaubaillon & Withers 2008; Christou et al. 2012) and observed (Selsis et al. 2005; Christou, Vaubaillon & Withers 2007; Hueso et al. 2010; Daubar et al. 2013) in other planets. The connection between asteroidal debris and bolides was first proposed by Halliday (1987), further explored by Williams (2002, 2004) and Jenniskens (2006) and first confirmed observationally by Trigo-Rodríguez et al. (2007). Since then, new examples of bolides associated with asteroids have been reported (Trigo-Rodríguez et al. 2009, 2010; Madiedo et al. 2013).
On 2013 February 15, 03:20:33 GMT a superbolide was observed in the skies near Chelyabinsk, Russia. The event is believed to have been caused by a relatively small impacting object (17–20 m) entering the Earth’s atmosphere at high speed and a shallow angle.[^1] Calculations by Adamo (2013), Borovicka et al. (Green 2013), Chodas & Chesley,$^{1}$ Emel’yanenco et al.,[^2] Lyytinen,[^3] Lyytinen & Matson,[^4] Nakano,[^5] Proud (2013), Zuluaga & Ferrin[^6] and Zuluaga, Ferrin & Geens (2013)[^7] revealed that the parent object was one of the Apollo asteroids that periodically cross the orbit of the Earth (see Table \[orbits\]). In this Letter, we assume that the meteoroid responsible for the Chelyabinsk event was the result of a relatively recent asteroid break-up event and use numerical analysis to single out candidates to be the parent body or bodies. Then we perform $N$-body calculations to further study any possible dynamical connection between the candidates and the superbolide. Our analysis indicates that the Chelyabinsk impactor was a small member of a not-previously-identified young asteroid family. The most probable pre-impact orbit is obtained in Section 2 using Monte Carlo optimization techniques. The candidate selection procedure is described and available information on the candidate bodies is presented in Section 3. The results of our $N$-body calculations together with the numerical model are shown in Section 4. The proposed new asteroid family is characterized in Section 5. Results are discussed and conclusions are summarized in Section 6.
0.1truecm
Authors $a$ (AU) $e$ $i$ ($^{\circ}$) $\Omega$ ($^{\circ}$) $\omega$ ($^{\circ}$) $P_{\rm 0.050 au}$ $P_{\rm 0.00004263 au}$ $\beta_{\rm max}$
------------------------- ------------------------ ------------------------ ------------------------ ------------------------- ------------------------- -------------------- ------------------------- -------------------
Adamo (2013) 1.60 0.53 4.07 326.46 109.36 0.9508 0.0000633 0.9862
Borovicka et al. 1.55$\pm$0.07 0.50$\pm$0.02 3.6$\pm$0.7 326.41 109.7$\pm$1.8 0.8891 0.0001696 0.9899
Chodas & Chesley$^1$ 1.73 0.57 4.2 - - - - -
Emel’yanenco et al.$^2$ 1.77 0.58 4.3 - - - - -
Lyytinen$^3$ 1.66 0.52 4.05 326.43 116.0 0.3364 0.0000018 0.8828
Lyytinen & Matson$^4$ 1.660079129 0.52409458 4.235664 326.457642 114.669325 0.3302 0.0000014 0.8765
Nakano$^5$ 1.6223665 0.5311191 3.87128 326.42524 109.70844 0.8999 0.0001183 0.9858
Proud (2013) 1.47$^{+0.03}_{-0.13}$ 0.52$^{+0.01}_{-0.05}$ 4.61$^{+2.58}_{-2.09}$ 326.53$^{+0.01}_{-0.0}$ 96.58$^{+2.94}_{-1.72}$ 0.7681 0 0.0582
Zuluaga & Ferrin$^6$ 1.73$\pm$0.23 0.51$\pm$0.08 3.45$\pm$2.02 326.70$\pm$0.79 120.62$\pm$2.77 0.3416 0.000002 0.7430
Zuluaga et al. (2013) 1.27$\pm$0.05 0.44$\pm$0.02 3.0$\pm$0.2 326.54$\pm$0.08 95.1$\pm$0.8 0.9783 0 0.4513
Zuluaga et al.$^7$ 1.368$\pm$0.006 0.470$\pm$0.010 4.0$\pm$0.3 326.479$\pm$0.003 99.6$\pm$1.3 1 0 0.2030
This work 1.62375$\pm$0.00014 0.53279$\pm$0.00011 3.817$\pm$0.005 326.4090$\pm$0.0007 109.44$\pm$0.03 1 0.020446 0.9997
\[orbits\]
Before impact: a Monte Carlo approach
=====================================
If the Chelyabinsk superbolide was the result of the decay of a larger asteroid and we want to identify the putative parent body or bodies, the first step is having a well-defined, statistically robust impactor orbit prior to its collision. Unfortunately, the range of orbital parameters from the solutions provided by the various authors (see Table \[orbits\]) is too wide to be useful in a systematic search. The impactor came from the direction of the rising Sun and no pre-impact observations have been released yet. The well-known and undeniable facts are that the pre-impact path of the parent body of the Chelyabinsk impactor intersected that of the Earth on 2013 February 15, 03:20:33 GMT and a collision took place. Obviously, for two objects moving in arbitrary Keplerian elliptical orbits to collide, the two paths must intersect and both objects must be in the same spot at the same time. The solution to this problem is not trivial, it does not have an analytical form and is connected with that of finding the minimal distance between two Keplerian orbits (see e.g. Kholshevnikov & Vassiliev 1999; Gronchi & Valsecchi 2013). Given two arbitrary orbits, the problem of finding whether or not they intersect and when is, however, well suited for a brute-force Monte Carlo approach in which the two orbits are extensively sampled in phase space and the distance between any two points on the orbits is computed so the minimal distance is eventually found. Let us consider the particular case of an actual collision between a large body with a well-known elliptical orbit and a small object moving in a relatively or completely unknown orbit. If the collision time and the dimensions of the large body are well known, a Monte Carlo calculation can, in principle, help us to determine the trajectory of the small object prior to the collision. It is just a matter of computing the minimal distance between an ensemble of Keplerian test orbits and a set of points compatible with the volume of space occupied by the large body at the collision time. In our implementation, we use the two-body problem expressions in Murray & Dermott (1999) to generate the orbits with orbital elements obtained, Monte Carlo style, assuming meaningful ranges for the semimajor axis, $a$, the eccentricity, $e$, the inclination, $i$, and both the longitude of the ascending node, $\Omega$, and the argument of perihelion, $\omega$. This method is computer-intensive but if the results can be properly ranked, then an optimal solution for the minimum distance and its associated orbit can be found. Using this Monte Carlo approach, we compute the minimum orbit intersection distance (MOID) for billions of test orbits using a sampling resolution of a few million points per orbit. Neglecting gravitational focusing and in order to have a physical collision, the MOID must be $<$ 0.00004263 au (one Earth’s radius in au) and the true anomaly associated with the orbit of the Earth must match the expected value at the collision time from the ephemerides. Assuming Gaussian errors and in order to rank the computed candidate solutions, we use the following estimator: $$\beta = {\Large e}^{-\frac{1}{2}\left[
\left(\frac{d - d^{*}}{\sigma_{d^{*}}}\right)^{2} +
\left(\frac{f - f^{*}}{\sigma_{f^{*}}}\right)^{2}
\right]} \,,
\label{estimate}$$ where $d$ is the MOID of the test orbit in au, $d^{*}$ = 0 au is the minimum possible MOID, $\sigma_{d^{*}}$ is assumed to be the radius of the Earth in au, $f$ is Earth’s true anomaly used in the computation, $f^{*}$ is Earth’s true anomaly at the collision time and $\sigma_{f^{*}}$ is half the angle subtended by the Earth from the Sun (0$\fdg$00488). If $\beta > 0.368$, a collision is possible. This technique gives the most probable orbit, statistically speaking, but degenerate solutions are possible, i.e. two or more most probable orbits, but fairly different, can be found for a given collision event. This is consistent with the fact that, in theory, our planet may suffer multiple, simultaneous impacts. In order to validate the method, we use data from the first and only case in which an asteroid impact on the Earth has been accurately predicted, the Almahata Sitta event caused by the meteoroid 2008 TC$_{3}$ (Jenniskens et al. 2009; Oszkiewicz et al. 2012). For this event, both the collision time (2008-Oct-07 02:46 UTC) and the pre-collision orbit ($a$ = 1.308201 au, $e$ = 0.312065, $i$ = 2$\fdg$54220, $\Omega$ = 194$\fdg$101138 and $\omega$ = 234$\fdg$44897) are well known[^8] so a detailed comparison can be made. MOID results for 2008 TC$_{3}$ are displayed on Fig. \[moid\], right-hand panels. The value of the MOID in au is colour coded following the associated colour box; the best orbit appears as a green $\times$ sign. For this case, our algorithm gives a best orbit with $\beta = 0.987$ and parameters $a$ = 1.30807203 au, $e$ = 0.31187919, $i$ = 2$\fdg$5445889, $\Omega$ = 194$\fdg$091701 and $\omega$ = 234$\fdg$400944. If we average the best 10 orbits ranked by $\beta$, we obtain $a$ = 1.3079$\pm$0.0006 au, $e$ = 0.3119$\pm$0.0004, $i$ = 2$\fdg$542$\pm$0$\fdg$006, $\Omega$ = 194$\fdg$0918$\pm$0$\fdg$0011 and $\omega$ = 234$\fdg$44$\pm$0$\fdg$02 which agrees well with the orbit determined by Steven R. Chesley above. Therefore, our primitive yet effective technique is robust enough to reproduce a well-established result that is based on actual observations even if our method is purely geometrical. Using the same approach for the Chelyabinsk impactor, we obtain a best orbit with $\beta = 0.9997$ and parameters $a$ = 1.62394517 au, $e$ = 0.53274620, $i$ = 3$\fdg$8210787, $\Omega$ = 326$\fdg$408990 and $\omega$ = 109$\fdg$466797 and the average of the best 10 orbits gives $a$ = 1.62375$\pm$0.00014 au, $e$ = 0.53279$\pm$0.00011, $i$ = 3$\fdg$817$\pm$0$\fdg$005, $\Omega$ = 326$\fdg$4090$\pm$0$\fdg$0007, $\omega$ = 109$\fdg$44$\pm$0$\fdg$03 and $M$ = 21$\fdg$99$\pm$0$\fdg$02, around the time of impact. This result matches well the orbit calculated by S. Nakano and included in Table \[orbits\]. Colour-coded MOIDs for the Chelyabinsk impactor are displayed in Fig. \[moid\], left-hand panels. The same technique can be used to evaluate the statistical significance of the candidate solutions compiled in Table \[orbits\]. If we calculate the probability of obtaining a MOID under 0.05 au ($P_{\rm 0.050 au}$) and 0.00004263 au ($P_{\rm 0.00004263 au}$) at the impact time (see above) and the highest $\beta$ rank for the various candidate orbits (see Table \[orbits\], last three columns), all of them are statistically less robust than the one obtained here. In these calculations, the errors associated with the orbital elements as provided by the respective authors (see Table \[orbits\]) have been used when known; if unknown, the errors in Zuluaga et al. (2013) have been used. From now on, we will assume that the orbit followed by the Chelyabinsk impactor prior to its collision was the averaged one.
![Results from the Monte Carlo approach described in the text for the Chelyabinsk impactor (left-hand panels) and 2008 TC$_{3}$ (right-hand panels). The colours in the colour maps are proportional to the value of the MOID in au for the associated test orbit following the associated colour box. Our preferred solutions appear as green $\times$ signs. The ranges for the various orbital parameters are indicated in the figures; both the longitudes of the ascending node, $\Omega$, and the arguments of perihelion, $\omega$, $\in$ (0, 360)$^{o}$.[]{data-label="moid"}](fmoid.eps){width="\linewidth"}
Candidate selection
===================
Now that the most probable orbit of the Chelyabinsk impactor has been established, the next step is finding candidates for the parent body of the impactor as we assumed that it was the result of a relatively recent asteroid break-up event. Groups of rocky fragments resulting from the disruption of an asteroid are called asteroid families. Trigo-Rodríguez et al. (2007) used the D-criterion of Southworth & Hawkins (1963), $D_{\rm SH}$, to investigate the connection between the asteroid 2002 NY$_{40}$ and the FN300806 bolide. Alternatives to $D_{\rm SH}$ are the D-criterion of Lindblad & Southworth (1971), $D_{\rm LS}$, which is based on the previous one and the $D_{\rm R}$ from Valsecchi, Jopek & Froeschlé (1999). In order to investigate a possible association between the meteoroid responsible for the Chelyabinsk superbolide as characterized by the orbital solution in Table \[orbits\] and any known asteroid, we carried out a search among all the objects currently catalogued by the JPL Small-Body Database[^9] using the three D-criteria. The lowest values of the various $D$s are found for 2011 EO$_{40}$ ($D_{\rm SH}$ = 0.12, $D_{\rm LS}$ = 0.011, $D_{\rm R}$ = 0.0084), followed by 2011 GP$_{28}$ (1.0, 0.015, 0.011), 2010 DU$_{1}$ (1.0, 0.020, 0.04), 2008 FH (1.0, 0.040, 0.049) and 2007 BD$_{7}$ (0.93, 0.047, 0.041) among those with relatively well known orbits. The best candidate, 2011 EO$_{40}$, was discovered on 2011 March 10 by Kowalski et al. (2011), has an absolute magnitude, $H$, of 21.7 (size of 150-330 m if $G$ = 0.15 and the albedo range is 0.04–0.20) and it has been observed 20 times with an arc length of 34 d. The orbital elements of this Apollo asteroid are $a$ = 1.6539$\pm$0.0004 au, $e$ = 0.54039$\pm$0.00014, $i$ = 3$\fdg$3638$\pm$0$\fdg$0008, $\Omega$ = 50$\fdg$310$\pm$0$\fdg$011 and $\omega$ = 17$\fdg$055$\pm$0$\fdg$013. Apollo asteroid 2007 BD$_{7}$ was discovered on 2007 January 23, has $H$ = 21.1 and its orbit is based on 185 observations with a data-arc span of 14 d. Its orbital elements are $a$ = 1.5624$\pm$0.0012 au, $e$ = 0.4980$\pm$0.0005, $i$ = 4$\fdg$849$\pm$0$\fdg$004, $\Omega$ = 343$\fdg$627$\pm$0$\fdg$008 and $\omega$ = 219$\fdg$875$\pm$0$\fdg$007. The other candidates have shorter arcs and larger $H$: 2008 FH (12 d, $H$ = 24.4, $a$ = 1.582$\pm$0.012 au, $e$ = 0.504$\pm$0.005, $i$ = 3$\fdg$45$\pm$0$\fdg$03, $\Omega$ = 5$\fdg$207$\pm$0$\fdg$013 and $\omega$ = 264$\fdg$09$\pm$0$\fdg$04), 2010 DU$_{1}$ (4 d, $H$ = 26.6, $a$ = 1.687$\pm$0.006 au, $e$ = 0.539$\pm$0.002, $i$ = 3$\fdg$704$\pm$0$\fdg$012, $\Omega$ = 147$\fdg$832$\pm$0$\fdg$002 and $\omega$ = 74$\fdg$250$\pm$0$\fdg$010) or 2011 GP$_{28}$ (1 d, $H$ = 29.4, $a$ = 1.591$\pm$0.004 au, $e$ = 0.5199$\pm$0.0015, $i$ = 4$\fdg$048$\pm$0$\fdg$009, $\Omega$ = 16$\fdg$397$\pm$0$\fdg$003 and $\omega$ = 252$\fdg$19$\pm$0$\fdg$02). All these objects are classified as Apollo asteroids, near-Earth objects (NEOs) and potentially hazardous asteroids (PHAs). Unfortunately, even the best known orbits are based on rather short arcs. In the following, we perform $N$-body calculations to further study any possible dynamical connection between some of the candidates and the superbolide.
{width="\linewidth"}
Dynamical evolution
===================
The orbital evolution of meteoroid orbits following the osculating elements in Table \[orbits\] (averaged orbit), those of the four most promising candidate objects pointed out above and several others, were computed for 0.25 Myr backwards in time using the Hermite integration scheme described by Makino (1991) and implemented by Aarseth (2003). The standard version of this serial code is publicly available from the IoA web site.[^10] Results from this $N$-body code have been discussed in de la Fuente Marcos & de la Fuente Marcos (2012). Our direct integrations include the perturbations by the eight major planets, the Moon, the barycentre of the Pluto-Charon system and the three largest asteroids. For accurate initial positions and velocities, we used the heliocentric ecliptic Keplerian elements provided by the Jet Propulsion Laboratory online Solar system data service[^11] (Giorgini et al. 1996) and based on the DE405 planetary orbital ephemerides (Standish 1998) referred to the barycentre of the Solar system. In addition to the calculations completed using the nominal orbital elements pointed out above, we have performed 50 control simulations for each object with sets of orbital elements obtained from the nominal ones within the accepted uncertainties (3$\sigma$). Meteoroid orbits have been treated similarly. Fig. \[f1\] summarizes the results of our backwards integrations for the parent candidate asteroids. The orbital evolution of 2011 EO$_{40}$ matches well that of the Chelyabinsk impactor. Giving the uncertainties in the initial conditions (orbital elements) for both candidates and impactor, the agreement is good and suggests that these bodies were formed in a single (or a sequence of) break-up event(s) 20-40 kyr ago. The orbits of these objects are strongly perturbed as they experience periodic close encounters not only with the Earth–Moon system but also with Mars, Ceres and, in some cases, Venus. The objects studied here are part of a genetic family not a dynamical one, like the NEO family recently identified by de la Fuente Marcos & de la Fuente Marcos (2013).
The Chelyabinsk asteroid family
===============================
So far, our numerical results are somewhat consistent with 2011 EO$_{40}$ and other minor bodies being members of a young asteroid family but, can we identify additional members tracing a putative Chelyabinsk asteroid complex? and, what is more critical, can we reasonably conclude that they could be the result of a break-up event? An analysis based on the various D-criteria shows about 20 candidates to be part of the proposed Chelyabinsk asteroid family. Unfortunately, most of them have $H >$ 25 and very short arcs (a few days) so the actual characterization of the family is rather speculative although only objects with $D_{\rm R} < 0.05$ have been tentatively selected. With this restriction, the orbital parameters of the proposed family and their spreads are $a$ = 1.66$\pm$0.08 au, $e$ = 0.54$\pm$0.02, $i$ = 3.7$\pm$1.3$^{\circ}$, $\Omega$ = 162$\pm$114$^{\circ}$ and $\omega$ = 173$\pm$96$^{\circ}$. In order to check if these numbers are compatible with a gentle break-up event, we start a simulation with 100 test particles moving in orbits similar to that of 2011 EO$_{40}$ but with negligible spread in their orbital elements at aphelion. This is equivalent to having a smoothly disintegrating rubble-pile asteroid in which the relative velocities of the resulting fragments are basically zero. Fig. \[cluster\] shows the standard deviation of the various elements of the test particles as a function of the time. Although this calculation gives no obvious constraint on the age of the family, the long-term values of their standard deviations match well the values obtained above even if we do not consider non-gravitational effects that could be important for small objects.
![Time evolution of the dispersion of the orbital elements of a set of particles resulting from the smooth break-up of a rubble-pile asteroid as described in the text. []{data-label="cluster"}](fclu.eps){width="\linewidth"}
Discussion and conclusions
==========================
Although the numerical and statistical evidence in favour of a Chelyabinsk asteroid family or complex is quite encouraging, the ultimate proof of a truly genetic relationship between all these objects requires spectroscopy or, much better, sample-return (e.g. Barucci et al. 2012). The analysis of the Chelyabinsk meteorites shows that they are chondrite breccias (Bischoff et al. 2013) so the parent meteoroid may be of the S class. Our calculations did not include the Yarkovsky effect (see e.g. Bottke et al. 2006) which may have a non-negligible role on the medium, long-term evolution of objects as small as the ones studied here. Proper modeling of the Yarkovsky force requires knowledge on the physical properties of the objects involved (for example, rotation rate, albedo, bulk density, surface conductivity, emissivity) which is not the case for the objects discussed here. Detailed observations during future encounters with the Earth should be able to provide that information. On the short term, the Yarkovsky force mainly affects $a$ and $e$ but within the dispersion range found here. Its effects are negligible if the objects are tumbling or in chaotic rotation. The non-inclusion of this effect has no major impact on the assessment completed.
In this Letter, we have obtained a statistically robust solution for the pre-impact orbit of the Chelyabinsk superbolide. Assuming that such a meteoroid could be the result of the decay of an asteroid, we have singled out some candidates for membership in a putative Chelyabinsk asteroid cluster or family and tested, using $N$-body simulations, their possible dynamical connection to the parent body of the Chelyabinsk event. Our calculations suggest a dynamical link between some of the candidates and the superbolide but, unfortunately, the current orbits of all the candidates are not reliable enough to claim a conclusive connection although the avaliable evidence is certainly encouraging. The situation is similar to that of PHA 2008 XM$_{1}$ recently studied by Madiedo et al. (2013).
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank the referee, J. M. Trigo-Rodríguez, for his constructive and helpful report and S. J. Aarseth for providing a code used in this research. This work was partially supported by the Spanish ‘Comunidad de Madrid’ under grant CAM S2009/ESP-1496. We thank M. J. Fernández-Figueroa, M. Rego Fernández and the Department of Astrophysics of the Universidad Complutense de Madrid (UCM) for providing computing facilities. Most of the calculations and part of the data analysis were completed on the ‘Servidor Central de Cálculo’ of the UCM and we thank S. Cano Alsúa for his help during this stage. In preparation of this Letter, we made use of the NASA Astrophysics Data System, the ASTRO-PH e-print server and the MPC data server.
Aarseth S. J., 2003, Gravitational N-body simulations. Cambridge Univ. Press, Cambridge, p. 27 Adamo D. R., 2013, Horizons Newsletter, March/April 2013, p. 28 (http://www.aiaahouston.org/Horizons/Horizons\_2013\_03\_and\_04.pdf) Adolfsson L. G., Gustafson B. A. S., Murray C. D., 1996, Icarus, 119, 144 Barucci M. A. et al., 2012, Experimental Astronomy, 33, 645 Bischoff A., Horstmann M., Vollmer C., Heitmann U., Decker S., 2013, M&PS, 48, 5171 Bland P. A., Smith T. B., 2000, Icarus, 144, 21 Bottke W. F., Jr., Vokrouhlický D., Rubincam D. P., Nesvorný D., 2006, ARE&PS, 34, 157 Ceplecha Z., Spalding E. R., Jacobs C., Revelle O. D., Tagliaferri E., Brown P., 1999, in Baggaley W. J., Porubcan V., eds, Meteoroids 1998, Astron. Inst. Slovak Acad. Sci., 37 Christou A. A., 2004, Earth Moon Planets, 95, 425 Christou A. A., 2010, MNRAS, 402, 2759 Christou A. A., Beurle K., 1999, Planet. Space Sci., 47, 1475 Christou A. A., Oberst J., Elgner S., Flohrer J., Margonis A., McAuliffe J. P., Koschny D., 2012, Planet. Space Sci., 60, 229 Christou A. A., Vaubaillon J., Withers P., 2007, A&A, 471, 321 Christou A. A., Vaubaillon J., Withers P., 2008, Earth Moon Planets, 102, 125 Daubar I. J., McEwen A. S., Byrne S., Kennedy M. R., Ivanov B., 2013, Icarus, 225, 506 de la Fuente Marcos C., de la Fuente Marcos R., 2012, MNRAS, 427, 728 de la Fuente Marcos C., de la Fuente Marcos R., 2013, MNRAS, 434, L1 Domokos A., Bell J. F., Brown P., Lemmon M. T., Suggs R., Vaubaillon J., Cooke W., 2007, Icarus, 191, 141 Dycus R. D., 1969, PASP, 81, 399 Ens T. A., Brown P. G., Edwards W. N., Silber E. A., 2012, JJ. of Atmos. Sol.-Terr. Phys., 80, 208 Foschini L., 2001, A&A, 365, 612 Giorgini J. D. et al., 1996, BAAS, 28, 1158 Green D. W. E., 2013, CBET, 3423, 1 Gronchi G. F., Valsecchi G. B., 2013, MNRAS, 429, 2687 Halliday I., 1987, Icarus, 69, 550 Hueso R. et al., 2010, ApJ, 721, L129 Jenniskens P., 2006, Meteor Showers and Their Parent Comets, Cambridge Univ. Press, Cambridge Jenniskens P. et al., 2009, Nat, 458, 485 Jopek T., J., Williams I. P., 2013, MNRAS, 430, 2377 Kholshevnikov K. V., Vassiliev N. N., 1999, Celest. Mech. Dyn. Astron., 75, 75 Kowalski R. A. et al., 2011, MPEC Circ., MPEC 2011-E59 Lindblad B. A., Southworth R. B., 1971, in Gehrels T., ed., Physical Studies of Minor Planets, NASA SP-267, 337 Madiedo J. M., Trigo-Rodríguez J. M., Williams I. P., Ortiz J. L., Cabrera J., 2013, MNRAS, 431, 2464 Makino J., 1991, ApJ, 369, 200 Murray C. D., Dermott S. F., 1999, Solar System Dynamics. Cambridge Univ. Press, Cambridge Oszkiewicz D., Muinonen K., Virtanen J., Granvik M., Bowell E., 2012, Planet. Space Sci., 73, 30 Proud S. R., 2013, Geophys. Res. Lett., 40, 3351 Selsis F., Lemmon M. T., Vaubaillon J., Bell J. F., 2005, Nat, 435, 581 Southworth R. B., Hawkins G. S., 1963, Statistics of Meteor Streams. Smithson. Contr. Astrophys., 7, 261 Standish E. M., 1998, JPL Planetary and Lunar Ephemerides, DE405/LE405. Interoffice Memo. 312.F-98-048, NASA JPL Tóth J., Vereš P., Kornoš L., 2011, MNRAS, 415, 1527 Trigo-Rodríguez J. M. et al., 2007, MNRAS, 382, 1933 Trigo-Rodríguez J. M., Borovicka J., Llorca J., Madiedo J. M., Zamorano J., Izquierdo J., 2009, M&PS, 44, 175 Trigo-Rodríguez J. M., Llorca J., Madiedo J. M., Tancredi G., Edwards W. N., Rubin A. E., Weber P., 2010, M&PS, 45, 383 Valsecchi G. B., Jopek T. J., Froeschlé C., 1999, MNRAS, 304, 743 Williams I. P., 2002, in Murad E., Williams I. P., eds, Meteors in the Earth’s Atmosphere, Cambridge Univ. Press, Cambridge Williams I. P., 2004, WGN, J. Int. Meteor Organ., 32, 11 Zuluaga J. I., Ferrin I., Geens S., 2013, E&PSL, preprint (arXiv:1303.1796)
[^1]: http://neo.jpl.nasa.gov/news/fireball\_130301.html
[^2]: http://www.inasan.ru/eng/asteroid\_hazard/chelyabinsk\_bolid\_new.html
[^3]: http://www.amsmeteors.org/2013/02/large-daytime-fireball-hits-russia/
[^4]: http://www.projectpluto.com/temp/chelyab.htm
[^5]: http://www.icq.eps.harvard.edu/CHELYABINSK.HTML
[^6]: http://arxiv.org/abs/1302.5377
[^7]: http://astronomia.udea.edu.co/chelyabinsk-meteoroid/
[^8]: http://ssd.jpl.nasa.gov/sbdb.cgi?sstr=3430291
[^9]: http://ssd.jpl.nasa.gov/sbdb.cgi
[^10]: http://www.ast.cam.ac.uk/$\sim$sverre/web/pages/nbody.htm
[^11]: http://ssd.jpl.nasa.gov/?planet\_pos
|
---
abstract: |
Preserving the privacy of individual databases when carrying out statistical calculations has a long history in statistics and had been the focus of much recent attention in machine learning In this paper, we present a protocol for computing logistic regression when the data are held by separate parties without actually combining information sources by exploiting results from the literature on multi-party secure computation. We provide only the final result of the calculation compared with other methods that share intermediate values and thus present an opportunity for compromise of values in the combined database. Our paper has two themes: (1) the development of a secure protocol for computing the logistic parameters, and a demonstration of its performances in practice, and (2) and amended protocol that speeds up the computation of the logistic function. We illustrate the nature of the calculations and their accuracy using an extract of data from the Current Population Survey divided between two parties.\
[**Keywords:**]{} Distributed analysis; Logistic regression; Privacy-preserving computation; Secure multiparty computation.
author:
- 'Stephen E. Fienberg[^1] and Robert J. Hall[^2] and Yuval Nardi[^3]'
title: Achieving Both Valid and Secure Logistic Regression Analysis on Aggregated Data from Different Private Sources
---
Introduction
============
Privacy concerns are becoming more and more acute, especially in the digitized world where new supercomputers with an increasing processing capacities appear almost every day. These new machines together with impressive new technologies make the process of data collection, data storing and data analysis as easy as ever. This “ease of use,” may be manipulated by untrustful elements, whose aim is to deliberately cause harm by, for example, identifying and exposing sensitive data. It is the goal of privacy preserving methods to prevent or at least lessen the chances of such harmful actions from happening. In this paper we present a novel way to achieve the goal when a certain statistical analysis is required.
Preserving the privacy of individual databases when carrying out statistical calculations has a long history in statistics and had been the focus of much attention in machine learning, e.g., see [@aggarwal]. Once data are merged across sources, however, privacy becomes far more complex and a number of privacy issues arise for the linked individual files that go well beyond those that are considered with regard to the data within individual sources. When the goal is the production of the results of some statistical calculation, such as a regression analysis, c.f. Karr et al. [@securereg-jcgs05; @securereg-springer06], we can often exploit results from the cryptography literature, borrowing tools such as secure multi-party computation, e.g., see [@lindell_pinkas_1; @ppdm_book]. Secure multi-party protocols are concerned with distributed computation where each participating party, holding a private input, learns nothing but the result (see Section \[sec:smpc\]).
This paper has two main themes. In both themes we conceptualize the existence of a single combined database containing all of the information for the individuals in the separate databases and for the union of the variables. We propose an approach that gives full statistical calculation on this combined database without actually combining information sources, see [@lindell_pinkas_2; @lindell_pinkas_1]. We focus mainly on logistic regression, but our methods and tools are essentially adaptable to other statistical models, as indicated in Section \[sec:extension\]. Our approach provides only the final result of the calculation compared with other methods that share intermediate values and thus present an opportunity for compromise of values in the combined database, c.f. [@fien_secure; @fien_secure2]. We remark that our problem differs from the one studied by Chaudhuri and Monteleoni [@Chaudhuri] using differential privacy, since they are concerned with information leakage by the output of the computation, whereas we are concerned with leakage from the computation itself ! The first theme is the development of a novel approach to perform the calculations required for fitting logistic regression models when the data are distributed among several parties. In our settings the parties are unwilling or are simply forbidden (by law regulations) to share their respective data. They acknowledge the fact that pooling their private data into a conceptual global database, and running the logistic regression on the pooled data, rather than on their own data, can only lead to a better statistical analysis. We develop a secure protocol to compute the maximum likelihood estimates of the logistic parameters. Throughout the paper we make repeated use of what is known as random secret sharing, which enables us to keep intermediate parameter values secret. The first theme aims at performing the required calculations by using operations which are restricted to a linear algebra type. Note that the fitting process requires computing the logistic function which is highly non-linear. In principle, we may perform any computation securely, by making use of Yao’s general protocol [@yao82]. Nonetheless, this is in essence a theoretical construction which will often be inefficient for large computations [@ppdm_book]. Instead, we craft a specially designed approximation to the logistic function, which can be securely evaluated using the machinery of random shares and Yao’s millionaire protocol.
We establish the theoretical validity of the secure protocol for computing the logistic parameters, and show its performances in practice. In high dimensional problems with large number of cases our protocol may require faster computing resources. This is mainly because our approximation requires computing the predicate “greater-than,” which may take many encryptions. Indeed, evaluating this predicate by a reduction to Yao’s protocol takes roughly $O(b)$ encryptions where $b$ is the number of bits used to represent the numbers (this becomes dauntingly large due to the secret sharing scheme).
This leads us to the second theme, which tries to amend the protocol is a way that speeds up the computation of the logistic function. The main idea here is to avoid special circuit sub-protocols, such as Yao’s protocol. To that end, we show that we can perform the fitting process using only sums and products. The advantage to this is that these computations are very well studied primitives in secure multiparty computation and thus we can instantiate our method in a different secure multiparty computation scheme (e.g., [@ppdm_book; @goldreich]), depending on the security demands of the data holders. We propose to approximate the vector of logistic function values, by repeatedly Taylor expanding around the current value and stepping along the gradient. Operations other than sums and products, are not needed here. In principle, the approach represents the logistic function as the solution to a ordinary differential equation, then applies Euler’s method to approximate the solution. As with the first theme, we show that we can make the approximation arbitrarily accurate, at the expense of computational efficiency, and we present an illustrative empirical result.
We close the introduction with a brief description of logistic regression, mainly for the purpose of setting notation. Logistic regression is used for predicting binary outcomes or class membership given a set of explanatory variables or predictors. We can use the fitted model to predict class membership for a newly obtained record consisting of only the values of the predictors. The basic framework of logistic regression treats binary responses $y_1, \ldots, y_n$ as realizations of $n$ independent Bernoulli random variables, $Y_1, \ldots, Y_n$, whose mean depends on a set of predictors $x_i\in\mathbb{R}^d$, as follows: $$\mathbb{E}Y_i = \sigma( x_i^T\beta) \; ,$$ where $\sigma(a) = (1+\exp(-a))^{-1}$, is the sigmoid (or the logistic) function, and $\beta$ is a $d$-dimensional parameter vector. This makes the log odds, $\log(\mathbb{E}Y_i/(1-\mathbb{E}Y_i))$, linear in the predictors.
A standard method for computing the maximum likelihood estimates of $\beta$ is Newton-Raphson’s method, since closed form expressions do not exist. The fitting process requires the user to supply the log-likelihood function associated with logistic regression, along with its first two derivatives. Suppressing dependence on the data and vector of parameters, we let $\ell$ be the log-likelihood, i.e., $\ell=\sum_i\{y_ix_i^T\beta - \log(1+e^{x_i^T\beta})\}$. We also put on record the first two derivatives:
$$\label{grad-hess}
\nabla\ell = \sum_i\{x_i y_i - x_i \sigma(x_i^T\beta)\} \quad ,\quad
\nabla^2 \ell = -\sum_i{\sigma(x_i^T\beta)(1-\sigma(x_i^T\beta))x_ix_i^T} \; .$$
The gradient and the Hessian are assembled together to produce an estimate of the logistic parameters through the iterative process: $$\label{eq:nr}
\beta_{t+1}=\beta_t - (\nabla^2 \ell)^{-1} \nabla \ell \; .$$ Our protocol will be structured in rounds, where each round corresponds to an iteration of Newton’s method (\[eq:nr\]) followed by a convergence check. Each round involves a loop through all the cases $x_i$ to compute the contribution to the gradient and Hessian. We keep intermediate values of $\beta_t$ unshared between the parties. This is made possible by representing $\beta_t$ as random shares (see Section \[sec:blocks\]).
The remainder of the paper is organized as follows. Section \[sec:smpc\] presents the multi-party setup. In section \[sec:blocks\] we provide several sub-protocols which we will need. Sections \[sec:protocol1\] and \[sec:protocol2\] describe our protocol and an approach for speeding up the calculation involved, respectively. Section \[sec:security\] describes implementation details. Section \[sec:experiment\] illustrates aspects of the computation on an extract of data from the Current Population Survey divided between two parties. Section \[sec:extension\] discusses possible extensions. We defer all technical details to Appendices \[sec:validity\_1\] and \[sec:validity\_2\].
### Setting {#setting .unnumbered}
We let $X$ denote the $n\times d$ design matrix, and $y$ the $n$-dimensional response vector. We assume the presence of $P \geq 2$ parties who are interested in computing logistic regression on the total of their data. We suppose that the union of the parties data corresponds to the $X$ and $y$ of the logistic regression. In particular, we suppose that party $j$ holds onto the pair $(X_j, y_j)$ with $X_j \in \mathbb{R}^{n\times d}$ and $y_j \in \{0,1\}^n$, where $X_j$ is the $j^{th}$ party design matrix, and $y_j$ is her (binary) response vector.
In this work we consider a setting where each party has an “additive share” of the dataset. That is, $\sum_j X_j = X$ and $\sum_j y_j=y$ where $X$ and $y$ correspond to the design matrix and response vector of the combined data on which the logistic regression is performed. This subsumes all the partitioning schemes for the database (e.g., vertical and horizontal partitioning which are the cases considered in [@ppdm_book]) as in these cases for each element, one party holds the value and the remaining parties hold zero. Furthermore this setup is applicable in a case where parties may have overlapping data, and the logistic regression is to be learned by using a linear function of the overlapping data (e.g., a weighted average) as a kind of measurement error model. We suppose that the union of the individual data sets gives the complete data. In cases where some data are missing, we can apply a privacy preserving imputation method such as in @jagannathan as a preprocess, and then run our protocol.
We note that our method is general in the sense that it is applicable to every partitioning scheme, but it is clearly possible to treat specific cases such as vertically partitioned data with more efficient specialized protocols.
Secure Multi-Party Computation {#sec:smpc}
==============================
Ideally we would like our method to provide only the output of the calculation to the parties involved, and reveal nothing more. This is a lofty goal without the aid of trusted third parties, however it is relaxed in a useful way in the cryptographic literature. First it is assumed that the parties are not able to quickly solve computationally hard problems (such as breaking RSA encryption). Then, a protocol is secure so long as intermediate values in the computation either contain almost no information (in the sense that the protocol would have to be re-run astronomically many times on the same input data in order to detect any information in the messages), or will only reveal information as the result of an intractable computation. We now briefly review the security model we intend to use.
We consider the “functionality” (see [@goldreich]) which maps the data of each party into the logistic regression parameter vector $\beta$:
$$\label{lr-fun} \{(X_1,y_1),(X_2,y_2),\cdots (X_P,y_P)\} \rightarrow \{\beta, \beta, \cdots \beta \}$$
The right hand side represents $P$ copies of the parameter, so that each party receives the same output. Note that each design matrix is of the same dimensions.
A protocol for computing the functionality is just a sequence of steps, consisting of parties performing local computations, and sending intermediate messages to each other. In this work we build up a protocol for computing (\[lr-fun\]) which is secure in the presence of “semi-honest” parties. That is, parties who obey the protocol (and do not try to e.g., inject malformed data) but keep a transcript of all the messages they receive. Intuitively, a protocol is secure in this setting whenever the intermediate messages give no information about the secret inputs of other parties. Formally, the “view” of the $j^{th}$ party during the protocol is:
$$\label{lr-view}
\text{view}_j((X_1,y_1),(X_2,y_2),\cdots (X_P,y_P)) = \{(X_j,y_j),r,m_1,\cdots m_{|m|} \}$$
where $r$ is a record of all the random draws made by party $j$, and $m_k$ is the $k^{th}$ message received by that party (we have dropped dependence of $m$ on $j$ for readability).
The protocol is secure so long as there exists a polynomial time algorithm which, when given only the input and output of party $j$, may output a random transcript of message which is *computationally indistinguishable* from $\text{view}_j$. See @goldreich for a definition and discussion of computational indistinguishability. In essence, if the distribution of the sequence of messages depends only on the private input and output of party $j$ then we can simulate messages by drawing from this distribution (so long as the random number generator returns samples which are computationally indistinguishable from draws from the distribution). The existence of a simulator shows that intermediate messages do not depend on the private input of other parties, and so the protocol is secure in the sense that parties gain no more information about each other’s private inputs than that revealed by the output of the protocol. Note that this type of security is “orthogonal” to that studied in @Chaudhuri, which seeks to prevent leakage of secret information in the parameter vector.
An example of a protocol which does not achieve this definition of security is one where all parties send their data to party 1, who computes the parameter locally on the combined data and then sends it back to all other parties. In this case the messages received by party 1 consist of the data of other parties, in general it is impossible to simulate these messages given only the input and output belonging to party 1.
In the next section we present a protocol for performing Newton’s method on the logistic regression objective in a way which is secure in the presence of semi-honest parties. Our protocol makes use of a specially designed approximation for the logistic function. Section \[sec:protocol2\] then describes a different approximation necessitating the operations of only sums and products, and thus speeding-up the computations.
Although we propose to use the cryptographic model for security, others exist and deserve a place in the theory of privacy preserving data analysis. The main alternatives we see are “weak” security, and [*perturbation*]{} of the data. The former comprises a body of literature summarized in @ppdm_book. The idea is that by giving weaker privacy guarantees, we can implement much more efficient protocols. Whether it is acceptable to have this weaker privacy guarantee is a question which one must consider on a case-by-case basis. Although we describe our protocol in terms of the cryptographic model, by replacing the primitive operations (in Section \[sec:blocks\]) with their weakly-secure counterparts, we convert our protocol into a weakly secure (but also computationally more efficient) one.
The second alternative is data perturbation or sanitization. The idea would be for each party to somehow perturb his data until he is happy to release it to the other parties (e.g., through the addition of random noise). Thereupon the parties would each have a noisy copy of all the data, and could locally compute whatever statistical method they wanted on the union of the data. The difficulty with this approach is that to protect privacy may require the addition of noise of such amplitude as to render the data itself useless.
Primitives for Secure Protocols {#sec:blocks}
===============================
In this section we lay out some primitives and sub-protocols which we will commbine to make a full logistic regression protocol. While, details of the implementation of these primitives are in the references cited, we also include some in the appendix.
Secret Sharing
--------------
In our construction we make extensive use of additive secret sharing. The idea is to divide a quantity of interest $a$ into $P$ random numbers $a_j$ (one for each party) so that $\sum_ja_j = a$. If the $a_j$ are distributed uniformly in the field (e.g., the entirety of $\mathbb{R}$) then any subset of the $a_j$ will reveal nothing about $a$. In fact the sum over any subset is a random variable, the distribution of which does not depend on the secret value.
We use this construction to keep all intermediate quantities secret during the evaluation of Newton’s method (i.e., the gradient, Hessian and intermediate parameter vectors). As long as we can construct sub-protocols which compute random shares of a quantity, from random shares of inputs, then we can compose these sub-protocols together to finally obtain random shares of the logistic regression estimate. With these in hand the parties can then exchange shares and reveal the vector itself.
Although the joint distribution of the $a_j$ concentrates on the linear subspace corresponding to the secret value, marginally the shares are uniformly distributed and do not depend on any parameters. Hence we can easily simulate messages based on these shares since the marginal distributions are known, and we achieve security as defined in Section \[sec:smpc\] . Next we show how to compute additive shares of all the intermediate quantities using the abstract definition of additive shares. Although this approach is intuitively appealing, computers would quickly run into problems representing samples drawn uniformly from $\mathbb{R}$. Therefore, in the appendix we show how to approximate arbitrarily well the same computations in modular arithmetic on $\mathbb{Z}_B = \{0,1,\cdots B-1\}$ for some large $B$.
Computing Sums and Products with Random Shares {#sec_la}
----------------------------------------------
To implement Newton’s method we must essentially perform linear algebraic operations on random shares, for example by computing shares of the Newton step from shares of the gradient and inverse Hessian. In this section we describe how to obtain random shares of sums and products of quantities that are themselves represented as random shares. Using these constructions, we compute inner and outer products of vectors of random shares, and hence also matrix multiplies.
Computing shares of the sum of two secret quantities $a=\sum_j a_j$ and $b=\sum_j b_j$ is direct, as it involves only the local computation $a_j+b_j$ for each party $j=1,\ldots, P$. That is, party $j$ simply adds his shares $a_j$ and $b_j$ together to get a random share of the quantity $a+b$. Obtaining random shares of the product of two secret quantities is more involved:
$$ab = \sum_ja_j\sum_kb_k = \sum_j{a_jb_j} + \sum_j\sum_{k\neq j}{a_jb_k}$$
The elements of the first sum on the right hand side can be computed locally by each party. The second (double) sum, however, requires products between random shares held by different parties. To obtain these terms while maintaining the security of the protocol, we turn to *oblivious function evaluation*. That is, we pose the problem of computing the product as evaluating a function so that one party only knows the function and the other party only knows his input and the value of the function applied to that input.
The function set up by party $j$ and evaluated by party $k$ on his input, $b_k$, is: $$\label{prod-share-fn}
f_{(j \rightarrow k)}(x\, ; a_j) = a_jx + r_{j,k} \; ,$$
where $r_{j,k}$ is a quantity generated uniformly at random by party $j$. Evaluation is done in a manner so that party $j$ learns nothing of the output (and thus only learns about $r_{j,k}$ which he generated) and party $k$ learns only the output. Since party $k$ does not know the value of the random variable $r_{j,k}$ he has learned potentially nothing about the true value of the product. Taking $m_{j,k} = -r_{j,k}$ and $n_{j,k} = f_{(j\rightarrow k)}(b_k\,;a_j)$ we have that $m_{j,k} + n_{j,k} = a_jb_k$, and thus they form random shares of the product $a_jb_k$.
Once parties compute random shares for all the terms $a_jb_k$, they can locally compute random shares of $c=ab$ as: $$\label{eq:prod_shares}
c_j = a_jb_j + \sum_{k\neq j}(n_{k,j} + m_{j,k}) = a_jb_j + \sum_{k\neq j}(f_{(k \rightarrow j)}(a_j\,;b_k) - r_{j,k}) \; .$$ Summing up these quantities, and utilizing the definition of the linear function set up in (\[prod-share-fn\]), we easily obtain: $$\sum_j c_j = \sum_j\big\{a_jb_j + \sum_{k\neq j}(a_jb_k + r_{k,j} - r_{j,k})\big\} = \sum_j{a_j\sum_k{b_k}} \; , \nonumber$$ which shows that the $c_j$’s in (\[eq:prod\_shares\]) are indeed (additive) shares of the product.
This protocol generates random shares of the product even if the original shares weren’t themselves random, e.g., if they were due to the partitioning of the data. A method which implements these encrypted multiplications using fixed-point arithmetic is given in [@fhn:10].
We also note that dividing one secret value into another securely is much more difficult than dealing with products and requires more elaborate (and computationally demanding) protocols. Below we show how matrix inversion can be performed without any divisions.
Evaluating Interval Membership {#sec:yaosgt}
------------------------------
We suppose we are able to evaluate the following predicate in a secure way: $$1\{a \geq b\}$$ Where $a,b$ are secret values held by separate parties. This is known as Yao’s “millionaires problem,” since he described it in the context of determining which millionaire has the most money, without disclosing actual bank balances.
An example of a protocol which computes this predicate is given by @gt_proto. We can also trivially extend it so that each party receives a random share of the output bit (i.e., each party receives a random bit, the “xor” of which yields the correct output bit). Using this technique we can also check whether a secret value (i.e., a sum of random shares) is greater or less than some constant:
$$\label{yaos-gt-eqn} 1\{a_1+a_2 \geq c\} = 1\{a_1 \geq c-a_2\},$$
where $a_1,a_2$ are the random shares of $a$ held by two parties.
Securely Inverting a Matrix {#sec:matrix_inversion}
---------------------------
We use a matrix inversion routine built up entirely of matrix multiplications and subtractions, thus allowing us to use the constructions of the preceding sections to implement it securely. We obtain the reciprocal of a number $a$ without necessitating any actual division by an application of Newton’s method to the function $f(x)=x^{-1}-a$. Iterations follow $x_{s+1}=x_s(2-ax_s)$, which requires multiplication and subtraction only.
It turns out that we can apply the same scheme to matrix inversion, e.g., see [@guo_higham] and references therein. A numerically stable, coupled iteration for computing $A^{-1}$, takes the form: $$\label{matrix-inv}
\begin{array}{llllll}
X_{s+1} &=& 2X_s-X_sM_s & \quad , \quad X_0 &=& c^{-1}I \; ,\\
M_{s+1}&=& 2M_s-M_s^2 & \quad , \quad M_0 &=& \; c^{-1}A,
\end{array}$$ where $M_s = X_s A$, and $c$ is to be chosen by the user. A possible choice, leading to a quadratic convergence of $X_s\rightarrow A^{-1}$ $(M_s \rightarrow I)$, is $c=\max_i \lambda_i(A)$. In our actual implementation we used instead the trace (which dominates the largest eigenvalue, as the matrix in question is positive definite), since we can compute shares of the trace from shares of the matrix locally by each party. To compute $c^{-1}$ we use the same iteration, with scalars instead of matrices. For this iteration we initialize with an arbitrarily small $\epsilon>0$ (as convergence depends on the magnitude of the initial value being lower than that of the inverse we compute). We use the constructions of section \[sec\_la\] to iterate through (\[matrix-inv\]) until convergence. As $M_s\rightarrow I$, we check for convergence by considering the absolute difference between the trace of $M_s$ and the data dimension $d$, and we can evaluate the function $1\{|\text{tr}(M_s)-d|>\epsilon\}$ on random shares of the trace of $M_s$ using the same form as (\[yaos-gt-eqn\]).
First Protocol for Logistic Regression {#sec:protocol1}
======================================
We recall the usual Newton-Raphson iteration expression (\[eq:nr\]). To perform the iteration we first compute random shares of the update direction: $\Delta_t = -(\nabla^2\ell(\beta_t))^{-1} \nabla\ell (\beta_t)$, via the formulation of matrix-vector products of random shares. We can then add these random shares to the current parameters $\beta_t$ to obtain random shares of $\beta_{t+1}$. To check convergence recall (from e.g., [@cvx_book]) we should end if: $$\label{convergence}
\lambda^2 = (\nabla\ell(\beta_t))^T\Delta_t \leq \epsilon \; .$$ We can compute (\[convergence\]) securely using the same form as (\[yaos-gt-eqn\]). The result is sharable among all the parties, and the protocol ends whenever the result is 0, i.e., when $\lambda^2$ is not greater than $\epsilon$.
By using the constructions of the previous section, we have the tools required to invert shares of the Hessian, and thus to compute the Newton step. All that we need to do is construct a secure protocol to evaluate the logistic (sigmoid) function. In principle, a specialized sub-protocol could be built up using the construction of Yao [@yao82]. The method would be to construct circuit that evaluates the sigmoid function in the same manner that the arithmetic logic unit in a CPU would. Then we could give this circuit the secure treatment and make it into a protocol following @goldreich. The disadvantage with this approach is that the circuit evaluation protocols are prohibitively expensive and thus they are not useful in practice except for trivial circuits, see e.g., @fairplay. Instead we use a specially crafted approximation to the logistic function in terms of indicator functions. We describe this next.
A Secure Approximation to the Logistic Function {#sec:RS_of_sigma}
-----------------------------------------------
The logistic function itself is the CDF of the logistic distribution. We propose to approximate this function with an “empirical CDF.” This is a function of a set of $L$ samples $z_l$, taken independently from a logistic distribution:
$$\label{sigma-appox}
\sigma(a) \approx F_L(a) = L^{-1}\sum_{l=1}^L{1\{a \geq z_l\}} \; .$$
Based on the Glivenko-Cantelli theorem, and later work by Dvoretzky, Kiefer and Wolfowitz, the rate at which the empirical CDF converges to the true CDF (i.e., the logistic function which is of interest) is known. Using these results, we obtain bounds on the maximum difference between the logistic function and our approximation, which hold with high probability. See the remark below in Section \[sec:ECDF\] about the accuracy of the approximation.
We now turn attention to obtaining random shares of the logistic function evaluated at random shares of $\beta^Tx_i$. We obtain random shares of $\beta^Tx_i$ by using the inner product construction for multiplying together random shares. If we denote shares of this inner product by $(\beta^Tx_i)_j$ for party $j$, we write: $$\label{eq:sigma-approx2}
\sigma(\beta^Tx_i) \approx L^{-1}\sum_l{1\{\beta^Tx_i \geq z_l\}} = L^{-1}\sum_l{1\{(\beta^Tx_i)_1 + (\beta^Tx_i)_2 \geq z_l\}} \; .$$ Thus the problem reduces to getting random shares of the sum of indicators. Note that we can re-write each indicator function as: $$\label{gt-shares}
1\{(\beta^Tx_i)_1 + (\beta^Tx_i)_2 \geq z_l\} = 1\{(\beta^Tx_i)_1 \geq z_l - (\beta^Tx_i)_2 \} \; .$$
If party 2 generates the logistic random variables then we have a trivial reduction to (\[yaos-gt-eqn\]). In order to restrict the view of either party to a random share, we restrict the output to random bits $o^l_1$, and $o^l_2$, such that $$o^l_1\oplus o^l_2 = \left\{
\begin{array}{cc}
1 & \mbox{if} \quad 1\{a\geq z_l\} \\
0 & \mbox{otherwise}
\end{array}\right. \; ,$$ where $\oplus$ is the exclusive or. The right-hand side of equation (\[eq:sigma-approx2\]) requires (random shares of) the fraction of outputs with $o^l_1\oplus o^l_2=1$. This can be established by noticing that $$\sum_{l=1}^L (o^l_1\oplus o^l_2)=\sum_{l=1}^L o^l_1+\sum_{l=1}^L o^l_2-2o_1^To_2 \; ,$$ where we denote $o_k=(o^1_k, \ldots, o^L_k)$ for $k=1,2$. Jagannathan and Wright [@jagannathan] use this method to convert xor shares into additive shares for a different privacy-preserving task.
In order for the output to behave this way, we can either use Yao’s protocol directly, or take a (more efficient) GT protocol and modify it to give a (xor) random share. In this work we use the protocol of @gt_proto. Having computed random shares of the logistic function, we can use the constructions of Section \[sec\_la\] to compute random shares of the gradient and Hessian, and hence build a full logistic regression protocol.
Quality of the Logistic Approximation {#sec:ECDF}
-------------------------------------
A comment about the accuracy of approximation (\[sigma-appox\]), and the resulting logistic parameter estimator is in order. The tail behavior of the sup-norm of the error is given, for every $\epsilon>0$, by: $$\label{dkw_ineq}
P \Big(\big\|\sigma(\cdot)-L^{-1}\sum_{l=1}^L{1\{\cdot \geq z_l\}}\big\|_\infty>\epsilon \Big)\leq 2e^{-2L\epsilon^2} \; ,$$ known as the Dvoretzky-Kiefer-Wolfowitz (DKW) inequality. One possible way to choose the number of Logistic variables $L$ in practice, is by ensuring that the above probability is no more than a prescribed level of accuracy, say $\alpha$. Solving for $L$ we obtain the (very conservative) bound $L\geq -\frac{1}{2}\epsilon^{-2} \log(\alpha/2)$. A less conservative bound might entail the maximum absolute error restricted to some interval (containing the origin). We relate error in the approximation of the sigmoid (and hence the gradient) to the error in the convergent parameter by the following inequality:
$$\label{parm_err_bound}
||\hat\beta-\beta||_2 \leq \frac{R [L^{-1}+\|\sigma(\cdot)-F_L(\cdot)\|_\infty ] }{\hat\lambda_{\text{min}}} \; ,$$
in which $\hat\beta$ is the optimizer of the exact log likelihood, and $\beta$ is the optimizer of our approximation, $\hat\lambda_{\text{min}}$ is the smallest eigenvalue of the Fisher information matrix $I(\beta) = -n^{-1}\nabla^2\ell(\beta)$ (on some interval, see Appendix \[sec:validity\_1\]), and $R$ is the radius of a ball which containing all the data vectors (i.e., $\forall i, ||x_i||_2 \leq R$). The proof of this inequality follows lemma 1 of [@Chaudhuri] in which the two convex functions are the exact log likelihood objective, and the difference between the exact and approximate objectives. See Appendix \[sec:validity\_1\] for detailed theoretical derivation.
Since we can use expression (\[dkw\_ineq\]) to bound the numerator of expression (\[parm\_err\_bound\]), the parameter output by our protocol, and that output by the exact (non-private) algorithm can be brought arbitrarily close (except on a set of negligible probability) by increasing the parameter $L$. Later, we perform an experiment to show how well the method performs with reasonably small $L$. Note that for Newton’s method to converge in this approximation, we must use the same sample of $L$ logistic random variables each time we approximate the sigmoid. Otherwise assessing convergence would be difficult as the objective function would be constantly shifting. We propose that the parties draw $L$ logistic variables ahead of time, and use these for all the computations.
Hessian Lower Bound Technique {#sec:hess_bound}
-----------------------------
Notice that the Newton Raphson method requires inverting a matrix (the Hessian of the log likelihood) at each iteration. In our setting, using our iterative inversion method this becomes very expensive. Therefore we propose to use a well-studied approximation [@minka], which replaces the iteration by:
$$\label{eq_hessian_lb} \beta_{t+1}=\beta_t - 4(X^TX)^{-1} \nabla \ell \; .$$
First note that under this technique the algorithm only ever needs a single matrix inversion, since $X^TX$ is constant throughout all the iterations. Second, this algorithm still eventually converges to the correct parameter value (modulo the other approximations we make in our protocol). The reason is that the inverse hessian is always greater than $4(X^TX)^{-1}$, in the sense that the difference is positive semi-definite, see e.g, Minka [@minka] for more details. What’s more, this technique ensures that progress towards the optimum is monotonic, and so assessing convergence may be simpler.
Computation and Communication Complexity {#sec:complexity_1}
----------------------------------------
First we count how many times we must run each of our primitives for each iteration of Newton’s method. The approximation of section \[sec:RS\_of\_sigma\] requires $nL$ instances of the GT protocol per round, as $L$ instances are required per case. Computing the gradient and the Hessian requires $n(1+d+d^2)$ multiplications. Inverting the Hessian takes $2d^3$ multiplies and one GT per iteration of (\[matrix-inv\]). Since this inner iteration is quadratically convergent, it takes $O(\log{d})$ iterations to converge, and thus takes $O(d^3\log{d})$ multiplies and $O(\log{d})$ instances of GT. In total then, each outer iteration takes $O(nd^2+d^3\log{d})$ multiplies, and $nL+O(\log{d})$ invocations of the GT protocol.
Each multiplication requires a number of encryptions and decryptions; this scales quadratically with the number of parties $P$ since they must exchange with one another. Thus the computational workload increases as the data are split into more pieces. Note that although repeated use of the cryptosystem is quite expensive, performance on normal hardware is relatively rapid. A machine dedicated to the computation and running multiple threads can do thousands of encryptions per second.
Each instance of GT using the protocol of [@gt_proto] requires $O(\log{B})$ encryptions and decryptions (and operations on encrypted values etc.). Therefore in total our approximation of section \[sec:RS\_of\_sigma\] requires $O(nL\log{B})$ encryptions per iteration. This may be too computationally demanding for large $L$. One way to reduce this cost is to run the scheme using a coarse approximation to the sigmoid (i.e., a small $L$) to convergence, then increase $L$, resample the logistic variables and then continue Newton’s method from the previous convergent parameter. Although the latter iterations will still be computationally burdensome, there will be fewer of them. Another way is to use a different approximation to the sigmoid function. This is outlined next in Section \[sec:protocol2\].
Note that the total amount of communication by all parties is also proportional to the number of multiples and GT invocations. For an invocation of either, a party must transmit $\log{N}$ bits to another party, and then receive a message of the same length. There are a total of $O(P^2(nd^2+d^3\log{d})+nL)$ messages which must be sent for each iteration. If the number of parties or cases, or the granularity of the approximation is large, running the protocol over a high speed local area network would make the communication overhead manageable.
Second Protocol for Logistic Regression {#sec:protocol2}
=======================================
As we mentioned above, the computation complexity of evaluating approximation (\[sigma-appox\]) to the logistic function scales linearly with $L$, since on each of Newton’s iteration we invoke Yao’s protocol to compute the GT predicate, and we do it for every case $i$. This may be prohibitively expensive even for a moderate $L$. A possible way to reduce this computational burden was briefly described in Section \[sec:complexity\_1\]. Here, we provide full details of a more structured approach, which is reminiscent of Euler’s method. The approach is built (again) on computing Newton’s iteration (\[eq:nr\]). It would be more natural in this section to treat the logistic function in a [*vectorized*]{} fashion, i.e., $\sigma(a)=(\sigma(a_1),\ldots, \sigma(a_n))$, for an $n$-dimensional vector $a=(a_1, \ldots, a_n)$. Therefore, we use different, albeit equivalent, representations for the gradient and Hessian: $$\nabla \ell = X^T\{y-\sigma(X\beta)\} \quad , \quad \nabla^2\ell=-X^T \text{diag}\{\sigma(X\beta)\circ (1-\sigma(X\beta))\}X \; .$$ Here $X$ is the design matrix whose rows are $x_i^T$, the units or feature vectors (see (\[grad-hess\])). The symbol “$\circ$” denotes the element-wise product, i.e., $u\circ v=v\circ u=\text{diag}(u)v$.
We modify the iteration so that we neither explicitly compute the logistic function $\sigma(\cdot)$ which is involved in both the gradient and Hessian, nor use the approximation in expression (\[sigma-appox\]). Note that throughout the procedure we may treat each unit $x_i$ as having an associated logistic function value $\sigma(\beta_t^T x_i)$. We propose to track a vector of approximate function values $\hat{\sigma}_t \approx \sigma(X\beta_t)$ which will be updated after each iteration. Then, these approximate values will be used to compute the next iteration of $\beta_t$. Note that the derivative of the logistic function is given by: $$\label{logistic_deriv} \sigma^\prime(a) = \sigma(a)(1-\sigma(a)) \stackrel{\text{def}}{=} g(\sigma(a)) \; .$$ Therefore, knowing the value $\sigma(a)$, we can determine the derivative of the logistic function around $a$ by a single multiplication. Linearizing around some value $a_0$ gives: $$\label{sigma_lin} \sigma(a) = \sigma(a_0) + (a-a_0)g(\sigma(a_0)) + 2^{-1}(a-a_0)^2\sigma^{\prime\prime}(\cdot)\big|_{a^\star} \approx \sigma(a_0) + (a-a_0)g(\sigma(a_0)) \; ,$$ where the second derivative is evaluated at some value $a^\star$ in the interval between $a$ and $a_0$. Denote by $\Delta_t = \beta_{t+1}-\beta_t$ as in Section \[sec:protocol1\], then may make use of the approximation: $$\label{sigma_approx} \hat{\sigma}_{t+1} = \hat{\sigma}_t + (X\Delta_t)\circ g(\hat{\sigma}_t) \; ,$$ where $g$ is applied element-wise to $\hat{\sigma}_t$. Over the course of the entire algorithm, the approximation $\hat{\sigma}_t$ is updated repeatedly, in a manner very similar to using Euler’s method to numerically integrate the differential equation (\[logistic\_deriv\]). It is well known that the error of this method decreases with the size of the “step” taken at each iteration. In the above, the steps are of size $X\Delta_t$, which will in general be different on each iteration, and will also be different for each unit. In order to control the error we amend this approximation by breaking down the step into $k$ smaller steps each of size $k^{-1}X\Delta_t$, and performing $k$ such updates. As we shall see, we may base our choice of $k$ on some aspect of the design matrix, $X$, in order to reach a desired level of error in the approximation. We write this approximation as:
$$\label{sigma_approx_k}
\hat{\sigma}_{t+1} = \hat{\sigma}_t + k^{-1}X\Delta_t \circ \sum_{i=1}^k g(\hat{\sigma}_i^\star) \stackrel{\text{def}}{=} \hat{\sigma}_t + X\Delta_t \circ \tilde{g}_k(\hat{\sigma}_t, X\Delta_t) \; ,$$
where the $\hat{\sigma}_i^\star$ are the intermediate values corresponding to the inner iterations, and we define $\tilde{g}_k$ as the function which gives the average value of $g$ evaluated on these values.
We summarize our method in the following coupled iteration: $$\begin{aligned}
\beta_0 &=& 0^{d\times 1} \nonumber\\\nonumber
\hat{\sigma}_{0} &=& 2^{-1}\cdot 1^{n\times 1} \\
\Delta_t &=& 4(X^TX)^{-1}X^T(y-\hat{\sigma}_{t}) \label{eq:coupled} \\ \nonumber
\beta_{t+1} &=& \beta_t + \Delta_t \\ \nonumber
\hat{\sigma}_{t+1} &=& \hat{\sigma}_{t} + X\Delta_t \circ \tilde{g}_k(\hat{\sigma}_t, X\Delta_t) \; , \nonumber\end{aligned}$$ where $0^{d\times 1}$ is the $d$-dimensional vector of zeros and $1^{n\times 1}$ is the $n$-dimensional vector of ones. The proposed iteration differs from the protocol of Section \[sec:protocol1\] (and from the usual Newton-Raphson method). The main difference is that we have replaced the logistic function approximation (\[sigma-appox\]) with our Taylor approximation. Note that we are using again the bound on the Hessian (see Section \[sec:hess\_bound\]), which would make computation easier. We use this technique in our method for this reason, and also since it interacts well with our Taylor approximation by ensuring that convergence towards the optimum is in a sense monotonic, as shown in Section \[sec:convergence\]. In keeping with our goal of using only sums and products, we recall that it is possible to invert a matrix with just these operations (see Section \[sec:matrix\_inversion\]).
We now present a bound on the distance from our approximated regression coefficients $\beta_t$, to the true optimizer of the log-likelihood which we denote by $\hat\beta$, as in (\[parm\_err\_bound\]). Since our iterations are guaranteed to converge (see Section \[sec:convergence\]), we can choose to run the iterations until $||X^T(y-\sigma_t)||_2$ is smaller than some threshold $b$ (i.e., by choosing $t$ accordingly): $$b \geq ||X^T(y-\hat{\sigma_t})||_2 \geq ||X^T(y-\sigma_t)||_2 - ||X^T(\hat{\sigma}_t - \sigma_t)||_2 \; .$$
Therefore we can bound the norm of the gradient of the logistic log-likelihood taken at our final parameter estimate: $$||\nabla\ell(\beta_t)||_2^2 \leq b + nRc\tau \; .$$ where $R$ is the radius of a ball containing all the data vectors, exactly as in (\[parm\_err\_bound\]), $c$ is some constant, and $\tau$ is a quantity upper bounding the maximal Euler’s step size.
We can use this to construct our main result about the quality of our approximation. Suppose we choose $b\leq nRc\tau$, then from the above we have: $$\label{eq:error_2}
||\beta_t-\hat\beta||_2 \leq \frac{2Rc\tau}{\hat{\lambda}_{\text{min}}} \; ,$$ where $\hat{\lambda}_{\text{min}}$ is the smallest eigenvalue of the Fisher information matrix $I(\cdot) = -n^{-1}\nabla^2\ell(\cdot)$ in the line segment between $\beta$ and $\hat\beta$. Note that $\hat{\lambda}_{\text{min}} = n^{-1}\lambda_{\text{min}}$ and the factors of $n$ cancel.
Therefore we can make the accuracy of our approximation arbitrarily good by decreasing $\tau$ although, as we shall see there is a tradeoff involved. A smaller $\tau$ usually means a higher $k$, resulting in increased computational demands. We refer the reader to Appendix \[sec:validity\_2\] for complete technical details.
Choice of $k$
-------------
Thus far we have that the error of the approximation decreases as $\tau$ is decreased; however, this last variable is not controlled directly (as $L$ was in protocol 1) but rather is a function of $k$, the number of steps taken for each outer iteration of the algorithm.
In principle, to get at a prescribed step size $\tau$, we can choose $k$ by noting that:
$$\label{eq_choose_k}\tau \leq ||k^{-1}X\Delta_t||_\infty \leq k^{-1} ||X(X^TX)^{-1}X(y-\hat{\sigma}_t)||_2 \leq k^{-1}\sqrt{n}$$
This leads to the overly conservative choice of $k=\tau^{-1}\sqrt{n}$. An alternative choice is to run the protocol with a small value of $k$, e.g., 10, and then to re-run with different values to assess the sensitivity of the computation. In Section \[sec:experiment\] we show that this technique performs wekk even with small $k$.
Computational Complexity
------------------------
We can measure the overall complexity of our method in terms of the number of products that are needed, since these are the most time-consuming operations we use. First note that to construct the matrix $X^TX$ takes $nd^2$ products, and inversion of this matrix takes $O(d^3)$ using (\[matrix-inv\]), where the constant is related to the condition of the matrix. Then on each iteration, to compute $\Delta_t$ takes $nd + d^2$ products. Our approximation to the logistic function takes $nk$ products, for a total of $n(k+d) + d^2$ products per iteration.
We compare this with the cost of a protocol which computes the logistic function via a specially designed sub protocol based on circuit evaluation, cf. @yao82. If the latter may be evalutated using $q$ encryptions, then the complexity would be $n(d+q) + d^2$ operations per iteration. As mentioned before this number would typically be much larger than $k$ (for example on the order of the number of bits used to represent the numbers). Therefore on each iteration we can save a multiple of $n$ operations, which may be especially important when $n$ is large.
Security Guarantees {#sec:security}
===================
Since our protocol runs until convergence, the number of rounds is variable and depends on the data itself. Furthermore a matrix inversion was performed by an iterative scheme which itself took some variable number of iterations to converge. Therefore we amend the protocol so that the output for each party is a triple consisting of the convergent parameter value, and the number of iterations it took to converge, and the number of iterations taken for the matrix inversion. This way the messages received from testing convergence are easily simulated (i.e. a zero on every round up until the number specified in the output, then a one on that iteration) and this clearly reveals no more information since the parties know “where they are” in the protocol at all times and could count these numbers of iterations. Having dealt with this technicality we will consider simulating the other intermediate messages in our simulator, and consider these convergence tests already taken care of.
In both of our protocols, the messages which are transmitted are always part of some sub-protocol, namely multiplication or evaluation of the “greater than” predicate. The only exception to this is the final messages which are sent immediately before the output is reconstructed. As those messages are themselves random shares they may be simulated easily (although they must be simulated in such a way that they sum to the correct output values, but this is trivial). The messages which are passed during the sub-protocols may be simulated based on their respective input and outputs so long as the sub-protocols are cryptographically secure. Since we take care to ensure that the intermediate values are random shares, the simulators for the sub-protocols “compose” to form a simulator for the main protocol (see [@goldreich]).
Illustrative Experiments {#sec:experiment}
========================
We provide two illustrative experiments to demonstrate our approach. The first aims at showing the performance of our protocol from Section \[sec:protocol1\]. Specifically, we examine the effect of approximation (\[sigma-appox\]) on the resulting parameter values, when small, and large number of logistic variables $L$ are being used. The second example takes a look at the altered protocol from Section \[sec:protocol2\], which uses the coupled iteration (\[eq:coupled\]) instead of approximation (\[sigma-appox\]), and reports its performances for different values of $k$, the number of Euler’s “steps”.
For both experiments we use an extract from the Current Population Survey (CPS) data (see <http://www.bls.gov/cps/>), which includes data on a sample of slightly more than 50,000 U.S. households. We focus on predicting whether household income is greater than 50,000 dollars. We converted $M$-category features into $M-1$ binary features, and divided age into 4 bins corresponding to 20 year intervals. Note that although we expressed our approach in terms of continuous covariates, it handles binary flags just as well, where said covariates take on e.g., $0.0$ and $1.0$.
Our protocol from Section \[sec:protocol1\] deviates from the exact computation in two ways, first we use an approximation to the gradient, and second we perform all the calculations in fixed-point arithmetic. Both of these approximations can be made arbitrarily tight but at the expense of computational efficiency. To demonstrate that our protocol can be implemented in an efficient manner and produce reasonably accurate results we implemented it in a simulator and compared the results to exact logistic regression on the CPS data.
For each of $L=100$ and $L=500$ we ran our first protocol 100 times. The table below shows the means and standard deviations of the resulting parameter values. Evidently, as $L$ gets bigger, the accuracy of the parameter values improves. Figures \[fig\_like1\] and \[fig\_like1a\] show how the likelihood of the estimate maintained by the protocol increases with the number of iterations. We computed the error bars by removing the 5 samples that deviated from the mean by the greatest amount, and plotting the minimum and maximum from the remaining ones. This then corresponds to an approximate 95% confidence interval, and would become an exact interval if we were to perform more and more simulations. For the purposes of comparison, we also plotted the likelihood achieved by the exact non-private Newton Raphson algorithm, and a non-private algorithm which we referred to as “hessian lower bound.” Both give upper bounds for what we hope to achieve, the latter is an algorithm where we just use the approximation of (\[eq\_hessian\_lb\]), and exact (i.e., non-private) logistic sigmoid values. We see that as $L$ increases, the first protocol more closely approximates the hessian lower bound technique, which converges more slowly than the exact Newton Raphson method.
For the second experiment, we ran our coupled iteration on the CPS data with $k=5,10$. Although each iteration of our algorithm may be cheap, all is for nought if we require many more iterations for convergence. To determine whether this happens we compared our method to the Hessian lower bound method of (\[eq\_hessian\_lb\]), since this represents our algorithm without the approximation. In figure \[fig\_like2\], we plot the likelihoods of the second protocol against the iteration number. Since there is no randomness in the second approximation, there are no error bars. Even for small values of $k$, much smaller than those suggested by (\[eq\_choose\_k\]), the approximation to the Hessian lower bound technique is quite good, and increasing $k$ further (e.g., to 50) results in curves which are exactly the same as that of the Hessian lower bound method. In table \[tab:protocol2\] we show the resulting parameter estimates for both methods.
NR P1, $L=100$ s.d. P1, $L=500$ s.d. P2, $k=5$ P2, $k=10$
------------------ ---------- ------------- -------- ------------- -------- ----------- ------------ --
Intercept -10.7536 -11.6306 1.0761 -11.5732 0.5339 -10.7944 -11.2262
Child Sup 0.0002 0.0002 0 0.0002 0 0.0002 0.0002
Property Tax 0.0003 0.0003 0 0.0003 0 0.0003 0.0003
Num in Household 0.9802 0.9916 0.0863 0.9881 0.0434 0.9259 0.9601
Num Children -1.056 -1.0721 0.0935 -1.0685 0.047 -1.0017 -1.0384
Num Married 0.0342 0.0343 0.0053 0.0343 0.0022 0.032 0.0333
Child Sup Ind. -0.0001 -0.0001 0 -0.0001 0 -0.0001 -0.0001
Education 0.3218 0.3276 0.0295 0.3265 0.0146 0.3058 0.3172
-4.6178 -3.9701 0.3451 -3.94 0.1638 -3.6202 -3.8011
-4.2368 -3.6782 0.3148 -3.6596 0.1504 -3.4817 -3.5823
-3.9608 -3.3355 0.2852 -3.3157 0.135 -3.1592 -3.2481
-4.9575 -4.4069 0.3795 -4.3814 0.1829 -4.1096 -4.2624
0.0064 0.0051 0.0282 0.0001 0.012 -0.0035 -0.0007
-0.5362 -0.5623 0.058 -0.5562 0.0279 -0.5205 -0.5404
-0.4378 -0.4718 0.0819 -0.4609 0.0374 -0.3934 -0.4321
-0.5855 -0.6096 0.0675 -0.6018 0.0309 -0.572 -0.5889
-0.9617 -1.0283 0.1146 -1.0116 0.0549 -0.957 -0.9874
-0.7496 -0.7888 0.0852 -0.7783 0.0401 -0.736 -0.7598
- 0.2417 -0.2588 0.0276 -0.2565 0.0135 -0.2401 -0.2491
-0.4981 -0.5167 0.05 -0.5128 0.0244 -0.4835 -0.4997
-0.1658 -0.1796 0.0196 -0.1785 0.0102 -0.1631 -0.1719
Sex -0.2763 -0.2802 0.0244 -0.2792 0.0125 -0.2613 -0.2712
: Estimates produced by the exact method (Newton Raphson), and the two protocols, for different parameter settings of the protocols.[]{data-label="tab:protocol2"}
![Log Likelihood vs iteration number for protocol 1 with $L=100$, and that of the “Hessian Lower bound” algorithm, which is the same as protocol 1 except with exact sigmoid evaluations. We also compare to the full newton raphson method, which inverts the hessian on each iteration.[]{data-label="fig_like1"}](protocol1_likelihood.pdf){width="7in"}
![Log Likelihood vs iteration number for protocol 1 with $L=500$, and that of the “Hessian Lower bound” algorithm, which is the same as protocol 1 except with exact sigmoid evaluations. We also compare to the full newton raphson method, which inverts the hessian on each iteration.[]{data-label="fig_like1a"}](protocol1a_likelihood.pdf){width="7in"}
![Log Likelihood vs iteration number for protocol 2 with $k=5,10$, and that of the “Hessian Lower bound” algorithm, which is the same as protocol 1 except with exact sigmoid evaluations. We also compare to the full newton raphson method, which inverts the hessian on each iteration.[]{data-label="fig_like2"}](protocol2_likelihood.pdf){width="7in"}
Beyond Logistic Regression {#sec:extension}
==========================
We can use the construction of Section \[sec:protocol1\] to build secure protocols for similar statistical calculations, e.g., the constructions for computing shares of outer products and matrix inverses naturally yield a secure algorithm for performing linear regression, for details see [@fhn:10]. Furthermore using the “ridge regression” penalty on the weights (i.e., computing a MAP estimate under a Gaussian prior) can naturally be added to the protocol for both linear and logistic regression. It is also possible to implement the coordinate ascent computation of the lasso (or sparse logistic regression) using these constructions (i.e., using the GT protocol to perform soft thresholding).
Our protocol generalizes to the class of Generalized Linear Models (GLMs) including logistic regression with other link functions. GLMs consist of a random component $Y_i$ from an exponential family, a systematic component with a linear predictor $\eta_i = x_i^T\beta$, and a link function $\eta_i=h(\mu_i)$, where $\mu_i=\mathbb{E} Y_i$. If $h$ makes the linear predictor $\eta_i=\theta_i$, where $
\theta_i$ is the natural parameter of the exponential family, $h$ is canonical.
For Poisson log-linear models with the canonical link, $\mu_i=\exp\{\eta_i\}$, we approximate the exponential function similarly. For Gamma models with the canonical link, $\mu_i=1/\eta_i$, and for inverse-Gaussian models with the canonical link, $1/\mu^2$, we can use the number inverting without division scheme. Our approach can also be extended to treat binary regression with non-canonical links, such as the *probit* link function, or more generally, inverse CDF link functions. The general form of the gradient is: $$\nabla \ell = \sum_i \frac{\{y_i x_i - x_i \mu_i\}}{\text{Var}(Y_i)} \frac{\partial \mu_i}{\partial \eta_i} \; .$$ Let $F$ denote a given CDF ($F=\Phi$ leads to the probit link function, while, of course, $F=F_L$ leads to the logit link function). Then, $\mu_i = F(\eta_i)$, and thus $\partial \mu_i/\partial \eta_i = f(\eta_i)$, where $f$ is the density. Therefore, we should find approximations for $f$ as well as for $F$ (approximation for $F$ will follow the same idea as for $F_L$, i.e., using the empirical CDF).
Conclusion
==========
We have demonstrated that a fully secure approach to logistic regression based on the cryptographic notion of security may be made practical for use on moderately large datasets shared between several parties. Although it is slower than methods with weaker security guarantees, it offers more rigorous guarantees with respect to the privacy of the input data. We emphasize that our protocol (like any cryptographic protocol) prevents leakage of information which may arise from the computation itself. It does not address any leakage which results from the output.
The problem of secure regression is far from solved however, we have yet to deal with the problem of record linkage, and have implicity assumed that the parties know how their respective datasets are aligned. Furthermore record linkage due to a statistical model may be incorrect and may result in errorful estimates of model parameters.
Theoretical Validity of the First Protocol {#sec:validity_1}
==========================================
Here we show how a bound on the error in the approximation (\[sigma-appox\]) to the logistic function leads to a bound on the quality of the convergent parameter vector output by the protocol. Specifically, we establish the validity of (\[parm\_err\_bound\]). Let $R$ denote a constant such that $||x_i||_2 \leq R$, for $i=1,\ldots, n$. Recall the expressions for the gradient $\nabla\ell$ and Hessian $\nabla^2\ell$ given in (\[grad-hess\]). Define the approximated gradient, by substituting $F_L$ for $\sigma$: $$\nabla\tilde{\ell}(\beta) = \sum_{i=1}^n{x_iy_i-x_iF_L(x_i^T\beta)} \; .$$ Rewriting $\nabla\ell(\beta) = \nabla\tilde{\ell}(\beta) + \sum_{i=1}^n{x_iF_L(x_i^T\beta) - x_i\sigma(x_i^T\beta)}$, and applying the triangle inequality we obtain a bound on the norm of the gradient of the logistic objective: $$\label{exactnorm_bound}
{||\nabla\ell(\beta)||}_2 \leq
{||\nabla\tilde{\ell}(\beta)||}_2 + nR{||F_L(\cdot)-\sigma(\cdot)||}_\infty \; .$$ Next we convert a bound in the norm of the gradient into a bound on the distance to the optimum.
\[lem\_meanval\] Let $\hat{\beta}$ be the optimizer of the logistic regression objective, and let $\lambda_{\text{min}}$ denote the smallest eigenvalue of the negative Hessian in the line segment between $\beta$ and $\hat{\beta}$. Then: $$\label{distance_bound}
||\beta-\hat{\beta}||_2 \leq \frac{||\nabla\ell(\beta)||_2}{\lambda_{\text{min}}} \; .$$
We use the mean-value theorem (for vector-valued functions) to write the difference between gradient vectors at $\beta$ and $\hat{\beta}$: $$\label{eq:mean-value}
\nabla\ell(\beta) - \nabla\ell(\hat{\beta}) = \nabla\ell(\beta) - 0 = \left(\int_0^1 \! \nabla^2\ell(a\beta +(1-a)\hat{\beta}) \ da \right)(\beta-\hat{\beta}) \; .$$ Now, for every (symmetric) matrix $B$, and a non-zero vector $e$, the Rayleigh quotient satisfies $e^T Be/e^T e\geq \lambda_{\text{min}}(B)$, where $\lambda_{\text{min}}(B)$ is the minimal eigenvalue of $B$. If $B=A^2$, for a positive definite (symmetric) matrix $A$, this reduces (after taking the square root on both sides) to $\|Ae\|_2/ \|e\|_2 \geq \lambda_{\text{min}}(A)$. Applying this to (\[eq:mean-value\]), and using Weyl’s inequality, we have: $$||\nabla\ell(\beta)||_2 = \left|\left|\left(\int_0^1 \! \nabla^2(a\beta +(1-a)\hat{\beta}) \ da \right)(\beta-\hat{\beta})\right|\right|_2 \geq \lambda_{\text{min}} ||\beta-\hat{\beta}||_2 \; .$$ This completes the proof.
\[lem\_gradmin\] Using the same notation we have: $$\label{approxnorm_bound}
\min_{\beta\in\mathfrak{B}} ||\nabla\tilde{\ell}(\beta)||_2 \leq nRL^{-1} \; ,$$ where $\mathfrak{B}$ is a (non-empty) set of logistic parameters defined in the proof.
Consider a continuous, monotonically non-decreasing function $g(\cdot)$ which satisfies $||g(\cdot)-F_L(\cdot)||_\infty \leq L^{-1}$. Such a function clearly exists, for example the smooth nondecreasing curve which goes through all points $(z_{(j)},jL^{-1})$ where $1\leq j\leq L$ (where $z_{(j)}$ is $j^{th}$ smallest logistic variable used in $F_L$). Since $g(\cdot)$ is nondecreasing, it is the derivative of some convex function: $$G(a) = \int_{-\infty}^a{\! g(b)\ db}$$ Consider the approximation to the logistic gradient which uses $g$ instead of $F_L$: $$\nabla\bar{\ell}(\beta) = \sum_{i=1}^n{x_iy_i-x_ig(x_i^T\beta)}$$ This is the derivative of a concave function: $$\bar{\ell}(\beta) = \sum_{i=1}^n{x_i^T\beta y_i-G(x_i^T\beta)} \; ,$$ which is indeed concave since it is a linear function minus a convex function. Hence $\bar{\ell}$ has a unique maximum somewhere. Consider the functions $g(\cdot)$ so that the maximum is in the interior of the space $\mathbb{R}^d$ (i.e., is not at infinity). Hence for each such $g$ we have a point $\bar{\beta} \in \mathbb{R}^d$ where the gradient is zero, i.e., $\nabla\bar{\ell}(\bar{\beta}) = 0$. Denote the set of such $\bar{\beta}$ by $\mathfrak{B}$, and note that $\mathfrak{B}$ is not empty. An argument similar to the one that led to (\[exactnorm\_bound\]) shows that: $$||\nabla\tilde{\ell}(\beta)||_2 = ||\nabla\bar{\ell}(\beta) + \sum_{i=1}^n{x_ig(x_i^T\beta)-x_iF_L(x_i^T\beta)}||_2 \leq ||\nabla\bar{\ell}(\beta)||_2 + nRL^{-1} \; .$$ Therefore: $$||\nabla\tilde{\ell}(\bar{\beta})||_2 \leq ||\nabla\bar{\ell}(\bar{\beta})||_2 + nRL^{-1} = nRL^{-1} \; ,$$ which completes the proof.
We now put this all together and state the main result about our approximation $F_L$.
If our approximation $\nabla\tilde{\ell}$ is used as an approximation to the gradient of the logistic log likelihood, and numerical optimization is performed until $||\nabla\tilde{\ell}(\beta)||_2 \leq nRL^{-1}$, then: $$||\beta-\hat{\beta}||_2 \leq \frac{R(L^{-1} + {||F_L(\cdot)-\sigma(\cdot)||}_\infty)}{\hat{\lambda}_{\text{min}}} \; ,$$ where $\hat{\beta}$ is the optimizer of the exact logistic regression objective, $\beta$ is the result of our numerical optimization, $R$ is the radius of a ball containing all the $x_i$, and $\hat{\lambda}_{\text{min}}$ is the smallest eigenvalue of the Fisher information matrix $I(\cdot) = -n^{-1}\nabla^2\ell(\cdot)$ in the line segment between $\beta$ and $\hat{\beta}$.
Notice that $||\nabla\tilde{\ell}(\beta)||_2 \leq nRL^{-1}$ is guaranteed in light of Lemma \[lem\_gradmin\]. The proof follows by substituting (\[exactnorm\_bound\]) into (\[distance\_bound\]), and by noticing that $\hat{\lambda}_{\text{min}} = n^{-1}\lambda_{\text{min}}$ and the factors of $n$ cancel.
Theoretical Validity of the Coupled Iteration {#sec:validity_2}
=============================================
Here we establish the convergence of the coupled iteration (\[eq:coupled\]), and the error in our Taylor approximation of the logistic function.
Monotonicity and Convergence {#sec:convergence}
----------------------------
We show that the update described in (\[eq:coupled\]) converges monotonically towards some final value $\beta$. We relate the size of the step taken at one iteration to the size of the step in the previous iteration. We aim to show that first, these steps are always in the same directions for each unit, and secondly, the steps are monotonically decreasing and eventually the iterations converge.
$X\Delta_{t+1}$ element-wise has the same sign as $X\Delta_t$, in the sense that $X\Delta_{t+1}\circ X\Delta_t \geq 0$.
If we define the idempotent matrix $M=X(X^TX)^{-1}X^T$, then we write: $$\begin{aligned}
X\Delta_{t+1} &=& 4X(X^TX)^{-1}X(y-\hat{\sigma}_t) \nonumber \\
&=& 4M(y-\hat{\sigma}_t) \nonumber\\
&=& 4M[y-\hat{\sigma}_{t-1}-(X\Delta_{t})\circ \tilde{g}_k(\hat{\sigma}_{t-1})] \nonumber\\
&=& 4MM(y-\hat{\sigma}_{t-1})-16M\, \text{diag}\, (\tilde{g}_k(\hat{\sigma}_{t-1})) M(y-\hat{\sigma}_{t-1}) \nonumber\\
&=& 4M\,\text{diag}\,(1-4\tilde{g}_k(\hat{\sigma}_{t-1}))M(y-\hat{\sigma}_{t-1}) \nonumber\\
&=& M\,\text{diag}\,(1-4\tilde{g}_k(\hat{\sigma}_{t-1}))X\Delta_t \; ,
\label{stepsize_rel}\end{aligned}$$ where we made use of the idempotency of $M$. Next considering the element-wise product as the diagonal of the outer product of these two matrices,
$$X\Delta_{t+1}(X\Delta_{t})^T = M\,\text{diag}\,(1-4\tilde{g}_k(\hat{\sigma}_{t-1}))X\Delta_t \Delta_t^TX^T$$
Since we clearly have that $1-4\tilde{g}_k(\hat{\sigma}_{t-1}) > 0$ no matter what value $\hat{\sigma}_{t-1}$ takes (due to the definition of $g_k$), we have that this matrix is the product of positive semi-definite matrices, and therefore is itself positive semi-definite. Therefore the diagonal elements are all non-negative, and we have proved the claim.
This result allows us to analyze our approximation to the logistic function as though we were using the forwards Euler method to integrate the differential equation (\[logistic\_deriv\]), since all the steps for any particular unit will be in the same direction.
As long as each step $k^{-1}|X\Delta_t| \leq \tau < 1$ (where the inequality is element-wise), then $0 < \hat{\sigma}_t < 1,\ \forall t$ (i.e., the approximate logistic values will remain between 0 and 1).
Suppose that the step is positive for all units and $\hat{\sigma}_t < 1$, then: $$\hat{\sigma}_{t+1}-\hat{\sigma}_t \leq \tau\hat{\sigma}_t(1-\hat{\sigma}_t^2) < 1-\hat{\sigma}_t \; ,$$ so we also have that $\hat{\sigma}_{t+1} < 1$. Likewise for units which are involved in a negative step, if they are greater than 0, then they remain so into the next iteration by an argument which is symmetric to the one above. Therefore we have that our logistic values never leave the interval $(0,1)$.
With this we also have that $0 < 4\tilde{g}_k(\hat{\sigma}_t) < 1$ for all $t$, from the definition of $g$ and $\tilde{g}_k$. Substitution into (\[stepsize\_rel\]), yields that: $$\label{approx_sigma_bound}
||X\Delta_{t+1}||_2 \leq ||M||_2\ ||\,\text{diag}\,(1-4\tilde{g}_k(\hat{\sigma}_{t-1}))||_2\ ||X\Delta_t||_2 < ||X\Delta_t||_2 \; ,$$ since $M$ has eigenvalues which are each either 0 or 1. This shows that the magnitude of the steps for the individual units is shrinking towards zero. Therefore we conclude that eventually, our approximations of the logistic values stop updating. If we assume that $X$ has $d$ linearly independent columns, then this also implies that $\Delta_t$ is going towards zero, and therefore our algorithm eventually converges.
Quality of the Logistic Approximation {#sec:p2error}
-------------------------------------
We now analyze the error in the approximation of the logistic function values. We then use this together with the convexity of the problem to yield a bound on the error in the convergent parameters (see (\[eq:error\_2\])). To aid the notation, in this section we consider the problem of estimating the logistic values for just a single case, and specifically one for which the steps are all positive. Due to the symmetry of the logistic function about 0, we will then have the same type of bounds on the error when the approximation updates in the negative direction. We first show a loose upper bound on the supremum of the error which would be encountered if the approximation was run for an infinite number of steps of size at most $\tau$, and then use this to bound the error after finitely many such steps.
As we have shown by the above monotonicity argument, our approximation to the logistic function is essentially analogous to using Euler’s method to integrate the derivative of the logistic function. Since we consider approximating a single value, we change the names of our variables to avoid confusion with the previous vector valued approximation. If we denote by $\hat{s}_t$ the approximated value after $t$ steps of various sizes, $\tau_0\ldots \tau_{t-1} < \tau$. Thus $\hat{s}_t \approx s_t = \sigma(a_t)$ where $a_t = \sum_{i=0}^{t-1}\tau_i$. We compare this approximation to the exact values and consider the error: $$\xi_t = \hat{s}_{t} - s_{t} \; .$$
Making use of the step (\[sigma\_lin\]), we evaluate the error in the next iteration: $$\begin{aligned}
\xi_{t+1} & = \hat{s}_{t+1} - s_{t+1} \\
& = \hat{s}_t + \tau_{t}g(\hat{s}_t) -s_t -\tau_tg(s_t) - 2^{-1}\tau_t\sigma^{\prime\prime}(\cdot)|_{a^\star_t} \\
& = \xi_t + \tau_t[g(\hat{s}_t) - g(s_t)] + \zeta_t \\
& = \xi_t + \tau_t(\hat{s}_t-s_t)g^\prime(\cdot)|_{s^{\star}_t} + \zeta_t \\
& = \xi_t(1+\tau_tg^\prime(\cdot)\big|_{s^{\star}_t}) + \zeta_t \\
& = \xi_t(1+\tau_t -2\tau_ts^{\star}_t) + \zeta_t\end{aligned}$$ Where we have defined $$\label{eqn:zeta}\zeta_t = - 2^{-1}\tau_t\sigma^{\prime\prime}(\cdot)\big|_{a^\star_t}$$
and $a_t \leq a^\star_t \leq a_{t+1}$ is some value in the interval about which the second derivative is taken. Likewise $s^\star_t$ is bounded between $s_t$ and $\hat{s}_t$. As we have seen from (\[approx\_sigma\_bound\]), as long as $\tau_t \leq \tau < 1$ then $0 < \hat{s}_t < 1$ for all $t$. Since we only consider positive steps $\tau_t > 0$ then we have that $2^{-1} \leq \hat{s}_t < 1$, and hence the same bound applies to $s_t^\star$. Therefore we have that: $$|\xi_{t+1}| \leq |\xi_t| + |\zeta_t|$$ Therefore we see that: $$\label{bound_sum}\sup_t|\xi_t| \leq \sum_{i=1}^\infty |\zeta_i|$$
Examining the form of $\sigma^{\prime\prime}(\cdot)$, we find it to be a function which is everywhere negative. Examining the third derivative, we find that the second derivative has exactly one stationary point in $[0,\infty)$ which is located at:
$$a^\star = -\log{\frac{6-\sqrt{12}}{6+\sqrt{12}}}, \quad \sigma(a^\star) = \frac{6+\sqrt{12}}{12}$$
Whats more, we see that $\partial^3\sigma(\cdot) < 0$ on $[0,a^\star)$, and $\partial^3\sigma(\cdot) > 0$ on $(a^\star,\infty)$. Therefore we have that $a^\star$ is the minimum of the function. Using this we bound the sum (\[bound\_sum\]) by an integral:
$$-\sum_{t=0}^\infty\sigma^{\prime\prime}(\cdot)|_{a^\star_t} \leq -\int_0^\infty \sigma^{\prime\prime}(a)\ da - 2\tau\sigma^{\prime\prime}(x^\star)
= 4^{-1} - 2\tau\sigma^{\prime\prime}(x^\star)
\; .$$ Substituting this into (\[bound\_sum\]) and (\[eqn:zeta\]) we have that: $$\label{euler_bound}\max_t|\xi_t| \leq 2^{-1}\tau(4^{-1} - 2\tau\sigma^{\prime\prime}(x^\star)) \stackrel{\text{def}}{=} c\tau + d\tau^2 \approx c\tau \; .$$ We can make the approximation arbitrarily tight by decreasing the step size.
[99]{}
Aggarwal, C. and Yu, P. S. eds. (2008). *Privacy Preserving Data Mining: Models and Algorithms*. Springer-Verlag, New York.
Blake, I. and Kolesnikov, V. (2004). Strong conditional oblivious transfer and computing on intervals. In *Advances in Cryptology—ASIACRYPT 2004*, 515–529.
Boyd, S. and Vandenberghe, L. (2004). *Convex Optimization.* Cambridge University Press, New York.
Chaudhuri, K. and Monteleoni, C. (2008). Privacy-preserving logistic regression. *NIPS 2008*, 289–296.
Fienberg, S. E., Fulp, W. J., and Slavkovic, A. B. and Wrobel, T. A. (2006). “[S]{}ecure” log-linear and logistic regression analysis of distributed databases. *Privacy in Statistical Databases: CENEX-SDC Project International Conference, PSD 2006*, 277–290.
Fienberg, S. E., Hall, R. and Nardi, Y. (2010). Secure multiple linear regression based on homomorphic encryption." Submitted for publication.
Fienberg, S. E., Slavkovic, A. B., and Nardi, Y. (2009). Valid statistical analysis for logistic regression with multiple sources. In P. Kantor and M. Lesk, eds., *Proc. Workshop on Interdisciplinary Studies in Information Privacy and Security—ISIPS 2008*, USA LNCS volume, Springer-Verlag, New York.
Goethals, B., Laur, S., Lipmaa, H., Mielikainen, T. (2004). On secure scalar product computation for privacy-preserving data mining. *ISISC, 2004*.
Goldreich, O. (2004). *Foundations of Cryptography: Volume 2 Basic Applications.* Cambridge University Press, New York.
Goldwasser, S. (1997). Multi-party computations: [P]{}ast and present. *Proceedings of the 16th Annual ACM Symposium on Principles of Distributed Computing*, 1–6.
Guo, C. and Higham N. J. (2006). A Schur-Newton method for the matrix p’th root and its inverse. *SIAM Journal on Matrix Analysis and Applications*, 28(3), 788–804.
Jagannathan, G. and Wright, R. (2008) Privacy-preserving imputation of missing data. *Data Knowl. Eng.*, 65(1), 40–56.
Karr A.F., and Lin, X., and Reiter, J.P. and Sanil, A .P. (2005) Secure regression on distributed databases. *Journal of Computational and Graphical Statistics*, 14(2), 263–279.
Karr A.F., and Lin, X., and Reiter, J.P. and Sanil, A .P. (2006) Secure analysis of distributed databases. In D. Olwell and A. G. Wilson and G. Wilson, eds., [*Statistical Methods in Counterterrorism: Game Theory, Modeling, Syndromic Surveillance, and Biometric Authentication*]{}, Springer-Verlag, New York, 237–261.
Lindell, Y. and Pinkas, B. (2002). Privacy preserving data mining. *Journal of Cryptology*, 15(3), 177–206.
Lindell, Y. and Pinkas, B. (2009). Secure multiparty computation for privacy-preserving data mining. *Journal of Privacy and Confidentiality*, 1(1), 59–98.
Malkhi, D. and Nisan, N. and Pinkas, B. and Sella, Y. (2004). Fairplay: A secure two-party computation system. *Proceedings of the 13th conference on USENIX Security Symposium —Volume 13*, 20–20.
Minka, T. (2003). A comparison of numerical optimizers for logistic regression. Unpublished manuscript.
Paillier, P. (1999). Public-key cryptosystems based on domposite degree residuosity classes. *EUROCRYPT 1999*, 223–238.
Vaidya, J. and Zhu, Y. and Clifton, C. (2005). *Privacy Preserving Data Mining*. Springer-Verlag, New York.
Yao, A. C. (1982). Protocols for secure computations. *Proceedings of the 23rd Annual IEEE Symposium on Foundations of Computer Science*, 160–164.
Yao, A. C. (1986). How to generate and exchange secrets. *Proceedings of the 27th Symposium on Foundations of Computer Science (FOCS), IEEE*, 162–167.
[^1]: Department of Statistics, Machine Learning Department and Cylab, Carnegie Mellon University, Pittsburgh, PA.
[^2]: Department of Statistics and Machine Learning Department, Carnegie Mellon University, Pittsburgh, PA.
[^3]: Faculty of Industrial Engineering and Management, Technion - Israel Institute of Technology, Haifa, Israel.
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---
abstract: 'Six recently developed exchange functionals for pairing with different two versions of van der Waals density functionals (vdW-DF) are tested for weakly bonded solids. The test, using 26 layered weakly bonded compounds, benchmarks the lattice constants against experimental data and the interlayer binding energies against reference data from the random-phase approximation (RPA). The investigated functionals tend to give interlayer binding energies higher than the RPA benchmark, and the overall performance for lattice constants is good. The exchange functionals optB86b and cx13 paired with the original vdW-DF and the B86R functional paired with vdW-DF2 are found to give particularly good results for equilibrium geometries.'
author:
- Torbjörn Björkman
title: Testing several recent van der Waals density functionals for layered structures
---
Introduction
============
The van der Waals density functional (vdW-DF) method of Dion et al.[@Dion2004] is emerging as one of the most successful methods for including attractive long-range dispersion forces in practical density functional theory calculations. The scheme describes the long-range interaction by letting the density at all points in space explicitly interact with all other points through a kernel function that locally models the dielectric response at each point by means of a plasmon-pole approximation[@Dion2004; @langreth2005; @Lee2010; @Berland2014a; @Berland2014b]. This is then combined with a generalized-gradient approximation (GGA) for the exchange energy and the local density approximation (LDA) for the local part of the correlation, a construction that ensures that the functional reverts to LDA in the homogenous electron gas limit. The original method was followed by a revised version by Lee et al.[@Lee2010] (vdW-DF2) of the interaction kernel that modifies the plasmon-pole approximation to a form better suited for molecules.
The performance of the functional depends critically on the choice of the GGA exchange functional to be paired with the non-local correlation. The original choice was the revPBE functional[@revPBE], selected on the basis that it gives very little binding from exchange, but which also consistently overestimates the bond lengths of van der Waals bonded systems. A new exchange functional was developed by Murray et al.[@Murray2009] for pairing with the vdW-DF2 kernel, a reparametrization of the PW86 functional[@pw86] labelled PW86R, constructed to accurately represent the long-range behaviour of the exact exchange. The vdW-DF2 kernel with PW86R exchange gives an improvement of bond lengths, but still significantly overestimates them in many cases[@Bjorkman2012b]. The first attempt to construct a functional directly for the vdW-DF non-local correlation was done by Cooper[@Cooper2010], who designed a functional, C09, that retains the long range behaviour of revPBE while decreasing the short-range repulsion. A different set of functionals was designed by Klime[š]{}, Michaelides and co-workers by optimization of different functional forms to match interaction energies for molecules[@Klimes2010; @Klimes2011], named by prefixing the parent functionals by ”opt”. In an attempt to construct an exchange pairing based on considerations similar to those that lead to the non-local vdW-DF kernel, Berland and Hyldgaard constructed the cx13 functional[^1]. This was based on matching the exchange gradient expansion to the internal functional that generates the plasmon-pole approximation inherent in the vdW-DF kernel and then using insights gained from extensive benchmarking[@Berland2013] to combine this with PW86R to form a practically working scheme[@Berland2014a]. Even more recently, Hamada constructed a GGA exchange pairing for vdW-DF2 based on B86b by somewhat similar design principles, based on considerations of suitable forms for the small and large reduced density gradient regions (here designated revB86b-DF2)[@Hamada2014]. These functionals have been demonstrated to yield good results for small molecules[@Cooper2010; @Berland2013; @Klimes2010; @Klimes2011; @Hamada2014], solids[@Berland2013; @Klimes2010; @Klimes2011; @Hamada2014], some layered systems[@Cooper2010; @Graziano2012; @Berland2014a; @Ding2012; @Mirhosseini2014; @Peelaers2014] and the optB88 and optB86b functionals have been successfully applied to adsorption problems[@Mittendorfer2011; @Luder2014a]. In relation to layered compounds can also be mentioned that promising results for graphite has been reported with the rVV10 functional[@rVV10] as well as with a pairing of the C09 functional with the vdW-DF2 kernel[@Hamada2010; @Mapasha2012].
The present author and co-workers have in two previous papers investigated vdW interaction in layered, weakly bonded systems[@Bjorkman2012a; @Bjorkman2012b] in terms of their equilibrium geometries, compared with experimental structures, and the interlayer binding energy, compared to RPA. This data was also used to construct two revised versions of VV10 non-local correlation functional[@vv10], AM05-VV10sol and PW86r-VV10sol[@Bjorkman2012c], of which the former was shown to perform well both for equilibrium geometries and interlayer binding energies in layered materials. These studies did not include the later developments discussed above, and here an attempt is made to fill this gap in the knowledge of the performance of these methods for layered vdW solids. The present work investigates C09, cx13, revB86b-DF2 and the three ”opt” functionals optPBE, optB88 and optB86b by the same benchmark suite used earlier for the development of the VV10sol functionals. The interlayer binding energy is compared with RPA values and the equilibrium geometries are compared to experimental values.
Methods
=======
All calculations were carried out using the projector-augmented wave (PAW) method as implemented in the [vasp]{} software package[@vasp1; @vasp2; @vasppaw] with an in-house implementation of vdW-DF using an adaptive real-space grid technique[@gulans2009]. The cx13 functional was implemented in [vasp]{} for the present work[^2], while all other functionals correspond to specific choices of parameters of previously implemented functionals. The basis set plane wave cutoff was set to 1.5 times the default given in the PAW library and Brillouin zone integrations were carried out on a k-space mesh with spacing of 0.2Å$^{-1}$ and a gaussian broadening of 0.1 eV. As the vdW-DF implementation does not include stress tensor calculation, equilibrium geometries were determined by minimizing the energy with respect to the crystallographic parameters using a downhill simplex method, while fully relaxing the internal coordinates in each step. The internal coordinates were relaxed until residual forces were smaller than 0.01eV/Å and the downhill simplex method was terminating when the relative change in lattice constants were smaller than 0.001 and the relative change in energy was smaller than 0.1.
The binding energy was determined by calculation of the total energy as function of interlayer separation of the layers with the intraplanar geometry fixed at the experimental structure, as was previously done in the reference RPA calculations[@Bjorkman2012a]. The choice of the RPA as a benchmark for the binding energies is a matter of computational constraints; it is still the only higher order method that is feasible to apply to a set of solids as large as the one used here. However, from basic theoretical considerations the RPA is expected to describe the vdW part of the interaction[@Dobson2012]. There are presently very few binding energy calculations for layered systems available going beyond RPA. To the best of my knowledge, the only cases are quantum Monte Carlo results for graphite[@Spanu2009] (stronger binding than RPA) and TD-DFT results from Olsen and Thygesen for bilayer graphene[@Olsen2013](weaker binding than RPA) and graphene on Ni[@Olsen2014] (chemisorption as strong as RPA and physisorption weaker than RPA). The scarcity of data and the very special electronic structure of graphene planes make these of limited value for establishing the accuracy of RPA for layered systems, but at least they are close to the RPA in value and do not show any systematic errors.
Since the shape of the interlayer binding energy curve is rather asymmetric, the lowest vibrational energy eigenvalues are sufficiently high up from the bottom of the potential well for anharmonic effects to be apparent already in the ground state. The results should therefore be corrected for the resulting zero-point anharmonic expansion (ZPAE) In order to get an accurate comparison to experimental values of the out-of-plane $c$ lattice constants. Following Bauer and Wu[@Bauer1956], we may estimate the effect of zero point vibrations by fitting the binding energy curve near its minimum to a Morse potential, $$V(d) = D \left(1-e^{-a(d-d_0)}\right)^2 + C,$$ which has known analytic solutions for the vibrational spectrum. The additional constant $C$ is a convenience to get an accurate fit near the minimum and it should be noted that the normal interpretation of the parameter $D$ as the binding energy of the system is not correct when fitting the binding energy curve in this way. This is due to the asymptotic behavior of the Morse potential, which is exponential rather than polynomial, as would be correct for van der Waals bonded systems. We then use Equations (5) and (9) of Bauer and Wu to find $\Delta d$, the average displacement from equilibrium of a low lying Morse oscillator energy level $n$, $$\Delta d_n = \frac{3\hbar\omega}{4 a D}\left(\frac{1}{2}+n\right).$$ For the Morse oscillator, $\omega=a\sqrt{\frac{2D}{\mu}}$, with $\mu$ being the reduced mass of the system, which for the ground state gives of a system of layers of similar masses yields, $$\label{deltad}
\Delta d_n = \frac{0.068576}{\sqrt{D\mu}}\left(\frac{1}{2}+n\right) \text{\AA},$$ where $\mu$ is measured in atomic mass units per unit cell and layer, and $D$ is measured in eV per layer. Similarly, the correction to the binding energy for the lowest vibrational levels is given by the normal harmonic oscillator expression[@Bauer1956] with a suitable substitutions of Morse potential parameters made for $\omega$, $$\label{vibrenergy}
E_n = \hbar\omega \left( \frac{1}{2}+n\right)= 0.091435\cdot a \sqrt{\frac{D}{\mu}} \left(\frac{1}{2}+n\right)\text{eV,}$$ where $a$ is measured in Å$^{-1}$. The method of fitting the binding curve to a Morse potential and then applying equations and should provide useful estimates lowest vibrational states for many other systems, such as adsorption on surfaces and layered heterostructures, as long as a reduction of the out-of-plane motion to a single degree of freedom can be done.
Results and discussion
======================
-------------- ---------------- ----------------- --------- ---------- ---------- ---------- ---------- ----------
Functional
ME (meV/Å$^2$) MAE (meV/Å$^2$) MRE (%) MARE (%) MRE (%) MARE (%) MRE (%) MARE (%)
C09 8.9 8.9 **-0.6** 1.0 -1.6 1.6
cx13 4.4 4.4 22 22 **-0.4** **0.9** **-0.6** **0.9**
optPBE 0.8 2.3 **6.3** **12** 1.3 1.4
optB88 4.1 4.2 23 23 1.0 1.0 1.8 1.9
optB86b 5.1 5.1 **0.1** **0.6** **0.6** 1.0
revB86b-DF2 2.6 2.9 **14** 16 **0.2** **0.6** **0.3** **0.9**
AM05-VV10sol **5.2** **11** **-0.1** 1.6 -1.2 1.4
vdW-DF -4.1 4.4 -18 20 2.5 2.5
vdW-DF2 -3.1 3.5 **-13** **15**
-------------- ---------------- ----------------- --------- ---------- ---------- ---------- ---------- ----------
The results of all the investigated functionals are given in Tables \[energies\], \[c-constants\] and \[a-constants\]. For the $c$-axes lengths in Table \[c-constants\], the ZPAE have been individually calculated and added to the result using the same functional as was used in the relaxation. The errors of the different functionals are summarized in Table \[summarytable\] in terms of the mean relative errors (MRE) and mean absolute relative errors (MARE) of the binding energies and crystallographic parameters. The mean errors and absolute mean errors compared to the RPA reference are also listed for the binding energies[^3]. For reference is also included results for the previously investigated functionals vdW-DF[@Dion2004], vdW-DF2[@Lee2010] and AM05-VV10sol[@Bjorkman2012c]. The smallest average deviations have been highlighted by boldface and the largest deviations by underlining. A graphical illustration of the deviations is also given in Figure \[errors\], where compounds have been sorted from the smallest to largest value of deviation.
![Deviations from RPA binding energies and experimental lattice constants, sorted by the size of the deviation. Note that the index along the $x$-axes do not correspond to the same compound for the different functionals.[]{data-label="errors"}](bindenerrs2.eps){width="49.00000%"}
The convergence of these settings was tested by recalculating the revB86b-DF2 with a 15-20% basis set cutoff and on a k-space mesh with a tighter spacing of 0.15Å$^{-1}$. This induced fluctuations of the $c$ lattice constants at most by $\pm0.3$%, which is a good measure of the uncertainty of the individual compounds. The convergence errors of the $a$ lattice constants are about one order of magnitude smaller, 0.01%, and for $E_B$ they were found to be $\pm 0.02$meV/Å$^2$ or 0.1%. These errors showed no drift and do not alter the average values listed in Table \[summarytable\]. We may also note the excellent agreement of the values for optPBE and optB88 functionals for graphite and BN compared with those of Graziano et al.[@Graziano2012], using the same electronic structure code, but a different implementation of the vdW-DF framework. However, the agreement for cx13 with the calculations of Berland and Hyldgaard[@Berland2014a], which were performed using the Quantum Espresso package[@quantumespresso], is not as good, despite the fortran subroutine of the present implementation of the cx13 functional being verified to give the same results as that of Berland and Hyldgaard to machine precision. The most probable source of this difference appears to be the different pseudopotential libraries employed.
The functionals all give ZPAE of similar sizes for the different compounds. Indeed, within the accuracy of the present study, the correction could have been calculated with any one of the functionals and then applied instead to the experimental values. For reference, ZPAE corrected experimental values are given in parentheses in Table \[c-constants\], and zero point energy corrected RPA binding energies are given in Table \[energies\], with zero-point corrections taken from the revB86b-DF2 functional. The overall effect of ZPAE is to shift the average errors in $c$ axis length by approximately 0.1% for all functionals, with effects being very small in all compounds except graphite and h-BN, where it is approximately 1%, in agreement with a previous estimate of Graziano et al.[@Graziano2012]. The reason is obvious from the form of Equation , which has the square root of the reduced mass in the denominator, thus greatly reducing the effect for all but the lightest elements.
Table \[summarytable\] and Figure \[errors\] gives a rather clear gradation of the functionals’ performance for the different properties. The C09 functional is clearly overbinding, with binding energies far above the RPA values and a systematic underestimate of the lattice constants. The optPBE functional comes closest to the RPA binding energies, but has rather too large lattice constants, with average in-plane $a$ lattice constants being very much too large. The optB88 functional, which was shown to very closely reproduce RPA for the binding behaviour of graphene on Ni[@Mittendorfer2011], is here found to consistently give much higher binding energies, yet still produces too large lattice constants. The three functionals optB86b, cx13 and revB86b-DF2 all perform very well for the investigated equilibrium geometries, with binding energies higher than the RPA benchmark, in the case of optB86b as high as 27% higher.
Summarizing the results into practical recommendations, the rather strong overbinding of C09 and the large overestimate of in-plane lattice constants of optPBE speak against the use of these, while any of the three functionals optB86b, cx13 or revB86b-DF2 could be expected to perform well in layered systems, primarily based on the excellent agreement with the experimental geometries.. They all have higher interlayer binding energies than the reference RPA calculations, but it should also be noted that RPA is not an exact method, merely the best available benchmark. Furthermore, correct equilibrium geometries is expected to be of greater importance than binding energies in most typical applications.
---------- ------------- ------- ---- ------- ------ ------- ------ ------- ------ ------- ----- ------- -----
Compound $E_B^{RPA}$ $E_B$ RE $E_B$ RE $E_B$ RE $E_B$ RE $E_B$ RE $E_B$ RE
Graphite 18.3 (17.2) 26.9 47 23.0 25 22.7 24 25.1 38 24.8 35 21.0 15
BN 14.4 (13.4) 26.7 84 22.7 56 22.3 54 24.3 68 24.5 69 20.5 41
HfS$_2$ 16.1 (15.9) 22.7 40 18.7 16 18.9 17 20.9 30 20.8 29 18.3 13
HfSe$_2$ 17.0 (16.9) 23.2 36 19.3 13 18.3 7.5 20.7 21 20.9 22 18.6 9
HfTe$_2$ 18.6 (18.5) 27.3 46 23.4 25 19.0 2.2 22.2 19 23.4 25 21.5 15
MoS$_2$ 20.5 (20.2) 26.6 30 22.0 7.3 20.9 2.1 23.6 15 23.9 16 21.6 5
MoSe$_2$ 19.6 (19.4) 25.9 32 21.6 10 19.4 -1.1 22.2 14 22.8 16 20.6 5
MoTe$_2$ 20.8 (20.6) 27.5 32 23.7 14 19.0 -8.1 22.2 6.8 23.6 13 21.7 4
NbSe$_2$ 19.5 (19.3) 32.3 65 27.3 39 22.2 14 26.1 34 27.6 41 25.5 31
NbTe$_2$ 23.0 (22.9) 35.7 55 31.4 36 22.9 -0.1 27.3 19 30.1 30 28.2 23
PbO 20.2 (20.1) 24.6 22 20.0 -1.0 16.1 -20 20.1 -0.6 20.7 2.5 16.4 -19
PdTe$_2$ 40.1 (39.9) 52.0 29 47.0 17 31.4 -22 38.0 -5.2 43.3 7.9 40.8 2
PtS$_2$ 20.5 (20.4) 29.5 44 23.4 14 19.5 -4.7 23.3 14 24.1 17 21.5 5
PtSe$_2$ 19.0 (18.9) 31.5 65 25.6 35 18.1 -4.5 22.5 18 24.6 29 22.0 16
TaS$_2$ 17.6 (17.4) 28.4 61 23.5 33 22.0 25 25.0 41 25.3 43 23.1 31
TaSe$_2$ 19.4 (19.2) 28.4 46 23.8 23 21.1 8.7 24.2 25 25.0 28 22.9 18
TiS$_2$ 18.8 (18.5) 28.8 53 23.9 27 21.8 16 24.9 32 25.5 35 22.9 21
TiSe$_2$ 17.3 (17.1) 29.6 71 24.8 43 20.7 19 24.2 40 25.4 46 23.1 33
TiTe$_2$ 19.7 (19.5) 33.2 68 28.9 46 21.2 7.6 25.4 29 27.9 41 26.0 31
VS$_2$ 25.6 (25.3) 31.9 25 26.6 3.7 23.3 -8.7 26.8 4.7 27.7 8.3 25.3 -1
VSe$_2$ 22.2 (22.0) 30.7 38 26.0 17 21.8 -2.0 25.2 13 26.5 19 24.3 9
WS$_2$ 20.2 (20.0) 25.8 28 21.6 6.4 20.7 2.6 23.3 15 23.5 16 21.2 5
WSe$_2$ 19.9 (19.7) 25.9 30 21.7 9.5 19.5 -2.0 22.3 12 22.9 14 20.8 4
ZrS$_2$ 16.9 (16.7) 22.8 35 18.9 11 18.9 12 21.0 24 20.9 23 18.4 8
ZrSe$_2$ 18.5 (18.3) 24.0 30 20.0 8.0 18.6 0.5 21.1 14 21.4 15 19.1 3
ZrTe$_2$ 16.3 (16.1) 29.8 83 25.9 59 20.5 26 24.0 47 25.7 57 23.7 45
MRE 46 22 6.3 23 27 14
MARE 46 22 12 23 27 16
---------- ------------- ------- ---- ------- ------ ------- ------ ------- ------ ------- ----- ------- -----
\[energies\]
---------- ----------------- ------- ------ ------- ------ ------- ----- ------- ------ ------- ------ ------- ------
Compound $c^{exp}$ $c$ RE $c$ RE $c$ RE $c$ RE $c$ RE $c$ RE
Graphite 6.696 (6.635) 6.54 -2.4 6.65 -0.8 6.98 4.3 6.76 1.0 6.72 0.4 6.72 -0.5
BN 6.690 (6.635) 6.42 -4.1 6.51 -2.7 6.84 2.2 6.64 -0.7 6.58 -1.6 6.60 -2.1
HfS$_2$ 5.837 (5.830) 5.77 -1.2 5.84 0.0 6.05 3.6 5.91 1.2 5.87 0.5 5.85 0.1
HfSe$_2$ 6.159 (6.153) 6.09 -1.2 6.15 -0.2 6.39 3.7 6.25 1.4 6.22 0.9 6.19 0.3
HfTe$_2$ 6.650 (6.645) 6.58 -1.0 6.63 -0.2 6.97 4.9 6.80 2.3 6.69 0.5 6.70 0.6
MoS$_2$ 12.302 (12.283) 12.18 -1.0 12.34 0.3 12.81 4.1 12.53 1.8 12.41 0.9 12.38 0.5
MoSe$_2$ 12.927 (12.912) 12.86 -0.5 12.98 0.4 13.51 4.5 13.22 2.3 13.10 1.4 13.06 1.0
MoTe$_2$ 13.973 (13.961) 13.87 -0.7 13.95 -0.2 14.56 4.2 14.34 2.6 14.11 1.0 14.12 1.0
NbSe$_2$ 12.547 (12.534) 12.29 -2.0 12.36 -1.5 13.05 4.0 12.74 1.6 12.55 0.1 12.49 -0.5
NbTe$_2$ 6.610 (6.606) 6.65 0.6 6.69 1.3 7.08 7.2 6.92 4.8 6.82 3.2 6.81 3.0
PbO 4.995 (4.990) 4.88 -2.2 4.96 -0.6 5.28 5.6 5.10 2.2 5.07 1.4 5.12 2.3
PdTe$_2$ 5.113 (5.110) 5.09 -0.4 5.12 0.2 5.27 3.2 5.25 2.8 5.15 0.8 5.16 0.9
PtS$_2$ 5.043 (5.036) 4.72 -6.5 4.78 -5.3 5.37 6.4 5.12 1.4 4.99 -1.1 4.96 -1.8
PtSe$_2$ 5.081 (5.074) 4.86 -4.3 4.90 -3.6 5.46 7.6 5.15 1.3 5.02 -1.3 5.01 -1.6
TaS$_2$ 5.897 (5.890) 5.83 -1.2 5.89 -0.2 6.15 4.2 6.01 1.9 5.97 1.2 5.94 0.6
TaSe$_2$ 6.272 (6.267) 6.20 -1.1 6.25 -0.4 6.51 3.7 6.37 1.5 6.32 0.7 6.29 0.1
TiS$_2$ 5.705 (5.696) 5.58 -2.2 5.66 -0.8 5.91 3.6 5.77 1.1 5.73 0.4 5.69 -0.4
TiSe$_2$ 6.004 (5.996) 5.89 -2.0 5.94 -1.1 6.26 4.2 6.09 1.4 6.00 -0.1 6.01 -0.1
TiTe$_2$ 6.498 (6.493) 6.40 -1.4 6.46 -0.5 6.74 3.7 6.64 2.1 6.51 0.3 6.52 0.3
VS$_2$ 5.755 (5.748) 5.69 -1.1 5.74 -0.3 6.05 5.1 5.93 3.0 5.83 1.3 5.81 0.8
VSe$_2$ 6.107 (6.101) 6.07 -0.6 6.13 0.3 6.44 5.4 6.25 2.3 6.20 1.5 6.19 1.2
WS$_2$ 12.323 (12.308) 12.27 -0.4 12.40 0.6 12.86 4.3 12.60 2.3 12.49 1.3 12.46 0.9
WSe$_2$ 12.960 (12.947) 12.95 -0.1 13.04 0.6 13.56 4.7 13.28 2.5 13.16 1.6 13.16 1.5
ZrS$_2$ 5.813 (5.805) 5.73 -1.5 5.80 -0.2 6.02 3.5 5.87 0.9 5.85 0.6 5.84 0.3
ZrSe$_2$ 6.128 (6.122) 6.05 -1.3 6.13 0.0 6.36 3.7 6.24 1.8 6.18 0.8 6.16 0.4
ZrTe$_2$ 6.660 (6.655) 6.57 -1.3 6.60 -0.8 6.89 3.4 6.75 1.3 6.67 0.2 6.66 0.0
MRE -1.6 -0.6 4.4 1.8 0.6 0.3
MARE 1.6 0.9 4.4 1.9 1.0 0.9
---------- ----------------- ------- ------ ------- ------ ------- ----- ------- ------ ------- ------ ------- ------
\[c-constants\]
---------- ----------- ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
Compound $a^{exp}$ $a$ RE $a$ RE $a$ RE $a$ RE $a$ RE $a$ RE
Graphite 2.456 2.47 0.6 2.47 0.4 2.47 0.6 2.47 0.6 2.47 0.6 2.47 0.4
BN 2.510 2.51 0.0 2.52 0.2 2.52 0.4 2.52 0.4 2.52 0.4 2.52 0.2
HfS$_2$ 3.635 3.59 -1.2 3.60 -0.8 3.66 0.7 3.65 0.4 3.62 -0.4 3.62 -0.3
HfSe$_2$ 3.748 3.71 -1.0 3.72 -1.2 3.79 1.1 3.77 0.6 3.74 -0.2 3.74 -0.1
HfTe$_2$ 3.957 3.89 -1.7 3.90 -1.1 4.01 1.3 3.99 0.8 3.94 -0.4 3.94 -0.4
MoS$_2$ 3.162 3.15 -0.4 3.15 -0.3 3.20 1.2 3.20 1.2 3.17 0.3 3.17 0.2
MoSe$_2$ 3.289 3.28 -0.3 3.29 -0.3 3.35 1.9 3.34 1.6 3.31 0.6 3.30 0.5
MoTe$_2$ 3.518 3.50 -0.5 3.51 -0.2 3.59 2.0 3.58 1.8 3.54 0.6 3.54 0.7
NbSe$_2$ 3.442 3.44 -0.1 3.44 0.0 3.51 2.0 3.50 1.7 3.46 0.5 3.47 0.7
NbTe$_2$ 3.680 3.65 -0.8 3.66 -0.6 3.74 1.6 3.72 1.1 3.68 0.0 3.68 0.1
PbO 3.964 4.01 1.2 4.02 1.4 4.08 2.9 4.06 2.4 4.03 1.7 4.03 1.8
PdTe$_2$ 4.024 4.05 0.6 4.06 0.6 4.13 2.6 4.11 2.1 4.08 1.4 4.08 1.5
PtS$_2$ 3.542 3.58 1.1 3.58 1.0 3.60 1.6 3.60 1.6 3.59 1.4 3.59 1.2
PtSe$_2$ 3.727 3.77 1.2 3.78 1.4 3.80 2.0 3.80 2.0 3.77 1.2 3.79 1.7
TaS$_2$ 3.364 3.32 -1.3 3.33 -1.0 3.40 1.1 3.39 0.8 3.36 -0.1 3.36 -0.1
TaSe$_2$ 3.476 3.44 -1.0 3.45 -0.7 3.52 1.3 3.51 1.0 3.47 -0.2 3.47 -0.1
TiS$_2$ 3.409 3.36 -1.4 3.36 -1.2 3.43 0.6 3.41 0.0 3.39 -0.6 3.39 -0.7
TiSe$_2$ 3.536 3.48 -1.6 3.50 -1.1 3.57 1.0 3.55 0.4 3.52 -0.5 3.52 -0.3
TiTe$_2$ 3.777 3.72 -1.5 3.73 -1.3 3.82 1.1 3.80 0.6 3.76 -0.5 3.76 -0.4
VS$_2$ 3.221 3.14 -2.5 3.15 -2.2 3.21 -0.3 3.20 -0.7 3.17 -1.6 3.17 -1.6
VSe$_2$ 3.358 3.29 -2.0 3.30 -1.6 3.37 0.4 3.36 0.1 3.32 -1.1 3.33 -0.9
WS$_2$ 3.153 3.15 -0.1 3.16 0.2 3.21 1.8 3.20 1.5 3.17 0.5 3.17 0.7
WSe$_2$ 3.282 3.28 -0.1 3.29 0.1 3.35 2.1 3.34 1.8 3.31 0.9 3.31 0.8
ZrS$_2$ 3.662 3.63 -0.9 3.64 -0.6 3.70 1.0 3.68 0.5 3.66 -0.1 3.66 -0.1
ZrSe$_2$ 3.770 3.73 -1.1 3.74 -0.8 3.82 1.3 3.79 0.5 3.76 -0.3 3.77 -0.1
ZrTe$_2$ 3.952 3.89 -1.6 3.90 -1.6 4.01 1.5 3.99 1.0 3.94 -0.3 3.94 -0.3
MRE -0.6 -0.4 1.3 1.0 0.1 0.2
MARE 1.0 0.9 1.4 1.0 0.6 0.6
---------- ----------- ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
\[a-constants\]
Acknowledgements
================
Helpful discussions with Kristian Berland concerning the implementation of the cx13 exchange functional are gratefully acknowledged. The research was supported by the Academy of Finland through grant number 263416 and the COMP centre of excellence. Computational resources supplied by the Finnish IT Center for Science (CSC).
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[^1]: The name of cx13 is intended to denote ”consistent exchange” and 13 is the year of its construction. It is so not an acronym for the authors, hence the lower case letters.
[^2]: A patch for inclusion of the cx13 functional in [VASP]{} version 5.3 is available from the author upon request.
[^3]: Since the systems under consideration are not homogenous in shape, the deviations of the lattice constants cannot be directly compared since, unlike for the interlayer binding energies which are measured per layer and unit area, there is no sensible normalization common to all cases.
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bibliography:
- 'msc.bib'
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abstract: 'We carried out a target survey for Lyman break galaxies (LBGs) and Lyman $\alpha$ emitters (LAEs) around QSO SDSS J0211-0009 at $z=4.87$. The deep and wide broadband and narrowband imaging simultaneously revealed the perspective structure of these two high-$z$ populations. The LBGs without Ly$\alpha$ emission form a filamentary structure including the QSO, while the LAEs are distributed around the QSO but avoid it within a distance of $\sim4.5$Mpc. On the other hand, we serendipitously discovered a protocluster with a significant concentration of LBGs and LAEs where no strongly UV ionizing source such as a QSO or radio galaxy is known to exist. In this cluster field, two populations are spatially cross-correlated with each other. The relative spatial distribution of LAEs to LBGs is in stark contrast between the QSO and the cluster fields. We also found a weak trend showing that the number counts based on Ly$\alpha$ and UV continuum fluxes of LAEs in the QSO field are slightly lower than in the cluster field, whereas the number counts of LBGs are almost consistent with each other. The LAEs avoid the nearby region around the QSO where the local UV background radiation could be $\sim100$ times stronger than the average for the epoch. The clustering segregation between LBGs and LAEs seen in the QSO field could be due to either enhanced early galaxy formation in an overdense environment having caused all the LAEs to evolve into LBGs, or local photoionization due to the strong UV radiation from the QSO effectively causing a deficit in low-mass galaxies like LAEs.'
author:
- 'Nobunari Kashikawa, Tetsu Kitayama, Mamoru Doi, Toru Misawa, Yutaka Komiyama, and Kazuaki Ota'
title: 'The Habitat Segregation between Lyman Break Galaxies and Lyman $\alpha$ Emitters around a QSO at $z\sim5$'
---
Introduction
============
It is well established that QSOs and/or radio galaxies (RGs) are good markers for galaxy clusters in the local universe [@yee87; @hal88]. Although still very rare, some examples have been definitely established of galaxies strongly clustering around QSOs/RGs even in the high-$z$ universe beyond $z=3$ (e.g., [@zhe06; @ove06a; @ove06b; @ven05; @ven04; @ven02]), suggesting that such fields are indeed special regions of early galaxy formation. This trend is naively expected from a generic prediction of essentially every model of structure formation because the luminous objects tend to sit in the local high-density peaks where objects are clustered. A search for high-$z$ galaxy clustering around QSOs/RGs thus provides an important test of our basic idea about the biased galaxy formation [@kai84].
In the context of the hierarchical galaxy formation, it is easy to speculate that the merging process triggers the activity of the central massive galaxy/AGN. @kau00 asserts that a QSO represents a brief phase in the early life of every galaxy and that the QSO activity is triggered by the major mergers that a galaxy experiences during its assembly. Such co-evolution of the QSO and galaxy is also suggested by the tight correlation between the mass of the central supermassive black hole and the bulge stellar mass [@mag98] or velocity dispersion [@geb00], and that the downsizing trend in the luminosity functions of AGNs and galaxies are similar [@ued03]. The overdense region of galaxies around high-$z$ QSOs, which deserves to be called a protocluster, might be the first site for the co-evolution of QSOs and galaxies.
Beyond $z=3$, the detection of star-forming galaxies basically relies on two major diagnostics in their spectral energy distribution (SED): the redshifted Ly$\alpha$ emission line and the strong Lyman break in the continuum. Both characteristics have been promising tools for identifying high-$z$ galaxies, called Lyman $\alpha$ emitters (LAEs) and Lyman break galaxies (LBGs), respectively. LAEs have often been found to be associated with high-$z$ QSOs/RGs or to be tracing large-scale structures, especially at $2<z<3$ (e.g., [@pas96; @pen00; @kur00; @pal04]). @ven05 found $31$ LAEs around the luminous radio galaxy MRC0316-257 at $z=3.13$. Their narrowband (NB) imaging and very deep spectroscopy revealed a structure of at least $3.3\times3.3$ Mpc$^2$. @hu96 discovered two LAEs associated with the QSO BR2237-0607 at $z=4.55$, which was the most distant QSO known at the time. LBGs have also been found to be clustered around high-$z$ QSOs/RGs. @zhe06 found an overdensity of galaxies, whose very red color was consistent with that of LBGs, around a radio-loud QSO, SDSS J0836+0054 at $z=5.8$. The fraction of LAEs among their color-selected red galaxies is expected to be small based on their SED model simulation. @ove06a found that LBGs are more strongly clustered than LAEs around the most distant known RG, TN J0924-2201 at $z=5.2$. @ove06b confirmed the trend that LBGs are more strongly associated with a RG than LAEs in the case of TN J1338-1942 at $z=4.1$. On the other hand, some discoveries of high-$z$ protoclusters have occurred in which no QSO/RG activity is observed. @shi03 discovered a clustering region of LAE candidates at $z=4.86$ in the Subaru Deep Field, which was not a survey targeted for QSOs/RGs. @ouc05 also discovered two overdensity regions of LAEs at $z=5.7$ in the Subaru/XMM-Newton Deep Field. Interestingly, no evidence for either QSO/AGN or giant Ly$\alpha$ nebulae is seen in these structures. By definition, the Lyman break method should detect both LBGs without Ly$\alpha$ emission and LAEs when their continuum fluxes are sufficiently bright. The physical and/or evolutionary connection between LBGs and LAEs and the cause of the presence or absence of Ly$\alpha$ emission in these high-$z$ star-forming galaxies are still open questions. @sha01 proposed a plausible scenario of two Ly$\alpha$ bright phases: a galaxy appears as a LAE during its initial starburst epoch when it is still dust free, and then becomes a dusty LBG having Ly$\alpha$ absorption after the ISM metal enrichment. It becomes less dusty, and hence a LAE, again following the onset of a superwind when it becomes more than a few $10^8$ yr old. To address the connection between LBGs and LAEs, we must investigate both populations simultaneously at the same place and at the same redshift; a few studies [@ste00; @ove06a; @ove06b] have been conducted with this motivation in mind in high-$z$ protocluster regions. Galaxies in a high-density environment are likely to start forming earlier than in the general field, and studies of galaxy formation in the field only may have missed possible rare active spots associated with rich protoclusters.
In addition, another unsolved issue about the nature of high-$z$ galaxies is the environmental effects, especially for low-luminosity galaxies. Low-luminosity galaxies are closely bound up with the surrounding IGM. Photoionization heating by strong UV background radiation evaporates the collapsed gas in a galactic halo and inhibits gas cooling [@bar99]. This effect is inefficient for bright ($L>L^*$) galaxies residing in deep potential wells, but heavily suppresses star formation in lower mass objects [@ben02]. This could be an important feedback process from AGNs to galaxies. At $z\sim5$, when the reionization of the universe has completed, the formation of low luminosity galaxies can be suppressed by strong UV background radiation, which will be more intense with the presence of nearby strong UV ionizing sources such as QSOs. Consequently, the luminosity function of galaxies around a QSO is expected to be flatter than that of the general field. This problem, which concerns a low-mass population as a \`\`building block" in the high-$z$ universe, can only be addressed using 8-10m telescopes.
We carried out a targeted search for galaxy clustering in the vicinity of QSO SDSS J0211-0009 at $z\sim5$, using it as a marker for early galaxy formation sites. We searched simultaneously for LAE and LBG clusterings around the QSO over a relatively large field of view. Although the Lyman break method samples a broader range of redshift than NB-imaging, it can be speculated that there is an associated structure with the QSO if any overdensities are found around it (e.g., [@zhe06]). We found a remarkable contrast between the spatial distribution of LAEs and LBGs in the strong UV background field.
The outline of the paper is as follows. In § 2, we describe the observations, data reduction, and LBG/LAE sample selection. We compare the clustering trends and luminosity functions of the LBG/LAE in fields with and without strong UV background sources in § 3. In § 4, we discuss possible interpretations of the results. Throughout the paper, we analyze in the flat $\Lambda$CDM model: $\Omega_m=0.3$, $\Omega_\Lambda=0.7$, and $H_0=70h_{70}$kms$^{-1}$Mpc$^{-1}$. These parameters are consistent with recent CMB constraints [@spe06]. Magnitudes are given in the AB system.
The Data
========
Observation and Data Reduction
------------------------------
The target field to search for a protocluster contains QSO SDSS J0211-0009 at $z=4.874$ [@fan99]. We choose this QSO to coincide its redshifted Ly$\alpha$ emission line with the central wavelength of a NB filter \`\`$NB711$” ($\lambda_c=7126$Å: $\delta\lambda=73$Å). The $NB711$ narrowband filter was carefully designed to avoid strong OH night sky emission lines, and was used in the study of field LAEs [@shi03]. The imaging data were taken with Suprime-Cam [@miy02] on the Subaru telescope. Observations were made on the nights of UT $2004$ September and October in Johnson $V$, SDSS $i'$,$z'$, and $NB711$ filters. Suprime-Cam has ten $2$k $\times 4$k MIT/LL CCDs, and covers a contiguous area of $34'\times27'$ with a pixel scale of $0.''202$ pixel$^{-1}$. We could sample both LAEs at $z=4.86\pm0.03$ with this NB filter and LBGs at $z\sim5$ with broadband imaging. The broadband imaging also helped to discriminate LAEs at $z=4.87$ from low-$z$ H$\alpha$, $[$O [iii]{}$]$ and $[$O [ii]{}$]$ emitters.
There is another QSO, SDSS J0210-0018 ($z=4.77$) [@fan01] with only a $10.^\prime36$ separation from SDSS J0211-0009. A single Suprime-Cam’s FOV can sample both these QSOs simultaneously, although the redshift of SDSS J0210-0018 is out of the corresponding range of the wavelength coverage of $NB711$. Thus, we could only sample LBGs in the SDSS J0210-0018 field. The actual pointing center of the survey field is $02^h 10^m 53.^s0, -00^{\circ} 13^{\prime} 44.^{\prime \prime}4$ (J2000), which is almost the midpoint between the two QSOs. Our original motivation for targeting the field was to find a coherent large-scale structure of LBGs bridging the two QSOs (e.g., [@fuk04]); however we found no evidence of such a structure.
The integration time was $12000$-$16800$ sec in each band. For the broadband filters, the typical unit exposure time was $900$ s for $B$ and $240$ s for $z'$; shorter exposures were used for redder filters because of the increase in sky brightness with wavelength. For the $NB711$ filters, the typical exposure time was $1800$ s. We adopted a common dithering circle pattern of a full cycle of dithering consisting of $8$ pointings. The sky condition was fairly good with a seeing size of $0.5$-$0.9$ arcsec.
The data were reduced in almost the same manner as in @kas04, but we used a semi-automatic reduction system, Distributed Analysis System Hierarchy (DASH) [@yag02]. We performed object detection and photometry by running SExtractor version 2.1.6 [@ber96] on the images. Object detection was made in the $z'$-band and $NB711$-band images for LBG and LAE selections, respectively. For all objects detected in a given band pass, the magnitudes and several other parameters were measured in the other band passes at exactly the same positions as in the detection-band image, using the double image mode of SExtractor. We detected objects that had $6$ connected pixels above $2\sigma$ of the sky background RMS noise and took photometric measurements at the $2\sigma$ level. The size of the final images was $7330\times 9900$ pixels (corresponding to $24'.7 \times 33'.3$) after removal of very low-$S/N$ regions near the edges of the images. Object detection and photometry is significantly less efficient and reliable close to very bright stars due to bright halos and saturation trails. A similar degradation occurs near the edges of the images because of low $S/N$ ratios. Unfortunately, the target field contained some bright stars (see Figure \[fig\_skyall\]). We carefully defined \`\`masked regions" corresponding to these low-quality areas, and removed all objects falling within the masked regions. Aperture photometry was performed with a $2\arcsec$ diameter aperture to derive the colors of the detected objects. Photometric calibration was made with standard stars in the Landolt field SA 92 [@lan92] for the $V$-band magnitudes, spectroscopic standard stars GD50 for the $i'$ and $NB711$-bands, and Feige110 for the $z'$ band. The target field comprised several stars with SDSS (DR4) identification and photometry, with which we checked the zero-point consistency of the $i'$ and $z'$-band magnitudes. The $3\sigma$ limiting magnitudes in $2\arcsec$ apertures were $27.84$, $27.18$, $26.13$, and $26.76$ in $V$, $i'$, $z'$, and $NB711$, respectively. The final co-added image had an effective area of $742$ arcmin$^2$.
The LBG sample
--------------
The LBG sample was constructed using a color selection technique that distinguishes high-$z$ galaxies having a strong Lyman break in their SEDs. We used the ($V-i'$) versus ($i'-z'$) two-color plane to identify LBGs at $z\simeq5$. The exact selection criteria of the LBG sample were
$$\begin{aligned}
V-i' & \geq & 1.2, \nonumber\\
i'-z' & \leq & 0.7, \nonumber\\
V-i' & \geq & 1.8(i'-z')+1.7.
\label{eq-lbg}\end{aligned}$$
These criteria were almost the same as @yos06 except that we did not apply the criterion, $B>3\sigma$ (where 3$\sigma$ means a 3$\sigma$ limiting magnitude), which did not greatly affect the sample selection. Therefore, the completeness and contamination of the LBG sample were expected to be almost the same as evaluated by @yos06, who estimated the completeness of their LBG sample to be $60\%$—$50\%$ at $z'=24.15$—$26.15$. According to the estimate by @yos06, this $Vi'z'$-selected LBG sample had a redshift distribution at $3.7 \lesssim z \lesssim 5.5$ with a peak around $z\sim4.8$ and FWHM $\Delta z=0.8$. Our LBG criteria were incomplete for red continuum LBGs (i.e., larger $i'-z'$ color) due to dust extinction, while the contamination of galactic stars or low-$z$ galaxies are expected to be negligible ($<20\%$) [@yos06]. Figure \[fig\_lbgsel\] shows the distribution of objects in the ($i'-z'$) vs ($V-i'$) plane; we also show the evolutionary track of a model galaxy SED. We used the synthetic galaxy SED templates of Bruzal & Charlot model [@bc03]. We assumed the Miller & Scalo IMF from $0.1M_\odot$ to $125M_\odot$, solar metallicity $Z=Z_\odot=0.02$, ages ranging from $10^6$yr to $2\times10^{10}$yr, a constant star formation rate, no dust extinction, and Madau (1995)’s formula for continuum depression due to the intergalactic absorption blueward of Ly$\alpha$ emission. For cases in which the magnitude of an object was fainter than the $1\sigma$ limiting magnitude, it was replaced with the $1\sigma$ limiting magnitude. $550$ objects down to $26.13$ mag.($3\sigma$) in $z'_{tot}$, the $z'$-band total magnitude, met our criteria. We adopted the [MAG\_AUTO]{} value derived by SExtractor for the total magnitude.
The LAE sample
--------------
The LAE sample was constructed using a NB-excess criterion due to Ly$\alpha$ emission and a red color criterion due to the strong Lyman break in their SEDs. LAEs were selected by the criteria that met both equation (\[eq-lbg\]) and
$$\begin{aligned}
Vi - NB711 & > & 0.5,\end{aligned}$$
where $Vi=V\times0.25+i'\times0.75$ is defined as a rough estimate of the continuum flux at $7126$Å wavelength. The second criterion corresponds to a NB-excess with a rest-frame equivalent width (EW) of $>10$Å. Note that the LAE sample contained objects that have a fainter $z'$ magnitude than $26.13$ mag($3\sigma$), although all the LAE objects were detected at above $2\sigma$ in the $i'$-band. The first criterion as in equation (\[eq-lbg\]) excludes foreground galaxies with emission lines such as H$\alpha$ at $z=0.09$, $[$O [iii]{}$]$ at $z=0.42$, $[$O [ii]{}$]$ at $z=0.91$, and so on. Figure \[fig\_laesel\] shows the distribution of objects in the ($V-i'$) vs ($Vi - NB711$) plane as well as the model SED tracks. We used the same model tracks as in LBG selection, although we added emission line fluxes with a rest equivalent width of $70$Å as H$\alpha$, $[$O [iii]{}$]$, $[$O [ii]{}$]$ emitters, and LAEs for each corresponding redshift to $NB711$. As in the case of LBG selection, the magnitude was replaced with the $1\sigma$ limiting magnitude when it was fainter than the $1\sigma$ limiting magnitude. $221$ objects down to $NB711_{tot}=26.76$ ($3\sigma$) met our criteria. The detection completeness as a function of $NB711$ magnitude was estimated in the same way as in @kas06b by counting detected artificial objects distributed on the real $NB711$ image. The detection completeness was thus found to be $>90\%$ at $NB711<25.5$ and $50\%$ at the limiting magnitude $NB711=26.5$.
The LAA and LAE classification
------------------------------
There is a sample overlap between the LBGs and LAEs that we selected as above. Some LBGs had Lyman $\alpha$ emission lines that also categorized them as LAEs. Some LAEs did not meet the criteria of LBGs because their $z'$-band magnitudes were fainter than $3\sigma$ limiting magnitude. We hereafter distinguish between galaxies with and without Lyman $\alpha$ emission, and we define the \`\`Lyman $\alpha$ absorbers (LAAs)“ as \`\`Lyman break galaxies at $z\sim5$ with almost no Lyman $\alpha$ emission line (EW$<10$Å) at $\sim7126$Å ”, and the \`\`Lyman $\alpha$ emitters (LAEs)“ as \`\`Lyman $\alpha$ emission (EW$\geq 10$Å) line galaxies at $z=4.86\pm0.03$”. Therefore, the overlapping samples ($89$ objects) satisfying both LBG and LAE criteria were all categorized as LAEs, and these objects were removed from the LAA sample. Strictly speaking, the definition of a LAA also includes the galaxy population at $z\sim5$ but not at $z=4.86\pm0.03$. In the following analysis, we cannot distinguish whether we happen to see superposition on the sky, but any overdensities at specific regions, like the QSO environment or the protocluster of LAEs, are speculated to be physically associated with the region. Future spectroscopy could enable to clearly determine their accurate redshifts; however, most of the LAA/LAE galaxies are too faint to be identified with current spectroscopic capability.
We have $463$ LAA samples down to $z'_{tot}=26.13$ ($3\sigma$), and $221$ LAE samples down to $NB711_{tot}=26.76$ ($3\sigma$). Based on the redshift distribution of LBGs at $z\sim5$ estimated by @yos06, the expected probability of falling within the narrow redshift range $z=4.86\pm0.03$ for a LAA sample is $8.8\%$, which gives the expected number of LAAs to have $z=4.86\pm0.03$ to be $\sim63.95$ down to $z'_{tot}=26.0$ after correcting for completeness. On the other hand, the number of LAEs down to $z'_{tot}=26.0$ is $103.5$ after correcting for detection completeness. Therefore, the fraction of NB excess objects to all Lyman break selected objects is $61.8\%$, which is larger than the $20-25\%$ evaluated by @ste00 for LBGs at $z=3.1$ down to a similar absolute magnitude $M\simeq-20$. The LAE fraction of our estimate decreases to $22.6\%$ only when the sample is restricted to a bright LAA/LAE sample with $z'\leq24.5$. The difference might be caused by either real evolution of the LAE fraction among Lyman break selected galaxies, or by the LAE sample having a fairly high contamination of low-$z$ interlopers. @shi06 noted from the rest UV LF of the $z=5.7$ LAE sample that $\sim80\%$ or more of the LBG population would have strong Ly$\alpha$ emission at $z\sim6$.
Results
=======
Sky distribution
----------------
![The LAA (blue circle) and LAE (red triangles) distributions over the survey field plotted over the $i'$-band image. The small solid square is the QSO SDSS J0211-0009. The solid square (orange) is the QSO field, and dashed square (green) is the cluster field. The shaded regions are the masked regions around bright stars and saturated pixels, in which detected objects were rejected due to low S/N. There is a very bright star ($mR=9$ mag.) at the top of the figure where we had to mask over a relatively large region because of ghost features, especially in the NB-image. \[[*See the electronic edition of the Journal for a color version of this figure.*]{}\] \[fig\_skyall\]](f3_s2.eps)
Figure \[fig\_skyall\] shows the LAA/LAE distributions in the overall survey field. The distribution of LAEs gives a visual impression to have more inhomogeneous structure than that of LAAs. We should note that NB searches exploring only a small redshift coverage are sensitive to the large-scale structure, while the LAA distribution is diluted by the sky projection. We focused on a $12\times12$ arcmin$^2$ ($16\times16 h_{70}^{-1}$Mpc$^2$ in comoving scale) region around SDSS J0211-0009, which we call the \`\`QSO field". Figure \[fig\_qcont\] shows the LAA/LAE distributions in the QSO field. We also show smoothed surface number density contours for LAAs and LAEs. The surface number density was measured in a circle of $60\arcsec$ radius with a Gaussian smoothing of $\sigma=10\arcsec$. The mean value and the dispersion of the surface number density in the circle for each population were derived from the statistics of $10,000$ randomly chosen positions over the entire image. We confirmed that these mean and dispersion values for LBGs are almost consistent with those measured in a general blank field using the LBG sample at $z\sim5$ [@kas06a], restricting the LBG sample only with $z'<26.31$, the same limiting magnitude as in this study. The thin and thick contours in Figure \[fig\_qcont\] denote $1\sigma$ and $3\sigma$ excess from the mean density, respectively. The distributions of LAAs and LAEs around the QSO were in marked contrast with each other. The QSO SDSS J0211-0009 lies on a filamentary structure of LAAs extending from the northeast to the southwest, while the LAEs are distributed around the QSO but avoid its vicinity. Although the Lyman break method samples a broader range of redshifts than NB imaging, the most probable redshift by this $Vi'z'$-color selection is around $z\sim4.8$, almost corresponding to the same redshift of the NB-excess samples. However, we cannot completely rule out the possibility that the filamentary structure of LAAs around the QSO is attributed to a mere chance of alignment or is incidentally caused by the highly clustered contaminant populations at low-$z$, until spectra are obtained for the LAAs to confirm that they have similar redshifts to the QSO.
We found a galaxy overdensity region $\sim9$ arcmin east of the SDSS J0211-0009 where both LAAs and LAEs apparently have strong clustering. We hereafter call this region the \`\`cluster field". This region has a $6\sigma$ excess of local surface number densities for both LAAs and LAEs. The projected size of the overdense region is approximately $15h_{70}^{-1}$Mpc$\times20h_{70}^{-1}$Mpc, which is smaller than the $NB$ sampling depth $\sim33h_{70}^{-1}$Mpc. No QSOs, AGNs, or radio galaxies are known to exist in the cluster field. Note that the QSO field and the cluster field slightly overlap when an extent of a $12\times12$ arcmin$^2$ is adopted for both fields. Figure \[fig\_ccont\] shows the LAA/LAE distributions as well as their smoothed surface number density contours in the cluster field. In sharp contrast to Figure \[fig\_qcont\], the LAAs and LAEs follow a similar distribution in Figure \[fig\_ccont\]. They have high number density peaks above $3\sigma$ significance in almost the same position. The coincidence probability that two independent $\sim2'\times2'$ surface-overdensity regions will overlap within $24'.7 \times 33'.3$ is only about $0.001$.
To observe the trend more clearly, in Figure \[fig\_ccf\] we show the angular cross-correlation functions (CCFs) $w_{AE}(\theta)$ between LAAs and LAEs for the entire survey field, the QSO field, and the cluster field. The CCFs were derived using the same estimator by @ade03 $$\begin{aligned}
w_{AE}(\theta)=\frac{D_AD_E-D_AR-D_ER+RR}{RR},\end{aligned}$$ where the $D_AD_E$, $D_AR$, $D_ER$, and $RR$ denote the number of LAA-LAE, LAA-random, LAE-random, and random-random pairs having angular separations between $\theta$ and $\theta+\delta\theta$, respectively. We generated $100,000$ random points to reduce the Poisson noise in random pair counts and normalized $D_AD_E$, $D_AR$, $D_ER$, and $RR$ to the total number of pairs in each pair count. The random points were created with exactly the same boundary conditions as each field avoiding the mask regions where saturated stars dominate. The relative amplitude of the CCF among different regions was essential in this study, irrespective of the absolute amplitude of the CCF for each region. For example in the LAA case, the uncertainty of the CCF due to contamination by foreground galaxies was expected to amount to $20\%$ at most, and we did not correct the incompleteness of the derived CCF in any way. However, the contamination and completeness should not be much different among the QSO, cluster, and entire fields. We estimated the Poissonian errors only [@ls93] for the CCF as
$$\begin{aligned}
\sigma_w(\theta)=\sqrt\frac{1+w_{AE}(\theta)}{D_AD_E}.\end{aligned}$$
The CCF in the entire field has a slight amplitude excess at $< 250$arcsec ($\sim9.3 h_{70}^{-1}$Mpc comoving scale), indicating that LAAs and LAEs generally follow nearly identical structures. The CCF in the cluster field shows a higher amplitude than that of entire field, suggesting that LAAs and LAEs closely coexist in this overdense region. On the contrary, the CCF in the QSO field shows almost no correlation between LAAs and LAEs on all scales, which matches the visual impression in Figure \[fig\_qcont\].
Number count
------------
Figure \[fig\_lf\] shows comparisons of number counts between QSO and cluster fields. The number counts were investigated with respect to the $z'$-band magnitude corresponding to the rest-frame UV continuum flux ($\sim1540$Å ) for LAAs and LAEs, and the $NB711$-band magnitude corresponding to the Ly$\alpha$ flux for LAEs, respectively. The number count for each field is denoted as shaded regions indicating the $\pm1\sigma$ Poissonian scatter of each count. We corrected the completeness using the estimation by @yos06 for LAAs, and the detection completeness evaluated in § 2.3 for LAEs, respectively. We should again note here that no significant difference in the completeness between of the QSO and cluster fields is expected, even if slight uncertainties occur in the estimate procedures. The black regions show the number counts $\pm1\sigma$ scatter for the entire field, which are regarded as the average number count for each magnitude and each population. Indeed, the number count on the $z'$-mag for LAAs in the entire field was found to be consistent with @yos06 for a $z'$-band magnitude distribution of field LBGs at $z\sim5$. The number count on the $NB711$-mag for LAEs in the entire field was also determined to be almost consistent with @ouc03 for a $NB711$-band magnitude distribution of field LAEs at $z=4.86$ and with @shi04 for a $NB704$-band magnitude distribution of field LAEs at $z=4.79$. Note that the sampling field, $12\times12$ arcmin$^2$, is somewhat larger than the central overdense region in the cluster field, and therefore, the absolute number count in the cluster field might not be significantly higher than the average.
The number counts of the $z'$-mag for the QSO and cluster fields are almost consistent with each other for LAAs, while the $z'$-mag and $NB$-mag counts for LAEs differ slightly between the two fields. The $z'$-mag number count of LAEs in the cluster field is higher than in the QSO field, although the significance is as small as $1-2\sigma$. The difference becomes more significant if we take a narrower sampling field for the QSO and cluster fields. The LAE $NB$-mag number count also shows the same trend, but it is not pronounced. These differences could be interpreted as that either the number density of LAEs is high in the cluster field and low in the QSO field, or the Ly$\alpha$ (and probably rest-UV continuum) luminosity of LAEs is high in the cluster field and low in the QSO field. This trend was not seen in the rest-UV magnitudes of the LAA population. A photometric difference could exist between LAEs in the QSO and cluster fields, although the trend was not sufficiently significant, and the sample in this study was so small that the effect was not firmly established. We should also note that the discrepancy in the number count of LAEs between the QSO and cluster fields could be caused by the difference of the detection effeciency with $NB$-band for slightly different central redshift in each overdensity.
Discussion
==========
As seen in § 3.1, the relative spatial distribution of LAEs and LBGs is in stark contrast between the QSO and the cluster fields. Why do LAEs avoid the vicinity of the QSO ? We here consider two possible interpretations of the observational results. One is that all LAEs have already evolved to LBGs in the overdensity region. The second is that LAEs cannot form stars due to the photoionization effect of strong radiation from the QSO.
Evolutionary sequence from LAEs to LBGs
---------------------------------------
@sha01 proposed a plausible scenario of two Ly$\alpha$ bright phases: a galaxy appears as a LAE during its initial starburst epoch when it is still dust free, and then becomes a dusty LBG having Ly$\alpha$ absorption after ISM metal enrichment. It becomes less dusty, and hence a LAE again, with the onset of a superwind as it gets older than a few $10^8$ yr. The first phase of this evolutionary sequence is consistent with observational evidence that LAEs are generally younger and lower in stellar mass than LBGs, which have ages of $>10^9$ yr and stellar mass of $>10^{10}M_\odot$ at $z=3-6$ [@sha01; @sta06; @eyl06; @yan06]. @ove06b found that LAEs around a RG at $z=4.1$ are generally younger and less stellar-massive (a few $\times10^8M_\odot$) than LBGs. @gaw06 also suggested that field LAEs at $z=3.1$ have a lower stellar mass ($\sim5\times10^8M_\odot$) than LBGs. The stellar mass of LAEs at a higher redshift, $z\sim5$ [@pir06], was also found to be much smaller ($<10^8M_\odot$) than LBGs. Recently, @pen07 concluded that LBGs with Ly$\alpha$ emission are much younger and less massive than the LBGs without Ly$\alpha$ emission based on a systematic comparison of spectroscopic sample. @ven05 [@dow06] indicate that LAEs have, on average, smaller size than LBGs, on the basis of HST/ACS high-resolution imaging. The transition from LAEs to LBGs is also suggested in a numerical simulation by @mu06. In addition, the tendency for an overdense environment to favor early galaxy formation has been suggested by @gao05, and thus, an overdense region in a QSO field could contain a more evolved population. @ste05 showed a direct evidence suggesting that virialization redshift is higher for galaxies in the protocluster region than field galaxies. Therefore, one interpretation might be that all the galaxies around SDSS J0211-0009 have already passed the first starburst phase as LAEs and are now observed as LBGs forming a filamentary structure around the QSO.
However, some evidence contradicts this interpretation. First, in the cluster field, which shows a more significant overdensity than the QSO field, LBGs and LAEs coexist. The interpretation cannot expain the difference of LBG/LAE clustering property in the QSO and the cluster fields. Second, if LAEs were in previous times clustered around QSOs, some examples might be discovered at $z>4.8$; however, the observational evidence appears to suggest exactly the opposite. LAEs are usually found clustered around QSOs/RGs at $z<3$ [@pas96; @pen00; @ven05], while only a few examples [@hu96] of them are seen around QSOs/RGs at higher redshifts. Instead, LBGs are often found around QSOs/RGs at $z>3$. @ven04 found a LAE overdensity around RG TN J0924-2201 at $z=5.2$, although @ove06a reported that LBGs are more strongly clustered than LAEs in this region. @ove06b confirmed the trend that LBGs are more strongly associated with a RG than LAEs in the case of TN J1338-1942 at $z=4.1$. This inconsistency between the trend observed and a possible evolutionary sequence from LAEs to LBGs might be caused by observational bias; NB-imaging is difficult for a high-$z$ galaxy survey, which has usually been carried out with HST/ACS. More studies, especially on estimates of age and stellar mass of LAEs at all epochs and their variation with environment, are required to validate this hypothesis.
Photoionization by a QSO
------------------------
As discussed above, LAEs are generally revealed to be younger and have less stellar mass than LBGs; however, their dynamical mass has not been clearly determined. @ham04 estimated a relatively high halo mass of LAEs at $z=4.86$ of $\sim10^{12} M_\odot$, which was, however, calculated from their clustering strength in a protocluster region. While @shi06 concluded that LAEs at $z=5.7$ are distributed almost homogeneously in the same survey area. Large uncertainties still exist in the estimated dynamical mass of LAEs against the cosmic variance. In contrast, the dynamical mass of LBGs at $z\sim5$ has been well determined as $10^{11-12} M_\odot$ based on their clustering strength and the UV luminosity - halo mass relation predicted by semianalytic model combined with high-resolution N-body simulation [@kas06a].
It has been recognized that the formation of low mass galaxies is suppressed in the presence of an external UV field because of photoionization (e.g., [@ui84; @rees86; @bab92; @efs92; @tho96]). If LAEs are generally less massive than LBGs, their formation can be prevented to a greater extent in the vicinity of such radiation sources. This effect may explain our finding a different LAE distribution in the QSO field and the cluster field.
We estimated quantitatively the local UV radiation strength around the QSO. The local flux density at the Lyman limit, $\nu_L$, at a radius $r$ from the QSO is
$$\begin{aligned}
F^Q_\nu(\nu_L)=\frac{L(\nu_L)}{4\pi r^2},\end{aligned}$$
where $L(\nu_L)$ is the QSO luminosity at the Lyman limit. For SDSS J0211-0009, it can be derived by measurements of the continuum AB magnitude in the rest frame $1450$Å corrected for Galactic extinction, $AB_{1450}=19.93$, and the continuum slope $\beta=-0.99\pm0.70$ [@fan01], assuming $F^Q_\nu(\nu) \propto \nu^\beta$. In Figure \[fig\_qcont\], LAEs distributed outside $r>770$ physical kpc (corresponds to $>4.5$ comoving Mpc at $z=4.87$). The local flux density at $\nu_L$ inside a radius of $r=770$ physical kpc was estimated to be
$$\begin{aligned}
F^Q_\nu(\nu_L)=9.0\times10^{-19} {\rm ergs~cm}^{-2} {\rm s}^{-1} {\rm Hz}^{-1}.
\label{eq-fqso}\end{aligned}$$
This value corresponds to an isotropic UV intensity at the Lyman limit of
$$\begin{aligned}
J_{21}\sim72,
\label{eq-j21}\end{aligned}$$
when the UV intensity assumes a power-law spectrum of $J(\nu)=J_{21}(\nu/\nu_L)^\alpha\times 10^{-21}$ ergs cm$^{-2}$ s$^{-1}$ Hz$^{-1}$ sr$^{-1}$. The intensity of the background radiation at $z=4.5$ was evaluated to be log$(J(\nu_L))\sim-21.2$ by the QSO proximity effect measurements of @coo97 [@wil94]. Assuming that the background radiation at $z=4.87$ is almost the same as this value, the relative strength of the local QSO ionizing radiation was estimated to be $\sim100$ times the background radiation at LAEs located $4.5$Mpc distant from the QSO. Therefore, the local UV radiation is actually enhanced by the QSO at the position where the deficit of LAEs is observed in the QSO field.
We have investigated how this locally enhanced UV radiation influences the formation of LAEs by means of radiation-hydrodynamic simulations. The simulations are an improved version of those described in @kit00 [@kit01] with an addition of helium atoms in the chemical reaction network. Under the assumption of spherical symmetry, they solve self-consistently the dynamics of dark matter and baryons, non-equilibrium chemistry of nine species (e, H, H$^+$, H$^-$, He, He$^+$, He$^{++}$, H$_2$ and H$_2^+$), and the radiative transfer of ionizing photons. Self-shielding of H$_2$ against photo-dissociation is also taken into account. As an initial density contrast for the dark matter and baryons, we adopt a spherical top-hat perturbation that has just turned around from the Hubble expansion at $z=8.3$ and is destined to collapse (in the absence of thermal pressure) at $z=4.87$. We assume for simplicity that the gas is exposed to constant and isotropic external UV radiation with a spectral index $\alpha=-1$, which is close to the observed mean for SDSS J0211-0009. In order to cover a wide range of the UV source distance, we vary the UV intensities over the range $J_{21}=0.1 \sim 10^4$. Given large uncertainties as to when the radiation source turns on, we first explore the case in which the gas has been exposed to the UV radiation since the turn-around epoch, $z=8.3$; the impact of photoionization is likely to be maximal in this case. We will also discuss separately the case in which the radiation source turns on later.
In general, the enhanced UV radiation leads to a considerable delay in star formation in low-mass halos. Figure \[fig\_model\_tj\] illustrates the delay time of star formation as a function of radiation intensity for a given virial mass of a halo, $M_{\rm vir}$. The delay time is defined as the time between the epochs at which $1/10$ of the baryon mass in a halo has cooled with and without the external UV radiation. The gas is regarded as “cooled” when its temperature drops below $1000$K or its density exceeds $10^6$cm$^{-3}$. Figure \[fig\_model\_tj\] shows that star formation is completely suppressed in a halo with $M_{\rm vir}<10^{9} M_\odot$ under the background UV field with $J_{21} \sim 1$. In contrast, it remains almost unaffected in a massive halo with $M_{\rm vir}>10^{11} M_\odot$ even in the vicinity of radiation sources. This is mainly because photoionization can raise the gas temperature only to several times $10^4$ K. In an intermediate mass halo, the star formation is suppressed during the initial stage of the collapse and the delay time depends sensitively on the radiation strength. For example, the UV intensity corresponding to equation (\[eq-j21\]) will suppress the star formation for $\sim 10^8$yr in a halo with $M_{\rm vir}=10^{10} M_\odot$. Such a halo will be able to host bright galaxies $\sim 10^8$yr later than those placed in the weaker UV field.
The above results imply that if the observed LAE deficiency around the QSO is due to photoionization and the age of the LAE is $10^6 - 10^8$ yr [@gaw06; @ove06b; @pir06], the LAE mass is $M_{\rm vir} \sim 10^{10}M_\odot$. This is somewhat smaller than previous observational estimates [@ham04]. As mentioned already, the dynamical masses of the LAEs are still poorly determined from observations and we do not consider this difference to be critical. Our results further suggest that photoionization cannot delay the star formation in massive LBGs with $M_{\rm vir}\sim10^{11-12} M_\odot$.
Note that our model is restricted to the case in which stars form at the center of a nearly spherical halo. If star formation takes place after a disk-like collapse or in substructures (e.g., [@su00; @mu06]), the impact of photoionization will be greater and the inferred mass of the host halo can be larger.
Another important aspect of the photoionization efficiency by a local ionizing source is the time difference between the QSO activity phase and collapse time of galaxies. Even low-mass galaxies could collapse if they form before the QSO activity has grown. To examine this possibility, we also performed runs in which halos that have already collapsed at $z=4.87$ with a gas density profile of $\propto r^{-2}$ are exposed to an external UV field. In this case, star formation cannot be delayed even in a halo as small as $M_{\rm vir}=10^{9} M_\odot$. Hence, another possible interpretation of the observational result is that LAAs have already formed, or are in the process of formation, at the time of the active phase of the QSO host galaxy, whereas LAEs are young, having formed much later than the QSO/LAA formation. This scenario is supported by previous results that the stellar mass of LAEs is smaller than LBGs and is almost unchanged at all the epochs [@ove06b; @gaw06; @pir06], although more accurate estimates of the stellar mass of high-$z$ LAEs should be accumulated in the future. This picture is also consistent with the idea that a galaxy first appears as a LAE during its initial dust-free starburst phase [@sha01]. This interpretation also requires accurate estimates of the formation epochs of LAAs/LAEs relative to the evolutionary timescale of QSO activity, which have not yet been determined. @kaw03 proposed a QSO evolution model with a prediction of a timescale for a QSO active phase to be $\sim$ a few $10^8$ yr. @she07 estimated the timescale of the luminous accreation phase to be $\sim$ a few $10^7 - 10^8$yr for QSOs at $z>3.5$. These timescales are comparable to the LAE age $\sim10^8$ yr [@gaw06] and is shorter than the LBG age $\gtrsim10^9$ yr [@sha01] estimated by multicolor SED fitting. @ove06b concluded that LAEs prefer regions that are devoid of UV-bright LBGs in their protocluster region at $z=4.1$. Their result is inconsistent with ours in the cluster field, where we found that LBGs and LAEs coexist in an overdense region. The LAEs in their study might not have avoided LBGs but rather the much brighter central radio galaxy, which has a UV luminosity of $6L^*_{z=4}$, although the photoionization effect was less significant than for QSOs.
The importance of radiative feedback from a local strong UV ionizing source is suggested by several lines of observational evidence. The QSO proximity effect emerged in nearby QSOs, where the relative strength of the local QSO ionizing radiation is only $>0.1$ times the background (e.g., [@baj88; @lu91]), which is much more sensitive to the QSO intensity than we found in this study. @ade03 found that LBGs are associated with H [i]{} underdensities at a separation of $<0.5h^{-1}$Mpc, based on the statistics of several LBG samples and IGM absorption lines in the spectra of background QSOs. This result was interpreted as being caused by either photoionization from LBGs or by the LBG galactic wind. @fra04 detected no Ly$\alpha$ emission within $1$Mpc of a luminous QSO at $z=2.2$, and the neutral hydrogen clouds near the QSO may have been photoevaporated. Along with these examples, the observational result suggests that the photoionization effect from a local strong UV ionizing source could play an important role in making a possible \`\`habitat segregation" between LAAs and LAEs in the high-$z$ universe.
Conclusions
===========
In this paper, we have presented a possible clustering segregation between LAAs and LAEs around a high-$z$ QSO. We carried out a wide-field target survey for LAAs and LAEs simultaneously around a QSO SDSS J0211-0009 at $z=4.87$. Our main conclusions can be summarized as follows:
1\. LAAs form a filamentary structure including the QSO, while LAEs distribute around the QSO but avoid its vicinity.
2\. LAAs and LAEs are spatially cross-correlated in a protocluster field where no strongly UV ionizing source such as a QSO or radio galaxy is known to exist.
3\. We found a weak trend that number counts based on Ly$\alpha$ and UV continuum fluxes of LAEs in the QSO field are slightly lower than in the cluster field, whereas number counts of LAAs are almost consistent with each other.
4\. The LAEs avoid the region around the QSO, where the UV background radiation could be $100$ times greater than the average intensity at the epoch. The clustering segregation between LAAs and LAEs seen in the QSO field might be caused by either all the LAEs having already evolved into LAAs in the QSO field, or photoionization around the strong UV source effectively causing a dearth of low-mass galaxies such as LAEs.
These results can only be achieved by wide-field imaging targeting QSO/RG fields. For example, in the J0211-0009 field, small FOV imaging revealed only the filamentary structure of LAAs around the QSO, without showing the larger surrounding structure formed by LAEs (it \`\`fails to see the forest for the trees"). Recently, similar large FOV imaging revealed a cosmic web of LBGs around the RG TN J1338-1942 at $z=4.1$ [@int06]. Although our sample cannot, in isolation, allow the derivation of a general picture of the properties of high-$z$ galaxies in an overdense field around known high-$z$ objects, it provides a potentially important perspective for understanding the nature of high-$z$ galaxies in different environments. Future comparative studies of different fields and different UV background fluxes (different QSO/RG luminosity) would permit more quantitative discussions.
We acknowledge Hisanori Furusawa for his helpful support of the observations. We are very grateful to Kazuhiro Shimasaku and Masami Ouchi for kindly providing the data of their LAE sample and for their valuable comments on the manuscript. We thank the referee for helpful comments that improved the manuscript. The research was supported by the Japan Society for the Promotion of Science through Grant-in-Aid for Scientific Research 16740118.
Adelberger, K.L., Steidel, C.C., Shapley, A.E.,, & Pettini, M. 2003, , 584, 45 Babul, A., & Rees, M.J. 1992, , 255, 346 Bajtlik, S., Duncan, R.C., & Ostriker, J.P. 1988, , 327, 570 Barkana, R. & Loeb, A. 1999, , 523, 54 Benson, A.J., Lacey, C.G., Baugh, C.M., Cole, S., & Frenk, C.S. 2002, , 333, 156 Bertin, E. & Arnouts, S. 1996, , 117, 393 Bruzual, A.G. & Charlot, S. 2003, , 344, 1000 Cooke, A.J., Espey, B., Carswell, R.F. 1997, , 284, 552 Dow-Hygelund, C.C. et al. 2006, astro-ph/0612454 Efstathiou, G. 1992, , 256, 43 Eyles, L.P., Bunker, A.J., Ellis, R.S., Lacy, M., Stanway, E.R., Stark, D.P., & Chiu, K. 2006, astro-ph/0607306 Fan, X. et al. 1999, , 118, 1 Fan, X. et al. 2001, , 121, 31 Francis, P.J. & Bland-Hawthorn, J. 2004, , 301, 2004 Fukugita, M. et al. 2004, , 603, L65 Gao, L., Springel, V., & White, S.D.M. 2005, , 363, L66 Gawiser, E. et al., 2006, astro-ph/0603244 Gebhardt, K. et al. 2000, , 543, 5 Hall, P.B. & Green, R.F. 1998, , 507, 558 Hamana, T. 2004, , 347, 813 Hu, E.M. & McMahon, R.G. 1996, , 382, 231 Intema, H.T. et al. astro-ph/0606298 Kaiser, N. 1984, , 284, L9 Kashikawa, N. et al. 2004, , 56, 1011 Kashikawa, N. et al. 2006a, , 637, 631 Kashikawa, N. et al. 2006b, , 648, 7 Kauffmann, G. & Haehnelt, M.G. 2000, , 311, 576 Kawakatu, N., Umemura, M., & Mori, M. 2003, , 583, 85 Kitayama, T. et al. 2000, , 315, L1 Kitayama, T. et al. 2001, , 326, 1353 Kurk, J.D. et al. 2000, A&A, 358, L1 Landolt, A.U. 1992, , 104, 340 Landy, S.D., & Szalay, A.S. 1993, , 412, 64 Lu, L., Wolfe, A.M., & Turnshek, D.A. 1991, , 367, 19 Madau, P. 1995, , 441, 18 Magorrian, J. et al. 1998, , 115, 2285 Miyazaki, S. et al. 2002, , 54, 833 Mori, M. & Umemura, M. 2006, , 440, 644 Ouchi, M. et al. 2003, , 582, 60 Ouchi, M. et al. 2005, , 620, L1 Overzier et al. 2006a, , 637, 58 Overzier et al. 2006b, astro-ph/0601223 Palunas, P., Teplitz, H.I., Francis, P.J., Williger, G.M., & Woodgate, B.E. 2004, , 602, 545 Pascarelle et al. 1996, , 383, 45 Pascarelle et al. 2001, , 560, 101 Pentericci et al. 2000, , 361, L25 Pentericci et al. astro-ph/0703013 Pirzkal et al. 2006, astro-ph/0612513 Rees, M.J. 1986, , 218, 25 Shapley, A.E., Steidel, C.C., Adelberger, K.L., Dickinson, M., Giavalisco, M., & Pettini, M. 2001, , 562, 95 Shen, Y. et al. astro-ph/0702214 Shimasaku, K. et al. 2003, , 586, 111 Shimasaku, K. et al. 2004, , 605, L93 Shimasaku, K. et al. 2006, , 58, 313 Spergel, D.N. et al. 2006, astro-ph/0603449 Stark, D.P., Bunker, A.J., Ellis, R.S., Eyles, E.P., & Lacy, M. 2006, astro-ph/0604250 Steidel, C.C. et al. 2000, , 532, 170 Steidel, C.C. et al. 2005, , 626, 44 Susa, H. & Umemura, M. 2000, , 537, 578 Thoul, A.A. & Weinberg, D.H. 1996, , 465, 608 Ueda, Y. et al., 2003, , 598, 886 Umemura, M. & Ikeuchi, S., 1984, Prog. Theor. Phys., 72, 47 Venemans, B.P. et al. 2002, 569, 11 Venemans, B.P. et al. 2004, , 424, L17 Venemans, B.P. et al. 2005, , 793, 812 Williger, G.M. et al. 1994, , 428, 574 Yee, H.K.C. & Green, R.F. 1987, , 319, 28 Yagi, M. et al. 2002, SPIE, 4847, 322 Yan, H., Dickinson, M., Giavalisco, M., Stern, D., Eisenhardt, P.R.M., Ferguson, H.C. 2006, , 651, 24 Yoshida, M., et al. 2006, , 653, 988 Zheng, W. et al. 2006, , 640, 574
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abstract: 'Metal nanoparticles are attractive for plasmon-enhanced generation of hot carriers, which may be harnessed in photochemical reactions. In this work, we analyze the coherent femtosecond dynamics of photon absorption, plasmon formation, and subsequent hot-carrier generation via plasmon dephasing using first-principles simulations. We predict the energetic and spatial hot-carrier distributions in small metal nanoparticles and show that the distribution of hot electrons is very sensitive to the local structure. Our results show that surface sites exhibit enhanced hot-electron generation in comparison to the bulk of the nanoparticle. While the details of the distribution depend on particle size and shape, as a general trend lower-coordinated surface sites (e.g., corners, edges, {100} facets) exhibit a higher proportion of hot electrons than higher-coordinated surface sites (e.g., {111} facets). The present results thereby demonstrate how hot-carrier distributions can be tailored by careful design of particle size, structure, and composition.'
author:
- 'Tuomas P. Rossi'
- Paul Erhart
- Mikael Kuisma
bibliography:
- 'main.bib'
title: 'Hot-carrier generation in plasmonic nanoparticles: Atomic-scale structure matters'
---
=1
Plasmon-enhanced technologies enabled by metal provide promising avenues for, e.g., harvesting and converting sunlight to chemical energy [@AslRaoCha18] and driving photochemical reactions [@LinAslBoe15]. The underlying processes rely on the decay of plasmonic excitations and the subsequent generation of non-equilibrium carrier distributions [@BroHalNor15]. The latter are often collectively referred to as *hot carriers*, even though the actual distributions vary substantially with time after excitation [@SaaAseGar16; @LiuZhaLin18]. generated by plasmon decay can in principle be transferred to a chemically attached acceptor such as a semiconductor or a molecule, a process that is potentially useful for technologies such as photovoltaics [@AtwPol10], photo-detection [@KniSobNor11; @ChaSchBro14], photon up-conversion,[@NaiWelBri17] and photocatalysis [@LinAslBoe15; @MukLibLar13; @MubLeeSin13; @KalAvaChr14; @SweZhaZho16], and that is potentially relevant for growth processes [@ZhaDuCWan16].
There are, however, substantial gaps in our understanding that limit the exploitation of plasmon-generated hot carriers in applications. To resolve these questions, a purely experimental approach is limited both due to time constraints and the difficulty associated with disentangling different contributions [@ZhoSweZha18; @SivBarUn19; @DubSiv19]. In the context of photocatalysis, in particular, it is paramount to discern processes at the atomic scale. Here, theoretical and computational approaches can provide highly valuable insight as they enable us to scrutinize the relevant microscopic processes on the electronic and atomic scale.
Earlier theoretical studies on plasmonic generation have mostly focused on flat metal surfaces [@SunNarJer14; @BerMusNea15] or jellium neglecting the underlying atomic structure. [@ManLiuKul14; @SaaAseGar16; @BesGov16; @YanWanMen16; @LiuZhaLin18; @DalRanLis18; @RanDalLis18; @RomHesLis19] While atomic-scale effects in nanoplasmonics has been increasingly addressed in recent years, [@Zhang2014; @KuiSakRos15; @Rossi2015Quantized; @Marchesin2016; @DonLinAik17; @SenLinKud19] an atomic scale description of plasmonic generation is only emerging. [@MaWanWan15; @DouBerSan16; @DouBerFra19] In this work, we directly analyze the effect of local structure on generation and demonstrate that the distribution of after plasmon decay is in fact very sensitive to the atomic scale details. We quantitatively resolve the effect of surface orientation, step edges, corner sites, and ultimately shape and size on distribution. To this end, we develop a fully atomistic, parameter-free, and generally applicable description of plasmonic generation based on - [@Hohenberg1964; @Kohn1965] and . [@Runge1984]
Results {#results .unnumbered}
=======
**Real-time dynamics of localized surface plasmons.** We start with a comprehensive description of plasmon formation and subsequent dephasing. For illustration, we consider an icosahedral silver with a clear plasmon resonance (a) [@RosKuiPus17]. The ground-state electronic structure of the is with calculated using the GLLB-SC potential [@KuiOjaEnk10] for an improved d-band description [@Yan2011First; @KuiSakRos15] and the response is calculated with using the or the adiabatic GLLB-SC [@KuiSakRos15] (see Methods for details).
We consider the real-time dynamics of the electrons under a monochromatic ultrafast Gaussian light pulse $$\begin{aligned}
\mathcal{E}(t + t_0) = \mathcal{E}_{0} \cos(\omega_0 t) \exp(-t^2/\tau_0^2) ,
\label{eq:pulse}\end{aligned}$$ where the pulse frequency $\omega_0=\unit[3.6]{eV}$ is tuned to the plasmon resonance, the pulse duration is determined by $\tau_0 = \unit[3]{fs}$, and the pulse is centered at $t_0 = \unit[10]{fs}$ (b). The pulse strength is weak, $\mathcal{E}_{0}=\unit[51]{\text{\textmu{}V/\angstrom}}$, putting the response in linear response regime. In frequency space, the pulse is wide enough to cover the whole plasmon resonance (a).
We use the dipole approximation, where the light pulse creates a time-dependent external potential $v_{\mathrm{pulse}}(t) = z \mathcal{E}(t)$ that causes the time evolution of the states $\ket{\psi_n(t)}$ and excitation of the . The light pulse induces a strong dipole-moment response \[c(1–3)\]. The corresponding electron density oscillations \[d(1–3)\] are composed of a surface-to-surface component associated with delocalized valence electrons near the Fermi energy and atom-localized contributions that correspond to screening due to virtual excitations from the d-band. [@RosKuiPus17] As time proceeds to $t\approx \unit[17]{fs}$, the excited electrons start to lose their collective plasmonic motion via a dephasing process commonly referred to as Landau damping, which takes places due to the presence of multiple excitation eigenstates forming the broadened plasmon peak in the photoabsorption spectrum. [@Yannouleas1992] As the plasmon dephases, the dipole moment decays \[c(4–5)\] corresponding to vanishing surface-to-surface density oscillation \[d(4–5)\]. [@MaWanWan15]
**Time-dependent energy contributions.** Since the pulse is tuned to the , the electronic system absorbs energy from the incident light and remains in an excited state after the pulse has vanished. To analyze the distribution of this energy, consider the total time-dependent energy of the system given by $$\begin{aligned}
E_\mathrm{tot}(t) = E^{(0)}_\mathrm{tot} + \Delta E(t) + E_\mathrm{pulse}(t),\end{aligned}$$ where $E^{(0)}_\mathrm{tot}$ is the ground-state energy, $\Delta E(t)$ is the time-dependent energy stored in the excited state (e, black line), and $E_\mathrm{pulse}(t) = - \mu(t) \mathcal{E}(t)$ is the potential energy of the system under the external electric field.
The incident light pulse pumps energy into the system, i.e, it does work on the system as $\Delta \dot E(t) = \delta\dot\mu(t) \mathcal{E}(t)$, where dots indicate time derivatives and $\delta\mu(t) = \mu(t) - \mu^{(0)}$ is the induced dipole moment. Thus, the total accumulated electronic energy can be written as $$\begin{aligned}
\Delta E(t) = \int_{0}^{t} \delta\dot\mu(\tau) \mathcal{E}(\tau) \dee \tau.\end{aligned}$$ The electronic energy increases through absorption in a step-wise manner following the pulse intensity (e, $t\approx 5 \dots 15\unit{fs}$). After the pulse has ended ($t \gtrsim \unit[15]{fs}$), the absorbed energy remains in the system and the total energy has attained a new constant value given by the photoabsorption cross section \[a; Eq. in Methods\].
While the total energy remains constant, the electronic energy does not stay equally distributed among the electron-hole transitions $i \to a$ excited by the light pulse. To quantify this effect, we consider a decomposition in electron-hole transitions that is based on the linear response of the density matrix, $\delta\rho_{ia}(t)$, expressed in the basis of the eigenstates $(\epsilon_n, \psi^{(0)}_n)$ of the ground-state Hamiltonian. The electron-hole decomposition of energy is (see Supplementary Note 1 for derivation) $$\begin{aligned}
\Delta E(t) = \sum_{ia}^{f_i > f_a} \omega_{ia} P_{ia}(t) + E^{\mathrm{C}}_{ia}(t) ,
\label{eq:E}\end{aligned}$$ where the sum is restricted by ground-state occupation numbers $f_n$ so that the indices $i$ and $a$ correspond to the created hole and electron states, respectively. Here, $\omega_{ia} = \epsilon_{a} - \epsilon_{i}$ is the transition energy (the KS eigenvalue difference), $P_{ia}(t)$ is the transition probability defined as $$\begin{aligned}
P_{ia}(t) = \left|\frac{\delta\rho_{ia}(t)}{\sqrt{f_{i}-f_{a}}}\right|^2 ,
\label{eq:tprob}\end{aligned}$$ and $E^{\mathrm{C}}_{ia}(t)$ is the Coulomb energy given by the Hartree–exchange-correlation kernel (defined in Supplementary Note 1).
Plasmon formation and dephasing are scrutinized by considering the energy stored in the electronic system in terms of the electron-hole transition energy $\omega_{ia}$ with respect to the pulse energy (Supplementary Fig. 1). The plasmon is formed by constructive coupling of low-energy transitions \[$\omega_{ia} \lesssim\!\unit[2]{eV}$; see time instances (1–3) in b\]. [@Yannouleas1992; @Bernadotte2013; @RosKuiPus17] Simultaneously, high-energy virtual d-electron transitions ($\omega_{ia} \gtrsim\!\unit[4]{eV}$) screen the plasmonic density oscillation. These non-resonant transitions carry most of the energy during plasmon excitation (e, purple line). As the plasmon dephases, the absorbed energy is redistributed to electron-hole transitions that are resonant with the pulse (e, orange line; corresponding to the diagonal in the transition contribution maps in b; see time instances (4–5)\]. Thus the transitions comprising the plasmon are active during absorption (a), and subsequently, the plasmon decays into hot holes and hot electrons, instead of the transitions absorbing the light directly (shown in detail in Supplementary Fig. 2).
After plasmon dephasing, the energy remains almost exclusively stored in the resonant transitions \[b(5)\], which constitute the plasmon-generated hot carriers. At larger time scales, the electronic system would dissipate the absorbed energy to the environment via radiation, atomic motion, or other processes, but such decay pathways are not included within the description used here, and there is no significant dynamics at time scales beyond $t \gtrsim \unit[30]{fs}$ due to the strong dephasing in (Supplementary Fig. 3). However, the dynamics can be very different in small clusters with discrete excitation spectrum. For example, in cluster individual electron-hole transitions couple strongly to plasmon, [@RosKuiPus17] which is often referred to as plasmon fragmentation. [@Yannouleas1989; @Yannouleas1991] Correspondingly, the time-domain response can exhibits occasional energy transfer back from the resonant transitions to the plasmon [@MaWanWan15; @YouRamSei18] (Supplementary Fig. 4) due to the incomplete Landau damping enabling the re-emergence of coherence between plasmonic transitions.
Since the coupling of transitions via Coulomb interaction is recognized as an essential characteristic of plasmonic excitations, [@Yannouleas1992; @Bernadotte2013; @Zhang2017; @RosKuiPus17] it is instructive to consider the Coulomb energy $E_\mathrm{C}(t) = \sum_{ia} E^\mathrm{C}_{ia}(t)$. This energy exhibits strong oscillations (e, grey line) analogous to the dipole moment (c) as only the electron density oscillation contributes to the Coulomb energy. At the maxima of the surface-to-surface density oscillation \[time instances (1) and (3) in Figs. \[fig:energy\]d–e\], the Coulomb contribution is a significant part of the plasmon energy, but at the minima in between \[e.g., time instance (2)\] the Coulomb energy is vanishing as the electronic energy is stored in the electron current flowing through the particle.
**Temporal evolution of distributions.** Now that we have established the real-time picture of plasmon formation and decay, we are in the position to analyze the distributions of electrons and holes during the process. The probabilities for creating a hole in the initially occupied state $i$ or an electron in the initially unoccupied state $a$ are given directly by the transition probability of Eq. as $$\begin{aligned}
P^\mathrm{h}_{i}(t) = \sum_{a}^{f_i > f_a} P_{ia}(t) \quad\text{ and }\quad
P^\mathrm{e}_{a}(t) = \sum_{i}^{f_i > f_a} P_{ia}(t),
\label{eq:occ}\end{aligned}$$ respectively. $P^\mathrm{h}_{i}$ and $P^\mathrm{e}_{a}$ determine exactly the diagonal elements of the second order response of the density matrix (Supplementary Note 1); in other words, they correspond to the increase of the occupation of the initially unoccupied state $a$ and the decrease of the occupation of the initially occupied state $i$, respectively.
The occupation probabilities given by Eq. show strong oscillations during the time evolution \[c(1–3); dashed lines\]. These oscillations are explained by the oscillation of Coulomb energy. As the Coulomb energy contribution is carried only by non-resonant transitions, the occupation probabilities of the electron and hole states contributing to these non-resonant transitions oscillate analogously to the Coulomb energy. The oscillations are especially visible in the occupations of electron and hole states that form the plasmon, i.e., the states near the Fermi energy, often referred to as Drude carriers [@HarBesJoh17]. The oscillatory population and depopulation of these states indicate that they would not likely be individually separable while they are a part of the plasmon as Coulomb interaction is an essential part of the excitation [@AizBalBau19].
The resonant transitions have zero Coulomb energy contribution and the occupations of the corresponding electron and hole states grow steadily as the plasmon decays (c; solid lines). At the end of the dynamic evolution considered here \[c(5)\], electrons and holes are still coupled in the form of electron-hole transitions, and the distribution at $t=\unit[30]{fs}$ corresponds to the initial non-thermal distributions. At longer time scales (not described here) these carriers would separate and thermalize via electron–electron and electron–phonon scattering processes. [@LiuZhaLin18] The slight asymmetry between the hole and electron distributions is caused by a non-zero width of the pulse in frequency space (a).
**Energetic and spatial distributions of hot carriers: Role of structure.** We can now analyze the distribution of plasmon-generated hot carriers in space and the impact of local structure. We start by considering the series of icosahedral silver , , and , the photoabsorption properties of which we have described in detail in earlier work [@KuiSakRos15; @RosKuiPus17] (see Supplementary Fig. 5 for photoabsorption spectra and densities of states). The light pulse is tuned to the plasmon resonance of the and the initial distributions are analyzed after the plasmon has dephased, i.e. at time $t=\unit[30]{fs}$. The distributions show a very pronounced dependence on size (a) and local structure (b–c) as discussed in the following.
As particle size increases, the distributions are increasingly dominated by interband d-electron transitions (hole $\sim\!\!\unit[-4]{eV}$ $\to$ electron $\sim\!\!\unit[0]{eV}$) converging toward the distributions obtained for flat surfaces. [@SunNarJer14; @BerMusNea15] In contrast to extended systems, geometry confinement effects are significant for plasmonic generation in nanoscale systems. [@BroSunNar16] Due to the broken crystal symmetry in , additional “intraband” transitions are available for generation in comparison to extended systems. This allows the population of higher-energy electron and hole states (a; electrons $> \unit[0.5]{eV}$, holes $>\unit[-3.5]{eV}$). The relative contribution of these states is most pronounced in smaller (, ) but they are even non-negligible in . Similar size-dependent trends are also present in silver of other shapes, while the detailed relative contributions of different transitions vary (Supplementary Fig. 6).
Considering the icosahedral , the calculated spatial probability distributions (see Methods) reveal that plasmon-generated holes and electrons are distributed spatially differently (b–c): Holes are localized at atomic sites throughout the particle, which is expected as the majority of holes originates from the atom-localized d-states. As a result, their energy distribution is very similar for core and surface sites. Hot electrons, on the other hand, are more delocalized and primarily reside in the surface region. The surface contribution is even more pronounced for higher-energy hot electrons ($>\unit[1]{eV}$ electrons in c). Especially at low-coordinated edge and corner sites the probability density for hot electrons with more than is strongly enhanced compared to sites in the core and on flat surfaces (b).
A more quantitative view is obtained by considering the total per-atom occupation probability of hot electrons at a particular atomic site in comparison to the total per-atom occupation probability throughout the system (). Hot electrons with more than are almost three times more likely to be found in the vicinity of a corner site than on any other site in a icosahedral on average. We emphasize that these are per-atom considerations, that is, as the 12 corner atoms constitute only around 2% of the atoms in the particle, it is expected that 6% of the electrons with more than would be generated in the vicinity of the corner atoms. Hot electrons with more than constitute 30 to 60% of all hot electrons depending on system (Supplementary Fig. 6). The absolute total number of generated depends on the light energy that is absorbed, which is in turn determined by photoabsorption cross-section and light intensity.
Similarly to icosahedral shapes, hot electrons in the cuboctahedral and regularly truncated octahedral are more likely to be generated at surface sites than in the core (). The preference for surface sites is even more pronounced for hot electrons with more than . For the cuboctahedral and regularly truncated octahedral the corner sites do, however, not show an as extreme proportion of hot electrons as in the icosahedral particle. This further underlines the sensitivity of generation to atomic scale details and the exact electronic structure of the and site(s) in question. As a general trend, lower-coordinated sites exhibit an enhancement of hot electrons compared to higher-coordinated sites. This is observed for corner and edge sites, but one also finds that more hot electrons are generated on the lower-coordinated {100} surface than on the {111} surface. In contrast to strong spatial variation of hot electrons, plasmon-generated holes do not show strong spatial dependence (Supplementary Fig. 7).
Discussion {#discussion .unnumbered}
==========
For practical utilization, need to be transferred to the environment. In general, transfer can occur *indirectly*, i.e. carriers are first produced in the metal and subsequently transferred to the acceptor [@BroHalNor15; @LinAslBoe15; @NarSunAtw16; @ChrMos17], or *directly*, i.e., plasmon dephasing leads directly to the injection of in empty acceptor (and occupied donor) states [@WuCheMcB15; @TanArgRen17; @TanDaiZha18; @LiDiSMur17; @BoeCamMor16; @BoeAslLin16]. (The direct transfer process has also been referred to as coherent charge transfer [@TanArgRen17; @TanDaiZha18], plasmon-induced interfacial charge-transfer transition when the acceptor is a semiconductor [@WuCheMcB15], and chemical interface damping in the case of adsorbed molecules [@LinAslBoe15; @FoeJopKae17; @KalChr15; @DouBerSan16].) Both experiments [@RatDunVur17; @TanLiuDai17; @BroSunNar17] and calculations [@ManLiuKul14; @BroSunNar16; @BroSunNar17; @BesKonWan17] indicate that generated in the metal can quickly relax back to the Fermi level via electron-electron scattering, which renders the indirect pathway inefficient. The direct-transfer process, on the other hand, can overcome the problem of fast relaxation and presents an opportunity to obtain more efficient plasmonic devices [@ChrMos17; @FoeJopKae17; @KalChr15]. Moreover, in principle it enables the injection of in particular acceptor states, which is of special interest for applications in catalysis.
In this work, we have quantitatively analyzed the impact of local atomic structure on plasmon decay and generation using a predictive and material specific first-principles framework without resorting to empirical parameters. The simulations allow us to follow the real-time dynamics of localized surface plasmons during the first or so after excitation, which includes plasmon formation and decay into the initial non-thermal distribution. The latter is immediately relevant for direct transfer processes either across interfaces [@KumRosMar19; @MaGao19] and onto molecules [@KumRosKui19]. This distribution also represents the initial condition for longer-time scale dynamics, which are governed by electron-electron ($\unit[100]{fs} \lesssim t \lesssim \unit[1]{ps}$) as well as electron-phonon ($\unit[100]{ps} \lesssim t \lesssim \unit[10]{ns}$) scattering processes [@BroHalNor15; @LiuZhaLin18] and are relevant for indirect excitation transfer.
We find that in the silver considered here the distribution of hot *electrons* is very sensitive to local structure whereas the distribution of hot *holes* is relatively homogeneous. These features can be traced to the electronic structure as hole and electron states exhibit localized d and delocalized s-type character, respectively. We therefore expect the present insight to be qualitatively transferable to other late transition metals, which exhibit similar electronic structure. By studying a series of of increasing size and considering different particle shapes, we were able to correlate the local structure with the distribution. Specifically, we observe that hot *electrons* are more likely to be present on lower coordinated sites. The details are shape dependent. For example on icosahedral , the proportion of hot electrons at corner and edge sites can be strongly enhanced compared to flat surface and bulk sites, whereas in cuboctahedral and regularly truncated octahedral particles, more hot electrons are generated per site on lower-coordinated {100} than on denser {111} facets.
The prevalence of hot electrons on lower-coordinated surface sites described here facilitates their utility in direct transfer processes. It is, however, crucial to also consider the hybridization of the surface electronic states with acceptor states, where the latter can originate e.g., from an adsorbed molecule [@KumRosKui19] or a semiconductor [@KumRosMar19; @MaGao19]. To maximize the efficiency for direct excitation transfer the emitting (surface) receiving (acceptor) states should be energetically aligned and spatially overlap. Going forward, the framework introduced here enables one to analyze and quantify these transfer processes at the atomic scale with material specificity without resorting to empirical parameters. This provides the opportunity to identify promising surface-acceptor combinations and design future experiments.
In conclusion, we have presented a comprehensive first-principles account of the real-time dynamics of plasmon formation and its dephasing into incoherent . In larger particles, the density usually decays away from the surface due to propagation of electronic modes. In smaller , as demonstrated in this work, are, however, well defined electronic resonances within the particle apt to assist for example in heterogeneous photocatalytic redox reactions. The present results further show that the energetic distribution of hot electrons is very sensitive to the local structure and higher energy electrons are preferentially generated at sites with low coordination. First-principles predictions of plasmonic generation, as presented here, can thereby open avenues for tuning and optimizing photocatalytic systems down to the atomic scale.
Methods {#methods .unnumbered}
=======
**Computational details.** The ground-state electronic structures were calculated with - [@Hohenberg1964; @Kohn1965] using the GLLB-SC exchange-correlation potential. [@KuiOjaEnk10] The time-domain responses were calculated with [@Runge1984] starting from the ground state. The dynamical response was described with for the data in Figs. \[fig:energy\]–\[fig:snapshots\] and in Supplementary Figs. 1–4, while all the other data was calculated with the adiabatic GLLB-SC. [@KuiSakRos15] The two response kernels yield very similar results (Supplementary Fig. 8), but the GLLB-SC potential is not suitable for obtaining the total energies.
All the calculations are carried out with the open-source GPAW code package [@Enkovaara2010] using localized basis sets [@Larsen2009] and the real-time propagation implementation. [@KuiSakRos15] We used 11-electron projector augmented-wave [@Blochl1994] setups for Ag, treating the remaining electrons as frozen core. We used an extended “p-valence” basis set that includes diffuse 5p functions, which are important for describing plasmon resonances [@Rossi2015Nanoplasmonics]. The basis set is similar to the ones used in Refs. and .
The photoabsorption spectra were calculated using the $\delta$-kick technique [@Yabana1996] yielding linear impulse response. The photoabsorption of icosahedral particles is isotropic and the electric field was aligned along the $x$ direction. The resulting do not exhibit a strong variation between different sites (c). For the time propagation, we used a time step of and total propagation time of at least . The spectra were broadened using Gaussian damping with $\sigma = \unit[0.07]{eV}$ corresponding to a full width at half-maximum of . The real-time response to a pulse was calculated as a post-processing step via convolution as described below in detail. In the convolution Fourier transforms or time-domain response there is no artificial damping.
A grid spacing parameter of $h=\unit[0.3]{\angstrom}$ was chosen to represent densities and potentials and the nanoparticles were surrounded by a vacuum region of at least . The Hartree potential was evaluated with a Poisson solver using the monopole and dipole corrections for the potential. Fermi-Dirac smearing was applied to the occupation numbers to facilitate convergence. The electron-hole basis included electron-hole pairs with occupation number difference $f_i - f_a \geq 10^{-3}$.
Before the response calculations, all geometries were relaxed using the BFGS optimizer in the open-source ASE package. [@Larsen2017] The relaxation calculations used the Perdew-Burke-Ernzerhof (PBE) [@Perdew1996] functional, double-$\zeta$ polarized (dzp) basis sets, and $h=\unit[0.2]{\angstrom}$.
**Pulse response from convolution.** The photoabsorption spectrum can be efficiently calculated from real-time propagation using the $\delta$-kick perturbation [@Yabana1996] as in the linear-response regime all the frequencies are independent of each other. We utilize this property in the present work for calculating the linear response of the density matrix to the Gaussian pulse of Eq. as a post-processing step. First, the time-propagation is carried out for perturbation $v_\mathrm{kick}(t) = z K_0 \delta(t)$ yielding the impulse response of the system and the corresponding time-dependent density matrix $\delta\rho^\mathrm{kick}_{ia}(t)$. Then, in the linear-response regime, the response to the pulse of Eq. is obtained via convolution $$\begin{aligned}
\delta\rho_{ia}(t) = \frac{1}{K_0} \int_{0}^{\infty} \delta\rho^\mathrm{kick}_{ia}(\tau) \mathcal{E}(t - \tau) \dee \tau,\end{aligned}$$ which can be very efficiently calculated in frequency space via the convolution theorem and inverse Fourier transformation $$\begin{aligned}
\delta\rho_{ia}(t) = \frac{1}{2 \pi K_0} \int_{-\infty}^{\infty} \delta\rho^\mathrm{kick}_{ia}(\omega) \mathcal{E}(\omega) e^{-i \omega t} \dee \omega ,
\label{eq:conv:f}\end{aligned}$$ where $\delta\rho^\mathrm{kick}_{ia}(\omega)$ and $\mathcal{E}(\omega)$ are Fourier transforms of the respective time-domain quantities. Here, $\delta\rho^\mathrm{kick}_{ia}(\omega)$ can be efficiently calculated from the impulse response by using the computational framework developed in Ref. . The time derivatives required for calculating the energy (Supplementary Note 4) are obtained similarly as $$\begin{aligned}
\delta\dot\rho_{ia}(t) &= - \frac{i}{2 \pi K_0} \int_{-\infty}^{\infty} \omega \delta\rho^\mathrm{kick}_{ia}(\omega) \mathcal{E}(\omega) e^{-i \omega t} \dee \omega, \\
\delta\ddot\rho_{ia}(t) &= - \frac{1}{2 \pi K_0} \int_{-\infty}^{\infty} \omega^2 \delta\rho^\mathrm{kick}_{ia}(\omega) \mathcal{E}(\omega) e^{-i \omega t} \dee \omega .\end{aligned}$$ In practice $\mathcal{E}(\omega)$ is non-vanishing only on a finite frequency interval (see, e.g., a), which narrows the integration limits.
It should be emphasized here that the time-dependent density matrix $\delta\rho_{ia}(t)$ is a complex quantity in time domain, so in practical calculations it is convenient to carry out Fourier transformations for the real $\Re\delta\rho_{ia}(t)$ and imaginary $\Im\delta\rho_{ia}(t)$ parts separately to utilize the properties of Fourier transformations of real quantities.
We also note in passing that the impulse response $\delta\rho^\mathrm{kick}_{ia}(\omega)$ can be equivalently calculated from the Casida linear-response frequency-space formalism [@Casida1995; @RosKuiPus17]. Hence, the linear real-time response to any pulse can also be calculated from the Casida solutions via convolution of Eq. .
**Total absorbed energy.** By invoking Fourier transformation, the total absorbed energy after the pulse has vanished is obtained as $$\begin{aligned}
\int_{0}^{\infty} \delta\dot\mu(t) \mathcal{E}(t) \dee t
= \frac{1}{2} \int_{0}^{\infty} S(\omega) \left| \mathcal{E}(\omega) \right|^2 \dee \omega ,
\label{eq:totabs}\end{aligned}$$ where $S(\omega)=\frac{2\omega}{\pi} \Im[\alpha(\omega)]$ is the dipole strength function, which equals the photoabsorption cross section safe for a constant multiplier.
**distributions.** The hot-electron energy distributions corresponding to the state occupation probabilities $P^\mathrm{e}_{a}$ of Eq. are obtained as (time-dependence is not explicitly marked) $$\begin{aligned}
P_\mathrm{e}(\epsilon) &= \sum_{a} P^\mathrm{e}_{a} \delta(\epsilon - \epsilon_{a})
= \frac{1}{2} \sum_{ia}^{f_i > f_a} (q^2_{ia} + p^2_{ia}) \delta(\epsilon - \epsilon_{a}) ,
\label{eq:hcdist}\end{aligned}$$ where $q_{ia}$ and $p_{ia}$ correspond to scaled real and imaginary parts of $\delta\rho_{ia}$ (see Supplementary Note 2 for definitions). For visualization purposes, Gaussian smoothing (convolution) is applied with respect to the $\epsilon$ axis.
The spatial probability density of hot electrons is obtained by using the full electron-electron part of the second-order density matrix as (see Supplementary Note 1 and note that only the real part contributes due to the hermiticity of the density matrix) $$\begin{aligned}
P_\mathrm{e}(\V r) &= \frac{1}{2} \sum_{iaa'}^{\substack{f_i > f_a\\f_a = f_{a'}}} (q_{ia} q_{ia'} + p_{ia} p_{ia'}) \psi^{(0)}_{a}(\V r) \psi^{(0)}_{a'}(\V r) .
\label{eq:hcdns}\end{aligned}$$ The diagonal and degenerate states dominate the spatial density contributions, which allows us to define a spatio-energetic distribution $$\begin{aligned}
P_\mathrm{e}(\epsilon, \V r) &= \frac{1}{2} \sum_{iaa'}^{\substack{f_i > f_a\\\epsilon_a = \epsilon_{a'}}} (q_{ia} q_{ia'} + p_{ia} p_{ia'}) \psi^{(0)}_{a}(\V r) \psi^{(0)}_{a'}(\V r) \delta(\epsilon - \epsilon_a),
\label{eq:hcdistdns}\end{aligned}$$ which is used to calculate the spatial density of hot electrons with e.g., more than as $P_\mathrm{e}^{>\unit[1]{eV}}(\V r) = \int_{\unit[1]{eV}}^{\infty} P_\mathrm{e}(\epsilon, \V r) \dee \epsilon$, and the energy distribution of hot electrons in a spatial volume $V$ as $P_\mathrm{e}^{V}(\epsilon) = \int_{V} P_\mathrm{e}(\epsilon, \V r) \dee \V r$. The distribution at a specific atomic site (e.g., corner atoms) is obtained by integration over the Voronoi cell associated with the site.
The spatial and energetic distributions of hot holes are calculated analogously to the electrons.
**Software used.** The GPAW package [@Mortensen2005; @Enkovaara2010] with mode [@Larsen2009] was used for calculations. The real-time propagation - implementation in GPAW [@KuiSakRos15] was used for the calculations. Density-matrix-based analysis tools in frequency space [@RosKuiPus17] and in real time (present work) were used for analysis. The ASE library [@Larsen2017] was used for constructing atomic structures and geometry relaxation. The NumPy [@Walt2011] and Matplotlib [@Hunter2007] Python packages and the VMD software [@Humphrey1996; @Stone1998] were used for processing and plotting data.
Acknowledgements {#acknowledgements .unnumbered}
================
We acknowledge financial support from the Knut and Alice Wallenberg Foundation and the Swedish Research Council. T.P.R. acknowledges support from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk[ł]{}odowska-Curie grant agreement No 838996 and also thanks the Adlerbert Research Foundation and the Wilhelm and Martina Lundgren Foundation for support. M.K. acknowledges funding from Academy of Finland under grant Nr. 295602. We acknowledge generous computational resources provided by the Swedish National Infrastructure for Computing (SNIC) at PDC (Stockholm), NSC (Linköping), and C3SE (Gothenburg) as well as by the CSC – IT Center for Science (Finland).
|
---
abstract: 'The dynamics of the nuclear-spin quantum computer with large number ($L=1000$) of qubits is considered using a perturbation approach, based on approximate diagonalization of exponentially large sparse matrices. Small parameters are introduced and used to compute the error in implementation of entanglement between remote qubits, by applying a sequence of resonant radio-frequency pulses. The results of the perturbation theory are tested using exact numerical solutions for small number of qubits.'
address:
- '$^\dag$ Theoretical Division and CNLS, Los Alamos National Laboratory, Los Alamos, NM 87545'
- '$^\ddag$ IDS Department, Polytechnic University, Six Metrotech Center, Brooklyn, New York 11201'
author:
- 'G.P. Berman$^\dag$, D.I. Kamenev$^{\dag}$, and V.I. Tsifrinovich$^\ddag$'
title: |
Perturbation Approach\
for a Solid-State Quantum Computation
---
Introduction
============
Different solid-state quantum systems are considered now as candidates for quantum bits (qubits). They include: nuclear spins [@10; @11; @12], electron spins [@13; @14; @15], electron in a quantum dot [@16], and Josephson junctions [@17; @18]. For the most effective quantum information processing the quantum computer should operate with large number of qubits, $L$. For understanding how quantum computer works and for optimizing parameters of quantum computation it is necessary to model and simulate quantum logic operations and fragments of quantum algorizms. For numerical simulation of quantum computer dynamics one must solve a large set of $2^L$ linear differential equations on a long enough time-interval, or to diagonalize many large matrices of the size $2^L\times 2^L$. Hence, it is important to develop a consistent perturbation theory for quantum computation which allows one to predict the probability of correct implementation of quantum logic operations involving large number of qubits, in the real physical systems.
In this paper we describe the procedure which allows one to estimate the errors in implementation of quantum logic operations in the one-dimensional solid-state system of nuclear spins. Our approach does not require exact solution of quantum dynamical equations and direct diagonalization of large matrices. We assume that our quantum computer operates at zero temperature, and the error is generated only as a result of internal decoherence (non-resonant processes). Our approach provides the tool for choosing optimal parameters for operation of the scalable quantum computer with large number of qubits.
Dynamics of the spin chain quantum computer
===========================================
Application of Ising spin systems for quantum computations was first suggested in Ref. [@ber3]. Today, similar systems are used in liquid nuclear magnetic resonance (NMR) quantum computation with small number of qubits [@chuang]. The register (a 1D chain of $N$ identical nuclear spins) is placed in a magnetic field, $$\label{1}
{\rm\bf B}^{(n)}(z,\,t)=
\left(b^{(n)}_\perp\cos\left[\nu^{(n)} t+\varphi^{(n)}\right],
-b^{(n)}_\perp\sin\left[\nu^{(n)} t+\varphi^{(n)}\right], B^z(z)\right),$$ where $t^{(n)}\le t\le t^{(n+1)}$, $n=1,...,M$, $B^z(z)$ is a slightly non-uniform magnetic field oriented in the positive $z$-direction, $b^{(n)}_\perp$, $\nu^{(n)}$ and $\varphi^{(n)}$ are, respectively, the amplitude, the frequency, and the initial phase of the circular polarized (in the $x-y$ plane) magnetic field. This magnetic field has the form of rectangular pulses of the length (time duration) $\tau^{(n)}=t^{(n+1)}-t^{(n)}$. The total number of pulses which is required to perform a given quantum computation (protocol) is $M$. The chain of spins makes an angle $\theta$, where $\cos(\theta)=1/\sqrt 3$, with the direction of the magnetic field $B^z(z)$ to suppress the dipole interaction between the spins [@1].
The Hamiltonian of the spin chain in the magnetic field is, $$H^{(n)}=-\sum_{k=0}^{L-1}\omega_kI_k^z-2J\sum_{k=0}^{L-1}I_k^zI_{k+1}^z-$$ $$\label{H}
\Theta^{(n)}(t)(\Omega_n/2)\sum_{k=0}^{L-1}\left\{
I_k^-\exp\left[-i\left(\nu^{(n)} t+\varphi^{(n)}\right)\right]+
I_k^+\exp\left[i\left(\nu^{(n)} t+\varphi^{(n)}\right)\right]\right\}=
H_0+V^{(n)}(t),$$ where the index $k$ labels the spins in the chain, $J$ is the Ising interaction constant, $\Omega_n=\gamma b^{(n)}_\perp$ is the precession (Rabi) frequency, $\gamma$ is the gyromagnetic ratio. The function $\Theta^{(n)}(t)$ equals $1$ only during the $n$th pulse and equals to zero otherwise.
In order to remove the time-dependence from the Hamiltonian (\[H\]) we present the wave function, $\Psi(t)$, in the time-interval of the $n$th pulse, in the laboratory system of coordinates in the form, $$\label{Psi1}
\Psi(t)=
\exp\left[i(\nu^{(n)}t+\varphi^{(n)})\sum_{k=0}^{L-1}
I_k^z\right]\Psi_{rot}^{(n)}(t)=
\sum_pA_p(t)|p\rangle\exp(-i\chi_p^{(n)}t+\xi_p^{(n)}),$$ where $\Psi_{rot}^{(n)}(t)$ is the wave function in a reference frame rotating with the frequency, $\nu^{(n)}$, $\chi_p^{(n)}=-(\nu^{(n)}/2)\sum_{k=0}^{L-1}\sigma_k^p$, $\xi_p^{(n)}=\varphi^{(n)}\sum_{k=0}^{L-1}\sigma_k^p$, $\sigma_k^p=-1$ if the $k$th spin of the state $|p\rangle$ is in the position $|1\rangle$ and $\sigma_k^p=1$ if the $k$th spin is in the position $|0\rangle$, $|p\rangle$ is the eigenfunction of the Hamiltonian $H_0$. Below we take $\varphi^{(n)}=\xi_p^{(n)}=0$ for all $n$.
The dynamics during the $n$th pulse is described by the following Schrödinger equation for the coefficients $A_p(t)$, $$\label{Sch2}
i\dot A_p(t)=(E_p-\chi_p^{(n)})A_p(t)-\frac\Omega 2\sum_{p'}A_{p'}(t),$$ where we put, $\hbar=1$, for the Planck constant, and the sum is taken over the states $|p'\rangle$ connected by a single-spin transition with the state $|p\rangle$, $E_p$ is the eigenvalue of the Hamiltonian $H_0$. In the representation (\[Psi1\]) each spin flip is accompanied by a change of the phase of the wave function by the value $\pm\nu^{(n)} t$, which compensates the time-dependent phase in the perturbation, $V^{(n)}(t)$, in the Hamiltonian (\[H\]). In this case, the dynamics of the coefficients, $A_p(t)$, is governed by the effective time-independent Hamiltonian, ${\cal H}^{(n)}$, and Eq. (\[Sch2\]) can be written in the form, $i\dot A_p(t)={\cal H}^{(n)}_{pp'}A_{p'}(t)$.
The dynamics of the coefficients, $A_{p}(t)$, generated by the Hamiltonian, ${\cal H}^{(n)}$, can be computed using the eigenfunctions, $A_{m}^{q\,(n)}$, and the eigenvalues, $e_q^{n}$, of this Hamiltonian by, $$\label{dynamics}
A_m(t_n)=\sum_{m_0}A_{m_0}(t_{n-1})
\sum_{q}A_{m_0}^{q\,(n)}A_m^{q\,(n)}\exp(-ie_q^{n}\tau_n).$$ Since the pulses of the protocol are different, we should operate in the laboratory frame and make the transformation of the wave function to the rotating frame before each pulse, and the transformation to the laboratory frame after each pulse. If we write the wave function in the laboratory frame in the form, $$\label{Psi}
\Psi(t)=\sum_pC_p(t)|p\rangle\exp(-iE_pt),$$ then the coefficients, $C_p(t)$, in the laboratory frame are expressed through the coefficients, $A_p(t)$, in the rotating frame as, $$\label{AC}
C_p(t)=\exp\left(i{\cal E}_p^{(n)}t\right)A_p(t).$$ Here ${\cal E}_p^{(n)}=E_p-\chi_p^{(n)}$ are the diagonal elements of the Hamiltonian matrix ${\cal H}_{pp'}^{(n)}$, $t=t_{n-1}$ before the $n$th pulse and $t=t_n=t_{n-1}+\tau_n$ after the $n$th pulse.
The two-level approximation
===========================
We explain below how selective (resonant) transitions, which realize a quantum logic gate, can be implemented in the system described by the Hamiltonian (\[H\]). For this, we consider the structure of the effective time-independent Hamiltonian matrix, ${\cal H}_{pp'}$, (here and below we omit the upper index, $n$.) All non-zero non-diagonal matrix elements are the same and equal to $-\Omega/2$. At $\Omega\ll\delta\omega$ the absolute values of the diagonal elements in general case are much larger than the absolute values of the off-diagonal elements, and the resonance is coded in the structure of the diagonal elements of the Hamiltonian matrix, ${\cal H}_{pp'}$.
Suppose that we want to flip the $k$th spin in the chain. To do this, we choose the frequency of the pulse to be equal to the difference, $\nu=E_p-E_m$, between the energies of the states which are related to each other by flip of the $k$th spin. In this case the energy separation between the $p$th and the $m$th diagonal elements of the matrix, ${\cal H}_{pp'}$, related by the flip of the resonant $k$th spin is much less than the energy separation between the $p$th diagonal elements and diagonal elements related to other states, which differ from the state $|p\rangle$ by a flip of a non-resonant $k'$th ($k'\ne k$) spin. In this situation, in some approximation, one can neglect the interaction of the $p$th state with all states except for the state, $|m\rangle$, and the Hamiltonian matrix, ${\cal H}_{pp'}$, breaks up into $2^L/2$, approximately independent $2\times 2$ matrices, $$\label{2x2}
\pmatrix{{\cal E}_{l} & V \cr
V & {\cal E}_{m}},$$ where ${\cal E}_{p}={\cal E}_{m}+\Delta_{pm}$, $|\Delta_{pm}|\sim J$ or $|\Delta_{pm}|=0$ is the detuning from the resonance, which depends on the positions of $(k-1)$th and $(k+1)$ spins, $V=-\Omega/2$.
Suppose, for example, that $L=5$ and the third spin ($k=3$) has resonant ($|\Delta_{pm}|=0$) or near-resonant ($|\Delta_{pm}|\sim J$) frequency (we start enumeration from the right spin with $k=0$). Then, the block $2\times 2$ will be organized, for example, by the following states: $|01010\rangle$ and $|00010\rangle$; $|01111\rangle$ and $|00111\rangle$; $|00001\rangle$ and $|01001\rangle$, and so on. In order to find the state, $|m\rangle$, which forms a $2\times 2$ block with a definite state $|p\rangle$, one should flip the resonant spin of the state $|p\rangle$. In other words, positions of $N-1$ (non-resonant) spins of these states are equivalent, while position of the resonant spin is different. This approximation can be called the two-level approximation, since in this case we have $2^{L}/2$ independent two-level systems.
We now obtain the solution in the two-level approximation. The dynamics is given by Eq. (\[dynamics\]). Since we deal only with a single $2\times 2$ block of the matrix ${\cal H}_{pp'}$, (but not with the whole matrix), the dynamics in this approximation is generated only by the eigenstates of one block.
The eigenvalues $e^{(0)}_q$, $e^{(0)}_Q$, and the eigenfunctions of the $2\times 2$ matrix (\[2x2\]) are (we put $\Delta_{pm}=\Delta$), $$\label{q01}
\hspace{-9mm}e^{(0)}_q={\cal E}_m+\frac\Delta 2-\frac\lambda 2, \qquad
\pmatrix{
A^{q\,(0)}_m \cr A^{q\,(0)}_p}=
{1\over\sqrt{(\lambda-\Delta)^2+\Omega^2}}
\pmatrix{\Omega\cr \lambda-\Delta},$$ $$\label{q02}
e^{(0)}_Q={\cal E}_m+\frac\Delta 2+\frac\lambda 2, \qquad
\pmatrix{
A^{Q\,(0)}_m\cr A^{Q\,(0)}_p}=
{1\over\sqrt{(\lambda-\Delta)^2+\Omega^2}}
\pmatrix{
-(\lambda-\Delta)\cr \Omega},$$ where $\lambda=\sqrt{\Omega^2+\Delta^2}$. Suppose that before the pulse the system is in the state $|m\rangle$, i.e. the conditions, $$C_m(t_0)=1,\qquad C_p(t_{0})=0,$$ are satisfied. After the transformation (\[AC\]) to the rotating frame we obtain, $$A_m(t_{0})=\exp(-i{\cal E}_mt_{0})C_m(t_{0})
=\exp(-i{\cal E}_mt_{0}),\qquad A_p(t_{0})=0.$$ The dynamics is given by Eq. (\[dynamics\]), which in our case takes the form: $$A_m(t)=
A_m(t_{0})\left[\left(A_m^{q\,(0)}\right)^2\exp\left(-ie^{(0)}_q\tau\right)+
\left(A_m^{Q\,(0)}\right)^2\exp\left(-ie^{(0)}_Q\tau\right)\right]=$$ $${\exp\left[-i[{\cal E}_mt-(\Delta/2)\tau]\right]\over
\Omega^2+(\lambda-\Delta)^2}\left\{\Omega^2 e^{-i\lambda\tau/2}+
(\lambda-\Delta)^2e^{i\lambda\tau/2}\right\},$$ where $t=t_0+\tau$. Applying the back transformation, $$C_m(t)=\exp(i{\cal E}_mt)A_m(t),$$ and taking the real and imaginary parts of the expression in the curl brackets we obtain, $$\label{C1}
C_m(t_0+\tau)=\left[\cos(\lambda\tau/2)+
i(\Delta/\lambda)\sin(\lambda\tau/2)\right]
\exp(-i\tau\Delta/2).$$ For the amplitude, $A_p(t)$, we have, $$A_p(t)=A_m(t_{0})
\left[A_m^{q\,(0)}A_p^{q\,(0)}\exp\left(-ie^{(0)}_q\tau\right)+
A_m^{Q\,(0)}A_p^{Q\,(0)}\exp\left(-ie^{(0)}_Q\tau\right)\right]=$$ $$i{\Omega\over\lambda}
\exp\left\{-i\left[{\cal E}_mt-(\Delta/2)\tau\right]\right\}
\sin(\lambda\tau/2).$$ Applying the back transformation, $$C_p(t)=\exp[i({\cal E}_m+\Delta)t]A_p(t),$$ we obtain, $$\label{C2}
C_p(t_0+\tau)=i(\Omega/\lambda)\sin(\lambda\tau/2)
\exp(it_0\Delta+i\tau\Delta/2),$$
When $\Delta=0$ (the resonance case) and $\lambda\tau=\pi$ ($\pi$-pulse) Eqs. (\[C1\]) and (\[C2\]) describe the complete transition from the state $|m\rangle$ to the state $|p\rangle$. In the near-resonance case, when $\Delta\ne 0$ the transition probability is (here we again put the index $n$ indicating the pulse number), $$\label{epsilon}
\varepsilon_n=(\Omega_n/\lambda_n)^2\sin^2(\lambda_n\tau_n/2).$$ One can suppress the near-resonant transitions and to make the probability, $\varepsilon_n$, equal to zero by choosing the Rabi frequency in the form (see $2\pi k$-method in Ref. [@1]), $$\label{2pi_k}
\Omega^{(k)}_n=|\Delta_n|/\sqrt{4k^2-1}.$$
The solutions (\[C1\]) and (\[C2\]) can also be derived without transformation to the rotating frame (see Ref. [@1]). However, as will be shown below, our description allows us to introduce small parameters and to build a consistent perturbation theory. Using our perturbation approach we will compute the dynamics up to different orders in small parameters, and will test our approximate results using exact numerical solution for small number of qubits. Below we apply the perturbation theory for analysis of the quantum dynamics during implementation of a simple quantum logic gate in the spin chain with large number ($L=1000$) of qubits.
Protocol for creation an entangled state\
between remote qubits
=========================================
Here we schematically describe the protocol (the sequence of pulses) which allows one to create the entangled state for remote qubits in the system described by the Hamiltonian (\[H\]). The initial state of the system is the ground state, $|00\dots00\rangle$. The first pulse in our protocol, described by the unitary operator $U_1$, creates the superposition of the states $|00\dots00\rangle$ and $|10\dots00\rangle$ from the ground state, $$\label{U0}
U_1|00\dots00\rangle={1\over \sqrt 2}(|00\dots00\rangle+i|10\dots00\rangle).$$ Other pulses create from this state the entangled state for remote qubits. This procedure is described by the unitary operator, $U'$, $$\label{U}
U'\frac 1{\sqrt 2}(|00\dots00\rangle+
i|10\dots00\rangle)=\frac 1{\sqrt 2}(e^{i\varphi_1}(|00\dots00\rangle+
e^{i\varphi_2}|10\dots 01\rangle),$$ where $\varphi_1$ and $\varphi_2$ are known phases [@1].
Now, we describe the procedure for realization of the operator $U=U'U_1$ by applying a sequence of resonant pulses. Each pulse is described by the corresponding unitary operator, $U_n$, where $n=1,\,2,\,\dots,\,2L-2$. (The total number of pulses in our protocol is $M=2L-2$.) The unitary operator of the whole protocol is a product of the unitary operators of the individual pulses, $U=U_{2L-2}U_{2L-4}\dots U_2U_1$. The first pulse, described by the operator $U_1$, is resonant to the transition between the states $|00\dots00\rangle$ and $|10\dots00\rangle$. If we choose the duration of this pulse as $\tau_1=\pi/(2\Omega_1)$ (a $\pi/2$-pulse) then from Eqs. (\[C1\]) and (\[C2\]) we obtain Eq. (\[U0\]). In order to obtain the second term in the right-hand side of Eq. (\[U\]), we choose a sequence of resonant $\pi$-pulses which transforms the state $|10\dots 00\rangle$ to the state $|10\dots 01\rangle$ by the following scheme: $
|1000\dots 0\rangle\rightarrow|1100\dots 0\rangle\rightarrow
|1110\dots 0\rangle\rightarrow
$ $
|1010\dots 0\rangle\rightarrow|1011\dots 0\rangle\rightarrow
|1001\dots 0\rangle\rightarrow
$ $
\dots\rightarrow|100\dots 11\rangle\rightarrow|100\dots 01\rangle.
$ The frequencies of pulses which realize this protocol are: $\nu^{(2)}=\omega_{2L-2}$, $\nu^{(3)}=\omega_{2L-3}$, $\nu^{(4)}=\omega_{2L-2}-2J$, $\nu^{(5)}=\omega_{2L-4}$, …, $\nu^{(2L-3)}=\omega_0-J$, $\nu^{(2L-2)}=\omega_1$. If we apply the same protocol to the ground state, then with large probability the system will remain in this state because all transitions are non-resonant to the ground state.
Since the values of the detuning are the same for all pulses, $\Delta_n=\Delta=2J$ (except for the fourth pulse, where $\Delta_4=4J$), in our calculations we take the values of $\Omega_n$ to be the same, $\Omega_n=\Omega$ ($n\ne 4$), and $\Omega_4=2\Omega$. In this case, the probabilities of excitation of the ground state (near-resonant transitions), $\varepsilon_n$, are independent of $n$: $\varepsilon_n=\varepsilon$, since $\varepsilon_n$ depends only on the ratio $|\Delta_n/\Omega_n|$. (Here $\varepsilon$ and $\Omega$ are the numerical parameter used in simulations presented below.) One can minimize the probability of the near-resonant transitions choosing $\varepsilon=0$.
Errors in creation of an entangled state\
for remote qubits
=========================================
Our matrix approach allows us to estimate the error in the logic gate (\[U\]) caused by flips of non-resonant spins (non-resonant transitions). Consider a transition between the states $|l\rangle$ and $|l'\rangle$ connected by a flip of the non-resonant $k'$th spin. The absolute value of the difference between the $l$th and $l'$th diagonal elements of the matrix ${\cal H}^{(n)}_{pp'}$ is of the order or greater than $\delta\omega$, because they belong to different $2\times 2$ blocks. Since the absolute values of the matrix elements which connect the different blocks are small, $|V|\ll\delta\omega$, one can write, $$\label{psi10}
\psi_{q}=\psi^{(0)}_{q}+
{\sum_{q'}}'{v_{qq'}\over e^{(0)}_q-e^{(0)}_{q'}}
\psi^{(0)}_{q'},$$ where prime in the sum means that the term with $q'=q$ is omitted, $\psi_{q}$ is the eigenfunction of the Hamiltonian ${\cal H}$, the $q$th eigenstate is related to the $l$th diagonal element and the $q'$th eigenstate is related to the $l'$th diagonal element, $v_{qq'}=2V\langle\psi^{(0)}_{q}|I_{k'}^x|\psi^{(0)}_{q'}\rangle$ is the matrix element for the transition between the states $\psi^{(0)}_{q}$ and $\psi^{(0)}_{q'}$, the sum over $q'$ takes into consideration all possible one-spin-flip non-resonant transitions from the state $|l\rangle$. Because the matrix ${\cal H}$ is divided into $2^{N-1}$ relatively independent $2\times 2$ blocks, the energy, $e^{(0)}_q$ ($e^{(0)}_{q'}$), and the wave function, $\psi^{(0)}_{q}$ ($\psi^{(0)}_{q'}$), in Eq. (\[psi10\]) are, respectively, the eigenvalue and the eigenfunction with the amplitudes given by Eqs. (\[q01\]) and (\[q02\]) of the effective Hamiltonian, ${\cal H}$, in which all elements are equal to zero except the elements related to a single $2\times 2$ block.
The probability of non-resonant transition from the state $|l\rangle$ to the state $|l'\rangle$ connected by a flip of the non-resonant $k'$th spin is, $$\label{psi19}
P_{ll'}=\left|\langle l'|\psi_q\rangle\right|^2.$$ Only one term in the sum in Eq. (\[psi10\]) contributes to the probability $P_{ll'}$. When the block (\[2x2\]) is related to the near-resonant transition ($\Delta\sim J$), then from Eqs. (\[q01\]) and (\[q02\]) the eigenfunctions of this block are, $$\psi^{(0)}_{q'}\approx
[1-(\Omega^2/32J^2)]|l'\rangle+
(\Omega/4J)|m'\rangle\approx |l'\rangle,$$ $$\label{appr}
~~~~\psi^{(0)}_{Q'}\approx -(\Omega/4J)|l'\rangle+
[1-(\Omega^2/32J^2)]|m'\rangle\approx |m'\rangle.$$ On the other hand, if this block is related to the resonant transition ($\Delta=0$), we have $$\label{appr1}
\psi^{(0)}_{q'}=(1/\sqrt 2)(|l'\rangle+|m'\rangle), \qquad
\psi^{(0)}_{Q'}=(1/\sqrt 2)(|l'\rangle-|m'\rangle).$$ In both cases $v_{qq'}\approx V$, so that $$\label{psi9}
P_{ll'}\approx
\left({V\over {\cal E}_l-{\cal E}_{l'}}\right)^2\approx
\left({V\over |k-k'|\delta\omega}\right)^2,$$ where we put $e^{(0)}_q\approx {\cal E}_l$, $e^{(0)}_{q'}\approx {\cal E}_{l'}$; $|k-k'|$ is the “distance” from the non-resonant $k'$th spin to the resonant $k$th spin.
The total probability, $\mu_{N-1}$ (here the subscript of $\mu$ stands for the number of the resonant spin), of generation of all unwanted states by the first $\pi/2$ pulse in the result of non-resonant transitions is, $$\label{Pnr10}
\mu_{N-1}=\mu
\sum_{k'=0}^{N-2}\frac 1{|N-1-k'|^2},\qquad
\mu=\left(\Omega\over 2\delta\omega\right)^2.$$ The probability of error after applying $2N-2$ pulses is, $$\label{Pnr}
P=1-\frac 12\prod_{n=1}^{2L-2}(1-\mu_n)-
\frac 12\prod_{n=1}^{2L-2}(1-\mu_n-\varepsilon_n),$$ where $\varepsilon_1=0$ (operation (\[U0\])) and the last two terms in the right-hand side of Eq. (\[Pnr\]) are related to the last two terms in the right-hand side of Eq. (\[U\]).
Improved perturbation theory {#sec:improved}
============================
In the consideration presented above, we used the approximate solutions (\[appr\]) and (\[appr1\]) for the wave functions (\[q01\]) and (\[q02\]). A more advanced approach, which also does not require a diagonalization of large matrices, we use the explicit forms (\[q01\]) and (\[q02\]) to express the wave functions, $\psi^{(0)}_{q'}$ and $\psi^{(0)}_{Q'}$, of the $2\times 2$ blocks in Eq. (\[psi10\]), $$\label{psi100}
\psi^{(0)}_{q'}=A_{m'}^{q'\,(0)}|m'\rangle
+A_{l'}^{q'\,(0)}|l'\rangle,\qquad
\psi^{(0)}_{Q'}=A_{m'}^{Q'\,(0)}|m'\rangle
+A_{l'}^{Q'\,(0)}|l'\rangle.$$ Then, we put the functions, $\psi^{(0)}_{q'}$ and $\psi^{(0)}_{Q'}$, into Eq. (\[psi10\]), and obtain the coefficients, $A_{l}^{q}$, for the wave function $\psi_{q}$, $$\label{Amq}
\psi_{q}=\sum_m A_{m}^{q}|m\rangle,$$ where the sum in the right-hand side contains $2L$ terms. Using the functions, $A_{m}^{q}$, we solve the dynamical equations (\[dynamics\]) with the energies, $e_{q}$, computed up to the second order of our perturbation theory, $$\label{emq}
e_q=e^{(0)}_q+{\sum_{q'}}'{|v_{qq'}|^2\over e^{(0)}_q-e^{(0)}_{q'}},$$ where the prime in the sum means that the term with $q=q'$ is omitted, $e^{(0)}_q$ is defined by Eqs. (\[q01\]) or (\[q02\]).
We call the described above approach the “improved” perturbation theory to indicate the difference from the approach considered in the previous section. In this approximation each eigenfunction, $\psi_q$, of the Hamiltonian, $\cal H$, is expanded over $2L$ (see Eq. (\[Amq\])) basis functions, $|m\rangle$, with all other coefficients, $A_{m'}^q$, being equal to zero. Here we use all possible transitions between different $2\times 2$ blocks which include the two-spin-flip transitions: a flip of the resonant spin and a flip of a non-resonant spin. The number of non-zero coefficients in this approximation is $2^L\times 2L$. It still can be large for large $L$ and can require large computer memory for simulation. As will be shown below, under the condition, $\Omega\ll J\ll\delta\omega$, this approach (the “improved” perturbation theory) gives the results which practically coincide with the exact solution.
Numerical results
=================
All frequencies in this section are measured in units of the Ising interaction constant, $J$. Assume that we are able to correct the errors with the probability less than $P_0=10^{-5}$. Our perturbation theory allows us to calculate the region of parameters for which the probability of error, $P$, in realization of the logic gate (\[U\]) is less than $P_0$. In Figs. 1 (a) and (b) we plot the diagrams obtained by solution of Eq. (\[Pnr\]) and using the improved perturbation theory. In the hatched areas the probability of generation of unwanted states is less than $P_0$. One can see that two approaches yield similar results. They become practically identical at large values of the distance between the neighboring qubits, $\delta\omega$.
In almost all quantum algorithms the phase of the wave function is important. We numerically compared the phase of the wave function on the boundaries of the hatched regions in Figs. 1(a,b) with the phase in the centers of these regions, where $\Omega$ satisfies the $2\pi k$-method, and the expression for the phase can be obtained analytically [@1]. The deviation in phase is only $\sim 0.15\%$. This is much less that the corresponding change in the probability, $P$, of errors (by several orders).
We now analyze the probability of errors as a function of $\delta\omega$. When the value of $\delta\omega$ is large enough, the probability of error (and the widths of the hatched areas in $\Omega$) becomes practically independent of $\delta\omega$. This is because at $\delta\omega\gg 1$ and at $\varepsilon\gg\mu$ the error is provided mostly by $\varepsilon$, which is independent of $\delta\omega$. As a consequence, one can, for example, estimate the widths of the hatched areas at $\delta\omega\gg 1$ taking into account only the near-resonant transitions. To do this we put in Eq. (\[Pnr\]) the value $\mu_n=0$ for all $n$ and obtain $$\label{Pnr1}
P_B=\frac 12\left(1-\prod_{n=1}^{2L-3}(1-\varepsilon_n)\right)=
\frac 12\left(1-(1-\varepsilon)^{2L-3}\right)\approx
\frac {2L-3}2\varepsilon,$$ where $\varepsilon_n=\varepsilon\ll 1$, for all $n$. The positions of the boundaries in $\Omega$ in Figs. 1 (a,b) can be obtained from the equation $P_B=P_0$, where $P_B$ is given by Eq. (\[Pnr1\]) and $\varepsilon$ is a function of $\Omega$ (see Eq. (\[epsilon\])).
From Figs. 1 (a) and (b) one can see that even when the condition of $2\pi k$-method is satisfied, the error can be large when $\delta\omega$ (or the gradient of the magnetic field $B^z(z)$) is relatively small. Thus, at $\delta\omega<\delta\omega_A$, where $\delta\omega_A$ is the coordinate of the point A in $\delta\omega$ in Figs. 1 (a,b), the error is more than $P_0$. Since the position of the point A in $\Omega$ satisfies the $2\pi k$-method, this indicates that even in the case when the near-resonant processes are suppressed by the $2\pi k$ method ($\varepsilon=0$) the non-resonant transitions can make the error larger than the threshold, $P_0$. We can not suppress entirely the non-resonant transitions, defined by the values of $\mu=(\Omega/2\omega)^2$ and $L$, like we did with the near-resonant transitions. The value of $\Omega$ cannot be decreased considerably because decreasing of $\Omega$ makes the quantum computer very slow, so that the quantum state can be destroyed by decoherence due to possible influence of environment. Hence, one can make the value of $\mu$ small by increasing $\delta\omega$. In Fig. 2 we plot the minimum value of $\delta\omega=\delta\omega_A$ as a function of the number of qubits, $L$, which was computed using Eq. (\[Pnr\]).
One can see that $\delta\omega_A$ becomes large for large $L$. Thus, for example, for a protons with $J/(2\pi)\sim 100$ Hz (for estimations in this paragraph we use the dimensional units) with the distance between the neighboring spins $a=2$ nm, the value $\delta\omega/J=1000$ yield the gradient of the magnetic field $\delta\omega/(\gamma a\cos\theta)\sim 2\times 10^6$ T/m. From Fig. 2 one can see that this is the minimum value of the gradient of the magnetic field for $L_{max}\approx 155$ when $\Omega=\Omega^{(5)}$ and $L_{max}\approx 740$ when $\Omega=\Omega^{(11)}$ required to make the error less than $P_0=10^5$. At $L>L_{max}$, at a given gradient of the magnetic field, and at $\Omega\approx\Omega^{(k)}$, $k=5$ or $11$, the error will be always larger than $P_0$.
In Figs. 3 (a,b) we test our perturbation theory by using the exact numerical solution obtained by a diagonalization of $2^L\times 2^L$ matrices and using Eq. (\[dynamics\]). One can see that there is good correspondence with the exact numerical solution for the results obtained using Eq. (\[Pnr\]), and practically exact correspondence for the solution obtained using the improved perturbation theory. The similar correspondence can be demonstrated for other parameters ($\delta\omega,\,\Omega$).
Conclusion
==========
We developed the perturbation theory which allows us to estimate the errors in implementation of the quantum logic gates by the radio-frequency pulses in the solid-state system with large number (1000 and more) of qubits. Our perturbation approach correctly describes the behavior of the quantum system in the large Hilbert space (the Hilbert space with large number of states) and predicts the final quantum state of the system after action of the sequence of pulses with different frequencies. This is possible because in the system there are small parameters, characterizing the probabilities $\varepsilon$, of the near-resonant transitions, and probabilities, $\mu$, of the non-resonant transitions, which are small, $\varepsilon\ll 1$ and $\mu\ll 1$, when the conditions $\Omega\ll J\ll\delta\omega$ are satisfied. Our approach allows one to control the quantum logic operations in the system with large number of qubits and to minimize the error caused by the internal decoherence (non-resonant processes).
Acknowledgments
===============
The paper was supported by the Department of Energy (DOE) under contract W-7405-ENG-36, by the National Security Agency (NSA), and by the Advanced Research and Development Activity (ARDA).
B.E. Kane, Nature [**393**]{}, 133 (1998). F. Yamaguchi and Y. Yamamoto, Microelectron. Eng. [**47**]{}, 273 (1999). G.P. Berman, G.D. Doolen, P.C. Hammel, and V.I. Tsifrinovich, Phys. Rev. B [**61**]{}, 14694 (2000). G.P. Berman, G.D. Doolen, G.D. Holm, and V.I. Tsifrinovich, Phys. Lett. A [**193**]{}, 444 (1994). R. Vrijen, E. Yablonovitch, K. Wang, H.W. Jiang, A. Balandin, V. Roychowdhury, T. Mor, and D. DiVincenzo, Phys. Rev. A [**62**]{}, 2306 (2000). A. Imamoglu, D.D. Awschalom, G. Burkard, D.P. DiVincenzo, D. Loss, M. Sherwin, and A. Small, Phys. Rev. Lett. [**83**]{}, 4204 (1999). M. Sherwin, A. Imamoglu, and T. Montroy, quant-ph/9905096. A. Shnirman, G.Schoen, and Z. Hermon, Phys. Rev. Lett. [**79**]{}, 2371 (1997). D. V. Averin, Solid State Commun. [**105**]{}, 659 (1998). G.P. Berman, G.D. Doolen, G.D. Holm, V.I. Tsifrinovich, [*Phys. Lett. A*]{} [**193**]{}, 444 (1994). I.L. Chuang, N.A. Gershefeld, M. Kubinec, [*Phys. Rev. Lett.*]{} [**80**]{}, 3408 (1998). G.P. Berman, G. D. Doolen, G. V. Lòpez, and V. I. Tsifrinovich, Phys. Rev. A [**61**]{}, 2305 (2000).
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abstract: 'We propose a new hadronization mechanism, jet-fluid string (JFS) formation and decay, to understand observables in intermediate to high-$p_{T}$ regions comprehensively. In the JFS model, hard partons produced in jet lose their energy in traversing the QGP fluid, which is described by fully three-dimensional hydrodynamic simulations. When a jet parton escapes from the QGP fluid, it picks up a partner parton from a fluid and forms a color singlet string, then it decays to hadrons. We find that high-$p_T$ $v_2$ values in JFS are about two times larger than in the independent fragmentation model.'
address:
- |
Department of Physics, Graduate School of Science, Osaka University,\
Toyonaka 560-0043, Japan\
[email protected]
- |
Institute of Physics, Graduate School of Arts and Sciences, University of Tokyo,\
Komaba, Tokyo 153-8902, Japan
- |
Department of Physics, Faculty of Science, Hokkaido University,\
Sapporo 060-0810, Japan
- |
Institut für Theoretische Physik, Johann Wolfgang Goethe-Universität,\
60438 Frankfurt am Main, Germany
author:
- 'M. Isse'
- 'T. Hirano[^1]'
- 'R. Mizukawa, A. Ohnishi, K. Yoshino'
- 'Y. Nara'
title: ' Jet-fluid string formation and decay in high-energy heavy-ion collisions '
---
Introduction
============
Models based on the color strings have been highly successful to describe high energy hadronic collisions. A hadron-string cascade picture in high energy heavy-ion collisions works well at AGS[@JAM] and SPS[@Isse] energies, and it also describes some low-$p_T$ observables such as $p_T$-distribution and $\eta$-distribution[@HINOY2005; @Sahu] at RHIC energies. However, hadron-string cascade models underestimate the elliptic flow $v_2$ at RHIC energies, therefore we need partonic pressure in the early stage.
![ Left panel: A sketch of JFS model. Produced jet parton traverses in the evolving fluid with energy loss. When it reaches to the QGP phase boundary, it picks up a fluid parton and forms a string and decays into hadrons. Right panel: Schematic view of simple Recombination,[@Duke] Thermal-Shower recombination,[@Oregon] and JFS fragmentation (See text). []{data-label="Fig:JFSform"}](fig/JFS-pic.eps "fig:"){width="6cm"} ![ Left panel: A sketch of JFS model. Produced jet parton traverses in the evolving fluid with energy loss. When it reaches to the QGP phase boundary, it picks up a fluid parton and forms a string and decays into hadrons. Right panel: Schematic view of simple Recombination,[@Duke] Thermal-Shower recombination,[@Oregon] and JFS fragmentation (See text). []{data-label="Fig:JFSform"}](fig/ModelComp.eps "fig:"){width="6cm"}
There are several observations explained as a signal of quark-gluon plasma (QGP) formation at RHIC. A large elliptic flow of bulk matter, suppression of hadron yield at high-$p_T$, and quark number scaling of elliptic flow at intermediate-$p_T$, are explained as consequences of hydrodynamical evolution in the early stage,[@HG2005] parton energy loss in deconfined matter,[@e-loss] and parton recombination,[@Duke] respectively. All of these explanations support the formation of a QGP. While these three pictures — hydrodynamics, fragmentation, recombination — have succeeded in explaining many data at RHIC, there are some problems left unsolved. One of the problems is that elliptic flow generated by radiative energy loss is always smaller than observed data, when we fit the nuclear modification factor.
In this proceeding, we consider jet-fluid string (JFS) formation and its decay processes, where a propagating jet parton picks up a partner parton from the fluid to form a string. If we take account of JFS formation at the end of QGP evolution, JFSs would have a large momentum anisotropy coming from both of the hydrodynamically evolved fluid parton and the jet parton which suffers the energy loss. Furthermore, we can take account of both quarks and gluons on the same footing in JFS formation. We show the basic concept of the JFS model in Fig.\[Fig:JFSform\]. We find that JFS decay is an efficient process to produce high-$p_T$ hadrons, and a large energy loss is required to describe the high-$p_T$ hadron spectra. As a result of this large energy loss, high-$p_T$ hadron elliptic flow becomes larger than in the independent fragmentation (IF) picture, while we slightly underestimate the observed $v_2$ of pions at high-$p_T$.
There are several attempts to include hadron formation processes from jet (or shower) and fluid (or soft/thermal) partons. Recombination processes of thermal-thermal (TT), thermal-shower (TS), and shower-shower (SS) partons have been considered to form hadrons, which are considered to dominate in a wide range of $p_T$ in Ref.[@Oregon]. In the quark coalescence model,[@Texas] coalescence of a soft parton and a quenched jet parton (soft-hard coalescence) is found to be important in intermediate-$p_T$ ($3<p_T<6$ GeV/$c$) hadron production. In a recombination model,[@Duke_New] several processes to combine soft and hard partons are investigated. In these models, quarks are considered to form hadrons (including resonances) directly. On the other hand, we consider [*gluons*]{} as well as quarks can form [*strings*]{}, which decay into several hadrons in this work.
JFS Model
=========
In the JFS model, we have four ingredients to describe high-$p_T$ hadron distribution; mini-jet production, jet parton evolution in the QGP fluid, jet-fluid string formation, and its decay. First, mini-jet partons are generated in the pQCD framework by using the <span style="font-variant:small-caps;">pythia</span> program (version 6.4).[@PYTHIA] We have tuned <span style="font-variant:small-caps;">pythia</span> parameters, $K$-factor ($K\sim 1.85$) and minimum $p_T$ of jet partons ($p_{{\scriptscriptstyle T}0} \sim 2$ GeV), to reproduce the high-$p_T$ pion spectrum in $p+p$ collisions at RHIC.[@pp] Secondly, the energy loss of generated jet partons in the QGP fluid is evaluated by using the first order Gyulassy-Lévai-Vitev (GLV) energy loss formula[@e-loss] with simplification[@HN2004]: $$\Delta E=C\times 3\pi\alpha_s^3 F_{\rm color}\int_{\tau_0}^\infty
d\tau \;\rho(\tau,\bm x(\tau)) \cdot (\tau-\tau_0)
\log\biggl(\frac{2E_0}{\mu^2L}\biggr) ,
\label{Eq:GLV}$$ where $F_\text{color}$ is a color factor (1 and 9/4 for $q(\bar{q})$ and $g$), $E_0$ is the initial energy, $\alpha_s=0.3$, $\mu=0.5$ GeV, and $L=3$ fm. We utilize the fluid evolution obtained by the 3D hydrodynamical model calculation[@HN2004; @HN2003] with the Glauber type initial condition[@HT2002] to evaluate parton density $\rho(\tau,\bm x)$, which can be obtained on the web.[@HiranoWWW] The overall energy loss factor $C$ is regarded as an adjustable parameter to reproduce $R_{AA}$, and fixed as $C \sim 6$ by fitting pion $p_T$-spectrum in mid-central Au+Au collisions as discussed later. Low-$p_T$ ($p_T < 2\ \mathrm{GeV}/c$) partons are considered to be absorbed in the QGP fluid. Next, a jet parton is assumed to pick up a fluid parton at the boundary of QCD phase transition to form a color singlet string. We here only consider a string formation ($\bar{q}q$ or $gg$) whose mass is larger than 2 GeV. Fluid parton momenta are obtained from Lorentz-boosted Fermi (Bose) distribution for $q$ or $\bar{q}$ ($g$). Light flavored ($u, d$ and $s$) fluid quarks are regarded as massless and the flavor is chosen with the same probability of $1/3$. Effects of hadronic resonance formations for a mass smaller than 2 GeV will be reported in the future. Finally, the decay of these strings producing multi-hadrons are evaluated in the Lund string fragmentation model <span style="font-variant:small-caps;">pythia</span>. We also show the results in the independent fragmentation (IF) for comparison.
Results and Discussions
=======================
In the left panel of Fig. \[Fig:JFS-pt\], we show the calculated $p_T$-spectrum of charged pions in mid-central Au+Au collisions at RHIC. The energy loss factor of $C \sim 6$ is found to fit the high-$p_T$ part of the spectrum. When JFS results are combined with the low momentum hydro dominant part parametrized in the exponential form $Ed^3N_{\rm Hydro}/dp^3=A_1\exp(-p_T/T_1)+A_2\exp(-p_T/T_2)$, the calculated spectrum agrees with experimental data from STAR[@pipstar] and PHENIX.[@pi0phen; @pipphen]
![ Left panel: Charged pion $p_T$-spectrum in mid-central ($b=7.4$ fm) Au+Au collisions. We show JFS results and those combined with hydrodynamic exponential component. Experimental data of $\pi^\pm$ are taken from STAR[@pipstar] and PHENIX[@pipphen], and $\pi^0$ multiplied by two are taken from PHENIX[@pi0phen] data, with corresponding centrality 20-30%. Right panel: Two types of fragmentation scheme, JFS and IF, are compared in $R_{AA}$ for mid-central collisions. We see saturating behavior at high-$p_T$ in JFS, and agrees with the data well. []{data-label="Fig:JFS-pt"}](fig/JFSpt-032.eps "fig:"){width="6cm"} ![ Left panel: Charged pion $p_T$-spectrum in mid-central ($b=7.4$ fm) Au+Au collisions. We show JFS results and those combined with hydrodynamic exponential component. Experimental data of $\pi^\pm$ are taken from STAR[@pipstar] and PHENIX[@pipphen], and $\pi^0$ multiplied by two are taken from PHENIX[@pi0phen] data, with corresponding centrality 20-30%. Right panel: Two types of fragmentation scheme, JFS and IF, are compared in $R_{AA}$ for mid-central collisions. We see saturating behavior at high-$p_T$ in JFS, and agrees with the data well. []{data-label="Fig:JFS-pt"}](fig/JFS_RAA-032.eps "fig:"){width="6cm"}
It would be instructive to compare the JFS results with those in a standard picture, in which high-$p_T$ hadrons are formed through the IF of jet partons.[@HN2004] In the right panel of Fig. \[Fig:JFS-pt\], we compare the nuclear modification factor $R_{AA}$ in different hadronization schemes (JFS and IF) by using the same fluid evolution profile. Experimental values are pion $p_T$-spectrum of Au+Au [@pipstar; @pi0phen; @pipphen] normalized by that of $p+p$[@pp] and $N_{\rm coll}$. The latter model (IF) is similar to the Hydro+Jet model by Hirano and Nara[@HN2004] at high-$p_T$, but we have not taken care of the Cronin and shadowing effects, which enhance the intermediate-$p_T$ hadrons.
In JFS and IF models, different values of energy loss factor $C$ are needed to fit high-$p_T$ spectrum, $C_{\rm JFS}\sim 6$ and $C_{\rm IF}\sim 2$. This is because high-$p_T$ hadron production is easier in JFS decay; when a high-$p_T$ jet parton picks up an approximately collinear low-$p_T$ fluid parton, a light mass string is formed and decays into a fewer hadrons. It is also due to the fragmentation scheme itself.[@NVC] We also find a different $p_T$ dependence of $R_{AA}$. While $R_{AA}$ saturates at around $p_T \sim 6\ \mathrm{GeV/c}$ in JFS, it slowly grows in the IF model. The latter dependence may be compensated at intermediate-$p_T$ by the Cronin and shadowing effects, but the high-$p_T$ dependence would remain.
Next, we evaluate the elliptic flow $v_2$ in the JFS model. In the left panel of Fig. \[Fig:v2\], we show the calculated $p_T$ dependence of charged pion $v_2$ in comparison with the experimental data of charged pions.[@STARv2b; @Sorensen] We find that the JFS results are around two times larger than IF results, and are close to the data at high-$p_T$; the data may be around 0.10, and JFS gives $\sim$ 0.08 at $p_T > 6$ GeV/$c$. The difference between JFS and IF mainly comes from the energy loss strength (parameter $C$ in Eq. (\[Eq:GLV\])), and the picked fluid parton $v_2$ also contributes to enhance the hadron $v_2$ by around 0.01. The high-$p_T$ $v_2$ values in JFS is comparable with the results in the Recombination+Fragmentation model[@Duke] ($v_2 \sim 0.10$ at $p_T\sim 5-10$ GeV/c), and Hydro+Jet model[@HN2004] ($v_2 \sim 0.10$ at $p_T\sim 3$ GeV/c). In the former, the parton energy loss is parameterized by the angle dependent path length, then a sharp edge density distribution is implicitly assumed. In the latter, $v_2$ decreases and underestimates the data at higher $p_T$ ($p_T > 5\ \mathrm{GeV}/c$). We also show the combined results of JFS and the Hydro component, whose relative weight is already fixed from the $p_T$-spectrum fit. Combined results well explain $v_2$ data up to around 2 GeV/$c$ with Hydro component having $v_2 \propto p_T$. We clearly find that we underestimate the data at intermediate-$p_T$ ($3-6$ GeV/$c$), where recombination processes would be important.
In the right panel of Fig.\[Fig:v2\], we plot the impact parameter dependence of charged hadron $v_2$ integrated in the range of $3<p_T<6\ \mathrm{GeV}/c$ in comparison with STAR data[@STARv2] obtained by four-particle cumulant method. We note that the calculated $v_2$ values are only around half of the data. However, we would like to point out that the present $v_2$ values are two times larger than those in a simple simulation with Woods-Saxon density distribution[@STARv2] and comparable to those in a simple calculation with hard-sphere density profile (maximum $v_2\sim 0.10$),[@STARv2] while we simulate the parton dynamics with time-evolving 3D hydrodynamics.
![ Left panel: Charged pion $v_2$ as a function of $p_T$. We compare JFS results with STAR data[@STARv2] of corresponding centrality 20-30%, and PHENIX and STAR preliminary data[@Sorensen] of minimum-bias events. Right panel: Integrated $v_2$ of charged hadrons in the range of $3<p_T<6$ GeV/c versus impact parameter $b$. We compare JFS results are compared with STAR data[@STARv2]. []{data-label="Fig:v2"}](fig/JFSv2-032.eps "fig:"){width="6cm"} ![ Left panel: Charged pion $v_2$ as a function of $p_T$. We compare JFS results with STAR data[@STARv2] of corresponding centrality 20-30%, and PHENIX and STAR preliminary data[@Sorensen] of minimum-bias events. Right panel: Integrated $v_2$ of charged hadrons in the range of $3<p_T<6$ GeV/c versus impact parameter $b$. We compare JFS results are compared with STAR data[@STARv2]. []{data-label="Fig:v2"}](fig/JFSbv2-032.eps "fig:"){width="6.1cm"}
Summary
=======
We have proposed a Jet-Fluid String (JFS) model as a mechanism to produce high-$p_T$ hadrons. In the JFS model, following components are combined; mini-jet production in pQCD, energy loss with simplified GLV formula, 3D hydrodynamic simulations, and Lund string decay. JFS decay is found to produce high-$p_T$ hadrons effectively, and we can utilize the 3D hydrodynamical expansion. After fitting high-$p_T$ spectrum in Au+Au collisions, we find that the calculated $v_2$ values roughly reproduce the data at high-$p_T$. At intermediate-$p_T$, JFS results of $v_2$ are about two times larger than in the independent fragmentation, but still they are around half of the data. This underestimate may be due to the lack of lower mass ($<2$ GeV) string or resonance, or the Fluid-Fluid String formation. From these results, we conclude that JFS would be a plausible fragmentation scheme to produce high-$p_T$ hadrons in relativistic heavy-ion collisions.
One of the authors (MI) is grateful to M. Asakawa for fruitful discussions. This work was supported in part by the 21st Century COE Program “Towards a New Basic Science; Depth and Synthesis”, Osaka University (MI), and by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research under the grant numbers, 18-10104 (TH), 15540243 (AO), and 1707005 (AO).
[0]{} Y. Nara, N. Otuka, A. Ohnishi, K. Niita and S. Chiba, Phys. Rev. C [**61**]{} (2000), 024901. M. Isse, A. Ohnishi, N. Otuka, P. K. Sahu and Y. Nara, Phys. Rev. C [**72**]{} (2005), 064908. T.Hirano, M.Isse, Y.Nara, A.Ohnishi and K.Yoshino, Phys. Rev. C [**72**]{} (2005), 041901. P. K. Sahu, A. Ohnishi, M. Isse, N. Otuka and S. C. Phatak, Pramana [**67**]{} (2006), 257. T. Hirano and M. Gyulassy, Nucl. Phys. A [**769**]{} (2006) 71. M. Gyulassy, P. Lévai and I. Vitev, Nucl. Phys. [**B 594**]{} (2001), 371; [*ibid.*]{} [**B 571**]{}, 197 (2000); Phys. Rev. Lett. [**85**]{}, 5535 (2000); in *Quark Gluon Plasma 3*, edited by R. C. Hwa and X. N. Wang (World Scientific, Singapore, 2004), p123 \[arXiv:nucl-th/0302077\]. R. J. Fries, J. Phys. G [**30**]{} (2004), S853; R. J. Fries, B. Müller, C. Nonaka and S. A. Bass, Phys. Rev. C [**68**]{} (2003), 044902. R. C. Hwa, Eur. Phys. J. C [**43**]{} (2005), 233; R. C. Hwa and C. B. Yang, Phys. Rev. C [**70**]{} (2004), 024904; [*ibid.*]{} [**70**]{} (2004), 024905. V. Greco, C. M. Ko and P. Levai, Phys. Rev. Lett. [**90**]{} (2003), 202302; Phys. Rev. C [**68**]{} (2003), 034904. R. J. Fries, S. A. Bass and B. Muller, Phys. Rev. Lett. [**94**]{} (2005), 122301. T. Sjöstrand [*et al.*]{}, Comp. Phys. Comm. [**135**]{} (2001), 238. S. S. Adler [*et al.*]{} \[PHENIX Collaboration\], Phys. Rev. Lett. [**91**]{} (2003), 241803; J. Adams [*et al.*]{} \[STAR Collaboration\], Phys. Lett. B [**637**]{} (2006), 161. T. Hirano and Y. Nara, Phys. Rev. C [**69**]{} (2004), 034908. T. Hirano and Y. Nara, Phys. Rev. Lett. [**91**]{} (2003), 082301. T. Hirano and K. Tsuda, Phys. Rev. C [**66**]{} (2002), 054905. Y. Nara, S. E. Vance and P. Csizmadia, Phys. Lett. B [**531**]{} (2002), 209. J. Adams [*et al.*]{} \[STAR Collaboration\], Phys. Rev. Lett. [**92**]{} (2004), 112301. S. S. Adler [*et al.*]{} \[PHENIX Collaboration\], Phys. Rev. Lett. [**91**]{} (2003), 072301. S. S. Adler [*et al.*]{} \[PHENIX Collaboration\], Phys. Rev. C [**69**]{} (2004), 034909. J. Adams [*et al.*]{} \[STAR Collaboration\], Phys. Rev. C [**72**]{} (2005), 014904. P. R. Sorensen, Nucl. Phys. [**A 774**]{} (2006), 247. J. Adams [*et al.*]{} \[STAR Collaboration\], Phys. Rev. Lett. [**93**]{} (2004), 252301.
[^1]: Present address: Department of Physics, Graduate School of Science, University of Tokyo, Tokyo 113-0033, Japan
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abstract: 'This work aims at investigating and quantifying the Urban Transport System (UTS) resilience enhancement enabled by the adoption of emerging technology such as Internet of Everything (IoE) and the new trend of the Connected Community (CC). A conceptual extension of Functional Resonance Analysis Method (FRAM) and its formalization have been proposed and used to model UTS complexity. The scope is to identify the system functions and their interdependencies with a particular focus on those that have a relation and impact on people and communities. Network analysis techniques have been applied to the FRAM model to identify and estimate the most critical community-related functions. The notion of [*Variability Rate*]{} (VR) has been defined as the amount of output variability generated by an upstream function that can be tolerated/absorbed by a downstream function, without significantly increasing of its subsequent output variability. A fuzzy based quantification of the VR on expert judgment has been developed when quantitative data are not available. Our approach has been applied to a critical scenario (water bomb/flash flooding) considering two cases: when UTS has CC and IoE implemented or not. The results show a remarkable VR enhancement if CC and IoE are deployed.'
author:
- Emanuele Bellini Paolo Ceravolo Paolo Nesi
bibliography:
- 'biblio10.bib'
title: Quantify resilience enhancement of UTS through exploiting Connected Community and Internet of Everything emerging technologies
---
<ccs2012> <concept> <concept\_id>10010520.10010553.10010562</concept\_id> <concept\_desc>Computer systems organization Embedded systems</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010520.10010575.10010755</concept\_id> <concept\_desc>Computer systems organization Redundancy</concept\_desc> <concept\_significance>300</concept\_significance> </concept> <concept> <concept\_id>10010520.10010553.10010554</concept\_id> <concept\_desc>Computer systems organization Robotics</concept\_desc> <concept\_significance>100</concept\_significance> </concept> <concept> <concept\_id>10003033.10003083.10003095</concept\_id> <concept\_desc>Networks Network reliability</concept\_desc> <concept\_significance>100</concept\_significance> </concept> </ccs2012>
This work is supported by the H2020-RESOLUTE project
Introduction
============
The effectiveness of the current risk and efficiency-based approaches in complex socio-technical systems safety and security management is affected by their weakness in addressing the so-called “unknown unknowns" [@park2013integrating]. This is caused by the continuous increment of the complexity of the systems and the emergent and unpredictable conditions such as climate change or man-made sabotages. According to [@field2012managing], the climatic extremes may intensify or become more frequent in regions that are not used to cope with such events. Moreover, so far no scientific method is available to precisely predict the long-term evolution and spatial distribution of critical events, nor the impacts on society’s critical infrastructures.\
Complex socio-technical systems cannot be managed under the assumption that accidents are produced by an uncontrolled and undesired release or transfer of energy between technical components [@leveson] and the large number of human/social, organisational and technical aspects, together with their fast pace changing behavior, imposes serious limitations on the ability to fully understand and monitor system operations. Therefore, complex socio-technical systems are today underspecified by nature [@Wilson2006] and a certain level of epistemic and aleatory uncertainty must be taken into account as a contribution to the critical events.
As [@owens] pointed out, accidents within complex environments tend to be the result of unpredicted interactions, rather than single failures of human or technical components. This produces unexpected cascade effects, which could rapidly reach unacceptable proportions. In order to face these unknown elements, building resilience becomes the best decision for socio-technical systems as the Urban Transport Systems (UTS)[@linkovnature]. There are many definitions and interpretations about resilience[^1]. In the context of RESOLUTE[^2], it refers to the capability of a system of continuously adapting to its operational environment in the pursuit of its intentions/purposes. Thus resilience can be defined as the ability of a system to sustain required operations in both expected and unexpected conditions by adjusting its functioning prior to, during, or following changes. According to the resilience engineering field, the potential for resilience to emerge from system performance may be assessed based on the [*four resilience cornerstones*]{} [@hollangel2011b], [@Hollnagel2015]:
a) Knowing what to do - corresponds to the ability to respond to disruptions by adjusting system performance to changing conditions.
b) Knowing what to look for - corresponds to the ability to monitor both the system and the environment.
c) Knowing what to expect - corresponds to the ability to anticipate opportunities for changes in the system and identify sources of disruption and pressure and their consequences for system operation.
d) Knowing what has happened - corresponds to the ability to learn from past experiences of both successes and failures.
In other words, the essence of resilience is the ability of the system to recognize when variability in its performance is unanticipated and fall beyond the usual range, and to dampen such variability through continuos adaptation. In order to cope with such a variability and to respond to different and possibly conflicting local operational needs, the limited resources of the system (humans, technologies and organization) should be managed and exploited effectively to achieve the right system synchronization and coordination level needed to ensure successful operation. However, it is clear that the variability and uncertainty need to be considered as intrinsic characteristics of complex socio-technical systems [@hollnagel2008fram].
According to this perspective, the new global trend of Internet of Everything (IoE) in general and the Connected Community (CC) in particular, can be exploited as resources of the socio-technical system to enhance its adaptive capacity and thus the resilience of the UTS dampening unwanted variability. The IoE can be considered a natural development of the IoT concept. In fact, while “Things” are related to connect physical-first objects, IoE extends this view comprising the following four key elements including all sorts of possible connections:
a) People: Considered as end-nodes connected across the Internet to share knowledge, information, opinions, decisions, behaviors and activities.
b) Things: Physical sensors, devices, actuators and other items generating data or receiving information from other sources.
c) Data: Raw data analyzed and processed into useful information to enable intelligent decisions and control mechanisms (e.g., Human behaviors on the ground).
d) Processes: Leveraging connectivity among data, things and people to add value.
Thus IoE establishes an end-to-end ecosystem of connectivity where people with their relationships, social collaborations and grouping dynamics represent an integral part. In particular, according to [@bpcc], Connected Communities are characterised, among the others, by weak ties that can symbolise a range of potential relationships among community members. These relations range from tight, long-lasting and static to temporary, real-time and dynamic relationships of different durations to location-specific connections. In fact, communities can be established on the base of same interests, skills or because of being at the same place and time in relation to some adverse event. Such a CC characteristics can be exploited to shift from a public awareness approach to one of community-individual safety altering the traditional top-down “command and control" relationships with the population. In fact, in RESOLUTE, the community is seen as an active participant to build the system resilience, rather than a passive recipient of services. Hence, the IoE and CC, if properly exploited, can be considered as means to achieve resilience in UTS because they could:
a) enhance the monitoring and control capability, improving the granularity and breadth of knowledge and awareness about the system status and dynamics continuously collecting Big Data from heterogeneous data sources/streams and sensors as people GPS position, concentration, behaviors and sentiment through smart devices and social networks (User Generated Data), Open Data, data from environmental sensors (e.g., traffic flows, hydrometry, air pollution, underpasses water level), mobile cell data, wifi access points, and real-time reports such as weather forecast, and so forth [@dms];
b) enhance the responding capability by providing detailed and timely information to authorities on one side, and to delivering personalised, real-time, context-aware, and ubiquitous advice to the community exploiting technologies such as IoE, Fast Wireless Connections (free wifi, 3G/4G), LoRaWAN, Smart Mobile Devices, Big Data Analytics, Semantic Computing, etc., that are crucial for augmenting situation awareness and enhancing decision making;
c) enhance the learning capability applying advanced analysis on Big Data (e.g., deep learning, data analysis and prediction, sentiment analysis) to extract knowledge;
d) enhance then anticipation capability continuously supporting the assessment of vulnerability and identifying when the system operates nearer to safety boundaries, predict behaviors and event dynamics, support evidence-based decisions at strategic, tactic and operation level moving ahead respect the current practices based on pre-simulated emergency scenarios [@Woltjer].
Unfortunately, even if several initiatives are ongoing at international levels such as the political UNISDR Sendai Framework[^3], cities and local communities are slow in becoming smart and resilient because of several factors such as budget restriction, cultural gaps, and by the difficulties to quantify the benefits for the community (e.g. Social Return Of Investment). In fact, because of resource scarcity, a priority rank for infrastructure improvement actions tends to be based on political opportunity or heuristics instead of a quantitative evaluation of the benefit of the system as a whole.\
To this end, the present article aims at demonstrating and quantifying the enhancement of UTS resilience obtained with the exploitation of IoE and CC as enabling technologies capable of significantly increasing the variability dampening capacity of those functions in UTS related to the human/social aspects.\
Defining a method for variability quantification enables also the development of the so-called Big KID–driven Decision Support System[^4]. A Decision Support System (DSS) [@dorasamy2013knowledge], [@tsekourakis2012decision], [@suarez2013improving], [@bartolozzi2015smart] is a computer-based information system that supports organizational decision-making activities. The objective of a DSS is to provide evidence for making decisions for a problem by compounding experts’ experiences and data and analyzing them in an intelligent and fast way a human cannot do in reasonable time.
Hence to achieve research intent, the work has been organised in the following 3 steps:
a) The complex socio-technical system (e.g., the UTS) and the role of the CC in daily operations has been analysed through the Functional Resonance Analysis Method [@hollnagel2008fram] perspective. The FRAM is a method to analyse how the activities daily take places daily in the complex system and introduces powerful concepts as functional variability, dampening, adaptive capacity, functional resonance, etc.as well as a specific notation to model the systems that is described in section 3. However the lack of an effective formalization of FRAM prevents to carry out the quantitative assessment of the impact of the IoE and Connected Communities in the UTS resilience building.
b) A new formalization and a method to quantify the FRAM functional variability and the dampening capacity has been defined
c) The new method to quantify FRAM has been tested in a case study comparing the potential variability in UTS with or without IoE and Connected Community exploitation. The benefit in terms of dampening capacity has been quantified.
This article is organised as follow: in Section 2, the role of people in the context of UTS is presented; in Section 3, we introduce the background work about the FRAM based Critical Infrastructure Reference Model published in the RESOLUTE European Resilience Management Guidelines; in Section 4, a new methodology to quantify the variability in FRAM is proposed; in Section 5, an example of the application of the methods to govern CC behavior during emergency exploiting IoE technologies is represented; in Section 6, conclusions and next steps are discussed.
Urban Transport System and people
=================================
In the UTS, operations have developed a prominent safety and business critical nature, in view of which current practices have shown the evidence of important limitations in terms of resilience management. Hence, enhancing resilience in UTS is considered imperative for two main reasons:
a) such systems provide essential support to every socio-economic activity and rescue, and
b) the paths that convey people, goods and information, are the same through which risks are propagated and resource are provided [@esrel].
Unfortunately, even if the UTS plays a critical role in the society, there is a general tendency to leave out from resilience strategy implementation, crucial aspects such as the coordination and synchronisation amongst several system functions and elements as the community preparedness and behavior and the need to account for a wide range of unknown scenarios and context dependent factors. In fact, humans do not have the time, the mental resources or the capability to be aware of every problem at the same time. They devote their energy to problems that involve them and for which they can make a difference - J. E. Grunig quoted in Leffler [@MerrilLef]. Thus the community members need to be enabled and engaged as an active participant in his/her own safety developing a self-resilience attitude. For instance, in a situation where the number of options to escape from a hazard are limited or absent because of the presence of constraints as bridges or tunnels, CC members can help each other or receive valuable information from the first responders to adopt specific behaviors to mitigate the impact of the event. This requires new technologies, new skills and new approaches to enable users in being connected anytime and everywhere to provide and receive lifesaver information and adapt their behavior accordingly. In particular, every aspect of the human factor (behavior, attitude, belief, sentiment, skill, heuristics, etc.) needs to be considered in a critical infrastructure like UTS, as key elements for resilience building, going beyond the engineering and operational approaches that tend to be focused on technologies and procedures. Moreover, the different nature of UTS users (i.e. cars, motorcycles bicycles, pedestrians, among others) and the wide diversity of purposes encompassed within urban transport, tend to generate highly dynamic interdependencies, both within the private transport system and with public transport. It is clear that, governing/directing CC behaviors during the UTS usage, is the basis of solutions towards enhanced resilience discussed in this article.\
Beyond the aspects of system complexity, the global scenario of resource scarcity and changes is also put forward as a cause for many of the serious safety and security threats currently faced by societies. In [@Boin], such threats are distinguished from [*routine emergencies*]{} such as fires and traffic accidents, and characterise them as [*low-chance*]{}, [*high-impact*]{} events that can compromise life sustaining systems and require governmental intervention under high uncertainty conditions. Both are the circumstances in which resilience is highlighted as a possible solution for the sustainability, reliability and safety of systems [@Boin] and [@jackson]. In fact, in resilience engineering field, there is not a difference between routine and big events, what change is the amplitude of the functional variability and thus the possibility of emerging resonance effect among the system functions.
Such an IoE enabled data-driven approach provides the means to assess the levels of criticality at evidence/quantitative level, while seeking to enable the capabilities of the complex system to take the appropriate decision at strategic, tactical and operational levels [@bellini2014km4city].\
Understanding UTS Behavior through FRAM
=======================================
The system analysis is based on the Critical Infrastructure reference model defined in the European Resilience Management Guidelines[^5] where the human aspect and the community management is included in the system description. The FRAM [@hollnagel2008fram] was used to support system analysis, aiming to identify interdependencies and system emergent behaviors potentially relevant for resilience. The FRAM approach is essentially a system-modelling tool that focuses on system interdependencies, their dynamics and complexity. It is grounded on Resilience Engineering principles and provides a fundamental support to such ends by supporting systems understanding. In particular, a system is considered a set of coupled or mutually dependent functions. FRAM is particularly relevant in describing nonlinear systems and the overall rule is to try achieving a description of the normal activities performed by the socio-technical system involving stakeholders in its definition. A FRAM model is illustrated in Figure \[fig:FunctionalUnitFRAM\] were a [*function*]{} $F$ is composed by a label (usually a verb) representing the action of the function and by six [*aspects*]{} $A$.
![Functional unit of FRAM (adapted from Hollnagel, 2008).[]{data-label="fig:FunctionalUnitFRAM"}](./img/FunctionalUnitFRAM){width="12cm"}
Hollnagel in [@hollnagel2008fram] defines the six [*aspects*]{} in the following terms:
1. **Input**: that which the function processes or transforms or that which starts the function.
2. **Preconditions**: that must exist before a function can be executed.
3. **Resources**: that which the function needs or consumes to produce the output.
4. **Time**: as temporal constraints affecting the function (with regard to starting time, finishing time, or duration).
5. **Control**: how the function is monitored or controlled.
6. **Output**: is the result of the function, either a specific output or product or a state change.
It is important to notice that the first five aspects (Input, Preconditions, Resources, Time, Control) are acting as [*inputs*]{} while the function [*outputs*]{} are represented only by the Output aspect. The characterisation of the functions, in terms of the six aspects, contains the potential couplings among functions. In fact, the input aspects of a downstream function can receive a qualified output from upstream functions. Such qualified output is a [*relationship*]{} $R$ labeled with a textual definition and representing the tangible or intangible outcome of the function of origin towards the function of destination. In fact, each output can be the input of another function.\
On the basis of FRAM approach, Figure \[fig:UTSmod\] reports the desired functions and interdependencies that a UTS needs to implements to be resilient[^6]. In Appendix Table \[tab:functions\] lists the functions composing the model with their relationships.
![RESOLUTE UTS Model from European Resilience Management Guidelines[^7] []{data-label="fig:UTSmod"}](./img/UTSmod){width="15cm"}
According to the ERMG, the human/social components of the socio-technical system are addressed, at least, by the following functions: [*Use of the service*]{}, [*Manage awareness and user behavior*]{}, [*Monitor user generated feedback*]{} which is defined as follow:
- Use of the service: This function represents the actual usage of the service like driving a car, goods movement and delivery, taking taxi, bus or metro, walking etc. in the UTS.
- Manage awareness & user behavior: as providers of fundamental public services, critical infrastructures tend to be significantly exposed to individual and collective behaviors, in many cases, not just the service end-users, but also of the wider public. Recent technological developments, in particular in relation to ICTs, offer a great potential for the enhancement of interactions with the public and the use of this potential towards an increased effectiveness in managing and deploying operational adjustments to various relevant events and circumstances.
- Monitor user-generated feedback: monitor feedbacks about service usage on a wide range of parameters and produce fundamental support to the deployment of operational adjustments. This function deals with the need for an integrated approach to the assessment of user generated feedback, mainly by placing this data and information in the context of operational monitoring.
Extending performance variability concept
-----------------------------------------
The FRAM approach is based on the principle of equivalence of successes and failures and the principle of approximate adjustments thus performance is therefore in practice always variable. As thoroughly explained by Hollnagel [@hollnagel2008fram] [*performance variability*]{}, i.e., the range of result in a function or an overall system’s performance, is highly dependent on the variability of the conditions under which the system/function is performing.
Starting from the generic six main sources of human and organisational performance variability defined in [@hollnagel2008fram], it is possible to identify the following in the UTS domain:
- Fundamental human physiological and/or psychological characteristics as driving fatigue, vigilance, attention, risk perception of UTS users, etc.
- Pervasive higher level psychological phenomena like adaptability as taking decisions within UTS knowledge uncertainty.
- Organisational conditions and requirements, as the need to meet external demands, stretching resources, substituting goals, etc.
- Social or team psychological factors, such as meeting expectations of oneself or of colleagues, complying with group working standards, etc.
- Context variability: roads conditions are too hot, too noisy, too crowded, etc.
- Environment variability induced by the unpredictability of the domain, e.g., weather conditions, technical problems, etc.
According to FRAM, performance variability is assessed through the eleven Common Performance Conditions (CPC), verifying if their performances are stable or variable but adequate, stable or variable but inadequate, or unpredictable [@hollnagel2012]. The variability about the way a function is carried out may show itself by the variability of its [*output*]{}. Since the results generated in [*output*]{} by a function can affect other aspects, namely [*input*]{}, [*precondition*]{}, [*resource*]{}, [*time*]{}, or [*control*]{}, of one or more downstream functions. Note that, the range of behaviors and effects captured by the [*output*]{} element is very broad and includes any exchange of matter, energy, or information. The [*output*]{} can be seen as representing a change of state in the system or in one or more aspects of downstream functions. But the [*output*]{} can also represent a decision or a signal that starts a downstream function. Moreover, in complex and non-linear systems predicting the specific outcomes of a function can be hard or unmeaning. For this reason, the literature concentrated on characterizing function variability in term of performances. The following dimensions, that are a combination of what proposed in the FRAM Glossary[^8], are considered relevant in UTS:
- Timing: too early, on time, too late, not at all.
- Duration: too little, too much, right duration.
- Distance: too close, too far, right distance.
- Magnitude: too strong, too weak, right magnitude.
- Speed: too fast, too slow, right speed.
- Force/power/pressure: too high, too low, right force.
- Precision: precise, imprecise, right precision.
- Volume: too much, too little, right volume.
- Costs: cost effective, costly, too much expensive.
In this work, we refer to the performance variability of a single function as [*Function Performance Variability*]{} (FPV). We also underline that the FPV of upstream functions may affect the FPV of downstream functions, and thereby lead to non-linear effects called functional resonance. A resonance phenomenon in physics usually results in a significant increase in the amplitude of the oscillations, which corresponds to a considerable buildup of energy within the stressed system. Similarly, functional resonance in the system emerges when the variability is spread through the interdependencies of the system functions causing the amplification of the effects until the system loses its capability to manage variability safely.\
Even if the variability in function execution performance can be derived by the variability of its [*output*]{}, the impact of such a variability over the system cannot be determined by observing the variability of output values only. In particular, we argue that it also depends on the variability acceptance supported by the function receiving inputs. In fact, the functional resonance effect is triggered by the rest of the variability of the upstream function output that is not absorbed by the downstream function. Moreover, the impact of variability is then intrinsically associated to relationships coupling outputs and inputs and can be expressed by the matching between output variability and input dumping capacity. This approach extends the current conceptualization of the FRAM providing a new concept useful for its formalization as discussed in the subsequent sections.
Formalising FRAM
================
In order to develop analytics over a FRAM representation, it is necessary to formalize the description of the target system. A similar attempt has been done by Cambrensis[^9] where FRAM has been formalised with a dependency model based on Bayesian Belief Network to quantify functional variability. Such approach presents several advantages. It allows a rigorous formalization and the automatic update of all the relevant interdependencies among the FRAM functions, iteratively. Even if the variability propagation can be modeled weighting the arcs in the BBN, the basic assumption behind this approach is that the entire variability of the upstream function output affects the downstream function performance. This means that any kind of adaptation capable of dampening input variability exhibited by a function, is not taken into account. For instance, the human resources usually engaged by a function could be incremented to absorb the arrival delay (variability) of an input in order to produce the expected output in due time\
The instantiation of a FRAM is usually depicted as a directed graph where nodes represent the functions with their six aspects, taking the shape of a hexagon, and edges represent qualified relationships among functions by interconnecting two aspects. This representation is essentially oriented to human readability and does not offer any support to quantitative analysis. In order to improve the current state of the art in executing quantitative analysis over a FRAM representation we are facing 4 objectives:
- [**O1**]{}. Representing dependencies among functions as well as qualified relationships.
- [**O2**]{}. Representing the matching between performance variability and damping capacity intrinsic to relationships.
- [**O3**]{}. Integrating quantitative and perception-based observations.
- [**O4**]{}. Test our method with a contingency plan by comparing a scenario with IoE and CC deployed in UTS and a scenario without such technologies.
Dependability analysis
----------------------
A typical analysis to be carried out on a FRAM model is related to the identification of the dependencies among functions. The aim is to look at the couplings among functions in order to identify whether they will lead to unwanted outcomes that may compromise the process.
In Systems Engineering and Risk Management [@thalmann2014integrated] Dependability is typically estimated by the number of originated errors, using metrics such as [*Mean Time To Failure*]{} [@delong2005dependability]. However, this approach imposes onerous observations and tests in a posteriori analysis and subjective observations in a priori analysis. For this reason, we propose to implement a quantitative analysis of the dependencies by representing the connections among functions and relationships using graph metrics [@hernandez2011classification]. As stated in [**O1**]{} we do not want to limit our attention to functions. Several relationships may be originated from a single function, thus to distinguish them and to measure their position in the graph we need to include them in the set of nodes considered by our analysis.
The most proper way to represent relationships $r \in R$ in a FRAM model is to use a quadruple $r = \{o, d, a, qn\}$, where $o \in F$ is the origin or upstream function, $d \in F$ the destination or downstream function, $a \in A$ specifies the FRAM aspects involved in the relationship, while $qn \in QNames$ is a qualified name for the relationship[^10]. Note that the triple $\{o, d, a\}$ does not represent a sufficient condition for identifying a relationship as multiple links may interconnect two functions along the same aspect. In fact, the set of origin functions is included in the set of functions with relationships along the [*output*]{} aspect, or more formally: $O \in F \times A | a = output$. Similarly, the set of destination functions is included in the set of functions having [*input*]{}, [*precondition*]{}, [*resource*]{}, [*control*]{} or [*time*]{} as aspects, more formally: $D \in F \times A | a \in \{input, precondition, resource, control, time\}$.
These notions can be exploited to inspect, with an analytical perspective, the dependencies characterizing a FRAM, using a matrix to encode the graph structure resulting by the unification of the connections between functions and relationships. The simplest approach is to generate an [*adjacency matrix*]{} of a bipartite graph, i.e., a matrix $M$ that records the connections between two classes of objects, in our case $F$ and $R$, such as its element $m_{i,j} \in (0,1),$ is $1$ if $i$ and $j$ are related and $0$ if they are not. The properties of the matrix $M$ can be specified by stating that
$$M = \left(\begin{array}{cc}0_{f,f} & B \\B^{T} & 0_{r,r}\end{array}\right),$$
where $B$ is an $F \cup R \times F \cup R$ matrix, $B^{T}$ is its transpose, and $0_{f,f}$ and $0_{r,r}$ represent the $F \times F$ and $R \times R$ zero matrices. Moreover, not all possible connections in $B$ and in $B^{T}$ are allowed because $F$ is the union of two disjoint sets $O$ and $D$; where, by definition, all $m_{i,j}$ with $i \in D$ and $j \in R$ or $i \in R$ and $j \in O$ are equal to $0$. For example, if we know that the [*output*]{} of function $F13$ gives [*input*]{} to function $F14$ with a relationship named [*User Behavior*]{}, we can express the following by encoding two connections: [*F13 $\rightarrow$ F13:UserBehavior:F14:Input*]{} and [*F13:UserBehavior:F14:Input $\rightarrow$ F14*]{}. The resulting network has been weighted according to the importance of the relationship in the system. The weights assignment task has been conducted within the RESOLUTE project translating workshops and stakeholders interviews with Civil Protection and City Council managers, firefighters and citizens.\
The union of these connections provides us with a graph. A broad variety of measures to characterise graphs are exploited in several scientific domains [@scott2012social]. The ratio between the number of vertices and edges reveals the [*Sparsity*]{} of a graph. The [*Clustering Coefficient*]{} is a measure of the degree to which nodes tend to cluster together. [*Node Centrality*]{} gives a measure of how central in the overall graph a node is. The method we adopted to quantify dependability of FRAM model is the Degree Prestige (DP) index [@Freeman], a metric accounting the number of inward connections entering in a node. We claim this is the right choice, because in FRAM a) the number of connections (explicit), b) the importance of connections (implicit), c) the direction of the relations (explicit) are critical elements characterizing the functions identified. Thus using a weighted and directed graph DP is the sum of weights of all connections ending at a given node, where nodes with higher DP are considered more prominent among others because they receive more inbound “heavy” connections. The largest the index is, the more prestigious/important the node is.\
Clearly, other approaches can be followed. For instance, [*Closeness Centrality*]{} measures node centrality by considering the geodesic distances a node has with all the other nodes of the graph; the [*Betweenness Centrality*]{} is calculated based on the number of shortest paths that pass through a node [@brandes2001faster]. However, as stated in [@borgatti2005centrality], the importance of a node in a network cannot be determined without reference to how traffic or information flows through the network. For example, in a package delivery process, the essence of closeness is time-until-arrival, in contrast, the essence of betweenness is frequency of arrival. These interpretations do not seem to be able to represent the FRAM characteristics properly, where peripheral nodes may also result critical.\
Figure \[fig:indegree\] shows the graph obtained by encoding the FRAM specified in Table \[tab:functions\] and Table \[tab:relations\] available in the Appendix and ordering nodes in concentric range based on their DP value. In Tables \[tab:FanInDC\] and \[tab:RelInDC\], node values are listed in decreasing order and we can observe that the functions exposing the highest values are [*F2:Coordinate service delivery*]{}, [*F16: Manage awareness and human behavior*]{}, [*F1: Delivery service*]{}, [*F24:Collect event information*]{} and [*F6: Coordinate emergency action*]{}.\
![A graph $G_{f,r,m}$ ordering nodes in concentric ranges based on their DP value.[]{data-label="fig:indegree"}](./img/indegree){width="13.5cm"}
It is worth to notice that function $F16$, that is devoted to managing community behavior and awareness is considered one of the most critical in the network. Such a result is not unexpected and confirms and formalizes what the stakeholder thoughts and the FRAM model are able to expresses only implicitly. Moreover, the DP centrality approach is able to represent the FRAM background functions[^11] assigning 0 to the DP index.\
Regarding the relations affecting connected communities let us focus our attention on: [*R106*]{} (F13:User Behavior:F14:Input), [*R107*]{} (F13:User Feedback:F14:Input), [*R108*]{} (F14:User Behavior data:F16:Resources), [*R109*]{} (F14:User generated critical event detection:F2:Input), [*R110*]{} (F14:User generated critical event detection:F6:Input), [*R108*]{} (F14:User generated service improvement suggestions:F24:Input); the analysis revels that R106, R107, R107 belong to the most important group, characterised by a DP equal to 0,64935; while the nodes R109 and R110 belongs to the second most important group, with DP equal to 0,58442, as reported in Table \[tab:RelInDC\].\
This analysis reveals that people-community related issues represent a critical aspect of the UTS resilience management. In fact, if the variability of the outputs of those functions are exceeding the dampen capacity of the downstream functions, such variability surplus is propagated in the system exhibiting a resonance behavior that can be preparatory for a disaster. It is then clear that exploiting the IoE technologies enhances the capacity of such functions of damping the performance variability that can be generated by information delivery delay, misunderstandings, etc. The IoE adoption may speed up the co-production and the dissemination of information within the CCs (e.g. created during an emergency) and between these CCs and the first responders.\
To reduce the size and complexity of the graph, one may also consider aggregating connections insisting on the same dimension, for instance, all the connections with the same origin function and destination function insisting on the same FRAM parameter. To manage size reduction consistently, standard approaches for multidimensional data such as OLAP Cubes [@ciferri2013cube] may be implemented, but ad hoc projection operators accounting data aggregation with a domain specific approach are also possible [@markines2009evaluating].
Quantify Functional Variability
===============================
As previously discussed, in FRAM several dimensions with qualitative degrees are characterizing the FPV. However, even considering invariant the FPV of an origin function, the impact of this variability on the resonance vary based on the [*dampening capacity*]{} of the destination function. As stated in [**O2**]{}, quantify such an impact is crucial. The current approaches do not offer any method to compare FPV over the capacity of the downstream functions of continuing to operate within normal variability in the face of varied inputs. Thus, we define the [*function dampening capacity*]{} (FDC) of a function $F$ as the capability of $F$, in a certain context, of absorbing the variability of the incoming input $I$ (changing conditions) maintaining its output $O$ within acceptable/expected variability.\
We also argue that the factors composing the FDC index are the four properties considered for resilience assessment at system level (buffer capacity, flexibility, margin and tolerance) and introduced in [@Woltjer]. Hence, the FDC in a certain instant $t$ for a specific input $i$ is given by its function buffer capacities (FBC), function flexibility (FF), function margin (FM) and function tolerance (FT). However, in which degree those functions contribute to the FDC require further analysis and will be matter of next researches.\
The formalisation we are proposing in this paper is aimed at quantifying the amount of FPV in upstream exceeding the FDC of a downstream function. In particular, we call this matching the [*Variability Rate*]{} (VR). The VR expresses the amount of input variability dampened or absorbed by the downstream function avoiding effects on its subsequent outputs.
Variability Rate {#Sec:QFV}
----------------
A naif solution to quantitatively measure FDC is to interpret it as the inverse of a correlation. We compare the distribution of the performances of two connected functions along with a specific dimension. If we observe a correlation this can be considered a clue for an amplification effect on the downstream function, generated by the upstream function. Thus we have to consider that the downstream function has poor FDC. For example, in Table \[tab:perf\] we list the performances of the [*output*]{} of functions F15, F2, and F6, measured on the timing dimension, using delay, expressed in hours, as value. Note that $F15$ is the upstream function of both $F2$ and $F6$. The Pearson correlation coefficient for $F15$ and $F2$ is $0.905$ while for $F15$ and $F6$ is $-0.153$. Thus we could conclude that $F6$ has a good FDC while $F2$ has not.
[p[.25]{} p[.03]{} p[.03]{} p[.03]{} p[.03]{} p[.03]{} p[.03]{} p[.03]{} p[.03]{} p[.03]{} p[.03]{}]{} $\mathbf{Function~Performances}$ &\
F15: Manage financial affaire & 0 & 24 & 36 & 168 & 24 & 24 & 24 & 36 & 72 & 0\
F2: Coordinate service delivery & 0 & 1 & 24 & 96 & 0 & 1 & 2 & 3 & 2 & 2\
F6: Coordinate emergency action & 1 & 1 & 2 & 1 & 3 & 1 & 1 & 1 & 2 & 2\
\
This approach is, however, too much influenced by the internal variability of a function. As a matter of fact, our purpose is not accounting the variability in general but the variability generated by those performances that bring the function outside a margin of regular operation. This means we are not interested in accounting those performances that are within the margin. The approach we are proposing is centred around the idea of computing how much a specific performance is differing form an expected value and comparing this value to the margin that delimits regular performances. Formally this can be defined as in Equation \[eq:dev\], where $dev$ is the deviation, $x$ the observed performance, $e$ is the expected or more representative performance value and $m$ the margin of regular operation. When this fraction ranges in the interval $[-1, 1]$ the difference between the observed and the expected value is within the margin. Note that $e$ and $m$ could be defined as constant values or as the result of a function, for example, in a pawer low distribution, $m$ cloud be obtained by a function of $e$.
$$\label{eq:dev}
dev = \Big| \frac{(x - e)}{m} \Big|$$
If we use the [*mean value*]{} as $e$ and the [*standard deviation*]{} as $m$, our $dev$ is equivalent to the $z$-score. Clearly, the assumption of normal distribution required by the $z$-score is too restrictive for complex systems such as the UTS. In Table \[tab:perfdev\] we computed the deviations of F15, F2 and F6 using two different approaches. In $dev^{a}$ we use the [*median value*]{} as $e$ and the [*median absolute deviation*]{}[^12] as $m$, because the median is more robust than the mean to bias in skewed distributions. While in $dev^{b}$ we use ad-hoc thresholds: in particular, $0$ as $e$ and $24$ as $m$ for $F15$ or $1$ as $m$ for $F2$ and $F6$. For example, when the delay of $F2$ is $24$ hours, because the median value of the series of observations is $2$ and the median absolute deviation is $1$, $ dev^{a} = \frac{(24 - 2)}{1} = 22$.
[p[.25]{} p[.03]{} p[.03]{} p[.03]{} p[.03]{} p[.03]{} p[.03]{} p[.03]{} p[.03]{} p[.03]{} p[.03]{}]{} $\mathbf{dev^{a}}$ &\
F15: Manage financial affaire & 2 & 0 & 1 & 12 & 0 & 0 & 0 & 1 & 4 & 2\
F2: Coordinate service delivery & 2 & 1 & 22 & 94 & 2 & 1 & 0 & 1 & 0 & 0\
F6: Coordinate emergency action & 0 & 0 & 1 & 0 & 2 & 0 & 0 & 0 & 1 & 1\
$\mathbf{dev^{b}}$ &\
F15: Manage financial affaire & 0 & 1 & 1.5 & 7 & 1 & 1 & 1 & 1.5 & 3 & 0\
F2: Coordinate service delivery & 0 & 1 & 24 & 96 & 0 & 1 & 2 & 3 & 2 & 2\
F6: Coordinate emergency action & 1 & 1 & 2 & 1 & 3 & 1 & 1 & 1 & 2 & 2\
\
Using performance deviations we can now compute the FPV of an upstream function and the FDC of a downstream function, we can then quantify a matching between them to measure the VR.
Since the variability of a function is exhibited in its [*output*]{} variability, the evaluation of the FDC of a downstream function can be performed by evaluating the variability of its output in relation to the variability of the input received.
In particular, if the function has received inputs with a certain level of variability and the output of the function exhibits the same or increased level of variability, this means that the current FDC of the function was not enough to dampen incoming variability. The result is the variability propagation effect in the system that is called functional resonance in FRAM. Formally, we compute FPV as in Equation \[eq:fpv\], where $[1, ..., n]$ is the set of observations considered, i.e. distinct executions of the process. While FDC is given by summing the differences between deviations of the upstream and downstream functions for the same observation, referred as origin function ($O$) and destination function ($D$) in Equation \[eq:fdc\]. The percentage of VR on a pair upstream, downstream function is then calculated as the ratio defined in Equation \[eq:vr\].
$$\label{eq:fpv}
FPV(F) = \sum_{i=1}^{n} dev_{i}~|~dev_{i} \geq 1.$$
$$\label{eq:fdc}
FDC(D | O:q) = \sum_{i=1}^{n} dev_{O,i} - dev_{D,i}~|~dev_{O,i} \geq 1.$$
$$\label{eq:vr}
\% VR_{O,D} = \frac{FDC(D|O)}{FPV(O)} \cdot 100.$$
For example, as the $FPV_{dev^{a}}(F15)$ is $22$ and the $FDC_{dev^{a}}(F2 | F15)$ is $-76$, then the $VR_{dev^{a}}(F15, F2)$ we observe is $-345\%$, indicating that the dumping capacity is negative, or, in other words, we have an amplification. On the contrary, the $FDC_{dev^{a}}(F6 | F15)$ is $18$, then the $VR_{dev^{a}}(F15, F6)$ we observe is $81\%$, indicating that $F6$ can operate regularly even in presence of delays on $F15$. Note that these results are very much influenced by the margin of regular operation chosen, the $e$ and $m$ values in Equation \[eq:dev\]. For example, we have $VR_{dev^{b}}(F15, F2) = -658\%$ and $VR_{dev^{b}}(F15, F6) = 41\%$. However, it is also clear that the same orientation, in terms of positive or negative dumping capacity, is returned with different specification of Equation \[eq:dev\].
In order to visually represent the $VR$ as a ratio between $FDC$ and $FPV$, we propose to exploit an adjacency matrix, i.e. a square matrix such that its element $m_{i,j}$ is representing a relationship among two objects of the same class $P$; having the same set of objects encoded in both the rows and the columns of the matrix. This implies that the relationships encoded in an element $m_{i,j}$ are directed from the object $p_{i}$ to the object $p_{j}$, while the inverse relationship, directed from $p_{j}$ to $p_{i}$, is encoded in the element $m_{j,i}$.
Let us, for example, encode in a square matrix the relationships in $O \cup D$, i.e., the set of origin and destination functions. Using a Chord Diagram [@holten2006hierarchical], we can display the relationships encoded in the matrix drawing arcs connecting the objects in $O \cup D$ arranged radially around a circle. This approach is particularly appreciated by the end user as the radial order allows to estimate the global inter-connection density of the system as well as to explore each single relationship within a compact space.
Figure \[fig:Chord\] illustrates the Chord Diagram that describes the FRAM specified in Table \[tab:functions\] and Table \[tab:relations\]. Note that this diagram allows interactive selection of relationships, to visualize their VR ratio, as illustrated in Figure \[fig:ChordZoom\].\
![A Chord Diagram illustrating the VR characterising each relationships in $O \cup D$.[]{data-label="fig:Chord"}](./img/ChordFuzzy){width="12.5cm"}
![Interacting with the Chord Diagram to observe the VR characterising a specific relationship.[]{data-label="fig:ChordZoom"}](./img/ChordZoom){width="12.5cm"}
Encoding FRAM Variability Observations with fuzzy logic
-------------------------------------------------------
It is worth to notice that the FPV of a function is the result of multiple observations, possibly generated from different sources, where different encoding procedures may be applied. In many situations, it is naive to believe deriving the VR based on pure quantitative measurements. Hence, we propose to integrate the method described in Section \[Sec:QFV\] with subjective reviews, using perception based definitions, as requested by [**O3**]{}. The idea is that: when a quantitative value is not available, this value may be determined through the collection of a set of perception based valuations from domain experts. The imprecise nature of this kind of information is mitigated by focusing on the most representative opinion.
Many models are available for representing uncertain knowledge [@damiani2009toward]. In order to clarify the notion of uncertainty, it is important to distinguish between degrees of truth and degrees of uncertainty in the information. A degree of truth can be defined as the degree of compatibility between a statement and a fact. The uncertainty of a statement arises when there is no sufficient information to decide if a statement is true or false. In our study, the problem we want to resolve is related to the identification of a representative majority, the strength of this majority and the valuation reflecting the judgment of the majority [@ceravolo2007bottom]. The typical approach is to compute a value which synthesizes the opinion of the experts involved in the evaluation, but because different majorities are possible, we prefer to consider all of them, weighting their relevance based on how close they are to our idea of a full majority. In other words, we need to express degrees of truth. A formal method to deal with it is offered by Fuzzy Logic [@klir1995fuzzy], which has been largely exploited for aggregating opinions [@herrera2014review], votes [@ceravolo2005adding] or imprecise information [@herrera1997aggregation], in general. In classical set theory, the characteristic function of a set $E \subseteq D$ is a function assigning $1$ or $0$ to each element of the domain $D$, depending on whether the element is in the subset $E$ or not. In Fuzzy Logic, the characteristic function of a set $\mu E$ returns for each element in the domain $D$ a value $\omega$ in the interval $[0, 1]$, which represents the grade of membership of this element in $E$. Fuzzy sets can be used to partition the distribution of a variable in ranges corresponding to perception-based quantifiers, for instance [*Very low*]{}, [*Low*]{}, [*Average*]{}, [*High*]{}, [*Very High*]{}. These concepts, usually referred as linguistic variables, are exploited during reasoning to transform perception based observations into membership degrees. To provide a formal framework to address this problem we adapted the approach followed in [@pasi2003modeling]. The general idea is to interpret the representative majority no longer as a single value, but as a fuzzy set. This fuzzy set includes all the possible subsets representatives of a majority within the collection of values expressing all the valuations. This require for the identification of both the strength of a majority and the synthesized value expressed by this majority.
Once multiple observations from experts are gathers in a bag of valuations $E = \{ e_{1}, ..., e_{n} \}$, we are requested to define a characteristic function for identifying similar values. For example, asking that two values $e_{i}$ and $e_{j}$ are similar if their difference $\delta$ is not too far from a point of realisation $\epsilon$. Let us call $\mathcal{S}$ the characteristic function implementing this idea. The input of this function is $\frac{\delta}{\epsilon}$, this value is accounted as the intensity of the similarity, in the interval $[0, 1]$, except for values $> \gamma$, an upper bound, that are accounted as $0$, as illustrated in Figure \[fig:majop\]. Note that $\mathcal{S}$ is not a proper similarity as it is symmetric but not transitive, nevertheless this approach offers a very simple test condition. Another requirement is we define a characteristic function $\mathcal{M}$ for identifying a majority. This function evaluates the cardinality of a subset $X_{i} \subseteq E$ to define how intensively it can be considered a majority. $\mathcal{M}$ has a lower bound $\zeta$ that defines subsets we cannot consider a majority, moreover, it defines the intensity of a majority by computing $\frac{|X|}{|E|}$, as illustrated in Figure \[fig:majop\]. Now we can identify a majority if $X_{i}$ contains elements that are similar and its cardinality satisfies our idea of being a majority. Let us formalise this notion by stating that a subset $X_{i} \subseteq E$ is a majority with degree defined by a function $Maj(X_{i})$, where:
$$Maj(X_{i}) = min( \mathcal{M}(X_{i}), \mathcal{S}(X_{i})), with~\mathcal{S}(X_{i}) = Min_{e_{i}, e_{j} \in X_{i} } [\mathcal{S}(e_{i}, e_{j}) ].$$
To consider all subsets of $E$ we have to generate the power set of $E$, i.e. we have to consider $2^{|E|}$ subsets. The intrinsic complexity of this approach may seem excessive, however note that the characteristic functions $\mathcal{S}$ and $\mathcal{M}$ act as filters for several subsets that are discarded. Let us, for example, consider the following bag of opinions where values are drawn from a scale from 0 to 10:
$$E = \{1, 4, 4, 5, 6\}.$$
We have $2^{5} = 32$ subsets. Nevertheless, taking as $\mathcal{M}$ the characteristic function illustrated in Figure \[fig:majop\] all subsets with two elements are discarded because $ \frac{|X|}{|E|} < 0.4$. Moreover, shaping $\mathcal{S}$ as illustrated in Figure \[fig:majop\], any subset having any of its elements with a difference $ \geq 3$ is also discarded[^13].
Thus the following are the only subsets for which $Maj(X_{i}) \neq 0: X_{1} = \{4, 4, 5\}, X_{2} =\{4, 4, 6\}, X_{3} =\{4, 5, 6\}, X_{4} =\{4, 5, 6\}, X_{5} =\{4, 4, 5, 6\}$. As illustrated in Table \[tab:majopex\], we can now compute $Majop(X_{i})$, i.e. the intensity of a majority, for each of them. The next step is computing the opinion expressed by each majority that we define as $Op(X_{i})$ = $AVG_{j}(x_{j} \in X_{i})$, i.e. the value averaging the opinions expressed in a majority. Based on $Maj(X_{i})$ we can weights $Op(X_{i})$. We define the weight of each majority as $W(X_{i})$, which is computed as:
$$W(X_{i}) = \frac{Maj(X_{i})}{\sum_{X_{j} \subseteq E} Maj(X_{j})}.$$
Where $\sum_{X_{i}} W(X_{i}) = 1$. Now, the most representative value for $E$ can be computed as a weighted mean of the values expressed by each majority in $X_{i} \subseteq E$. For example,
$$Majop(E) = \sum_{X_{i}} W(X_{i}) \times Op(X_{i})$$
Then, following our example, the proposed methodology determines the most representative value in $E$ as $4.75$, formally this is written as: $MajOp(E) = 4.75$. The method we presented can be applied to any value that is relevant for evaluating the FRAM. In Section \[sec:CS\] we are using this method to evaluate the Z-score characterising the FDC of the relationships analysed in the scenario we investigated.
[p[.15]{} p[.15]{} p[.15]{} p[.15]{} p[.15]{} p[.15]{} ]{} $\mathbf{ X_{i}}$ & $\mathbf{\mathcal{S}(X_{i})}$ & $\mathbf{\mathcal{M}(X_{i})}$ & $\mathbf{Maj(X_{i})}$ & $\mathbf{Op(X_{i})}$ & $\mathbf{W(X_{i})}$\
$X_{1}$ & 0.99 & 0.66 & 0.66 & 4.33 & 0.22\
$X_{2}$ & 0.66 & 0.66 & 0.66 & 4.66 & 0.22\
$X_{3}$ & 0.66 & 0.66 & 0.66 & 5 & 0.22\
$X_{4}$ & 0.66 & 0.66 & 0.66 & 5 & 0.22\
$X_{5}$ & 0.66 & 1 & 0.66 & 4.75 & 0.22\
$MajOp(E)$ & 4.75 & & & &\
\
![The characteristic functions $\mathcal{M}(X)$ and $\mathcal{S}(X)$.[]{data-label="fig:majop"}](./img/MajOp){width="13.5cm"}
Case Study {#sec:CS}
==========
In emergency it is very important to manage CC behaviors effectively in order to reduce the impact generated also by wrong decisions. To this end the UTS needs to properly address the community related system functions as: [*F16: Manage awareness and user behavior*]{} .
In this scenario, it is necessary to support community members in taking right decisions for their safety while addressing their mobility goals. The methodology considered in RESOLUTE is inspired by Bungay’s Directed Opportunism approach [@bungay] since it represents the main shift of the strategy, from centralised “command and Control" to “Mission Control" and it can be considered more appropriate to support the community self-resilience. The Bungay’s approach is a control loop composed by Outcome (e.g. reduction of the car presence in the affected area through redirection of the traffic flow towards other part of the road network), Plans (e.g. application of rerouting strategy), Actions (e.g. sets of traffic lights cycle, close streets, send recommendation message to city panels) steps. The objective of the approach is to left people free to take opportunistic decisions exploiting their local knowledge that is considered more accurate respect to the centralised one. However, to guarantee that such decisions are actually appropriate to the actual condition, the following gaps need to be crossed:
1. **Knowledge Gap**: the delta between what we would like to know and what we actually know.
2. **Alignment Gap**: the difference between what we want people to do and what they actually do.
3. **Effects Gap**: the difference between the expected and the actual results of our actions.
When these gaps are encountered, the intuitive response is to seek increasing the control on these areas by gathering more detailed information, providing more detailed instruction, and installing tighter controls. These intuitive responses typically have the opposite effect of their intent, creating greater confusion and entropy. In fact, in order to reduce these gaps it is necessary to apply the following actions:
1. Do not command more than is necessary or plan beyond the circumstances you can foresee (Knowledge gap).
2. Communicate to every one as much of the higher intent as is necessary to achieve the purpose (Alignment gap).
3. Being sure everyone is empowered to make decisions within bounds (Effects gap).
The data used for the analysis are directly derived by the 1st RESOLUTE stakeholder workshops held in Florence in December 2015, where several scenarios were analyzed (water bomb/flash flooding, large yard, river flooding, car accidents, etc.). For use case addressed in the present work, we take into account the flash flooding extreme event. This kind of event is charactersied by sudden, unpredictable and localised (radius of few kilometers) extreme heavy rain that is capable to badly affect UTS operations (cars speed reduction/ blocked, car accidents for reduced visibility or roads grip, fall trees, traffic lights out of order, underpasses flooded, etc.) in a specific part of a city.
This scenario is assessed considering cases in which: a) an UTS where the IoE and Connected Community are integrated in the system, and b) an UTS where such facilities are not exploited. In particular, as we explained, the IoE and Connected Community emerging concepts consider the people part of the system (e.g., Smart City). In this system, the people can be considered as always connected through their personal devices to informal, fluid and/or thematic communities exploiting different communication infrastructures (e.g. city free wifi or 4G/LTE).
This assumption is justified by the existence of WAN communication infrastructures with several overlaps (e.g., LoRaWAN) that are able to guarantee connectivity also in case of disaster. On the other hand, it is evident that in case of extreme disruption where all the communication and electrical infrastructures are damaged, the IoE and CC facilities cannot be exploited until such infrastructures have been restored.
The people want to access tailored and context aware data and services and stay in contact with the other community members every time everywhere to exchange information, suggestions, to share their opportunistic decisions, etc.. During the emergency such connections can be exploited to alert first responders about the position of the disaster, or can be used by the authorities to send personalized messages according to the 4R approach (right person at right time, in the right place, through the right channel) or generic messages for the entire community to speed up the dissemination of critical information. In the following tables, two scenarios mapped to the UTS FRAM function relations affected, are reported.
[p[.18]{} p[.30]{} p[.30]{} p[.10]{}]{}
Relations & Standard Scenario & CC Scenario & Gap\
[*F13:User Behavior:F14:input*]{} & User behaviors signals are basically captured by security/surveillance video systems with limits in view extensions, mass quantification, people velocity and direction, information processing, sharing capability, etc. During a water bomb people tends to reduce the speed up to block the local variability. Such event is very difficult to be propagated to the other drivers that are reaching the same area, thus they will be surprised by the event even if they arrive minutes after. In order to escape from the traffic jam, people tends to broke roads rules with U turns, reverse gear, double parking, etc. People tends to apply heuristics combined with local knowledge and risk perception to continue of addressing their movement objectives (e.g., go back home from work). & User as a sensor concept allow real-time people movement tracking through 3/4G, Public Wifi, bluetooth, etc. Positions, velocities, trajectories can be calculated and predicted in close to real-time. Such information can be shared with multi decision makers through dashboards. Providing personalised real-time and contextualised information and suggestions, allows people to take a decisions in due time (e.g. avoiding traffic jam if the driver is arriving in the affected area) [@fgcs]. & Knowledge, Effects Gap\
[*F13:User feedback:F14:input*]{} & People communicate with authorities or UTS operators through call centres, email or directly with the transport employees (e.g. bus drivers). Such channels are basically used for caring.
& People can post pictures and videos of a critical events on social networks in real-time with a impressive dissemination capability. Applications like WhatsApp, Telegram or XMPP allow instant multimedia messaging with groups and communities. & Effect\
[*F14:User behavior data:F16:input*]{} & Data about people behavior and movement are usually derived from slow dynamic data like seasonality of touristic visits, daily traffic pressure for work entrance/exit, etc. Such data are indicators that are useful for risk assessment while for real-time emergency decisions (e.g. directing first responders in a specific area) more dynamic and fresh information are necessary. & IoE and CC allow the possibility to deliver maps of people behaviors integrating different channels from social networks (georeferenced Tweets of pictures in Flickr), GPS signals from always connected smart devices, etc. & Knowledge, Alignment\
[*F14:User generated critical event detection:F6:input*]{} [*F14:User generated critical event detection:F2:input*]{} & During the emergency people alert first responders about a critical event basically through telephone call, but misunderstandings, partial descriptions of the scenario, wrong perception given by stressful situation, language gaps, etc. impact on the precision of the description and thus on the effectiveness of the emergency action. Moreover, in Italy there are several emergency numbers (113- Police, 118 Ambulance, 115 Fire brigades, etc.) that people can call for an emergency. This fragmentation determine multiple calls for the same event increasing the noise. & The possibility of taking and sharing pictures, videos, voice and text messages in real-time improve the quality of the information and the event understanding. & Knowledge\
[*F14:User generated service improvement suggestions:F24*]{} & Improvements are basically driven by post-event accident analysis and it is carried out by experts in the field without any participatory approach. & Opening a direct digital channel towards the users/citizens to collect their suggestions improve the understandability of the system usage and perception. Sentiment, clustering and statistical analysis on such a database can extract unexpected knowledge.
& Knowledge\
[*F16:User generated service improvement suggestions:F13*]{} & Authorities and UTS have a very partial idea in which place and what the people are doing when a critical event happen. Information is spread in broadcast using massive and pre registered telephone calls (if the people is registered to alerting service), megaphone, Tv and radio channels, variable message panels, etc.. The messages are usually synthetic with very few information and reach the intent is to reach more people as possible without any filter. Generally, people that are not prepared to cope with emergency, thus they tend to react according to their heuristics, emotion, past experience, local knowledge, etc. with an high level of uncertainty (e.g. people that try to save the car in the box during the flooding). & The capability of exploiting smart devices as a personal end point to communicate the right message to the right person at the right time through the right channel - The 4R approach - represents a relevant improvement offered by IoE to enhance UTS resilience addressing the human-social side. &\
\
Quantitative development of the case study
------------------------------------------
To develop a quantitative evaluation about the impact of migrating a system into a CC scenario we applied the methodology described in Section 5.2.\
The first step foresees the collection of the VR assessment provided by the experts.
In fact, the FRAM is basically a qualitative method, the VR can be evaluated looking at the expert judgment in [*input*]{} about the variability of function $F$ and the expert judgment about the variability of its subsequent [*output*]{}. Since, such linguistic variables express a variability range, the estimation of VR provided by the experts for each relations, can vary from one function to another even if the input and output variability judgments are similar. For instance, if the timing variability in [*input*]{} of a function F is evaluated as “too late" and the variability of $F$ [*output*]{} is evaluated as “in time", this does not mean that the VR estimated by the expert for $F$, should be equal to another function $G$ that has the same input and output variability judgments. In fact, VR estimation of a function $F$ should take into account its function buffer capacities (FBC), function flexibility (FF), function margin (FM) and function tolerance (FT). This means that, even if $F$[*output*]{} is judged “in time", the VR estimation is influenced by the assessment of the function aspects status at the instant $t$ considered. The VR estimation provided by 8 local experts (that has been selected carefully of the bases of their experience and decision level), against the flash flooding analysis in Florence is a number from 0 to 10 representing the VR percentage (for instance 1= 10%). The $MajOp(E)$ value is calculated based on the membership functions illustrated in Figure \[fig:majop\].\
[*F13:User Behavior:F14:input*]{}\
------------------------------------------------------------------------
\
[*Standard Scenario*]{}: $E = \{ 1, 0, 2, 2, 4, 5, 2, 2 \}$; $MajOp(E)=1.15$; $VR=11.5\% $\
[*CC Scenario*]{}: $E_{CC} = \{ 9, 8, 9, 7, 9, 8, 7, 7 \}$; $MajOp(E_{CC})=8$; $VR=80\%$\
[*F13:User feedback):F14:input*]{}\
------------------------------------------------------------------------
\
[*Standard Scenario*]{}: $E = \{ 1, 0, 2, 2, 1, 0, 4, 2 \}$; $MajOp(E)=1.15$; $VR=11.5\%$\
[*CC Scenario*]{}: $E_{CC} = \{ 9, 6, 6, 9, 8, 5,6, 7 \}$; $MajOp(E_{CC})=6.4$; $VR=64\%$\
[*F14:User behavior data:F16:resources*]{}\
------------------------------------------------------------------------
\
[*Standard Scenario*]{}: $E = \{ 1, 0, 0, 2, 0, 0, 2, 2 \}$; $MajOp(E)=0.87$; $VR=8.7\%$\
[*CC Scenario*]{}: $E_{CC} = \{ 10, 10, 10, 9, 10, 9, 9, 10 \}$; $MajOp(E_{CC})=9.62$; $VR=96.2\%$\
[*F14:User generated critical event detection:F6:input*]{}\
[*F14:User generated critical event detection):F2:input*]{}\
------------------------------------------------------------------------
\
[*Standard Scenario*]{}: $E = \{ 5, 4, 6, 2, 4, 5, 3, 6 \}$; $MajOp(E)=4.82$; $VR=48.2\%$\
[*CC Scenario*]{}: $E_{CC} = \{ 10, 10, 9, 9, 8, 10, 10, 6 \}$; $MajOp(E_{CC})=9$; $VR=90\%$\
[*F14:User generated service improvement suggestions:F24:input*]{}\
------------------------------------------------------------------------
\
[*Standard Scenario*]{}: $E = \{ 2, 0, 2, 0, 1, 2, 1, 2 \}$; $MajOp(E)=1.25$; $VR=12.5\%$\
[*CC Scenario*]{}: $E_{CC} = \{ 6, 6, 7, 8, 6, 5, 7, 7 \}$; $MajOp(E_{CC})=6.5$; $VR=65\%$\
[*F16:Warnings -Alerts:F13:resources*]{}\
------------------------------------------------------------------------
\
[*Standard Scenario*]{}: $E = \{ 1, 0, 2, 2, 4, 5, 2, 2 \}$; $MajOp(E)=1.65$; $VR=16.5\%$\
[*CC Scenario*]{}: $E_{CC}= \{ 10, 10, 10, 9, 10, 10, 9, 10 \}$; $MajOp(E_{CC})=9.75$; $VR=97.5\%$\
[*F16:Advice - Recommendation Alert:F13:resources*]{}\
------------------------------------------------------------------------
\
[*Standard Scenario*]{}: $E = \{ 1, 1, 3, 0, 1, 2, 0, 1 \}$; $MajOp(E)=0.99$; $VR=9.9\%$\
[*CC Scenario*]{}: $E_{CC} = \{ 10, 10, 10, 9, 10, 9, 9, 10 \}$; $MajOp(E_{CC})= 9.62$; $VR=96.2\%$\
The results show that, according the expert judgments, the percentage of the variability that can be potentially absorbed by the functions with the CC and IoE technologies, is significantly higher respect to the same functions that operate without such facilities (standard scenario). This means that thanks to the introduction of such new technologies the capacity to manage people and community during the emergency is inherently enhanced and the propagation of variability in the system is prevented or mitigated. In fact, VR is influenced by the FDC as well as the Output variability distribution. The VR score may result high also when the Output variability of the upstream is significantly reduced given the FDC of the downstream function. This could happen for instance when the communication processes managed by F16 becomes pervasive, ubiquitous and personalized thanks to the IoE and CC technologies. Thus the contribution of the CC and IoE is twofold: a) on FDC enhancement and b) on Output variability reduction. In this perspective, the evidence shows that introducing IoE and CC to enhance resilience in UTS represents an option whose benefits value from 6 to 8 times the VR increment.
Conclusions
===========
Connected Communities and the related enabling technologies (Personal Smart Devices, multiple communication networks as WiFi, Bluetooth, LTE, Smart Sensors, etc.) set the scene of a new class of emergency and decision support systems based on knowledge, real-time situational awareness and personalised communication. In the present article, the Connected Community concept has been applied to the UTS resilience scenario to demonstrate the capability of such a concept in addressing the human-social side of the emergency in a more effective way, enhancing the resilience of the system as a whole. To this end, we started from the RESOLUTE Resilience Management Guidelines, where a Critical Infrastructure reference model based on FRAM has been proposed. Then we have developed a new method to analyse and quantify function’s variability as a method to move towards resilience quantification. The application of a network science approach to the FRAM model, has revealed what are the most critical functions in the system, while a method based on deviation score, was used to define the general principle for variability quantification. Since in FRAM the assessment is based on qualitative judgment, a fuzzy logic based method was proposed to translate perception based observations into a quantification of the VR. In particular, a fuzzy notion of majority was adopted to guarantee a representative value. A scenario from those explored in RESOLUTE project, the water bomb, offered us the opportunity to compare and quantify the variability of those functions devoted to manage community aspects in UTS, considering two different contexts: where CC and IoE are deployed and where they are not (standard situation).\
The outcomes obtained from the expert judgments on VR estimation reveal a remarkable differences between the two cases. This result shows that a technological upgrade of the UTS community-related functions towards the IoE and CC, would have an impact on the system resilience as a whole. In fact, such a VR enhancement in a specific connection between two functions, may act as an adaptive levee through the reduction of output variability of the upstream function or the enhancement of the damping capacity of the downstream function. Thus, the propagation of the variability in the system through function interdependencies that may trigger the resonance effect is prevented or mitigated within a threshold of acceptance. We can conclude that the introduction of IoE and CC in UTS domain allows for the implementation of the next generation decision support systems, able to gather any kind of data generated by smart cities. The possibility of knowing where people are situated in a specific moment, their direction, velocity, and concentration, as well as, the possibility to reach them collectively or personally, in every time, everywhere, with tailored information, enhances the effectiveness of respond and recovery actions during emergencies. Nevertheless, the assessment of the global properties of a system, such as resilience, asks for the consistent integration of quantitative and perception based evaluations. Future researches will focus on these aspects by investigating alternative formalizations of the FDC, the quantification of the effects of mechanisms to simulate the variability propagation within the system, the development of a decision support system able to predict and provide recommendations on optimal resource allocation and technology upgrade to enhance VR in critical interdependencies.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work has been supported by the RESOLUTE project (www.RESOLUTE-eu.org) and has been funded within the European Commission H2020 Programme under contract number 653460. This paper expresses the opinions of the authors and not necessarily those of the European Commission. The European Commission is not liable for any use that may be made of the information contained in this paper.
Appendices
==========
The RESOLUTE FRAM Model
-----------------------
[p[.15]{} p[.55]{} ]{} ID & Function\
F1 & Deliver service\
F2 & Coordinate service delivery\
F3 & Manage human resources\
F4 & Training staff\
F5 & Supply resources\
F6 & Coordinate emergency action\
F7 & Repair/restore operations\
F8 & Maintain physical/cyber infrastructure\
F9 & Manage ICT resources\
F10 & Monitor safety and security\
F11 & Regulate domain and operation\
F12 & Define procedures\
F13 & Use of the service\
F14 & Monitor user generated feedback\
F15 & Manage financial affaire\
F16 & Manage awareness and user behaviour\
F17 & Develop strategic plan\
F18 & Provide adaptation and improvement insight\
F19 & Monitor operation\
F20 & Supply financial resources\
F21 & Perform risk assessment\
F22 & Monitor resource availability\
F23 & Provide risk warning\
F24 & Collet event information\
F25 & Fight the emergency\
\
[p[.06]{} p[.13]{} p[.40]{} p[.13]{} p[.15]{}]{}
ID & Origin Function & Qualified Name & Destination Function & Aspect\
R26 & F1 & Infrastructure performance & F 14 & Input\
R27 & F1 & Service & F13 & Resources\
R28 & F1 & Service performance & F19 & Input\
R29 & F1 & Service\_Safety\_Security\_performance & F 10 & Input\
R30 & F2 & Operation HR plan & F3 & Input\
R31 & F2 & Operation plan & F16 & Input\
R32 & F2 & Operation plan & F16 & Control\
R33 & F2 & Operation plan & F12 & Input\
R34 & F2 & Operation plan & F22 & Resources\
R35 & F2 & Operation Restore service request & F7 & Input\
R36 & F2 & Service delivery plan & F1 & Input\
R37 & F2 & Service delivery plan & F13 & Resources\
R38 & F2 & Service delivery plan & F16 & Resources\
R39 & F2 & Service improvement plan & F8 & Input\
R40 & F2 & Training staff requirements & F4 & Input\
R41 & F3 & human resources availability & F1 & Resources\
R42 & F3 & human resources availability & F6 & Resources\
R43 & F3 & human resources availability & F7 & Resources\
R44 & F4 & Staff trained & F1 & Resources\
R45 & F4 & Staff trained & F19 & Resources\
R46 & F4 & Staff trained & F19 & Precondition\
R47 & F4 & Staff trained & F6 & Precondition\
R48 & F4 & Training performance data & F24 & Input\
R49 & F5 & Supply resources & F1 & Resources\
R50 & F5 & Supply status & F2 & Resources\
R51 & F6 & Emergency HR request & F3 & Input\
R52 & F6 & Emergency response command & F25 & Input\
R53 & F6 & Emergency response data & F10 & Resources\
R54 & F6 & Emergency response data & F24 & Input\
R55 & F6 & Emergency response plan & F2 & Resources\
R56 & F6 & Emergency response plan & F10 & Input\
R57 & F6 & Emergency response status & F2 & Resources\
R58 & F6 & Emergency response status & F16 & Input\
R59 & F6 & Emergency response status & F10 & Resources\
R60 & F6 & Emergency response status & F19 & Resources\
R61 & F6 & Emergency response status & F1 & Resources\
R62 & F7 & Operation Restore service plan & F2 & Resources\
R63 & F7 & Operation Restore service plan & F16 & Input\
R64 & F7 & Operation restore/repair performance data & F24 & Input\
R65 & F7 & Operation restore/repair status & F2 & Resources\
R66 & F7 & Operation restore/repair status & F16 & Input\
R67 & F7 & Operation restored/repaired & F1 & Precondition\
R68 & F7 & Operation restored/repaired & F2 & Input\
R69 & F7 & Operation restored/repaired & F19 & Precondition\
R70 & F8 & Infrastructure installed maintained & F1 & Precondition\
R71 & F8 & Infrastructure resotore/repair performance data & F24 & Input\
R72 & F8 & Infrastructure resotore/repair plan & F2 & Resources\
R73 & F8 & Infrastructure restored repaired status & F2 & Resources\
R74 & F8 & Infrastructure restored/repaired & F1 & Precondition\
R75 & F8 & Infrastructure restored/repaired & F2 & Input\
R76 & F8 & Infrastructure restored/repaired & F2 & Resources\
R77 & F9 & ICT infrastructures & F1 & Precondition\
R78 & F9 & ICT infrastructures & F2 & Resources\
R79 & F9 & ICT infrastructures & F16 & Resources\
R80 & F9 & ICT infrastructures & F10 & Resources\
R81 & F9 & ICT infrastructures & F19 & Resources\
R82 & F9 & ICT infrastructures & F22 & Resources\
R83 & F9 & ICT infrastructures & F14 & Resources\
R84 & F9 & ICT infrastructures & F6 & Resources\
R85 & F9 & ICT resource performance & F22 & Input\
R86 & F10 & Safety Security control & F1 & Control\
R87 & F10 & Safety Security control & F13 & Control\
R88 & F10 & Safety Security control & F19 & Control\
R89 & F10 & Safety Security critical event detection & F6 & Input\
R90 & F10 & Safety Security performance data & F24 & Input\
R91 & F11 & Law & F1 & Control\
R92 & F11 & Law & F15 & Control\
R93 & F11 & Law & F2 & Control\
R94 & F11 & Law & F19 & Control\
R95 & F11 & Law & F7 & Control\
R96 & F11 & Safety regulation & F21 & Resources\
R97 & F11 & Safety regulation & F4 & Input\
R98 & F11 & Safety regulation & F12 & Resources\
R99 & F11 & Safety regulation & F10 & Control\
R100 & F11 & Standards & F1 & Control\
R101 & F11 & Standards & F2 & Control\
R102 & F11 & Standards & F19 & Control\
R103 & F11 & Standards & F7 & Control\
R104 & F12 & Procedure & F1 & Control\
R105 & F13 & Revenues & F15 & Resources\
R106 & F13 & User Behaviour & F14 & Input\
R107 & F13 & User feedback & F14 & Input\
R108 & F14 & User behaviour data & F16 & Resources\
R109 & F14 & User generated critical event detection & F2 & Input\
R110 & F14 & User generated critical event detection & F6 & Input\
R111 & F14 & User generated service improvement suggestions & F24 & Input\
R112 & F15 & Budget & F2 & Resources\
R113 & F15 & SLA(Service Level Agreement) & F2 & Control\
R114 & F15 & SLA(Service Level Agreement) & F22 & Resources\
R115 & F15 & SLA(Service Level Agreement) & F6 & Control\
R116 & F16 & Early warnings & F13 & Resources\
R117 & F16 & Service status & F13 & Resources\
R118 & F17 & Develop strategic plan & F15 & Input\
R119 & F17 & Strategic plan & F2 & Resources\
R120 & F17 & Strategic plan & F16 & Input\
R121 & F17 & Strategic plan & F8 & Input\
R122 & F17 & Strategic plan & F18 & Input\
R123 & F17 & Strategic plan & F24 & Control\
R124 & F18 & Event\_analysis\_insights & F21 & Input\
R125 & F18 & Knowledge base & F24 & Resources\
R126 & F18 & Service sustained adaptability improvement insights & F2 & Input\
R127 & F18 & System Sustained adaptability insights & F17 & Input\
R128 & F19 & Install Maintenance requirement & F8 & Input\
R129 & F19 & Install Maintenance requirement & F2 & Input\
R130 & F19 & Operation Critical event detection & F16 & Input\
R131 & F19 & Operation Critical event detection & F6 & Input\
R132 & F19 & Operation performance monitoring data & F16 & Resources\
R133 & F19 & Operation performance monitoring data & F24 & Input\
R134 & F19 & Operation requirements & F2 & Input\
R135 & F20 & Funds & F15 & Resources\
R136 & F21 & Risk assessment report & F4 & Input\
R137 & F21 & Risk assessment report & F2 & Resources\
R138 & F21 & Risk assessment report & F12 & Input\
R139 & F21 & Risk assessment report & F10 & Input\
R140 & F22 & Energy supply report & F2 & Input\
R141 & F22 & Resource supplied Critical event detection & F2 & Input\
R142 & F22 & Resource supplied Critical event detection & F6 & Input\
R143 & F23 & Official risk warning & F2 & Input\
R144 & F24 & Knowledge base & F18 & Resources\
R145 & F2 & Operation Plan & F9 & Input\
R146 & F5 & Supply Resources & F9 & Resources\
\
Analytics on the FRAM Model
---------------------------
[p[.15]{} p[.55]{} ]{} Function ID & Degree Prestige Centrality\
F2 & 10,13\
F16 & 5,3896\
F1 & 5,3247\
F24 & 4,2208\
F6 & 3,5714\
F19 & 3,3117\
F10 & 2,8571\
F13 & 2,7922\
F14 & 2,4675\
F22 & 1,6883\
F15 & 1,2338\
F8 & 1,1039\
F7 & 1,039\
F3 & 0,97403\
F12 & 0,84416\
F4 & 0,77922\
F21 & 0,71429\
F9 & 0,64935\
F25 & 0,64935\
F18 & 0,58442\
F17 & 0,45455\
F5 & 0\
F11 & 0\
F20 & 0\
F23 & 0\
\
[p[.20]{} p[.50]{} ]{} Relationship ID & Degree Prestige Centrality\
R26 & 0,64935\
R27 & 0,64935\
R28 & 0,64935\
R29 & 0,64935\
R49 & 0,64935\
R52 & 0,64935\
R68 & 0,64935\
R71 & 0,64935\
R74 & 0,64935\
R75 & 0,64935\
R76 & 0,64935\
R77 & 0,64935\
R89 & 0,64935\
R106& 0,64935\
R107 & 0,64935\
R108 & 0,64935\
R116 & 0,64935\
R117 & 0,64935\
R133 & 0,64935\
R67 & 0,58442\
R69 & 0,58442\
R70 & 0,58442\
R72 & 0,58442\
R73 & 0,58442\
R109 & 0,58442\
R110 & 0,58442\
R130 & 0,58442\
R131 & 0,58442\
R31 & 0,51948\
R51 & 0,51948\
R54 & 0,51948\
R57 & 0,51948\
R59 & 0,51948\
R64 & 0,51948\
R66 & 0,51948\
R78 & 0,51948\
R79 & 0,51948\
R80 & 0,51948\
R81 & 0,51948\
R82 & 0,51948\
R83 & 0,51948\
R84 & 0,51948\
R85 & 0,51948\
R111 & 0,51948\
R132 & 0,51948\
R141 & 0,51948\
R142 & 0,51948\
R30 & 0,45455\
R33 & 0,45455\
R35 & 0,45455\
R36 & 0,45455\
R53 & 0,45455\
R55 & 0,45455\
R58 & 0,45455\
R60 & 0,45455\
R61 & 0,45455\
R62 & 0,45455\
R65 & 0,45455\
R86 & 0,45455\
R87 & 0,45455\
R88 & 0,45455\
R90 & 0,45455\
R124 & 0,45455\
R125 & 0,45455\
R126 & 0,45455\
R127 & 0,45455\
R128 & 0,45455\
R129 & 0,45455\
R134 & 0,45455\
R34 & 0,38961\
R37 & 0,38961\
R38 & 0,38961\
R112 & 0,38961\
R118 & 0,38961\
R140 & 0,38961\
R32 & 0,32468\
R39 & 0,32468\
R40 & 0,32468\
R41 & 0,32468\
R42 & 0,32468\
R43 & 0,32468\
R50 & 0,32468\
R105 & 0,32468\
R120 & 0,32468\
R121 & 0,32468\
R122 & 0,32468\
R135 & 0,32468\
R146 & 0,32468\
R48 & 0,25974\
R114 & 0,25974\
R119 & 0,25974\
R143 & 0,25974\
R144 & 0,25974\
R145 & 0,25974\
R44 & 0,19481\
R45 & 0,19481\
R46 & 0,19481\
R47 & 0,19481\
R104 & 0,19481\
R113 & 0,19481\
R115 & 0,19481\
R123 & 0,19481\
R91 & 0,12987\
R92 & 0,12987\
R93 & 0,12987\
R94 & 0,12987\
R95 & 0,12987\
R96 & 0,12987\
R97 & 0,12987\
R98 & 0,12987\
R99 & 0,12987\
R136 & 0,12987\
R137 & 0,12987\
R138 & 0,12987\
R139 & 0,12987\
R56 & 0,064935\
R63 & 0,064935\
R100 & 0,064935\
R101 & 0,064935\
R102 & 0,064935\
R103 & 0,064935\
\
[^1]: RESOLUTE D2.2 Sate of the art
[^2]: RESOLUTE is an EC funded research project - http://www.resolute-eu.org
[^3]: UNISDR - http://www.unisdr.org/we/coordinate/sendai-framework
[^4]: KID stays for [*knowledge, information, data*]{}.
[^5]: E. Bellini, P. Ferreira, and E. Gaitanidou. 2016a. European Resilience Management Guidelines (h2020 RESOLUTE project ed.)
[^6]: E. Bellini, P. Ferreira, and E. Gaitanidou. 2016a. European Resilience Management Guidelines (h2020 reso- lute project ed.).
[^7]: E. Bellini, P. Ferreira, and E. Gaitanidou. 2016a. European Resilience Management Guidelines (h2020 resolute project ed.).
[^8]: FRAM Glossary - http://functionalresonance.com/a-fram-glossary.html
[^9]: http://www.cambrensis.org/wp-content/uploads/2012/05/Systemic-Interdependency-Modelling-GENSIM-0.1-docx.pdf
[^10]: For a definition of $QNames$ we refer the reader to [@weik2000computer]
[^11]: FRAM Glossary - http://functionalresonance.com/a-fram-glossary.html
[^12]: The [*median absolute deviation*]{} of a series of observations is the median value of all the absolute deviations of each observation from the median value of the series. Formally this can be expressed as $mad = median(\bigcup_{i}^{n}|X_{i} - median(X)|)$.
[^13]: When the distance equals to $3$ or more, $\frac{\delta}{\epsilon}$ is over $\gamma$.
|
---
abstract: |
\[sec:abstract\] Modern multi-core processors share cache resources for maximum cache utilization and performance gains. However, this leaves the cache vulnerable to side-channel attacks, where inherent timing differences in shared cache behavior are exploited to infer information on the victim’s execution patterns, ultimately leaking private information such as a secret key. The root cause for these attacks is mutually distrusting processes sharing the cache entries and accessing them in a deterministic and consistent manner. Various defenses against cache side-channel attacks have been proposed. However, they suffer from serious shortcomings: they either degrade performance significantly, impose impractical restrictions, or can only defeat certain classes of these attacks. More importantly, they assume that side-channel-resilient caches are required for the entire execution workload and do not allow the possibility to selectively enable the mitigation only for the security-critical portion of the workload.
We present a generic mechanism for a flexible and soft partitioning of set-associative caches and propose a hybrid cache architecture, called . can be configured to selectively apply side-channel-resilient cache behavior only for isolated execution domains, while providing the non-isolated execution with conventional cache behavior, capacity and performance. An isolation domain can include one or more processes, specific portions of code, or a Trusted Execution Environment (e.g., SGX or TrustZone). We show that, with minimal hardware modifications and kernel support, can provide side-channel-resilient cache only for isolated execution with a performance overhead of 3.5–5%, while incurring no performance overhead for the remaining execution workload. We provide a simulator-based and hardware implementation of to evaluate the performance and area overheads, and show how mitigates typical access-based and contention-based cache attacks.
author:
- |
[Ghada Dessouky, Tommaso Frassetto, Ahmad-Reza Sadeghi]{}\
Technische Universität Darmstadt, Germany\
}
bibliography:
- 'main\_bib.bib'
title: |
**: Hybrid Side-Channel-Resilient Caches\
for Trusted Execution Environments**
---
|
---
abstract: 'The Dirac procedure for dealing with constraints is applied to the quantization of gauge theories on the light front. The light cone gauge is used in conjunction with the first class constraints that arise and the resulting Dirac brackets are found. These gauge conditions are not used to eliminate degrees of freedom from the action prior to applying the Dirac constraint procedure. This approach is illustrated by considering Yang-Mills theory and the superparticle in a $2 + 1$ dimensional target space.'
author:
- 'D.G.C. McKeon'
- Chenguang Zhao
title: Light Front Quantization with the Light Cone Gauge
---
10.0in 9.0in -0.60in
email: [email protected]\
PACS No.: 11.10Ef\
KEY WORDS: Light front quantization; gauge fixing; Dirac Constraints
Introduction
============
Ever since Dirac introduced the idea of light front quantization \[1\], this approach has received attention. It is related to working in the infinite momentum frame \[2\], and has proved useful in such diverse areas as gauge theories \[3-11\], supersymmetry \[12\], general relativity \[13-16\] and superstrings \[17\].
Quite often, the light cone gauge is used to simply eliminate variables occurring in the original gauge invariant action and then the resulting reduced action is quantized on the light front. However, if one were to follow the Dirac procedure for quantizing gauge systems \[18-19\], one should first identify and then classify all constraints in a system and then introduce a gauge condition to accompany each of the first class constraints. This procedure can be applied when using light-front variables. It need not result in the same quantized theory that arises if the light cone gauge is used at the outset to eliminate “superfluous” degrees of freedom before applying the Dirac procedure. We will illustrate this by considering Yang-Mills theory and the superparticle. In both of these examples, strict adherence to the Dirac procedure yields Dirac brackets which are different from what is obtained by using gauge conditions to eliminate degrees of freedom prior to involving Dirac’s approach. This would indicate that it would be in order to re-examine conclusions reached by applying gauge conditions to eliminate degrees of freedom in light-front quantization at the outset.
In an appendix, we show how the generator of a local gauge transformation can be found from first class constraints in a model in which primary second class constraints occur, thereby extending the discussion of ref. \[23\].
Yang-Mills Theory and the Light-Cone
====================================
When a covariant vector $a^\mu \quad(\mu = 0, 1, \ldots , D-1$ with $g_{\mu\nu} = \rm{diag} (+, - \ldots))$ has light front coordinates $$\begin{aligned}
\begin{split}
a^{\pm} &= \frac{1}{\sqrt{2}} (a^0 \pm a^{D-1})\\
a^i &= a^\mu \quad(\mu = 1 \ldots D-2)
\end{split}\end{aligned}$$ so that $$a \cdot b = a^+ b^- + a^- b^+ - a^ib^i ,$$ then the Yang-Mills (YM) action is $$\begin{aligned}
S_{YM} &= \int d^dx \left( - \frac{1}{4} F_{\mu\nu}^a F^{a\mu\nu}\right)\nonumber \\
&= \int d^dx \left(\frac{1}{2} F^{a+-}F^{a+-} + F^{a+i} F^{a-i} - \frac{1}{4} F^{aij} F^{aij} \right)\end{aligned}$$ where $$F^{a\mu\nu} = \partial^{\mu} A^{a\nu} - \partial^\nu A^{a\mu} + \epsilon^{abc} A^{b\mu} A^{c\nu}.$$ This action, as well as ones in which $A^{a\mu}$ is coupled with spinor and/or scalar fields, has been analyzed in a number of papers \[3-11\], most often by reducing the number of independent fields in the initial action through imposition of a gauge condition $$A^{a+} = 0$$ and using any resulting equation of motion that is independent of the “time” derivative $$\partial^+ f \equiv \dot{f}.$$ We will instead apply the Dirac constraint formalism \[18-19\] to the action of eq. (3), imposing gauge conditions in conjunction with first class constraints that arise. This has been done when applying path integral quantization to the action of eq. (3) \[11\]. Dirac’s canonical procedure has been applied to the light-front formulation of the $U(1)$ limit of Yang-Mills theory in ref. \[28\]. In ref. \[15, 16\], this approach has been used to analyze the spin-two action (ie, linearized gravity). We will also show how the first class constraints arising from the action of eq. (3) lead to a generator of the usual gauge transformation $$\begin{aligned}
\delta A_\mu^a &= D_\mu^{ab} \theta^b\nonumber \\
& \equiv \left(\partial_\mu \delta^{ab} + \epsilon^{apb} A_\mu^p\right) \theta^b\end{aligned}$$ despite the presence of second class constraints.
We begin by computing the canonical momenta
$$\begin{aligned}
\pi^a_i &= \partial \mathcal{L}_{YM}/\partial \dot{A}^{ai} = F^{a-i}\\
\pi^a_+ &= \partial \mathcal{L}_{YM}/\partial \dot{A}^{a+} = 0\\
\pi^a_- &= \partial \mathcal{L}_{YM}/\partial \dot{A}^{a-} = F^{a+-}.\end{aligned}$$
Together, these result in the canonical Hamiltonian $$\mathcal{H}_c = \frac{1}{2}\pi_-^a \pi_-^a + \frac{1}{4} F^{aij} F^{aij} - A^{a+} \left( D^{abi} \pi_i^b + D^{ab-} \pi_i^b\right).$$ Eq. (8a) is obviously a second class primary constraint $$\theta_i^a = \pi_i^a - F^{a-i}.$$ From the primary constraint of eq. (8b)
$$\begin{aligned}
\phi_1^a &= \pi_+^a\\
\intertext {and the canonical Hamiltonian of eq. (9) we obtain the secondary constraint}
\phi^a_2 &= D^{abi} \pi_i^b + D^{ab-} \pi_-^b;\end{aligned}$$
it is evident that $\phi_1^a$ and $\phi_2^a$ are both first class and that no further constraints arise.
The constraints of eqs. (9,11) have the Poisson bracket (PB) algebra
$$\begin{aligned}
\left\lbrace \phi_2^a, \phi_2^b\right\rbrace &= \epsilon^{abc}\phi_2^c\\
\left\lbrace \phi_2^a, \theta_i^b\right\rbrace &= \epsilon^{abc}\theta_i^c\\
\left\lbrace \theta_i^a(x), \theta_j^b(y)\right\rbrace &= -2\delta_{ij} D^{ab-} \delta (x-y),\end{aligned}$$
and so, by eq. (12c), we can eliminate the second class constraint $\theta_i^a$ by defining the Dirac bracket (DB) $$\left\lbrace M, N\right\rbrace^* = \left\lbrace M,N \right\rbrace - \left\lbrace M, \theta_i^a (z) \right\rbrace \frac{-1}{2D_z^{ab-}} \delta(z-w) \left\lbrace \theta_i^b (w), N\right\rbrace .$$
The non-trivial DB of eq. (12c) leads to a non-trivial contribution to the measure of the path integral if one were to use path integral quantization \[11\]. In the $U(1)$ limit considered in ref. \[28\], this contribution to the measure of the path integral becomes trivial. We also note that the inverse operator $1/\partial^-$ arising in the $U(1)$ limit of eq. (13) is carefully defined in ref. \[28\].
As in eq. (A.7), we define the generator of the gauge transformation that leaves $S_{YM}$ of eq. (3) invariant to be $$G = \mu_1^a \phi_1^a + \mu_2^a\phi_2^a$$ with $\mu_1^a$ determined in terms of $\mu_2^a$ by those terms in eq. (A.11) at least linear in $\phi_2^a$, $$\left( \dot\mu_1^a \phi_1^a + \dot{\mu}_2 \phi_2^a \right) +\left\lbrace \mu_1^a \phi_1^a + \mu_2^a \phi_2^a, \mathcal{H}_c \right\rbrace - \delta \mu_1^a \phi_1^a = 0$$ which by eqs. (9, 12) leaves us with $$G = \left( \dot{\mu}_2^a + \epsilon^{abc} A^{b+} \mu_2^c\right) \phi_1^a + \mu_2^a \phi_2^a .$$ From eq. (16) we find the gauge transformation of eq. (7) with $\theta^a = \mu_2^a$, as expected.
As was done ref. \[11\], the first class constraints $\phi_I^a$ of eqs. (11a,b) are accompanied by gauge conditions $\gamma_I^a$ so that together $\phi_I^a$ and $\gamma_I^a$ form a set of second class constraints. Here we will use the same gauge conditions that were suggested in ref. \[11\], and will proceed to find the resulting DB.
The constraint of eq. (11a) naturally leads to the gauge condition $$\gamma_1^a = A^{a+}$$ while that of eq. (11b) suggests
$$\begin{aligned}
\gamma_{2I}^a &= A^{a-}\\
\intertext{or}
\gamma_{2II}^a &= \partial^i A^{ai}.\end{aligned}$$
(The gauge conditions of eq. (18) are distinct from the gauge condition $\partial_\mu A^\mu = 0$ considered in ref. \[28\].) Having already eliminated $\theta_i^a$ of eq. (10) by defining the DB of eq. (13), we can now eliminate $\phi_1^a$ and $\gamma_1^a$ by the “second stage” DB $$\left\lbrace M,N\right\rbrace^{**} = \left\lbrace M,N\right\rbrace^{*} -
\left[ \left\lbrace M, \pi_+^a(z)\right\rbrace^{*}\delta (z-w) \left\lbrace A^{a+} (w),N\right\rbrace^{*} - ( M \leftrightarrow N)\right].$$ (Unlike ref. \[28\], we eliminate second class constraints in stages.) In the same way $\phi_2^a$ and $\gamma_{2I}^a$ give rise to a “third stage” DB. This involves using constraints in stages
$$\begin{aligned}
\left\lbrace \gamma_{2I}^a, \phi_2^b \right\rbrace^{**} &= - D^{ab-}_x \delta(x-y)\\
\left\lbrace \phi_2^a, \phi_2^b\right\rbrace^{**} &= \epsilon^{abc} \phi_2^c - \left[ \epsilon^{apm}\theta_i^m (x) \right] \frac{-1}{2D^{pq}_x} \delta (x-y) \left[ -\epsilon^{bqn} \theta_i^n (y)\right].\end{aligned}$$
When forming the DB to eliminate $\gamma_{2I}^a$ and $\phi_2^a$, one can set $\phi_2^a$ and $\theta_i^a$ to zero in eq. (20b) and so our third stage DB is $$\begin{aligned}
\left\lbrace M,N\right\rbrace^{***}& = \left\lbrace M,N\right\rbrace^{**} - \left[ \left\lbrace M, \phi_2^a (z)\right\rbrace^{**} \frac{-1}{D_z^{ab-}} \delta(z-w)\right. \nonumber \\
& \left. \left\lbrace \gamma_{2I}^b(w),N\right\rbrace^{**} - (M \rightleftharpoons N) \right].\end{aligned}$$ Computing the third stage DB when using the gauge condition $\gamma_{2II}^a$ of eq. (18b) in conjunction with the first class constraint $\phi_2^a$ of eq. (11b) is more involved. Eq. (20b) still holds, but now we also have $$\left\lbrace \gamma_{2II}^a, \gamma_{2II}^b \right\rbrace^{**} = \frac{1}{2} \partial^k \frac{1}{D^{ab-}_x} \partial^k \delta (x-y)$$ as well as $$\left\lbrace \gamma_{2II}^a, \phi_2^b \right\rbrace^{**} = -\partial^i D^{abi} \delta (x-y) - \frac{1}{2} \partial^i \frac{1}{D^{aq-}_x} \delta(x-y)\epsilon^{bqr} \theta_i^r(y).$$ Again, in eqs. (20b, 22, 23) we can set $\phi_2^a = \theta_i^a = 0$ when forming the DB to eliminate $\gamma_{2II}^a$ and $\phi_2^a$; since $$\left(
\begin{array}{cc}
\frac{1}{2} \partial ^k \frac{1}{D^{ab-}} \partial^k & -\partial^i D^{abi}\\
- D^{abi} \partial^i & 0
\end{array}\right) =
\left(
\begin{array}{cc}
0 & \frac{-1}{D^{ab-}\partial^i}\\
\frac{-1}{\partial^iD^{abi}} &\quad - \frac{1}{2} \frac{1}{\partial^iD^{api}} \partial^k \frac{1}{D^{pq-}} \partial^k \frac{1}{D^{qbj}\partial^j}
\end{array}
\right)$$ we find that $$\begin{aligned}
\left\lbrace M, N \right\rbrace^{***}& = \left\lbrace M, N \right\rbrace^{**} - \bigg[
\left\lbrace M,\gamma_{2II}^a (z) \right\rbrace^{**} \frac{-1}{\partial^j D^{abj}_z} \delta(z-w) \nonumber \\
& \quad \left\lbrace \phi_2^b(w), N\right\rbrace^{**} - (M \rightleftharpoons )N \bigg]\nonumber\\
& \quad - \bigg[ \left\lbrace M, \phi_2^a (z)\right\rbrace^{**} \left( \frac{-1}{2}\right)\frac{1}{\partial^iD^{api}_z}\partial^k \frac{1}{D^{pq-}_z} \partial^k \frac{1}{D^{qbj}_z\partial^j}\delta(z-w)\nonumber \\
&\qquad \left\lbrace \phi_2^b (w), N\right\rbrace^{**}\bigg].\end{aligned}$$ For example, from eq. (25) it follows that we have the novel DB $$\begin{aligned}
&\left\lbrace A^{ai}(x), A^{bj}(y)\right\rbrace^{***} \nonumber \\
&\quad = \frac{1}{2} \bigg[ -\delta^{ij} \frac{1}{D^{ab-}} + \frac{1}{D^{ap-}} \partial^i \frac{1}{D^{pqk}\partial^k} D^{qbj} + D^{api} \frac{1}{\partial^k D^{pqk}}\partial^j
\frac{1}{D^{qb-}}\nonumber \\
&\quad - D^{api} \frac{1}{\partial^kD^{pqk}}\partial^m \frac{1}{D^{qr-}} \partial^m
\frac{1}{D^{rs\ell}\partial^\ell} D^{sbj}\bigg]\delta(x-y).\end{aligned}$$ We see from eq. (26) that $$\left\lbrace \partial^i A^{ai}, A^{bj}\right\rbrace^{***} = 0$$ which is consistent with the gauge condition of eq. (18b). In the $U(1)$ limit, eq. (26) reduces to $$\left\lbrace A^i(x), A^j(y) \right\rbrace^{***} = \frac{1}{2}\left( -\delta^{ij} + \frac{\partial^i\partial^j}{\partial^k\partial^k} \right) \frac{1}{\partial^-}\delta(x-y).$$
The form of eq. (26) ensures that only the transverse components of $A^{ai}$ contribute to the dynamics of Yang-Mills theory when using light-front quantization and the gauge condition of eq. (18b). When working in $3 + 1$ space-time dimensions, we see that the only two physical degrees of freedom are $A^{a-}$ and the transverse components of $A^{ai}$ when using the gauge condition of eq. (18b); if the gauge condition of eq. (18a) were used, then all of the degrees of freedom reside in $A^{ai} \quad (i = 1,2)$.
We thus see that applying the Dirac canonical analysis to YM theory right from the outset (ie, only introducing constraints after the first class constraints which follow from the initial YM action when written in light front coordinates) yields different DB than what arises when the light cone gauge is used to eliminate degrees of freedom from the YM action before employing the Dirac formalism.
The spin two action has been treated in a manner consistent with the approach used here with YM theory in ref. \[15\].
We now turn to examining the superparticle in the light cone gauge.
The Superparticle and the Light Cone
====================================
The superparticle \[20\] has Bosonic variables $x^\mu(\tau)$ and Fermionic variables $\theta(\tau)$; its action is $$S = \int d\tau \frac{1}{2e} \left( \dot{x}^\mu + i \dot{\overline{\theta}} \gamma^\mu \theta\right) \left( \dot{x}_\mu + i \dot{\overline{\theta}} \gamma_\mu \theta\right).$$ A discussion of its constraint structure appears in ref. \[21\] (see also ref. \[22\]). In ref. \[28\], considering application of the Dirac formalism to systems involving Fermionic degrees of freedom was not yet feasible as degrees of freedom that were Grassmann degrees of freedom had not yet been introduced into the Dirac canonical formalism. Quite often, the light cone gauge conditions
$$\begin{aligned}
x^+ &= p_+\tau\\
\gamma^+\theta &= 0\end{aligned}$$
are used \[17\] to eliminate degrees of freedom from the action of eq. (29) prior to applying Dirac’s formalism; below we will instead use the gauge conditions of eq. (30) in conjunction with the first class constraints arising from eq. (29).
The spinor $\theta$ has different properties in every dimension of the target space; we will restrict our attention to $2 + 1$ dimensions to simplify our discussion. The conventions of ref. \[21\] will be used, so that $$\begin{aligned}
\gamma^0 = \sigma_2 \quad \gamma^1 = i\sigma_3 \quad \gamma^2 = i\sigma_1\\
\gamma^\mu \gamma^\nu = \eta^{\mu\nu} + i\epsilon^{\mu\nu\lambda}\gamma_\lambda\nonumber\end{aligned}$$ $$\begin{aligned}
C &= -\gamma^0\\
\theta = C \overline{\theta}^T &= (-\gamma^0)(\theta^\dagger\gamma^0)^T\nonumber\end{aligned}$$ so that $$\theta = \left( \begin{array}{c}
u \\ d
\end{array} \right) =
\left( \begin{array}{c}
u^* \\ d^*
\end{array} \right).$$ With this, we find that eq. (29) becomes $$S = \int \frac{d\tau}{2e}\left[ \left( \dot{x}^0 + i (\dot{u} u + \dot{d}d)\right)^2 -
\left( \dot{x}^1 - i (\dot{u} d + \dot{d}u)\right)^2 -
\left( \dot{x}^2 + i (\dot{u} u - \dot{d}d)\right)^2 \right]$$ so that the momenta conjugate to $e$, $x^\mu$, $u$ and $d$ are
$$\begin{aligned}
p_e &= 0\\
p_\mu &= \frac{1}{e} \left( \dot{x}^0 + i (\dot{u} u + \dot{d}d),
-\dot{x}^2 + i (\dot{u} d + \dot{d}u),
-\dot{x}^2 - i (\dot{u} u - \dot{d}d)\right)\\
\pi_u &= -id p_1 + iu p_+\\
\pi_d &= i(dp_- - up_1)\end{aligned}$$
where $p_{\pm} \equiv p_0 \pm p_2$. We see that eqs. (34a,c,d) are primary constraints. Following ref. \[21\], we treat $\sigma_1 = \pi_u + idp_1 - iup_+$ as a second class constraint and eliminate it by defining the DB $$\left\lbrace M, N \right\rbrace^* = \left\lbrace M, N \right\rbrace - \left\lbrace M, \sigma_1 \right\rbrace \frac{1}{2ip_+} \left\lbrace \sigma_1, N \right\rbrace .$$ With this DB, the constraint $\sigma_1 = \pi_d - idp_- + iup_1$ satisfies $$\left\lbrace\sigma_2, \sigma_2 \right\rbrace^* = 2ip^2/p_+ .$$ Since the canonical Hamiltonian is $$H_c = \frac{e}{2}p^2,$$ we see that the primary constraint of eq. (35a) leads to the secondary first class constraint $$p^2 = 0,$$ and hence by eq. (37), we see that once $\sigma_1$ has been taken to be second class, $\sigma_2$ becomes first class. (The roles of $\sigma_1$ and $\sigma_2$ can be reversed.)
It is at this stage we introduce gauge conditions to accompany the first class constraints that have been derived. In conjunction with $$\phi_1 = p_e, \quad \phi_2 = p^2, \quad \phi_3 = \sigma_2 \tag{40a,b,c}$$ we introduce respectively $$\gamma_1 = e-1, \quad \gamma_2 = x^+ - p_+\tau, \quad \gamma_3 = \gamma^+\theta = u .
\tag{41a,b,c}$$ From the first class constraints of eq. (40), one can use the approach of ref. \[23\] to derive a generator of a set of Bosonic and Fermionic gauge transformations, the Fermionic ones being half of the so-called $\kappa$-symmetry transformations of ref. \[24\]. (The other half can be generated by reversing the rules of $\sigma_1$ and $\sigma_2$.)
Together, $\phi_I$ and $\gamma_I$ in eqs. (40, 41) constitute a set of second class constraints that can be eliminated by forming a “second stage” DB. This involves inverting the matrix $$\begin{aligned}
M &= \left\lbrace (\gamma_1, \phi_1, \gamma_2, \phi_2, \gamma_3, \phi_3)^T, (\gamma_1, \phi_1, \gamma_2, \phi_2, \gamma_3, \phi_3) \right\rbrace^* \tag{42}\\
&= \left( \begin{array}{cccccc}
0 & 1 & 0 & 0 & 0 & 0 \\
-1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 2p_- & -u/p_+ & -2iup_1/p_+ \\
0 & 0 & -2p_- & 0 & 0 & 0 \\
0 & 0 & u/p_+ & 0 & i/2p_+ & -p_1/p_+ \\
0 & 0 & 2iup_1/p_+ & 0 & -p_1/p_+ & 2ip^2/p_+ \end{array}
\right).\nonumber\end{aligned}$$ To find $M^{-1}$, we use the identity $$\left(\begin{array}{cc} A & B \\
C & D \end{array} \right)^{-1} =
\left(\begin{array}{cc} \Delta^{-1} & -\Delta^{-1}BD^{-1} \\
-D^{-1}C\Delta^{-1} & D^{-1}+D^{-1}C\Delta^{-1}BD^{-1} \end{array} \right)
(\Delta = A - BD^{-1}C)\nonumber$$ and $u^2 = 0$ (since u is Grassmann); we arrive at $$M^{-1} = \left( \begin{array}{cccccc}
0 & -1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1/2p_- & 0 & 0 \\
0 & 0 & 1/2p_- & 0 & -iu/p_- & 0 \\
0 & 0 & 0& -iu/p_+ & -2ip^2/p_- & -p_1/p_-\\
0 & 0 & 0 & 0 & -p_1/p_- & 1/2ip_- \end{array}
\right). \tag{43}$$ From the resulting DB, it follows, for example that $$\begin{aligned}
\left\lbrace x^1, x^2 \right\rbrace^{**} &= \left\lbrace x^1, x^2 \right\rbrace^* -
\left\lbrace x^1, \Phi^T \right\rbrace^* M^{-1} \left\lbrace \Phi , x^2 \right\rbrace\nonumber\\
&= \frac{p_1 \tau}{p_-} + \frac{iudp_0}{2p_+p_-} .\tag{44}\end{aligned}$$ where $\Phi^T = (\gamma_1, \phi_1, \gamma_2, \phi_2, \gamma_3, \phi_3)^T$. This result serves to illustrate how using the light cone gauge conditions of eq. (41) in conjunction with the first class constraints of eq. (40) (arrived at by applying Dirac’s canonical procedure to the initial action of eq. (29)) leads to results different from those obtained by using eq. (41) to eliminate fields from eq. (29) and only then applying the Dirac procedure (as is normally done).
These considerations can also be applied to string theories. For the Bosonic string, the action is \[25\] $$S = \int d\tau d\sigma \left( \frac{1}{2} \sqrt{-g}\; g^{ab} x^A_{,a} x_{A,b} \right).\tag{45}$$ The canonical momenta associated with $g^{ab}$ and $x_A$ are
$$\begin{aligned}
I\!\!P_{ab} &= 0\tag{46a}\\
p_A &= \sqrt{-g} \left(g^{00} x_{A,0} + g^{01}x_{A,1}\right)\tag{46b}\end{aligned}$$
which lead to the secondary first class constraints
$$\begin{aligned}
\Sigma_S &= \frac{1}{2} \left(p^2 + x_{,1}^2\right)\tag{47a}\\
\Sigma_p &= p_{A} x_{,1}^A\tag{47b}\end{aligned}$$
Both of these in principle should be accompanied by a suitable gauge condition. However, the usual practice \[29\] is to use a single gauge condition (the “light cone gauge”) and then using this to simplify the initial action of eq. (45). Only at this stage is the Dirac procedure invoked. A similar approach is generally used with the superstring. (A discussion of the canonical structure of the superstring appears in ref. \[26\].)
Discussion
==========
The Dirac procedure for treating the canonical structure of dynamical systems which have a local gauge invariance is well defined; all constraints are first obtained and then classified, and those which are first class are then paired with suitable gauge conditions. All superfluous degrees of freedom arising on account of there being a local gauge symmetry are then eliminated by replacing the PB by a DB defined using both the first and second class constraints and the gauge conditions. This procedure can be tedious, as can be seen by examining YM theory on the light front and the superparticle (as was done above). Both of these systems can be simplified by using a “light cone” gauge condition to eliminate superfluous degrees of freedom at the outset from the classical action, and then using Dirac’s procedure. However, the DB one arrives at after using these two approaches need not be the same. Being different would result in the two procedures leading to different quantum theories if one were to use the DB to define a quantum mechanical commutator.
When one employs path integral quantization, second class constraints can lead to highly non-trivial contributions to the measure of the path integral, as is discussed in ref. \[20\]. This is true when using the path integral to quantize the first order Einstein-Hilbert action \[31\] as well as when considering Yang-Mills theory on the light-front (see eq. \[12c\] and (43) above).
On the other hand, one often uses the Faddeev-Popov \[32\] approach to handling the path integral quantization of gauge theories. This involves a “factoring” of the divergence arising in the path integral resulting from the presence of a gauge invariance. In doing this, one (or more \[33\]) gauge conditions must be introduced, and “ghost” fields are used to cancel the contribution of non-physical degrees of freedom. However, we note that there need not be agreement between results obtained from the path integral when the formalisms of refs. \[30\] and \[32\] are employed, especially if non-trivial second class constraints occur (such as in eq. \[10\] above). Fortunately, in Yang-Mills theory the Faddeev-Popov approach to the path integral is equivalent to the canonical approach to the path integral when not using the light-front \[34\].
Acknowledgements {#acknowledgements .unnumbered}
================
Roger Macleod had useful advice.
[99]{} P.A.M. Dirac, *Rev. Mod. Phys.* **21**, 392 (1949). S. Weinberg, *Phys. Rev.* **150**, 1313 (1966). E. Tomboulis, *Phys. Rev.* **D8**, 2736 (1973). A. Casher, *Phys. Rev.* **D14**, 452 (1976). A. Bassetto, M. Dalbosco, I. Lazzizzera and R. Soldati, *Phys. Rev.* **D31**, 2012 (1985). P.J. Steinhardt, *Ann. of Phys.* **128**, 425 (1980). S.J. Brodsky, H.-C. Pauli and S.S. Pinsky, *Phys. Rep.* **301**, 299 (1998). D.M. Capper and M.J. Litvak, *J. Phys.* **A18**, 955 (1985). M. Kaku and P. Senjanovic, *Phys. Rev.* **D15**, 1019 (1977). H.C. Lee, *Can. J. Phys.* **64**, 264 (1986). D.G.C. McKeon, *Can. J. Phys.* **64**, 549 (1986)\
R. Casana, B.M. Pimentel and G.E.R. Zambrano,*Nucl. Phys.* (Proc. Supp.) **B199, B219** (2010)\
U. Kulshreshtha, J. P. Kulshreshtha and J. P. Vary, *European Physical Journal* **C75**, 174 (2015). S. Mandelstam, *Nucl. Phys.* **B213**, 149 (1983). R. Gambini and A. Restuccia, *Phys. Rev.* **D17**, 3150 (1978). R.G. Root, *Phys. Rev.* **D8**, 3382 (1973). D. Evans, G. Kunstatter and C. Torre, *Class. Quantum Grav.* **4**, 1503 (1987). C.G. Torre, *Class. Quantum Grav.* **3**, 773 (1986). M.B. Green, J.H. Schwarz and E. Witten, “String Theory” (Cambridge U. Press, Cambridge, 1987). P.A.M. Dirac, *Can. J. Math.* **2**, 129 (1950). M. Henneaux and C. Teitelboim, “Quantization of Gauge Systems”, (Princeton U. Press, Princeton, 1992). L. Brink and J.H. Schwarz, *Phys. Lett.* **B100**, 310 (1981). D.G.C. McKeon, *Can. J. Phys.* **91**, 604 (2013). J.M. Evans, *Class. Quantum Grav.* **7**, 699 (1990). M. Henneaux, C. Teitelboim and J. Zanelli,*Nucl. Phys.* **B332**, 169 (1990). W. Siegel,*Phys. Lett.* **B128**, 397 (1983). A.M. Polyakov, *Phys. Lett.* **103B**, 207, 211 (1981). F.A. Chishtie and D.G.C. McKeon, hep-th 1401.0783 (Can. J. Phys. to be published). L. Castellani, *Ann. of Phys.* **143**, 357 (1982). M. Huszar, *J. Phys.* **A9**, 1359 (1976). P. Goddard, J. Goldstone, C. Rebbi and C.B. Thorn, *Nucl. Phys.* **B56**, 109 (1973). P. Senjanovic, *Ann. of Phys.* **100**, 227 (1976). F.A. Chishtie and D.G.C. McKeon, *Class. and Quant. Gravity* **29**, 235016 (2012). L.D. Faddeev and V. Popov, *Phys. Lett.* **B58**, 29 (1967). F. Brandt, J. Frenkel and D.G.C. McKeon, *Phys. Rev.* **D76**, 105029 (2007). L.D. Faddeev, *Theor. and Math. Phys.* **1**, 1 (1970).
Appendix {#appendix .unnumbered}
========
In refs. \[19, 23 \] it is shown how to obtain the generator of a gauge transformation for systems involving exclusively first class constraints. Here we will extend this discussion to include the situation in which there are also primary second class constraints so that one can consider the light front formulation of Yang-Mills theory.
In the presence of primary second class constraints $\theta_\alpha$ and first class constraints $\phi_{A_{i}}$ (where $i$ denotes the generation of the constraint-primary is $i = 1$, secondary is $i = 2$ etc.), then suppose we have the PB algebra $$\left\lbrace \theta_\alpha , \theta_\beta \right\rbrace = \Delta_{\alpha\beta},\tag{A.1}$$ as well as $$\left\lbrace \phi_A , \phi_B \right\rbrace = C_{AB}^C \phi_C + C_{AB}^\alpha \theta_\alpha\tag{A.2}$$ and $$\left\lbrace \phi_A , \theta_\alpha \right\rbrace = C_{A\alpha}^\beta \theta_\beta + + C_{A\alpha}^B \phi_B .\tag{A.3}$$ We then can deine the DB $$\left\lbrace M, N \right\rbrace^* = \left\lbrace M,N \right\rbrace - \left\lbrace M, \theta_\alpha \right\rbrace \Delta_{\alpha\beta}^{-1} \left\lbrace \theta_\beta , N\right\rbrace . \tag{A.4}$$ Upon using the constraints $\theta_\alpha$ and $\phi_{A_{i}}$, the canonical Hamiltonian $H_C$ can be defined $$H_C = p_i \dot{q}^i - L(q^i, \dot{q}^i); \tag{A.5}$$ this leads to the extended Hamiltonian $$H_E = H_C + \sum_\alpha U_\alpha \theta_\alpha + \sum_{A_{i}} V_{A_{i}} \phi_{A_{i}} . \tag{A.6}$$ If the sum over $A_i$ in eq. (A.6) is restricted to having $i = 1$ (ie, just the primary constraints) then $H_E$ reduces to $H_T$, the total Hamiltonian.
We now can consider the generator $$G = \sum_{A_{i}} \mu_{A_{i}}\phi_{A_{i}}\tag{A.7}$$ of “gauge” transformations that leave the extended action $S_E$ invariant, that is the change induced by $G$ on a dynamical quantity $f$ is given by $$\delta f = \left\lbrace f, G \right\rbrace . \tag{A.8}$$ The change in the extended action is given by $$\begin{aligned}
\delta S_E &= \int dt \;\delta\left( p_i \dot{q}^i - H_E\right)\nonumber \\
&\int dt \bigg[ \delta p_i \dot{q}^i + p_i \delta \dot{q}_i - \left\lbrace H_C, G \right\rbrace \tag{A.9} \\
&\quad - \sum_\alpha \left( \delta U_\alpha \theta_\alpha + U_\alpha \left\lbrace \theta_\alpha , G \right\rbrace \right) \nonumber \\
& \qquad - \sum_{A_{i}} \left( \delta V_{A_{i}}\phi_{A_{i}} + V_{A_{i}} \left\lbrace \phi_{A_{i}} , G \right\rbrace \right) . \nonumber\end{aligned}$$ But now into eq. (A.9) we can substitute $$\begin{aligned}
\delta p_i\dot{q}^i +p_i &\delta \dot{q}^i = -\frac{\partial G}{\partial q^i}\dot{q}^i + \frac{d}{dt}\left( p_i \frac{\partial G}{\partial p_i}\right) - \dot{p}^i
\frac{\partial G}{\partial p_i} \tag{A.10} \\
& = \frac{d}{dt}\left( p_i \frac{\partial G}{\partial p_i}-G\right) + \left[
\left( \frac{\partial}{\partial t} + \dot{U}_\alpha \frac{\partial }{\partial U_\alpha} + \dot{V}_{A_{i}} \frac{\partial}{\partial V_{A_{i}}}\right) \mu_{B_{j}} \right]\phi_{B_{j}}\nonumber\end{aligned}$$ yielding $$\begin{aligned}
\delta S_E = \int dt & \bigg[ \left( \frac{D}{Dt}\mu_{B_{i}}\right) \phi_{B_{i}} +
\mu_{A_{i}} \left( D_{A_{i}}^{B_{j}} \phi_{B_{j}} + D_{A_{i}}^\gamma \theta_\gamma \right) \tag{A.11} \\
&- \sum_\alpha \left( \delta U_\alpha \theta_\alpha - U_\alpha \mu_{B_{j}} \left(
C_{B_{j\alpha}}^\gamma \theta_\gamma + C_{B_{j\alpha}}^C \phi_C\right)\right)\nonumber\\
& - \sum_{A_{i}} \left( \delta V_{A_{i}} \phi_{A_{i}} - V_{A_{i}} \mu_{B_{j}}
\left( C_{B_{j}A_{i}}^{C_{k}} \phi_{C_{k}} + C_{B_{j}A_{i}}^\gamma \theta_\gamma \right)\right) \bigg] .\nonumber\end{aligned}$$ In eq. (A.11), we have dropped all surface terms, defined $$\frac{D}{Dt} = \frac{\partial}{\partial t} + \dot{U}_\alpha \frac{\partial}{\partial U_\alpha} + \dot{V}_{A_{i}} \frac{\partial}{\partial V_{A_i}}\tag{A.12}$$ and have used the fact the $\phi_{A_{i}}$ are all first class so that $$\left\lbrace \phi_{A_{i}} , H_C \right\rbrace = D_{A_{i}}^{B_{j}} \phi_{B_{j}}+ D_{A_{i}}^\gamma
\theta_\gamma . \tag{A.13}$$ In eq. (11), we can arrange for $\delta S_E = 0$ by choosing $\delta U_\alpha$ so that all coefficients of $\theta_\alpha$ vanish, and by having the $\mu_{B_{i}}$ satisfy a differential equation that answers that the coefficients of $\phi_{B_{i}}$ sum to zero. Upon having \[19, 23 \] $\delta V_{A_{i}} = V_{A_{i}} = 0 ( i \geq 2)$, $S_E$ reduces to $S_T$, the total action, and $G$ becomes the generator of gauge transformations that leave $$S_C = \int dt L(q^i, \dot{q}^i ) \tag{A.14}$$ invariant, as $S_T$ and $S_C$ have the same dynamical content.
We can replace eq. (A.8) with $$\delta f = \left\lbrace f, G \right\rbrace^*\tag{A.15}$$ as by eq. (A.3), $\left\lbrace f, G \right\rbrace^*$ and $\left\lbrace f, G \right\rbrace$ differ by an expression that is at least linear in $\theta_\alpha$; in eq. (A.11) this term can be absorbed into $\delta U_\alpha$. The advantage of using the DB over the PB in finding $\delta f$ is that we can set $\theta_\alpha = 0$ at the outset of any calculation.
It would be interesting to see how the approach of ref. \[27\] to finding gauge symmetries could be adapted to the case in which primary second class constraints are present.
|
---
abstract: |
High Energy Nuclear Physics (HENP) collaborations’ experience show that the computing resources available at a single site are often neither sufficient nor satisfy the need of remote collaborators eager to carry their analysis in the fastest and most convenient way. From latencies in the network connectivity to the lack of interactivity, work at distant computing centers is often inefficient. Having fully functional software stack on local resources is a strong enabler of science opportunities for any local group who can afford the time investment. The situation becomes more complex as vast amount of data are often needed to perform meaningful analysis.
Prague’s heavy-ions group participating in STAR experiment at RHIC has been a strong advocate of local computing as the most efficient means of data processing and physics analyses. To create an environment where science can freely thrive, a Tier 2 computing center was set up at a Regional Computing Center for Particle Physics called “Golias”. It is the biggest farm in the Czech Republic fully dedicated for particle physics experiments.
We report on our experience in setting up a fully functional Tier 2 center leveraging the minimal locally available human and financial resources. We discuss the solutions chosen to address storage space and analysis issues and the impact on the farms overall functionality. This includes a locally built STAR analysis framework, integration with a local DPM system (a cost effective storage solution), the influence of the availability and quality of the network connection to Tier 0 via a dedicated CESNET/ESnet link and the development of light-weight yet fully automated data transfer tools allowing the movement of entire datasets from BNL (Tier 0) to Golias (Tier 2). We will summarize the impact of the gained computing performance on the efficiency of the local physics group at offline physics analysis and show the feasibility of such a solution that can used by other groups as well.
address:
- |
Nuclear Physics Institute\
Academy of Sciences of the Czech Republic\
Prague, Czech Republic
- |
Brookhaven National Laboratory\
Upton, USA
author:
- 'Petr Chaloupka, Pavel Jakl, Jan Kapitán and Michal Zerola'
- Jérôme Lauret for the STAR collaboration
title: 'Setting up a STAR Tier2 Site at Golias/Prague Farm'
---
Overview of STAR experiment computing
=====================================
The STAR (Solenoidal Tracker At RHIC)[@haris-STAR; @star_nim] experiment at Brookhaven National Laboratory (BNL) is a collaboration of over $500$ scientists from 55 institutions in 12 countries. It is the largest high energy nuclear experiment in the world currently taking data. One of RHIC’s goals is to study nuclear matter at extreme conditions (pressure and temperature) similar to those that existed in the very early universe by colliding heavy ion nuclei[@star_whitepaper]. In parallel with this program, with increasing importance, there is also a unique proton-proton program to study proton spin structure[@pp_rhic]. To achieve these ambitious tasks, STAR collected large volumes amount of data since its first run in year 2000 and the rate at which data are accumulated is increasing. Over the 9 years of running, STAR has collected $~7$PB of physics data in total.
![\[fig:storage2\] Projected disk storage capacity required by STAR software together with the relative portion of space allocated to central (NFS like) and distributed (cheap disks attached to processing nodes) storage as a function of year. ](storage2.eps){width="8cm"}
In order to remain on the forefront of nuclear research, the STAR and RHIC are undergoing upgrades heading toward high-luminosity beams and detectors focused to harvest the fruits of the most challenging physics topics[@decadal_plan]. The STAR upgrades will include new detector instrumentation targeted to enhance STAR’s acceptance, particle identification capability, and effective sampling of luminosity. In particular, new front end electronics for the STAR Time Projection Chamber (TPC) and data acquisition readout electronics will lead to an order of magnitude increase in the bandwidth for acquisition of minimum bias data and operation for rare triggers with nearly zero dead time, increasing the effective luminosity. These improvements will lead to a dramatic increase in the volume of data accessible for physics research with these probes by 60-70% in heavy ion and proton-proton interactions.
To capitalize on these investments, it is essential that the computing capability of the STAR experiment, now and into the future, be strategically positioned to receive and analyze the flood of data which the upgraded STAR detector will collect. Part of the plan to accomplish smooth delivery of physics results is to leverage the use of Tier 2 computing centers for user’s analyses. This approach utilizes locally available storage and computing power and increases the productivity of scientific work at a favorable hardware and human resource cost affordable by the local group.
STAR computing model
====================
Located at BNL, the RHIC Computing Facility (RCF) at BNL serves as a Tier 0 center for the STAR experiment. Its primary mission is to provide storage resources for mission critical data as well as at least one pass data reconstruction and some analysis bandwidth at a lower priority. The collected data are stored in the primary mass storage system and reconstructed later at the RCF. The final physics-ready data is then made available to users. At the present time the RCF has 350 computing nodes with 1500 CPUs and 500TB of disk space dedicated to STAR only. The STAR’s needs for CPU and storage space (Figure \[fig:storage2\]) are expected to steadily grow hand in hand with the planned increase of the volume of collected data.
Until recently STAR had a single Tier 1 center: the Parallel Distributed Systems Facility (PDSF) located at the National Energy Research Supercomputing Center (NERSC) at LBNL. PDSF’s main purpose is to provide STAR with supplemental user analysis cycles and simulation (aka embedding) cycles. In addition STAR will soon open a second Tier 1 center at the Korean Institute of Science and Technology (KISTI) in Daejon, South Korea, a center which has the capacity to serve as a regional data redistribution hub for STAR collaborators in Asia and provide resource boost for data production and simulations.
Although initially STAR mainly relied on centralized user analysis facilities (BNL and PDSF) to provide the bulk of the analysis power for STAR collaborators and scientists, the steadily growing amount of experimental data taken and increased demand for storage and processing power required to reconstruct data caused the computing model to naturally evolve toward multi-Tier structure, as shown in the Figure \[fig:triangle\]. As analyses become more complex, individual users compete for available resources along with simulation and reconstruction processes at the main Tier 0 and Tier 1 centers. Since there can be no analysis without reconstructed data, the requirement to allocate resources for reconstructing data in a timely fashion often takes precedence over user analyses. A squeeze-out at Tier 0 and Tier 1 centers motivates local groups with their own resources to set up Tier 2 centers. The STAR software and computing plan[@decadal_plan] partly relies on this migration of analysis toward Tier 2 centers serving as a buffer to resource constraints at primary facilities. Furthermore, it has been empirically observed in STAR that productivity seem to increase as local resources are utilized making such a computationally distributed model even more attractive.
![\[fig:triangle\]Current structure of the STAR Software and Computing (S&C) effort. In addition, a Tier-1 center in South Korea is in development.](tri3.eps){width="7cm"}
Paradigm of local computing: bringing data to users
===================================================
In world wide collaborations such as STAR, users connect to the main computing Tiers from distant locations, often over unreliable and slow networks. This combined with having to share computing resources and disk storage with reconstruction and simulation jobs renders their work inefficient. Especially, interactivity necessary to carry out analysis is hindered as well as the research objective. These problems are even more apparent before major conferences when more computing resources are required by users. However, at many places computing and storage resources are available at smaller local computing farms that can be utilized as Tier 2 centers for data processing and physics analyses of the local physics groups after some basic prerequisites are fulfilled and the STAR software stack is deployed. Although the resources available at Tier 2 sites are modest, these sites strongly leverage the capabilities at the STAR Tier 0 and Tier 1 sites with respect to scientific productivity due to the involvement of scientist end-users in data manipulation and handling. Such an approach allows running analyses and simulation on locally stored copies of data, either full data sets or specifically pre-filtered subsets. The short distance between the user and the computing center makes fast interactive work viable and quick retrieval of computed results becomes possible thus shortening overall turn-around time for scientific papers.
The advantage of a Tier 2 center is not only in increase of the efficiency of scientific work, but also in cost effectiveness. The setup of Tier 2 centers relies on building on already existing infrastructure, such as a network or cooling system, and sharing maintenance costs with other users of the computing farm. As the users are expected to work predominantly with copies of data and performance (number of users) is much lower than what is required for a Tier 0 facility, it is possible to use cheap retail class storage components such as SATA disk arrays to store the data, and further decrease the total computing costs. Example of above mentioned approach is a setup of a STAR Tier 2 site at the already existing infrastructure of Golias farm in Prague.
The Regional Computing Center for Particle Physics in Prague - the Golias farm
==============================================================================
The Regional Computing Center for Particle Physics (Figure \[fig:farma\]), named Golias, is the biggest site in the Czech Republic. It is mainly dedicated to simulations and data processing for particle physics experiments. The farm resides in the Institute of Physics of the Academy of Sciences of the Czech Republic and provides computing and storage services for particle physics experiments as well as for solid state physics and astrophysics communities. The computing and storage resources consisted of of about 450 CPUs and 50 TB of disc space in the end of 2008 and will be gradually expanded to about 1550 CPUs/3,3 MSI2k and 216 TB of disc space in 2009.
![\[fig:farma\]Regional Computing Center for Particle Physics, Prague](P2.ps){width="7cm"}
The network connectivity of the farm is maintained by CESNET (Czech Education and Scientific NETwork)[@cesnet], a non-profit association founded by universities and the Academy of Sciences of the Czech Republic. Its aim is to conduct research in advanced network technologies and provide network connection to educational and research institutions. Excellent network connectivity with other institutions, both national and worldwide (1 or 10 Gb/s) (Figure \[fig:cesnet\]) allows Golias to be integrated to other projects (LCG, EGEE, SAMGrid) through their experiments’ specific middle-ware components and offer its capacity to all qualified members of supported projects (ATLAS, ALICE and D0). The installed gLite grid middle-ware components include CE, SE, UI, MON box, site BDII and have been configured with YAIM, integrated into Cfengine used for the management of the local site changes. Since 2005, the site is a certified Tier 2 center of the LCG project and in 2008 signed the Memorandum of Understanding of the Worldwide LHC Computing Grid Collaboration (WLCG). Part of the farm’s capacity is available for local users from other projects such as AUGER and STAR and spare cycles can be used by anyone, making the facility a good example of efficient resource usage.
Job management at the Golias farm is performed using the workload management system PBSPro (current version 9.2)[@pbs_pro]. Golias is in the process of evaluating alternatives such as Torque[@torque] and the cluster scheduler MAUI[@maui]. The hardware, software and network status are extensively monitored using many standard packages including Nagios, Munin, MRTG, IPAC as well as a set of locally developed custom tools tailored to the facility’s need.
{width="98.00000%"}
\[fig:cesnet\]
Setup of the local STAR computing environment
=============================================
The local ultra-relativistic heavy ion collisions group from the Nuclear Physics Institute of the Academy of Sciences of the Czech Republic (NPI ASCR) has been participating in the STAR experiment since its very first collisions in 2000. It’s main activities have been data analyses in the area of correlations and high momentum physics as well as detector research and development activities.
The road towards productive work was however paved with difficulties. In the beginnings the group had struggled with running analyses on the major Tier centers from its remote location given the slow connectivity over the Atlantic. This was extremely inefficient, as then the network capabilities were an order of magnitude slower than the ones available today. The idea of developing local STAR computing environment has been tested since 2002 and a decisive advance was achieved by joining the Golias farm at 2005, leveraging the knowledge acquired over the years to set up the necessary components for a Tier 2 center. Since then, the STAR physics group has used the Golias farm for short-time and mid-scale analyses. Over this period of time we have tested multiple approaches and identified key components for the successful setup of local Tier 2 computing center. A schematic layout of the STAR Tier 2 at Golias in the configuration described by this article is shown in the Figure \[fig:scheme\].
{width="70.00000%"}
\[fig:scheme\]
Deployment of STAR software framework
-------------------------------------
First the STAR software framework must be deployed and adapted to the specifics the of local farm. In-situ compilation of the STAR software stack and import of user environment to emulate the RCF is a key aspect making the work at any facility interchangeable hence transparent (learn once, use many times). The deployment of the STAR software stack includes a version of ROOT and a set of add-ons customized by the STAR experiment called “root4star”. For the needs of the Golias farm, the ROOT framework was patched to support the RFIO protocol which is used to access locally stored data, as will be explained in the next section. To simplify the selection of data for analysis and subsequent job submission STAR has developed the “STAR Unified Meta Scheduler” (SUMS)[@sums] tool and middle-ware. Customizing SUMS for the Golias environment requires an adaptation of policies to make use of PBSpro and Torque for scheduling, but from a user stand point, the use of SUMS and its job syntax is identical regardless of the facility, allowing the user to simply migrate their work from the Tier 0 to the Tier 2 center using the same job description files.
Data storage,transfer and aggregation strategy
----------------------------------------------
Data management was found to be a key component of our local computing strategy. The overall aim is to allow physicists to easily download data sets from the RCF to local storage and also to provide them with a simple and fast way to connect their jobs running on computing nodes of the farm with the storage. As trivial as it may sound, at the same time one requirement was that the access to the data not strain internal network of the farm, as would mounted NFS. This combined with an additional requirement of a low purchase and maintenance cost resulted in the use of the SRM-based Disk Pool Manager (DPM) grid-aware solution developed at CERN[@dpm]. In order to keep the farm maintenance cost low DPM was actually located outside of the farm premises[@our_dpm] and is connected to Golias via a dedicated 1Gbit/s optical fiber link. The DPM offers various interfaces for data access (RFIO, XROOTD) used in the STAR computing framework. The DPM solution is very convenient for end-users since it affects their jobs in a minimal way and allows for easy scalability and maintenance by the administrators. The DPM manages about 20 TB of disk space and with sophisticated data-mover tools developed by the group at Prague and STAR software staff, physicists can download almost 1 TB of data per 24 hours from BNL.
Network connectivity
--------------------
Fast networking available to the users of the Golias farm makes the main difference between working remotely at major Tier centers and working locally at smaller farm for the end-users. As already stated above, the Golias has an excellent connection to other major scientific institutions via CESNET. In order to efficiently transfer data to and from the RCF, a dedicated 1Gbit/s intercontinental link between BNL and Golias was established through cooperation between CESNET, ESNet and BNL. More than a year of operations show that about 1TB of data per day can be easily transferred from BNL and written to DPM storage using SRM transfer tools. Since DPM storage lies outside of the farm the transfer of data to running jobs at working nodes is of a concern. At this point a 1Gbit optic fiber link is setup between the storage facility and computing the farm. Stress tests show that under conditions of typical physics analyses jobs run by the group the DPM is able to supply data to $~200$ running jobs without significant slow down of the jobs. Above this threshold, the IO is a limiting factor which may require the consolidation of the Local Area Network.
Local database mirror
---------------------
Database access to the main calibration, geometry and status information database at the local farm is a small, but very helpful addition to the local setup. Even though we have high throughput connection at our disposal, the large distance of jobs from main database located at the RCF caused major prolongations of running jobs due to latencies inherent to the communication protocol. Access to a local database mirror enables running detector simulations and embedding, if needed, at the same speed as at the Tier 0 facility. Since the database mirroring is a fully automatic process done through a MySQL master/slave replication, the local database mirror is a maintenance-free addition to the Tier 2 center.
Performance
===========
The setup of a STAR Tier 2 center at the Golias farm, in the above described configuration, has been tested by the Prague physics group for more then a year in conditions typical for small to mid-size physics working group. While such a group does not always need high processing power it needs intermittent on-demand access to computing resources. The most common tasks performed at the Golias site are physics analyses on locally stored, usually preprocessed, data sets on the order of $10-1000$GB. With the available connection to the RCF facility, where primary copies of data are stored, users are able to transfer data for their analyses with minimal time delays. Hence, switching between analyses and data sets is possible and a wide range of studies have been carried out locally.
Since the setup of a Tier 2 center must to be cost-effective and simple, it has limitations and future bottlenecks are predicted. We expect to increase the number of running jobs in future increasing the data transfer from the DPM storage. Our disk-array setup of DPM is currently based on a single bus/NIC which may cause a bottle neck for data-hungry jobs in the future. For this reason another separate storage server is being purchased to distribute the disk-space among several loosely coupled disk-servers and balance the overall load. Another limitation follows from the chosen low-cost solution. Our experience shows that this solution works well at a smaller site, like ours, however if Golias Tier 2 were to grow to the size of a regular Tier 1 center, such a solution would be rather human resource hungry and likely inefficient.
Conclusions and perspectives
============================
The experience with the Tier 2 center in Prague shows that a light-weight distributed computing approach, concentrating on the redistribution of data, can improve analysis viability, sustainability, and reliability and increase the local scientific opportunity by carrying out local analyses and leveraging local human and hardware resources. This approach not only relieves the load from main computing centers (BNL/RCF and NERSC/PDSF) but also takes advantage of faster data access and shorter waiting times for the local users.
In the near future STAR experiment has to move to grid-oriented computing. Such a need is driven by steadily increasing data volumes expected in the next 5-10 years. It is thus very likely that the Golias farm as well as the NPI ASCR local data storage will be used to process grid-submitted jobs.
Acknowledgements
================
The authors wish to thank the team of Golias farm for their support of the STAR Tier 2 project. We also thank the BNL computing team for continuous support and development of the off-site computing facilities. We would like to extend our gratitude to people from ESNet, and CESNET.
This work was supported in part by the Grant Agency of the Czech Republic, Grant 202/07/0079 and Ministry of Education of the Czech Republic, Grant LC 07048.
References {#references .unnumbered}
==========
[9]{} J.W. Harris and the STAR Collaboration: [*The STAR Experiment at the Relativistic Heavy Ion Collider*]{}, Nucl. Phys [**A566**]{} (1994) 277c. K. H. Ackermann and others: [*STAR detector overview*]{}, Nucl. Instrum. Meth. [**A499**]{} (2003) 624-632 J. Adams and others: [*Experimental and theoretical challenges in the search for the quark gluon plasma: The STAR collaboration’s critical assessment of the evidence from RHIC collisions*]{}, Nucl. Phys. [**A757**]{} (2005) 102-183; nucl-ex/0501009 N. Saito: [*Spin Physics Program at RHIC: The First Polarized-Proton Collider*]{}, Nucl. Phys. [**A698**]{} (2001) 275-286 http://drupal.star.bnl.gov/STAR/starnotes/public/csn0474 http://www.ces.net http://www.gridpp.ac.uk/wiki/Disk\_Pool\_Manager http://dfpch.ujf.cas.cz/dpm\_root/main.html http://www.star.bnl.gov/public/comp/Grid/scheduler/index.html http://www.clusterresources.com/products/maui/ http://www.clusterresources.com/pages/products/torque-resource-manager.php http://www.pbsgridworks.com
|
[10]{}
G. E. Hinton, S. Osindero, and Y.-W. Teh, “A fast learning algorithm for deep belief nets,” [*Neural Computation*]{}, vol. 18, no. 7, pp. 1527–1554, 2006.
Y. Bengio, A. Courville, and P. Vincent, “Representation learning: A review and new perspectives,” [*IEEE Transactions on Pattern Analysis and Machine Intelligence*]{}, vol. 35, no. 8, pp. 1798–1828, 2013.
I. Arel, D. C. Rose, T. P. Karnowski, [*et al.*]{}, “Deep machine learning-a new frontier in artificial intelligence research,” [*IEEE Computational Intelligence Magazine*]{}, vol. 5, no. 4, pp. 13–18, 2010.
Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” [*Nature*]{}, vol. 521, no. 7553, p. 436, 2015.
G. Hinton, L. Deng, D. Yu, G. E. Dahl, A.-r. Mohamed, N. Jaitly, A. Senior, V. Vanhoucke, P. Nguyen, T. N. Sainath, [*et al.*]{}, “Deep neural networks for acoustic modeling in speech recognition: The shared views of four research groups,” [*IEEE Signal Processing Magazine*]{}, vol. 29, no. 6, pp. 82–97, 2012.
T. N. Sainath, A.-r. Mohamed, B. Kingsbury, and B. Ramabhadran, “Deep convolutional neural networks for lvcsr,” in [*IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)*]{}, pp. 8614–8618, IEEE, 2013.
G. E. Dahl, D. Yu, L. Deng, and A. Acero, “Context-dependent pre-trained deep neural networks for large-vocabulary speech recognition,” [*IEEE Transactions on Audio, Speech, and Language Processing*]{}, vol. 20, no. 1, pp. 30–42, 2012.
R. Collobert, J. Weston, L. Bottou, M. Karlen, K. Kavukcuoglu, and P. Kuksa, “Natural language processing (almost) from scratch,” [*Journal of Machine Learning Research*]{}, vol. 12, pp. 2493–2537, 2011.
I. Sutskever, O. Vinyals, and Q. V. Le, “Sequence to sequence learning with neural networks,” in [*Advances in Neural Information Processing Systems*]{}, pp. 3104–3112, 2014.
R. Socher, C. C. Lin, C. Manning, and A. Y. Ng, “Parsing natural scenes and natural language with recursive neural networks,” in [*Proceedings of the 28th International Conference on Machine Learning (ICML-11)*]{}, pp. 129–136, 2011.
A. Krizhevsky, I. Sutskever, and G. E. Hinton, “Imagenet classification with deep convolutional neural networks,” in [*Advances in Neural Information Processing Systems*]{}, pp. 1097–1105, 2012.
C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich, “Going deeper with convolutions,” in [ *Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*]{}, pp. 1–9, 2015.
L. He, G. Wang, and Z. Hu, “Learning depth from single images with deep neural network embedding focal length,” [*IEEE Transactions on Image Processing*]{}, vol. 27, no. 9, pp. 4676–4689, 2018.
J. Gao, J. Yang, G. Wang, and M. Li, “A novel feature extraction method for scene recognition based on centered convolutional restricted boltzmann machines,” [*Neurocomputing*]{}, vol. 214, pp. 708–717, 2016.
M. M. Najafabadi, F. Villanustre, T. M. Khoshgoftaar, N. Seliya, R. Wald, and E. Muharemagic, “Deep learning applications and challenges in big data analytics,” [*Journal of Big Data*]{}, vol. 2, no. 1, p. 1, 2015.
G. Litjens, T. Kooi, B. E. Bejnordi, A. A. A. Setio, F. Ciompi, M. Ghafoorian, J. A. Van Der Laak, B. Van Ginneken, and C. I. S[á]{}nchez, “A survey on deep learning in medical image analysis,” [*Medical Image Analysis*]{}, vol. 42, pp. 60–88, 2017.
X. Mo, K. Tao, Q. Wang, and G. Wang, “An efficient approach for polyps detection in endoscopic videos based on faster R-CNN,” in [*2018 24th International Conference on Pattern Recognition (ICPR)*]{}, pp. 3929–3934, IEEE, 2018.
A. Elgammal, B. Liu, M. Elhoseiny, and M. Mazzone, “CAN: Creative adversarial networks, generating “art” by learning about styles and deviating from style norms,” [*arXiv preprint arXiv:1706.07068*]{}, 2017.
W. Xu, S. Keshmiri, and G. Wang, “Adversarially approximated autoencoder for image generation and manipulation,” [*IEEE Transactions on Multimedia*]{}, DOI: 10.1109/TMM.2019.2898777, 2019.
W. Xu, S. Keshmiri, and G. Wang, “Toward learning a unified many-to-many mapping for diverse image translation,” [*Pattern Recognition*]{}, https://doi.org/10.1016/j.patcog.2019.05.017, 2019.
W. Ma, Y. Wu, Z. Wang, and G. Wang, “MDCN: Multi-scale, deep inception convolutional neural networks for efficient object detection,” in [*2018 24th International Conference on Pattern Recognition (ICPR)*]{}, pp. 2510–2515, IEEE, 2018.
Z. Zhang, Y. Wu, and G. Wang, “Bpgrad: Towards global optimality in deep learning via branch and pruning,” in [*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*]{}, pp. 3301–3309, 2018.
L. Liu, W. Ouyang, X. Wang, P. Fieguth, J. Chen, X. Liu, and M. Pietik[ä]{}inen, “Deep learning for generic object detection: A survey,” [*arXiv preprint arXiv:1809.02165*]{}, 2018.
F. Cen and G. Wang, “Dictionary representation of deep features for occlusion-robust face recognition,” [*IEEE Access*]{}, vol. 7, pp. 26595–26605, 2019.
M. Everingham, L. Van Gool, C. K. Williams, J. Winn, and A. Zisserman, “The pascal visual object classes (voc) challenge,” [*International Journal of Computer Vision*]{}, vol. 88, no. 2, pp. 303–338, 2010.
T.-Y. Lin, M. Maire, S. Belongie, J. Hays, P. Perona, D. Ramanan, P. Doll[á]{}r, and C. L. Zitnick, “Microsoft coco: Common objects in context,” in [*European Conference on Computer Vision*]{}, pp. 740–755, Springer, 2014.
S. P. Bharati, S. Nandi, Y. Wu, Y. Sui, and G. Wang, “Fast and robust object tracking with adaptive detection,” in [*2016 IEEE 28th International Conference on Tools with Artificial Intelligence (ICTAI)*]{}, pp. 706–713, IEEE, 2016.
Y. Wu, Y. Sui, and G. Wang, “Vision-based real-time aerial object localization and tracking for UAV sensing system,” [*IEEE Access*]{}, vol. 5, pp. 23969–23978, 2017.
S. P. Bharati, Y. Wu, Y. Sui, C. Padgett, and G. Wang, “Real-time obstacle detection and tracking for sense-and-avoid mechanism in UAVs,” [*IEEE Transactions on Intelligent Vehicles*]{}, vol. 3, no. 2, pp. 185–197, 2018.
Y. Wei, X. Pan, H. Qin, W. Ouyang, and J. Yan, “Quantization mimic: Towards very tiny CNN for object detection,” in [*Proceedings of the European Conference on Computer Vision (ECCV)*]{}, pp. 267–283, 2018.
K. Kang, H. Li, J. Yan, X. Zeng, B. Yang, T. Xiao, C. Zhang, Z. Wang, R. Wang, X. Wang, [*et al.*]{}, “T-CNN: Tubelets with convolutional neural networks for object detection from videos,” [*IEEE Transactions on Circuits and Systems for Video Technology*]{}, vol. 28, no. 10, pp. 2896–2907, 2018.
W. Chu and D. Cai, “Deep feature based contextual model for object detection,” [*Neurocomputing*]{}, vol. 275, pp. 1035–1042, 2018.
S. Ren, K. He, R. Girshick, and J. Sun, “Faster R-CNN: Towards real-time object detection with region proposal networks,” in [*Advances in Neural Information Processing Systems*]{}, pp. 91–99, 2015.
J. Redmon, S. Divvala, R. Girshick, and A. Farhadi, “You only look once: Unified, real-time object detection,” in [*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*]{}, pp. 779–788, 2016.
K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in [*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*]{}, pp. 770–778, 2016.
T.-Y. Lin, P. Goyal, R. Girshick, K. He, and P. Doll[á]{}r, “Focal loss for dense object detection,” [*IEEE Transactions on Pattern Analysis and Machine Intelligence*]{}, p. 1, 2018.
Q. Zhao, T. Sheng, Y. Wang, Z. Tang, Y. Chen, L. Cai, and H. Ling, “M2det: A single-shot object detector based on multi-level feature pyramid network,” [*CoRR*]{}, vol. abs/1811.04533, 2019.
P. Felzenszwalb, D. McAllester, and D. Ramanan, “A discriminatively trained, multiscale, deformable part model,” in [*IEEE Conference on Computer Vision and Pattern Recognition*]{}, pp. 1–8, IEEE, 2008.
P. Sermanet, D. Eigen, X. Zhang, M. Mathieu, R. Fergus, and Y. LeCun, “Overfeat: Integrated recognition, localization and detection using convolutional networks,” [*arXiv preprint arXiv:1312.6229*]{}, 2013.
J. R. Uijlings, K. E. Van De Sande, T. Gevers, and A. W. Smeulders, “Selective search for object recognition,” [*International Journal of Computer Vision*]{}, vol. 104, no. 2, pp. 154–171, 2013.
R. Girshick, J. Donahue, T. Darrell, and J. Malik, “Rich feature hierarchies for accurate object detection and semantic segmentation,” in [ *Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*]{}, pp. 580–587, 2014.
W. Liu, D. Anguelov, D. Erhan, C. Szegedy, S. Reed, C.-Y. Fu, and A. C. Berg, “Ssd: Single shot multibox detector,” in [*European Conference on Computer Vision*]{}, pp. 21–37, Springer, 2016.
J. Redmon and A. Farhadi, “Yolov3: An incremental improvement,” [*arXiv preprint arXiv:1804.02767*]{}, 2018.
K. Simonyan and A. Zisserman, “Very deep convolutional networks for large-scale image recognition,” [*arXiv preprint arXiv:1409.1556*]{}, 2014.
X. Glorot and Y. Bengio, “Understanding the difficulty of training deep feedforward neural networks,” in [*Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics*]{}, pp. 249–256, 2010.
G. Huang, Z. Liu, L. Van Der Maaten, and K. Q. Weinberger, “Densely connected convolutional networks.,” in [*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*]{}, pp. 4700–4708, 2017.
Y. Chen, J. Li, H. Xiao, X. Jin, S. Yan, and J. Feng, “Dual path networks,” in [*Advances in Neural Information Processing Systems*]{}, pp. 4467–4475, 2017.
T.-Y. Lin, P. Doll[á]{}r, R. Girshick, K. He, B. Hariharan, and S. Belongie, “Feature pyramid networks for object detection,” in [*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*]{}, pp. 2117–2125, 2017.
D. G. Lowe, “Object recognition from local scale-invariant features,” in [ *The Proceedings of the Seventh IEEE International Conference on Computer Vision*]{}, vol. 99, no. 2, pp. 1150–1157, IEEE, 1999.
R. Girshick, “Fast R-CNN,” in [*Proceedings of the IEEE International Conference on Computer Vision*]{}, pp. 1440–1448, 2015.
J. Dai, Y. Li, K. He, and J. Sun, “R-FCN: Object detection via region-based fully convolutional networks,” in [*Advances in Neural Information Processing Systems*]{}, pp. 379–387, 2016.
Z. Cai and N. Vasconcelos, “Cascade R-CNN: Delving into high quality object detection,” in [*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*]{}, pp. 6154–6162, 2018.
J. Redmon and A. Farhadi, “Yolo9000: better, faster, stronger,” In [*Proceedings of the IEEE conference on computer vision and pattern recognition*]{}. pp. 7263-7271, 2017.
S. Ioffe and C. Szegedy, “Batch normalization: Accelerating deep network training by reducing internal covariate shift,” [ *Proceedings of the 32nd International Conference on Machine Learning*]{}, vol. 37, pp. 448–456, 2015.
C.-Y. Fu, W. Liu, A. Ranga, A. Tyagi, and A. C. Berg, “DSSD: Deconvolutional single shot detector,” [*arXiv preprint arXiv:1701.06659*]{}, 2017.
C. Szegedy, V. Vanhoucke, S. Ioffe, J. Shlens, and Z. Wojna, “Rethinking the inception architecture for computer vision,” in [*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*]{}, pp. 2818–2826, 2016.
H. Rezatofighi, N. Tsoi, J. Gwak, A. Sadeghian, I. Reid, and S. Savarese, “Generalized intersection over union: A metric and a loss for bounding box regression,” [*arXiv preprint arXiv:1902.09630*]{}, 2019.
P. Zhou, B. Ni, C. Geng, J. Hu, and Y. Xu, “Scale-transferrable object detection,” in [*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*]{}, pp. 528–537, 2018.
S. Zhang, L. Wen, X. Bian, Z. Lei, and S. Z. Li, “Single-shot refinement neural network for object detection,” in [*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*]{}, pp. 4203–4212, 2018.
B. Singh and L. S. Davis, “An analysis of scale invariance in object detection snip,” in [*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*]{}, pp. 3578–3587, 2018.
P. Zhu, L. Wen, X. Bian, L. Haibin, and Q. Hu, “Vision meets drones: A challenge,” [*arXiv preprint arXiv:1804.07437*]{}, 2018.
P. Zhu, L. Wen, D. Du, X. Bian, H. Ling, Q. Hu, Q. Nie, H. Cheng, C. Liu, X. Liu, [*et al.*]{}, [*VisDrone-DET2018: The Vision Meets Drone Object Detection in Image Challenge Results*]{}. 2018.
J. S. Bergstra, R. Bardenet, Y. Bengio, and B. K[é]{}gl, “Algorithms for hyper-parameter optimization,” in [*Advances in Neural Information Processing Systems*]{}, pp. 2546–2554, 2011.
B. Wu and R. Nevatia, “Detection of multiple, partially occluded humans in a single image by bayesian combination of edgelet part detectors,” in [ *Tenth IEEE International Conference on Computer Vision*]{}, vol. 1, pp. 90–97, 2005.
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abstract: 'The 2011 PhD thesis of Farris [@farris] demonstrated that the ECH of a prequantization bundle over a Riemann surface is isomorphic as a ${{\mathbb Z}}_2$-graded group to the exterior algebra of the homology of its base. We extend this result by computing the $\mathbb{Z}$-grading on the chain complex, permitting a finer understanding of this isomorphism. We fill in some technical details, including the Morse-Bott direct limit argument and some writhe bounds. The former requires the isomorphism between filtered Seiberg-Witten Floer cohomology and filtered ECH as established by Hutchings–Taubes [@cc2]. The latter requires the work on higher asymptotics of pseudoholomorphic curves by Cristofaro-Gardiner–Hutchings–Zhang [@CGHZ]. In a subsequent paper, we plan to extend these methods for Seifert fiber spaces and relate the ECH $U$-map to orbifold Gromov-Witten invariants of the base.'
author:
- 'Jo Nelson[^1] and Morgan Weiler'
title: Embedded contact homology of prequantization bundles
---
Introduction
============
Embedded contact homology (ECH) is an invariant of three dimensional contact manifolds, due to Hutchings [@Hu2], with powerful applications to dynamics and symplectic embedding problems. Most computations of ECH rely on enumerative toric methods, and a general Morse-Bott program for ECH does not presently exist. Following the framework given by Farris [@farris] and providing additional details in preparation for our future work, we show that for prequantization bundles, there is an appropriately filtered ECH differential which only counts cylinders corresponding to unions of fibers over Morse flow lines of a perfect Morse function on the base. We then make use of direct limits for filtered ECH, as established in [@cc2], to provide a Morse-Bott means of computing ECH for prequantization bundles over closed Riemann surfaces. This permits us to conclude that the ECH of a prequantization bundle over a Riemann surface is isomorphic as a ${{\mathbb Z}}_2$-graded group to the exterior algebra of the homology of this base. In a subsequent paper we will extend our work to Seifert fiber spaces and consider the U-map on the ECH of prequantization bundles and Seifert fiber spaces, enabling us to compute their ECH capacities.
Definitions and overview of ECH
-------------------------------
Let $Y$ be a closed three-manifold with a contact form $\lambda$. Let $\xi=\ker(\lambda)$ denote the associated contact structure, and let $R$ denote the associated Reeb vector field, which is uniquely determined by $$\lambda(R)=1, \ \ \ d\lambda(R, \cdot)=0.$$ A [*Reeb orbit*]{} is a map $\gamma:{{\mathbb R}}/T{{\mathbb Z}}\to Y$ for some $T>0$ such that $\gamma'(t)=R(\gamma(t))$, modulo reparametrization. A Reeb orbit is said to be *embedded* whenever this map is injective. For a Reeb orbit as above, the linearized Reeb flow for time $T$ defines a symplectic linear map $$\label{eqn:lrt}
P_\gamma:(\xi_{\gamma(0)},d\lambda) \longrightarrow (\xi_{\gamma(0)},d\lambda).$$ The Reeb orbit $\gamma$ is [*nondegenerate*]{} if $P_\gamma$ does not have $1$ as an eigenvalue. The contact form $\lambda$ is called nondegenerate if all Reeb orbits are nondegenerate; generic contact forms have this property. Fix a nondegenerate contact form below. A nondegenerate Reeb orbit $\gamma$ is [*elliptic*]{} if $P_\gamma$ has eigenvalues on the unit circle and [*hyperbolic*]{} if $P_\gamma$ has real eigenvalues. If $\tau$ is a homotopy class of trivializations of $\xi|_\gamma$, then the [*Conley-Zehnder index*]{} ${{\operatorname}{CZ}}_\tau(\gamma)\in{{\mathbb Z}}$ is defined; see the review in §\[cz-sec\]. The parity of the Conley-Zehnder index does not depend on the choice of trivialization $\tau$, and is even when $\gamma$ is positive hyperbolic and odd otherwise. We say that an almost complex structure $J$ on ${{\mathbb R}}\times Y$ is [*$\lambda$-compatible*]{} if $J(\xi)=\xi$; $d\lambda(v,Jv)>0$ for nonzero $v\in\xi$; $J$ is invariant under translation of the ${{\mathbb R}}$ factor; and $J(\partial_s)=R$, where $s$ denotes the ${{\mathbb R}}$ coordinate. We denote the set of all $\lambda$-compatible $J$ by ${\mathscr{J}}(Y,\lambda)$.
ECH is defined roughly defined as follows, with a complete description given in §\[basics\]. Given a closed 3-manifold $Y$ equipped with a nondegenerate contact form $\lambda$ and generic $\lambda$-compatible $J$, the *ECH chain complex* (with respect to a fixed homology class $\Gamma \in H_1(Y)$) is the ${{\mathbb Z}}_2$ vector space freely generated by finite sets of pairs $\alpha = \{ (\alpha_i, m_i) \}$ where the $\alpha_i$ are distinct embedded Reeb orbits, the $m_i$ are positive integers, the total homology class $\sum_i m_i[\alpha_i] = \Gamma$, and $m_i=1$ whenever $\alpha_i$ is hyperbolic. Let ${{{\mathcal M}}}^J(\alpha,\beta)$ denote the set of $J$-holomorphic currents from $\alpha$ to $\beta$. The *ECH differential* is a mod 2 count of ECH index 1 currents. The definition of the ECH index is the key nontrivial part of ECH [@Hindex], and under the assumption that $J$ is generic, guarantees that the curves are embedded, except possibly for multiple covers of *trivial cylinders* ${{\mathbb R}}\times \gamma$ where $\gamma$ is a Reeb orbit. Let $ECH_*(Y,\lambda,\Gamma,J)$ denote the homology of the ECH chain complex. It turns out that this homology does not depend on the choice of $J$ or on the contact form $\lambda$ for $\xi$, and so defines a well-defined ${{\mathbb Z}}/2$ module $ECH_*(Y,\xi,\Gamma)$. The proof of invariance goes through Taubes’ isomorphism with Seiberg-Witten Floer cohomology [@taubesechswf1].
There is a filtration on ECH which enables us to compute it via successive approximations, as explained in §\[subsec:PQBgens\] and §\[subsec:directlimit\]. The *symplectic action* or *length* of an orbit set $\alpha=\{(\alpha_i,m_i)\}$ is $$\mathcal{A}(\alpha):=\sum_im_i\int_{\alpha_i}\lambda$$ If $J$ is $\lambda$-compatible and there is a $J$-holomorphic current from $\alpha$ to $\beta$, then $\mathcal{A}(\alpha)\geq\mathcal{A}(\beta)$ by Stokes’ theorem, since $d\lambda$ is an area form on such $J$-holomorphic curves. Since $\partial$ counts $J$-holomorphic currents, it decreases symplectic action, i.e., $$\langle\partial\alpha,\beta\rangle\neq0\Rightarrow\mathcal{A}(\alpha)\geq\mathcal{A}(\beta)$$
Let $ECC_*^L(Y,\lambda,\Gamma;J)$ denote the subgroup of $ECC_*(Y,\lambda,\Gamma;J)$ generated by orbit sets of symplectic action less than $L$. Since $\partial$ decreases action, it is a subcomplex; we denote the restriction of $\partial$ to $ECC_*^L$ by $\partial^L$. It is shown in [@cc2 Theorem 1.3] that the homology of $ECC_*(Y,\lambda,\Gamma;J)$ is independent of $J$, therefore we denote its homology by $ECH_*^L(Y,\lambda,\Gamma)$, which we call *filtered ECH*.
Main theorem and outline of the proof {#ssec:introoutline}
-------------------------------------
Given a closed Riemann surface $(\Sigma_g, \omega)$ of genus $g \geq 0$, consider the the principal $S^1$-bundle ${\mathfrak{p}}: Y \to \Sigma_g$ with Euler class $e = -\frac{1}{2\pi}[\omega]$. A connection on $Y$ is an imaginary-valued 1-form $A \in \Omega^1(P, i{{\mathbb R}})$ that is $S^1$-invariant and satisfies $A(X) = i$, where the vector field $$X(p): = \frac{d}{d\theta}\bigg \vert_{\theta =0} p e^{i\theta}, \ \ \ \theta \in {{\mathbb R}}/2\pi {{\mathbb Z}},$$ is the derivative of the $S^1$-action of the bundle. Since the Euler class of $Y$ is $-\frac{1}{2\pi}[\omega]$, the connection can be chosen such that its curvature is $$F_A = dA = i {\mathfrak{p}}^*\omega;$$ see [@intro Thm. 2.7.4]. The 1-form $$\lambda:= -i A$$ is a contact form on $Y$ and satisfies $d\lambda = {\mathfrak{p}}^*\omega$. Since $\lambda$ is $S^1$-invariant and $\lambda(X) = 1$, the Reeb vector field associated to $\lambda$ is $R:=X$, and all Reeb orbits are degenerate and have period $2\pi$. This contact manifold $(Y,\lambda)$ is called the *prequantization bundle* of $(\Sigma_g, \omega)$, and the construction can be generalized to a symplectic base of arbitrary dimension. The Hopf fibration is an example of a prequantization bundle of Euler class -1 over the sphere, while the lens space $L(|e|,1)$ is an example of a prequantization bundle of arbitrary negative Euler class $e$ over the sphere. Our main result is the following computation of the ECH of prequantization bundles.
\[thm:mainthm\] Let $(Y, \xi = \mbox{\em ker} \lambda)$ be a prequantization bundle over $(\Sigma_g, \omega)$ of negative Euler class $e$. Then as ${{\mathbb Z}}_2$-graded ${{\mathbb Z}}_2$-modules, $$\bigoplus_{\Gamma\in H_1(Y;{{\mathbb Z}})}ECH_*(Y,\xi,\Gamma)\cong\Lambda^*H_*(\Sigma_g;{{\mathbb Z}}_2).$$ Moreover, each $\Gamma\in H_1(Y;{{\mathbb Z}})$ satisfying $ECH_*(Y,\xi,\Gamma)\neq0$ corresponds to a number in $\{0,\dots,-e-1\}$, and under this correspondence $$\label{eqn:Lambdacorresp}
ECH_*(Y,\xi,\Gamma)\cong\bigoplus_{d\in{{\mathbb Z}}_{\geq0}}\Lambda^{\Gamma+(-e)d}H_*(\Sigma_g;{{\mathbb Z}}_2)$$ as ${{\mathbb Z}}_2$-graded ${{\mathbb Z}}_2$-modules. When $\Gamma=0$, the ${{\mathbb Z}}_2$-graded isomorphism (\[eqn:Lambdacorresp\]) can be upgraded to an isomorphism of ${{\mathbb Z}}$-graded ${{\mathbb Z}}_2$-modules, where if $*$ denotes the grading on ECH and $\bullet$ the grading on the right hand side of (\[eqn:Lambdacorresp\]), then $$\label{eqn:0grading}
*=-ed^2+(\chi(\Sigma_g)+e)d+\bullet.$$ When $\Gamma\neq0$, we have a relatively ${{\mathbb Z}}$-graded isomorphism. Let $\alpha\in\Lambda^{\Gamma+(-e)d_\alpha}H_*(\Sigma_g;{{\mathbb Z}}_2)$ and $\beta\in\Lambda^{\Gamma+(-e)d_\beta}H_*(\Sigma_g;{{\mathbb Z}}_2)$, and let $\alpha$ and $\beta$ also denote the corresponding homogeneous ECH elements under (\[eqn:Lambdacorresp\]). Let $|\cdot|_*$ denote the ECH grading and $|\cdot|_\bullet$ the grading on the right hand side of (\[eqn:Lambdacorresp\]). Then $$|\alpha|_*-|\beta|_*=-e(d_\alpha^2-d_\beta^2)+(\chi(\Sigma_g)+2\Gamma+e)(d_\alpha-d_\beta)+|\alpha|_\bullet-|\beta|_\bullet.$$
\[ex:lensspc\] As a refinement of Theorem \[thm:mainthm\], we compute the ${{\mathbb Z}}$-graded ECH of the lens spaces $L(|e|,1)$, obtaining the ECH version of [@kmos Corollary 3.4]: $$ECH_*(Y,\xi,\Gamma)=\begin{cases}
{{\mathbb Z}}&\text{ if $*\in2{{\mathbb Z}}_{\geq0}$}
\\0&\text{ else}
\end{cases}$$
We now give a sketch of the proof, which follows much of Farris’ approach as outlined in [@farris §1], and provide the organization of the paper along the way. In §\[basics\], we provide a short primer on the ingredients that go into ECH. Then, in §\[sec:ECHI\] we give a detailed computation of the ECH index for certain orbit sets in a perturbation of the canonical contact form associated to a prequantization bundle by a perfect Morse function $H$. Our computation of the ECH index uses the blueprint given in [@farris §3] and fills in a number of details which were not previously available. Before we can give the computation, we need to perturb the degenerate canonical contact form $\lambda$.
As in [@jo2] we use the following perturbation in §\[sec:ECHI\] to employ Morse-Bott computational methods on action filtered ECH. We use a Morse-Smale function $H:\Sigma\to{{\mathbb R}}$ with $|H|_{C^2}<1$ to perturb the canonical connection contact form $\lambda$, $$\lambda_\varepsilon:=(1+\varepsilon\mathfrak{p}^*H)\lambda.$$ The perturbed Reeb vector field is $$R_\varepsilon=\frac{R}{1+\varepsilon\mathfrak{p}^*H}+\frac{\varepsilon\tilde X_H}{(1+\varepsilon\mathfrak{p}^*H)^2}$$ where $\tilde X_H$ is the horizontal lift of the Hamiltonian[^2] vector field $X_H$ to $\xi$. Note that if $p \in \mbox{Crit}(H)$ then $X_H(p)=0$. Fix $L>0$; there exists $ \varepsilon >0$ so that the orbit $\gamma$ of $R_\varepsilon$ satisfies the following dichotomy:
- if $\mathcal{A}(\gamma)<L$ then $\gamma$ is nondegenerate and projects to $p \in \mbox{Crit}(H)$;
- if $\mathcal{A}(\gamma)>L$ then $\gamma$ loops around the tori above the orbits of $X_H$, or is a larger iterate of a fiber above $p\in\mbox{Crit}(H)$.
We denote the $k$-fold cover projecting to $p \in \mbox{Crit}(H)$ by $\gamma_p^k$. We have the following expression for the Conley-Zehnder index (see [@vknotes], [@jo2 §4]): $$CZ_\tau(\gamma_p^k) = RS_\tau( \mbox{fiber}^k) - \frac{\mbox{dim}(\Sigma_g)}{2} + \mbox{ind}_p(H).$$ Using the constant trivialization $\tau$ of $\xi=\mathfrak{p}^*T\Sigma_g$, we have that the Robbin-Salamon index of the degenerate fiber is $RS_\tau( \mbox{fiber}^k) =0$. Thus $$CZ_\tau(\gamma_p^k) = \mbox{ind}_p(H) -1.$$ If $\mbox{ind}_p(H)=1$ then $\gamma_p$ is positive hyperbolic. Since $\mathfrak{p}$ is a bundle, all linearized return maps are close to the identity, thus there are no negative hyperbolic orbits. If $\mbox{ind}_p(H)=0,2$ then $\gamma_p$ is elliptic.
After taking $H$ to be perfect, we denote the index zero elliptic orbit by $e_-$, the index two elliptic orbit by $e_+$, and the hyperbolic orbits by $h_1,\dots,h_{2g}$. The critical points of a perfect $H$ form a basis for $H_*(\Sigma_g;{{\mathbb Z}}_2)$. The generators of $CC_*^L(Y,\lambda_{\varepsilon(L)},\Gamma)$ for some $L$, where $\lambda_\varepsilon(L)$ are of the form $e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+}$ where $m_i=0,1$, and hence form a basis for the exterior algebra $\Lambda^*H_*(\Sigma_g;{{\mathbb Z}}_2)$ when both are given a ${{\mathbb Z}}_2$ grading, under the map generated by sending each critical point to its Morse homology class.
For homologous orbit sets $\alpha$ and $\beta$, let $d$ denote the difference in the number of embedded orbits, counted with multiplicity, appearing in $\alpha$ and $\beta$, divided by $-e$: $$\label{intro:d}
d=\frac{M-N}{|e|},$$ where $M:=m_-+m_1+\cdots+m_{2g}+m_+$ and similarly $N:=n_-+n_1+\cdots+n_{2g}+n_+$.
We obtain the following formula for the ECH index, cf. Proposition \[prop:indexcalc\]:
\[intro:ECHindex\] Let $(Y,\lambda)$ be a prequantization bundle over a surface $\Sigma_g$ with Euler class $e\in{{\mathbb Z}}_{<0}$. The ECH index in $ECH_*^L(Y,\lambda_{\varepsilon(L)},\Gamma)$ satisfies the following formula for any $\Gamma, L$, and orbit sets $\alpha$ and $\beta$, where $L$ and $\varepsilon(L)$ are as in Lemma \[lem:efromL\], $$\label{introeqn:indexformula}
I(\alpha,\beta)=\chi(\Sigma_g)d-d^2e+2dN+m_+-m_--n_++n_-.$$
The above formula recovers the ECH index for perturbations of the standard $S^3$. The index formula in [@Hu2], when applied to the contact forms on the ellipsoids $E(1+\delta_n,1-\eta_n)$ where $\delta_n,\eta_n\to0$ and $\frac{1-\delta_n}{1-\eta_n}\in{{\mathbb R}}-{{\mathbb Q}}$, converges to our formula with $g=0$ and $e=-1$.
We prove that the filtered ECH differential $\partial$ only counts cylinders corresponding to Morse flows on $\Sigma_g$, therefore $\partial(e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+})$ is a sum over all ways to apply $\partial^{Morse}$ to $h_i$ or $e_+$. In §\[sssec:pfg>0\], we show that $\partial$ is zero because $H$ is perfect.
Before sketching the correspondence between Morse flows and the filtered ECH differential (the content of §\[sec:modspcs\]-\[handleslides\]), we say a few words about how we make the above Morse-Bott formalism precise. In §\[sec:ECHI\] we show that given $L$ there exists $ \varepsilon(L)>0$ so that the generators of $ECC^L_*(Y,\lambda_\varepsilon,J)$ consist solely of orbits which are fibers over critical points. We have:
\[intro:directlimitcomputesfiberhomology\] With $Y,\lambda$, and $\varepsilon(L)$ as in Lemma \[lem:efromL\], for any $\Gamma \in H_1(Y;{{\mathbb Z}})$, there is a direct system consisting of all the $ECH^L_*(Y,\lambda_{\varepsilon(L)},\Gamma)$. The direct limit $\lim_{L\to\infty}ECH^L_*(Y,\lambda_{\varepsilon(L)},\Gamma)$ is the homology of the chain complex generated by orbit sets $\{(\alpha_i,m_i)\}$ where the $\alpha_i$ are fibers above critical points of $H$ and $\sum_im_i[\alpha_i]=\Gamma$.
Then in §\[sec:finalcomp\], rather than considering degenerations of moduli spaces directly, we instead pass to filtered ECH and take direct limits by appealing to the isomorphism with filtered Seiberg-Witten theory [@cc2]. We prove:
\[intro:thm:dlisECH\] With $Y,\lambda,\varepsilon(L)$ as in Lemma \[lem:efromL\], and for any $\Gamma \in H_1(Y;{{\mathbb Z}})$, $$\lim_{L\to\infty}ECH_*^L(Y,\lambda_{\varepsilon(L)},\Gamma)=ECH_*(Y,\xi,\Gamma).$$
Next we sketch why the appropriately filtered differential $\partial^{L}$ only counts cylinders which are the union of fibers over Morse flow lines in $\Sigma_g$. Cylinder counts associated to the above perturbation permit the use of fiberwise $S^1$-invariant ${\mathfrak{J}}= {\mathfrak{p}}^*j_{\Sigma_g}$, even for multiply covered curves, by automatic transversality, cf. [@jo2]. However for somewhere injective curves with genus, we cannot obtain transversality using a $S^1$-invariant ${\mathfrak{J}}$, and following [@farris §6], we make use of $S^1$-invariant domain dependent almost complex structures ${\mathbb{J}}\in {\mathscr{J}}_{{\dot{\Sigma}}}^{S^1}$ in §\[sec:modspcs\] to achieve regularity. That the ECH differential does not admit noncylindrical contributions, follows in part from the following proposition, which demonstrates that domain dependent positive genus curves do not contribute to ECH index 1 moduli spaces.
\[intro-nogenus\] Let $\alpha$ and $\beta$ be nondegenerate orbit sets and ${\mathbb{J}}\in {\mathscr{J}}_{{\dot{\Sigma}}}^{S^1}$ be generic. If ${\operatorname}{deg}(\alpha,\beta) >0$ and $I(\alpha,\beta)=1$ then ${{{\mathcal M}}}^{{\mathbb{J}}}(\alpha,\beta) = \emptyset$.
In §\[degree\], we define the notion of degree of a completed projected pseudoholomorphic curve in $\Sigma_g$ and show that ${\operatorname}{deg}(\alpha,\beta) = d$, where $d$ is as in , c.f. Definition \[degree-gen\]. By Lemma \[lem:d0g0\], a curve $C$ which contributes nontrivially to the ECH differential has degree zero if and only if it is a cylinder. We note that Proposition \[intro-nogenus\] and its proof previously appeared in [@farris Corollary 6.2.3], and we repeat his arguments from which this corollary follows, including the regularity result for $S^1$-invariant domain dependent almost complex structures Theorem \[thm:ddacs\]; cf. Corollary \[nogenus\].
However, the definition of ECH relies on the choice of a generic $\lambda$-compatible domain independent almost complex structure
$J$, rather than a domain dependent ${\mathbb{J}}$. For higher genus curves, we cannot achieve transversality with $S^1$-invariant domain independent $J$. As observed by Farris in [@farris §7], one can obtain the time independent analogue of Proposition \[intro-nogenus\] by considering a generic one parameter family of almost complex structures $\{ {\mathcal{J}}_t\}_{t\in[0,1]}$ interpolating between ${\mathcal{J}}_0:={\mathbb{J}}\in {\mathscr{J}}_{{\dot{\Sigma}}}^{S^1} $ and ${\mathcal{J}}_1:=J \in {\mathscr{J}}(Y,\lambda)$. We obtain the following result in §\[handleslides\]:
\[intro-handleslides\] Let $\alpha$ and $\beta$ be admissible orbit sets with $I(\alpha,\beta)=1$ and ${\operatorname}{deg}(\alpha,\beta) >0.$ For generic paths $\{ {\mathcal{J}}_t\}_{t\in[0,1]}$, the moduli space $\mathcal{M}_t := \mathcal{M}^{{\mathcal{J}}_t}(\alpha, \beta)$ is cut out transversely save for a discrete number of times $t_0,...,t_\ell \in (0,1)$. For each such $t_i$, the ECH differential can change either by:
1. The creation or destruction of a pair of oppositely signed curves.[^3]
2. An “ECH handleslide."
In either case, the homology is unaffected.
From this, we obtain the desired corollary:
\[intro: nogenusJ\] Let $\alpha$ and $\beta$ be nondegenerate admissible orbit sets and $J \in {\mathscr{J}}(Y,\lambda)$ be generic. If ${\operatorname}{deg}(\alpha,\beta) >0$ and $I(\alpha,\beta)=1$ then the mod 2 count $\#_{{{\mathbb Z}}_2}{{{\mathcal M}}}^{J}(\alpha,\beta) = 0$. If $\alpha$ and $\beta$ are associated to $\lambda_\varepsilon$ as in Lemma \[lem:efromL\] and $\mathcal{A}(\alpha), \mathcal{A}(\beta) <L(\varepsilon)$ then $\langle \partial^{L(\varepsilon)} \alpha, \beta \rangle =0$.
This corollary completes the proof that if any moduli space of $J$-holomorphic currents contributes to $\partial^L$ then the currents must consist of trivial cylinders together with one cylinder, which is the union of fibers over a Morse flow line in the base $\Sigma_g$ contributing to the Morse differential. We obtain the main result, Theorem \[thm:mainthm\] as the counts of such cylinders equal the counts of the Morse flow lines which are their images under ${\mathfrak{p}}$, cf. Proposition \[cylinder-to-morse\]. We show in §\[subsec:pfmainthm\], under the assumption that the Morse function $H$ is perfect, that the latter counts are all zero.
We conclude with a brief discussion of Proposition \[intro-handleslides\], which is proven in §\[handleslides\]. In Proposition \[intro-handleslides\](a) the mod 2 counts of curves in ${{{\mathcal M}}}_{t_i-\varepsilon}(\alpha,\beta)$ and ${{{\mathcal M}}}_{t_i+\varepsilon}(\alpha,\beta)$ are the same. The differential can change at an ECH handleslide, at which a sequence of Fredholm and ECH index 1 curves $\{C(t)\}$ breaks into a holomorphic building in the sense of [@BEHWZ] into components consisting of an ECH and Fredholm index 0 curve, an ECH and Fredholm index 1 curve, and some “connectors," which are Fredholm index 0 branched covers of a trivial cylinder $\gamma \times {{\mathbb R}}$. In §\[sec:connectors\], we demonstrate that connectors cannot appear at the top most or bottom most level of the building via intersection theory arguments similar to [@dc §4], appealing to refined asymptotic estimates of Cristofaro-Gardiner–Hutchings–Zhang [@CGHZ]. The framework regarding this classification of connectors was previously indicated in [@farris §7]. We additionally explain how to invoke the obstruction bundle gluing theorems of [@obg1; @obg2] and then review how Farris’ inductive argument involving the degree of a completed projected curve finishes the proof of Proposition \[intro-handleslides\](b).
Finer points of the isomorphism and future work {#ssec:futurework}
-----------------------------------------------
In §\[sec:finalcomp\], as a step in the proof of Theorem \[thm:mainthm\], we prove the second, stronger conclusion of Theorem \[thm:mainthm\] in terms of the total $H_1(Y)$ classes $\Gamma$ and the total multiplicity of representatives of the ECH homology classes, namely that as ${{\mathbb Z}}_2$-graded ${{\mathbb Z}}_2$-modules $$\label{eqn:splitting}
ECH_*(Y,\xi,\Gamma)\cong\bigoplus_{d\in{{\mathbb Z}}_{\geq0}}\Lambda^{\Gamma+(-e)d}H_*(\Sigma_g;{{\mathbb Z}}_2).$$ Here we are abusing notation on the right hand side by considering $\Gamma$ as the element of $\{0,\dots,-e-1\}$ corresponding to its homology class (see Lemma \[lem:HYreps\] (i)). We illustrate the ${{\mathbb Z}}$-grading (\[eqn:0grading\]) in an example.
The following table organizes the first several generators in the case $g=2, e=-1, \Gamma=0$ by multiplicity (vertically) and ECH index (horizontally): $$\begin{array}{r|ccccccc}
&I=-2&I=-1&I=0&I=1&I=2&I=3&I=4
\\\hline
\\\vspace{-.9cm}
\\\Lambda^0&&&\emptyset
\\\Lambda^1&e_-&h_i&e_+
\\\Lambda^2&e_-^2&e_-h_i&e_-e_+, h_ih_j&h_ie_+&e_+^2
\\\Lambda^3&&&e_-^3&e_-^2h_i&e_-^2e_+, e_-h_ih_j&e_-h_ie_+, h_ih_jh_k&e_-e_+^2, h_ih_je_+
\\\Lambda^4&&&&&&&e_-^4
\end{array}$$ We use $I$ to denote $I(\cdot,\emptyset)$, and we use $h_i$ to denote any hyperbolic generator (there are at most four). Whenever any two or three $h_i$ appear together, they are all different.
Notice that for $*\geq1$, the $ECH_*$ groups are isomorphic to ${{\mathbb Z}}_2^8$. This is a special case of the following more general result, which is a corollary of Theorem \[thm:mainthm\]. It relies on the degree $-2$ map $$U:ECH_*(Y,\xi,\Gamma)\to ECH_{*-2}(Y,\xi,\Gamma)$$ induced by a chain map which counts index 2 $J$-holomorphic curves passing through a base point.
\[stability\] For $*$ sufficiently large and $g>0$, the groups $ECH_*(Y,\xi,\Gamma)$ are isomorphic to $\mathbb{Z}_2^{f(g)}$, where $f(g) = 2^{2g-1}$.
(For the computation of all of the ECH groups for $g=0$, see Example \[ex:lensspc\].)
Firstly we show that for $*$ sufficiently large, all of the homology groups are isomorphic. Seiberg-Witten Floer cohomology, denoted $\widehat{HM}^\bullet$ (see §\[subsec:directlimit\]), is an invariant of smooth closed three-manifolds $Y$ with spin-c structures $\mathfrak{s}$, for which Taubes [@taubesechswf1]-[@taubesechswf4] has shown that $\widehat{HM}^{-*}(Y,\mathfrak{s}_{\xi,\Gamma})\cong ECH_*(Y,\xi,\Gamma)$ (the notation $\mathfrak{s}_{\xi,\Gamma}$ denotes a spin-c structure naturally associated to $(\xi,\Gamma)$ as in [@cc2]). It fits into a long exact sequence with two other homology theories $$\cdots\leftarrow\widecheck{HM}^\bullet(Y,\mathfrak{s})\leftarrow\widehat{HM}^\bullet(Y,\mathfrak{s})\leftarrow\overline{HM}^{\bullet-1}(Y,\mathfrak{s})\leftarrow\cdots$$ where $\overline{HM}^\bullet$ contains $\widehat{HM}^\bullet$ as a subcomplex, and is constructed to satisfy the property $$U:\overline{HM}^\bullet(Y,\mathfrak{s}_{\xi,\Gamma})\to\overline{HM}^{\bullet+2}(Y,\mathfrak{s}_{\xi,\Gamma}) \text{ is an isomorphism}$$ where $U$ is a map on $\overline{HM}^\bullet$ equalling the image of the $U$-map in ECH under the isomorphism $\widehat{HM}^{-*}(Y,\mathfrak{s}_{\xi,\Gamma})\cong ECH_*(Y,\xi,\Gamma)$. Since $\widecheck{HM}^\bullet$ is zero for $\bullet$ sufficiently small as in the proof of [@KMbook Cor. 35.1.4], the isomorphism from $\overline{HM}$ descends to an isomorphism $$U:\widehat{HM}^{-*}(Y,\mathfrak{s}_{\xi,\Gamma})\to\widehat{HM}^{-*+2}(Y,\mathfrak{s}_{\xi,\Gamma})$$ thus to an isomorphism between $ECH_*(Y,\xi,\Gamma)$ and $ECH_{*-2}(Y,\xi,\Gamma)$ for $*$ sufficiently large.
We will compute the dimension of a group $ECH_i(Y,\xi,\Gamma)$ for a well-chosen value of $i$, where if $\Gamma\neq0$ we set $i$ to be the index relative to $e_-^\Gamma$. We first make two observations, both following directly from the index formula (\[introeqn:indexformula\]):
- Amongst all generators $e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+}$ with $m_-+m_1+\cdots+m_{2g}+m_+=M$, the highest value of $I(e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+},e_-^\Gamma)$ is realized by $e_+^M$, and the lowest by $e_-^M$. We also have $$I(e_+^M,e_-^M)=2M.$$
- We have $$I(e_-^{\Gamma-(d+1)e},e_+^{\Gamma-de})=\chi(\Sigma_g).$$
By (i) and (ii), we can take $d$ large enough compared to $g$ so that all generators $e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+}$ with $I(e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+},e_-^{\Gamma-de})=M$ have $$\label{eqn:allsamed}
m_-+m_1+\cdots+m_{2g}+m_+=M=\Gamma-ed.$$ Therefore, to complete the proof we will count all generators with $I(e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+},e_-^{\Gamma-de})=M$ which satisfy (\[eqn:allsamed\]). Because $e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+}$ satisfies (\[eqn:allsamed\]), we obtain $$\label{eqn:indexsamed}
I(e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+},e_-^{\Gamma-de})=m_+-m_-+\Gamma-de.$$ By (\[eqn:allsamed\]) and (\[eqn:indexsamed\]) it suffices to count all generators with $m_-=m_+$ and $2m_-+m_1+\cdots+m_{2g}=M$. We can choose $d$ even larger if necessary so that $d>2g$.
If $\Gamma-de$ is even, then the set of such generators can be grouped into those with $0,2,4,\dots,2k,\dots,2g$ hyperbolic generators. There are ${2g\choose 2k}$ generators with $2k$ hyperbolic generators, therefore there are $$\label{eqn:evenbinomial}
{2g\choose0}+{2g\choose 2}+{2g\choose 4}+\cdots+{2g\choose 2k}+\cdots{2g\choose2g}=2^{2g-1}$$ generators with $m_-=m_+$ and $2m_-+m_1+\cdots+m_{2g}=M$ in total. Similarly, if $\Gamma-de$ is odd, then the set of such generators can be grouped into those with $1,3,\dots,2k+1,\dots,2g-1$ hyperbolic generators, and there are $$\label{eqn:oddbinomial}
{2g\choose 1}+{2g\choose 3}+\cdots+{2g\choose 2k+1}+\cdots{2g\choose2g-1}=2^{2g-1}$$ such generators in total. Both (\[eqn:evenbinomial\]) and (\[eqn:oddbinomial\]) are elementary equalities following from the sum of binomial coefficients $$\label{eqn:binomialsum}
\sum_{k=0}^n{n\choose k}x^k=(1+x)^n$$ with $n=2g$. Setting $x=1$ shows that the sum of all ${2g\choose k}$ is $2^{2g}$, and adding (\[eqn:binomialsum\]) for $x=-1$ gives us (\[eqn:evenbinomial\]) with both sides doubled. Then (\[eqn:oddbinomial\]) follows by subtracting (\[eqn:evenbinomial\]) from (\[eqn:binomialsum\]) with $x=1$.
\[conj:Uperiodicity\] For $*$ sufficiently large, the $U$ map is an isomorphism on the chain level of the ECH of prequantization bundles, for the chain complex of Proposition \[intro:directlimitcomputesfiberhomology\].
When $g=0$, Conjecture \[conj:Uperiodicity\] can likely be proved using Example \[ex:lensspc\] and the methods of [@Hu2 §4.1], discussed below. Furthermore, we will investigate the non-stable part of the homology, e.g. prove the ECH analogue of the computation of the reduced Heegaard Floer homology of nontrivial circle bundles in [@knotHF Thm. 5.6].
We are further interested in the $U$-map at the chain level as it is crucial to the definition of the “ECH spectrum." This is a list where the $k^\text{th}$ term is the minimum action of a homologically essential ECH cycle in $ECH_*(Y,\xi,[\emptyset])$ which is homologous to the preimage under $U^k$ of the class of the empty orbit set. The ECH spectrum of a contact three-manifold governs the embedding properties of its symplectic fillings, and we plan to use the $U$-map of prequantization bundles and their generalization, Seifert fiber spaces, to understand the embedding properties of fillings of the simple singularities, among other examples. In particular, we expect to directly recover that $U$ is an isomorphism in sufficiently large degree, as is known to be true for Seiberg-Witten cohomology.
We expect to be able to compute the $U$-map for general prequantization bundles following the computation in the $g=0, e=-1$ case (i.e., $Y=S^3$) in [@Hu2 §4.1], which interprets $U$ as a count of index two gradient flow lines for a Morse function on the base $S^2$, together with a count of meromorphic multisections of the complex line bundle associated to the Hopf fibration. As discussed in [@farris §1.2], in the case of a general base a $J$-holomorphic curve in ${{\mathbb R}}\times Y$ which intersects each ${{\mathbb R}}\times\{\text{fiber}\}$ the same number of times can be interpreted as a meromorphic multisection of the line bundle associated to $Y\to\Sigma_g$ with its asymptotic ends on Reeb orbits interpreted as either zeroes or poles. We plan to make this perspective rigorous.
\[conj:u\] For prequantization bundles with negative Euler class $e$ over a closed surface of genus $g$, the $U$-map is given in terms of index two gradient flow lines and meromorphic multisections of the line bundle associated to $Y$, and for sufficiently large grading induces an isomorphism on ECH.
We will investigate an analogous result for Seifert fiber spaces. Here, the novel contribution will be to understand how to characterize the $U$-map in terms of the orbifold Gromov-Witten invariants of the base, and should follow ideas in [@RiGW], which computes the symplectic cohomology of negative line bundles via Gromov-Witten theory of the base orbifold.
In addition, we hope to extend our methods to compute the knot filtered ECH of Seifert fiber spaces considered as prequantization bundles over orbifolds in the sense of [@thomas]. Knot filtered ECH was introduced in [@HuMAC] to study disk symplectomorphisms via ECH, and extended in [@weiler] to study annulus symplectomorphisms via similar methods. Since periodic surface symplectomorphisms can be realized as the return map of open book decompositions on Seifert fibered contact manifolds [@CH], our computations will allow us to study the dynamics of such symplectomorphisms, which include all symplectomorphisms on genus zero surfaces. Along the way to knot filtered ECH, understanding the standard ECH of Seifert fiber spaces should allow us to recover the computation in [@moy Cor. 1.0.6] of the irreducible Seiberg-Witten Floer homology of Brieskorn manifolds.
**Acknowledgements.** We thank Michael Hutchings for numerous elucidating conversations and comments on an earlier draft of this paper.
Basics of ECH {#basics}
=============
Let $Y$ be a closed contact 3-dimensional manifold equipped with a nondegenerate contact form $\lambda$; let $\xi=\ker(\lambda)$ denote the associated contact structure. We say that a closed Reeb orbit $\gamma$ is *elliptic* if the eigenvalues of the linearized return map $P_{\gamma}$ are on the unit circle, *positive hyperbolic* if the eigenvalues of $P_{\gamma}$ are positive real numbers, and *negative hyperbolic* if the eigenvalues of $P_{\gamma}$ are negative real numbers.
An *orbit set* is a finite set of pairs $\alpha=\{(\alpha_i,m_i)\}$, where the $\alpha_i$ are distinct embedded Reeb orbits and the $m_i$ are positive integers. We call $m_i$ the *multiplicity* of $\alpha_i$ in $\alpha$. The homology class of the orbit set $\alpha$ is defined by $$[\alpha]=\sum_i m_i [\alpha_i] \in H_1(Y).$$ The orbit set $\alpha$ is *admissible* if $m_i=1$ whenever $\alpha_i$ is positive or negative hyperbolic.
The ECH chain complex is generated by admissible orbit sets. The differential counts ECH index one $J$-holomorphic currents in ${{\mathbb R}}\times Y$. The definition of the ECH index depends on three components: the relative first Chern class $c_\tau$, which detects the contact topology of the curves; the relative intersection pairing $Q_\tau$, which detects the algebraic topology of the curves; and the Conley-Zehnder terms, which detect the contact geometry of the Reeb orbits. Further details are provided in the subsequent sections.
Pseudoholomorphic curves and currents
-------------------------------------
Given a punctured compact Riemann surface $({{\dot{\Sigma}}}, j)$, with a partition of its punctures $\mathcal{S}$ into a positive subset $\mathcal{S}_+$ and negative subset $\mathcal{S}_-$ we consider *asymptotically cylindrical $J$-holomorphic* maps of the form $$C:({{\dot{\Sigma}}}, j) \to ({{\mathbb R}}\times Y, J), \ \ \ dC \circ j = J \circ dC,$$ subject to the following asymptotic condition. If $\gamma$ is a (possibly multiply covered Reeb orbit), a *positive end* of $C$ at $\gamma$ is a puncture near which $C$ is asymptotic to ${{\mathbb R}}\times \gamma$ as $s \to \infty$ and a *negative end* of $C$ at $\gamma$ is a puncture near which $C$ is asymptotic to ${{\mathbb R}}\times \gamma$ as $s \to -\infty$. This means there exist holomorphic cylindrical coordinates identifying a punctured neighborhood of $z$ with a respective positive half-cylinder $Z_+=[0, \infty) \times S^1$ or negative half-cylinder $Z_-=(-\infty, 0] \times S^1$ and a trivial cylinder ${\mathscr{C}}_{\gamma_z}: {{\mathbb R}}\times S^1 \to {{\mathbb R}}\times Y$ such that $$\label{ass1}
C(s,t) = \exp_{{\mathscr{C}}_{\gamma_z}(s,t)}h_z(s,t) \mbox{ for $|s|$ sufficiently large},$$ where $h_z(s,t)$ is a vector field along ${\mathscr{C}}_{\gamma_z}$ satisfying $|h_z(s,\cdot)| \to 0$ uniformly as $s \to \pm \infty$. Both the norm and the exponential map are assumed to be defined with respect to a translation-invariant choice of Riemannian metric on ${{\mathbb R}}\times Y$. To obtain a moduli space of $J$-holomorphic curves from the above asymptotically cylindrical maps, we mod out by the usual equivalence relation $({{\dot{\Sigma}}}, j, \mathcal{S}, C) \sim ({{\dot{\Sigma}}}', j', \mathcal{S}', C')$, which exists whenever there exists a biholomorphism $\phi: ({{\dot{\Sigma}}}, j) \to ({{\dot{\Sigma}}}', j')$ taking $\mathcal{S}$ to $\mathcal{S}'$ with the ordering preserved, i.e. $\phi(\mathcal{S}_+) =\mathcal{S}'_+$ and $\phi(\mathcal{S}_-) =\mathcal{S}'_-$, such that $C = C' \circ \phi.$
\[covered\] An asymptotically cylindrical pseudoholomorphic curve $$C: ({{\dot{\Sigma}}}, j) \to ({{\mathbb R}}\times Y, J)$$ is said to be *multiply covered* whenever there exists a pseudoholomorphic curve $$\overline{C}: ({{\dot{\Sigma}}}' , j') \to ({{\mathbb R}}\times Y, J),$$ and a holomorphic branched covering $\varphi: ({{\dot{\Sigma}}}, j) \to ({{\dot{\Sigma}}}', j') $ with $\mathcal{S}'_+ = \varphi(\mathcal{S}_+)$ and $\mathcal{S}'_- = \varphi(\mathcal{S}_-)$ such that $$C = \overline{C} \circ \varphi, \ \ \ \mbox{deg}(\varphi)>1,$$ allowing for $\varphi$ to not have any branch points. The *multiplicity* of $C$ is given by $\mbox{deg}(\varphi).$
An asymptotically cylindrical pseudoholomorphic curve $C$ is called *simple* whenever it is not multiply covered. In [@jo1 §3.2] we gave a proof of the folk theorem that that every simple asymptotically cylindrical curve is *somewhere injective*, meaning for some $z \in {{\dot{\Sigma}}}$, which is not a puncture, $$dC(z) \neq 0 \ \ \ C^{-1}(C(z))=\{z\}.$$ A point $z \in {{\dot{\Sigma}}}$ with this property is called an *injective point* of $C$.
Let $\alpha=\{(\alpha_i,m_i)\}$ and $\beta=\{(\beta_j,n_j)\}$ be orbit sets in the same homology class $\sum_i m_i [\alpha_i]=\sum_j n_j [\beta_j]=\Gamma\in H_1(Y).$ We define a *$J$-holomorphic current* from $\alpha$ to $\beta$ to be a finite set of pairs $\mathcal{C} = \{ (C_k,d_k) \}$, where the $C_k$ are distinct irreducible somewhere injective $J$-holomorphic curves in ${{\mathbb R}}\times Y$, the $d_k$ are positive integers, $\mathcal{C}$ is asymptotic to $\alpha$ as a current as the ${{\mathbb R}}$-coordinate goes to $+\infty$, and $\mathcal{C}$ is asymptotic to $\beta$ as a current as the ${{\mathbb R}}$-coordinate goes to $-\infty$. This last condition means that the positive ends of the $C_k$ are at covers of the Reeb orbits $\alpha_i$, the sum over $k$ of $d_k$ times the total covering multiplicity of all ends of $C_k$ at covers of $\alpha_i$ is $m_i$, and analogously for the negative ends.
Let $\mathcal{M}^J(\alpha,\beta)$ denote the set of $J$-holomorphic currents from $\alpha$ to $\beta$. A holomorphic current $\mathcal{C} $ is said to be *somewhere injective* if $d_k=1$ for each $k$. A somewhere injective current is said to be *embedded* whenever each $C_k$ is embedded and the $C_k$ are pairwise disjoint.
Let $H_2(Y,\alpha,\beta)$ denote the set of relative homology classes of 2-chains $Z$ in $Y$ such that $$\partial Z =\sum_i m_i \alpha_i - \sum_j n_j \beta_j,$$ modulo boundaries of 3-chains. The set $H_2(Y,\alpha,\beta)$ is an affine space over $H_2(Y)$, and every $J$-holomorphic current $\mathcal{C}\in {{{\mathcal M}}}^J(\alpha, \beta)$ defines a class $[{{\mathcal C}}] \in H_2(Y,\alpha,\beta)$.
The Fredholm index {#sec:freddie}
------------------
Let $C$ be an asymptotically cylindrical curve with $k$ positive ends at (possibly multiply covered) Reeb orbits $\gamma_1^+,...,\gamma_k^+$ and $\ell$ negative ends at $\gamma_1^-,...,\gamma_\ell^-$. We denote the set of equivalence classes by $${\mathcal{M}}^J (\gamma^+,\gamma^-): ={\mathcal{M}}^J(\gamma_1^+,...,\gamma_k^+;\gamma_1^-,...,\gamma_\ell^-)$$ Note that ${{\mathbb R}}$ acts on ${\mathcal{M}}^J(\gamma^+,\gamma^-)$ by translation of the ${{\mathbb R}}$ factor in ${{\mathbb R}}\times Y$.
The *Fredholm index* of $C$ is defined by $$\label{w-index}
{{{\operatorname}{ind}}}(C) = -\chi(C) + 2c_\tau(C) + \displaystyle \sum_{i=1}^k{CZ}_\tau(\gamma_i^+) - \sum_{j=1}^\ell {CZ}_\tau(\gamma_j^-).$$ The [*Euler characteristic*]{} of the domain ${{\dot{\Sigma}}}$ of $C$ is denoted by $$\chi(C) =(2-2g(\Sigma) - k - \ell).$$ The remaining terms are a bit more involved to define as they depend on the choice of a trivialization $\tau$ of $\xi$ over the Reeb orbits $\gamma_i^{+}$ and $\gamma_j^-$, which is compatible with $d\lambda$, and are defined in the following subsections.
The significance of the Fredholm index is that the moduli space of somewhere injective curves ${\mathcal{M}}^J(\gamma^+,\gamma^-)$ is naturally a manifold near $C$ of dimension $\mbox{ind}(C)$. In particular, we have the following results.
[*[@wendl-sft Theorem 8.1]*]{} \[si-thm1\] Fix a nondegenerate contact form $\lambda$ on a closed manifold $Y$. Let ${\mathscr{J}}(Y,\lambda)$ be the set of all $\lambda$-compatible almost complex structures on ${{\mathbb R}}\times Y$. Then there exists a comeager subset ${\mathscr{J}}_{reg}(Y,\lambda) \subset {\mathscr{J}}(Y,\lambda), $ such that for every $J \in {\mathscr{J}}_{reg}(Y,\lambda) $, every curve $C \in {\mathcal{M}}^J(\gamma^+,\gamma^-)$ with a representative $C \colon ({{{\dot{\Sigma}}}},j) \to ({{\mathbb R}}\times Y, J)$ that has an injective point $z\in \mbox{\emph{int}}({{{\dot{\Sigma}}}})$ satisfying $ \pi_\xi \circ dC(z) \neq 0 $ is Fredholm regular.
The above result also holds for the set of somewhere injective curves in completed exact symplectic cobordisms $(W, J)$; see [@wendl-sft Theorem 7.2]. Moreover, we have that Fredholm regularity implies that a neighborhood of a curve admits the structure of a smooth orbifold.
\[folk0\] Assume that $C\colon ({{{\dot{\Sigma}}}},j) \to (W, J)$ is a non-constant curve in $\mathcal{M}^J (\gamma^+,\gamma^-).$ If $C$ is Fredholm regular, then a neighborhood of $C$ in $\mathcal{M}^J (\gamma^+,\gamma^-)$ naturally admits the structure of a smooth orbifold of dimension $${\mbox{\em ind}}(C) = -\chi(C) + 2c_\tau(C) + \displaystyle \sum_{i=1}^k{CZ}_\tau(\gamma_i^+) - \sum_{j=1}^\ell {CZ}_\tau(\gamma_j^-),$$ whose isotropy group at $C$ is given by $$\mbox{\em Aut}:=\{ \varphi \in \mbox{\em Aut}({{{\dot{\Sigma}}}}, j) \ | \ C = C \circ \varphi \}.$$ Moreover, there is a natural isomorphism $$T_C\mathcal{M}^J (\gamma^+,\gamma^-) = \ker D{\bar{\partial}_{J}}(j,C)/\mathfrak{aut}({{{\dot{\Sigma}}}},j).$$
### The relative first Chern number
The trivialization $\tau$ determines a trivialization of $\xi|_C$ over the ends of $C$ up to homotopy. We denote the set of homotopy classes of symplectic trivializations of the 2-plane bundle $\gamma^*\xi$ over $S^1$ by $\mathcal{T}(\gamma)$. After fixing trivializations $\tau_i^+ \in \mathcal{T}(\gamma_i^+)$ for each $i$ and $\tau_j^- \in \mathcal{T}(\gamma^-_j)$, we denote this set of trivialization choices by $\tau \in \mathcal{T}(\alpha,\beta)$.
We are now ready to define the *relative first Chern number* of the complex line bundle $\xi|_C$ with respect to the trivialization $\tau$, which we denote by $$c_\tau(C) = c_1(\xi|_C,\tau).$$ Let $\pi_Y:{{\mathbb R}}\times Y\to Y$ denote projection onto $Y$. We define $c_1(\xi|_C,\tau)$ to be the algebraic count of zeros of a generic section $\psi$ of $\xi|_{[\pi_YC]}$ which on each end is nonvanishing and constant with respect to the trivialization on the ends. In particular, given a class $Z\in H_2(Y,\alpha,\beta)$ we represent $Z$ by a smooth map $f: S \to Y$ where $S$ is a compact oriented surface with boundary. Choose a section $\psi$ of $f^*\xi$ over $S$ such that $\psi$ is transverse to the zero section and $\psi$ is nonvanishing over each boundary component of $S$ with winding number zero with respect to the trivialization $\tau$. We define $$c_\tau(Z) : = \# \psi^{-1}(0),$$ where ‘\#’ denotes the signed count.
In addition to being well-defined, the relative first Chern class depends only on $\alpha, \beta, \tau,$ and $Z$. If $Z' \in H_2(Y,\alpha,\beta)$ is another relative homology class then $$c_\tau(Z) - c_\tau(Z') = \langle c_1(\xi), Z - Z' \rangle.$$
Our sign convention is that if $$\tau_1, \ \tau_2: \gamma^*\xi \to S^1 \times {{\mathbb R}}^2$$ are two trivializations then $$\label{degtrivs}
\tau_1 - \tau_2 = \mbox{deg} \left(\tau_2 \circ \tau_1^{-1} : S^1 \to \mbox{Sp}(2,{{\mathbb R}}) \cong S^1\right).$$ Thus, given another collection of trivialization choices $\tau' = \left( \{ {\tau'}_i^+ \}, \{ {\tau'}_j^-\} \right)$ over the orbit sets and the convention , we have $$\label{cherntriv}
c_\tau(Z) - c_{\tau'}(Z) = \sum_i m_i\left(\tau_i^+-{\tau}_i^{+'}\right) - \sum_j n_j \left(\tau_j^--{\tau}_j^{-'}\right).$$
### The Conley-Zehnder index in dimension 3 {#cz-sec}
Next we define the *Conley-Zehnder index* of an embedded nondegenerate Reeb orbit $\gamma$ with respect to the trivialization $\tau.$ Pick a parametrization $$\gamma: {{\mathbb R}}/T{{\mathbb Z}}\to Y.$$ The choice of trivialization $\tau$ of $\xi$ over $\gamma$ is an isomorphism of symplectic vector bundles $$\tau: \gamma^*\xi \stackrel{\simeq}{\longrightarrow} ({{\mathbb R}}/T{{\mathbb Z}}) \times {{\mathbb R}}^2.$$ Let $\{\varphi_t\}_{t\in {{\mathbb R}}}$ denote the one-parameter group of diffeomorphisms of $Y$ given by the flow of the Reeb vector field $R$. With respect to $\tau$, the linearized flow $(d\varphi_t)_{t\in[0,T]}$ induces an arc of symplectic matrices $P:[0,T]\to \mbox{Sp}(2)$ defined by $$\label{symparc}
P_{t} = \tau(t) \circ d\varphi_t \circ \tau(0)^{-1}.$$ To each arc of symplectic matrices $\{P_t\}_{t\in[0,T]}$ with $P_0=1$ and $P_T$ nondegenerate, there is an associated Conley Zehnder index $CZ(\{P_t\}_{t\in[0,T]})\in{{\mathbb Z}}$. We define the *Conley-Zehnder index* of $\gamma$ with respect to $\tau$ by $$CZ_\tau(\gamma) = CZ\left(\{P_t\}_{t\in[0,T]}\right).$$
[=2em ]{}
In the elliptic case each trivialization is homotopic to one whose linearized flow $\{\varphi_t\}$ can be realized as a path of rotations. If we take $\tau$ to be one of these trivializations so that each $\varphi_t$ is rotation by the angle $2 \pi \vartheta_t \in {{\mathbb R}}$ then $\vartheta_t$ is a continuous function of $t \in [0, T]$ satisfying $\vartheta_0=0$ and $\vartheta:=\vartheta_T \in {{\mathbb R}}\setminus {{\mathbb Z}}$. The number $\vartheta \in {{\mathbb R}}\setminus {{\mathbb Z}}$ is the *rotation angle* of $\gamma$ with respect to the trivialization and $${CZ}_\tau({\gamma}^k) = 2 \lfloor k \vartheta \rfloor + 1.$$
Let $v \in {{\mathbb R}}^2$ be an eigenvector of $\phi_T$. Then for any trivialization used, the family of vectors $\{ \varphi_t(v) \}_{t \in [0,T]}$, rotates through angle $\pi r$ for some integer $r$. The integer $r$ is dependent on the choice of trivialization $\Phi$, but is always even in the positive hyperbolic case and odd in the negative hyperbolic case. We obtain $$CZ_\tau({\gamma}^k) = k r.$$
The Conely-Zehnder index depends only on the Reeb orbit $\gamma$ and homotopy class of $\tau$ in the set of homotopy classes of symplectic trivializations of the 2-plane bundle $\gamma^*\xi$ over $S^1 = {{\mathbb R}}/ T{{\mathbb Z}}$. Given two trivializations $\tau_1$ and $\tau_2$ we have that $$\label{cztrivs}
CZ_{\tau_1}(\gamma^k) - CZ_{\tau_2}(\gamma^k) =2 k (\tau_1 - \tau_2 ),$$ maintaining our sign convention .
In our later expression of the ECH index, we will use the following shorthand for an orbit set $\alpha$: $$\label{CZ^I}
CZ^I_\tau(\alpha) = \sum_i \sum_{k=1}^{m_i}CZ_\tau(\alpha_i^k).$$ Another set of trivialization choices $\tau'$ for $\alpha$ yields $$CZ^I_\tau(\alpha) - {{\operatorname}{CZ}}^I_{\tau'}(\alpha) = \sum_i (m_i^2 + m_i)(\tau_i' - \tau_i).$$
We will also abbreviate the following Conley-Zehnder contributions to the Fredholm index and ECH index for a curve $C \in {{{\mathcal M}}}(\alpha,\beta;J)$ by: $$\label{CZshorthand}
\begin{array}{lcl}
CZ^I_\tau(C) &=&CZ^I_\tau(\alpha) - CZ^I_\tau(\beta) \\
CZ^{ind}_\tau(C) &=& \displaystyle \sum_{i=1}^k{CZ}_\tau(\alpha_i^{m_i}) - \sum_{j=1}^\ell {CZ}_\tau(\beta_j^{n_j}).\\
\end{array}$$
The relative intersection pairing and relative adjunction formula
-----------------------------------------------------------------
The ECH index depends on the relative intersection pairing, which is related to the asymptotic writhe and linking number. We review these notions now and summarize the relative adjunction formula.
### Asymptotic writhe and linking number {#s:writhe}
Given a somewhere injective curve $C \in {{{\mathcal M}}}^J(\gamma^+,\gamma^-)$, we consider the slice $u \cap ( \{s\} \times Y)$. If $s \gg 0$, then the slice $u \cap ( \{s\} \times Y)$ is an embedded curve which is the union, over $i$, of a braid $\zeta^+_i$ around the Reeb orbit $\gamma^+_i$ with $m_i$ strands. The trivialization $\tau$ can be used to identify the braid $\zeta_i^+$ with a link in $S^1 \times D^2$. We identify $S^1 \times D^2$ with an annulus cross an interval, projecting $\zeta^+_i$ to the annulus, and require that the normal derivative along $\gamma^+_i$ agree with the trivialization $\tau$.
We define the *writhe* of this link, which we denote by $w_\tau(\zeta_i^+) \in {{\mathbb Z}}$, by counting the crossings of the projection to ${{\mathbb R}}^2 \times \{ 0 \} $ with (nonstandard) signs. Namely, we use the sign convention in which counterclockwise rotations in the $D^2$ direction as one travels counterclockwise around $S^1$ contribute positively.
Analogously the slice $u \cap ( \{s\} \times Y)$ for $s \ll 0$ is the union over $j$ of a a braid $\zeta^-_j$ around the Reeb orbit $\beta_j$ with $n_j$ strands and we denote this braid’s writhe by $w_\tau(\zeta^-_i) \in {{\mathbb Z}}$.
The writhe depends only on the isotopy class of the braid and the homotopy class of the trivialization $\tau$. If $\zeta$ is an $m$-stranded braid around an embedded nondegenerate Reeb orbit $\gamma$ and $\tau' \in \mathcal{T}(\gamma)$ is another trivialization then $$w_\tau(\zeta) - w_{\tau'}(\zeta) = m(m-1)(\tau' - \tau)$$ because shifting the trivialization by one adds a full clockwise twist to the braid.
If $\zeta_1$ and $\zeta_2$ are two disjoint braids around an embedded Reeb orbit $\gamma$ we can define their *linking number* $\ell_\tau(\zeta_1,\zeta_2) \in {{\mathbb Z}}$ to be the linking number of their oriented images in ${{\mathbb R}}^3$. The latter is by definition one half of the signed count of crossings of the strand associated to $\zeta_1$ with the strand associated to $\zeta_2$ in the projection to ${{\mathbb R}}^2 \times \{0\}$. If the braid $\zeta_k$ has $m_k$ strands then a change in trivialization results in the following formula $$\ell_\tau(\zeta_1,\zeta_2) - \ell_{\tau'}(\zeta_1,\zeta_2) = m_1m_2 (\tau'-\tau).$$ The writhe of the union of two braids can be expressed in terms of the writhe of the individual components and the linking number: $$\label{eqn:writhelinking}
w_\tau(\zeta_1 \cup \zeta_2) = w_\tau(\zeta_1) + w_\tau(\zeta_2) + 2\ell_\tau(\zeta_1,\zeta_2).$$ If $\zeta$ is a braid around an embedded Reeb orbit $\gamma$ which is disjoint from $\gamma$ we define the *winding number* to be the linking number of $\zeta$ with $\gamma$: $$\eta_\tau(\zeta) :=\ell_\tau(\zeta,\gamma) \in {{\mathbb Z}}.$$
We have the following bound on writhe [@Hu2 §5.1].
\[prop:writheCZbound\] If $C \in {{{\mathcal M}}}(\alpha,\beta;J)$ is somewhere injective then, $$w_\tau(C) \leq CZ^I_\tau(C) - CZ^{ind}_\tau(C),$$ where the Conley-Zehnder shorthand is given by .
### Admissible representatives
In order to speak more “globally" of writhe and winding numbers associated to a curve, we need the following notion of an admissible representative for a class $Z \in H_2(Y,\alpha,\beta)$, as in [@Hrevisit Def. 2.11]. Given $Z\in H_2(Y,\alpha,\beta)$ we define an *admissible representative* of $Z$ to be a smooth map $f: S \to [-1,1] \times Y$, where $S$ is an oriented compact surface such that
1. The restriction of $f$ to the boundary $\partial S$ consists of positively oriented covers of $\{ 1\} \times \alpha_i$ whose total multiplicity is $m_i$ and negatively oriented covers of $\{ -1\} \times \beta_j$ whose total multiplicity is $n_j$.
2. The projection $\pi_Y: [-1,1] \times Y \to Y$ yields $[\pi(f(S))]=Z$.
3. The restriction of $f$ to $\mbox{int}(S)$ is an embedding and $f$ is transverse to $\{-1,1\} \times Y$.
If $S$ is an admissible representative for any $Z\in H_2(Y,\alpha,\beta)$ then we say $S$ is an *admissible surface*.
The utility of the notion of an admissible representative $S$ for $Z$ can be seen in the following. For $\varepsilon >0$ sufficiently small, $S \cap ( \{ 1 - \varepsilon \} \times Y)$ consists of braids $\zeta_i^+$ with $m_i$ strands in disjoint tubular neighborhoods of the Reeb orbits $\alpha_i$, which are well defined up to isotopy. Similarly, $S \cap ( \{ -1+ \varepsilon \} \times Y)$ consists of braids $\zeta_j^-$ with $n_j$ strands in disjoint tubular neighborhoods of the Reeb orbits $\alpha_i$, which are well defined up to isotopy.
Thus an admissible representative of $Z\in H_2(Y;\alpha,\beta)$ permits us to define the *total writhe* of a curve interpolating between the orbit sets $\alpha$ and $\beta$ by $$w_\tau(S) = \sum_i w_{\tau_i^+}(\zeta_i^+) - \sum_j w_{\tau_j^-}(\zeta_j^-).$$ Here $\zeta_i^+$ are the braids with $m_i$ strands in a neighborhood of each of the $\alpha_i$ obtained by taking the intersection of $S$ with $\{ s \} \times Y$ for $s$ close to 1. Similarly, the $\zeta_j^-$ are the braids with $n_j$ strands in a neighborhood of each of the $\beta_j$ obtained by taking the intersection of $S$ with $\{ s \} \times Y$ for $s$ close to $-1$. Bounds on the writhe in terms of the Conley-Zehnder index are given in [@dc §3.1], which relates the asymptotic behavior of pseudoholomorphic curves, extensively explored by Hutchings, cf. [@Hu2 §5.1]. We review and make use of these writhe bounds as well as the refinements [@CGHZ] in §\[handleslides\].
Taking a similar viewpoint with regard to the linking number results in the following formula. If $S'$ is an admissible representative of $Z' \in H_2(Y,\alpha',\beta')$ such that the interior of $S'$ does not intersect the interior of $S$ near the boundary, with braids $\zeta_i^{+ '}$ and $\zeta_j^{- '}$ we can define the *linking number of $S$ and $S'$* to be $$\ell_\tau(S,S') := \sum_i \ell_\tau(\zeta_i^+,\zeta_i^{+ '}) - \sum_j \ell_\tau(\zeta_j^-,\zeta_j^{- '}).$$ Above the orbit sets $\alpha$ and $\alpha'$ are both indexed by $i$, so sometimes $m_i$ or $m_{i}'$ is 0, similarly both $\beta$ and $\beta'$ are indexed by $j$ and sometimes $n_j$ or $n_{j}'$ is 0. The trivialization $\tau$ is a trivialization of $\xi$ over all Reeb orbits in the sets $\alpha, \alpha', \beta,$ and $\beta'$.
### The relative intersection pairing
The relative intersection pairing can be defined using an admissible representative, which is more general than the notion of a $\tau$-representative [@Hindex Def. 2.3], as the latter uses the trivialization to control the behavior at the boundary. Consequently, *we see an additional linking number term appear in the expression of the relative intersection pairing when we use an admissible representative.* Let $S$ and $S'$ be two surfaces which are admissible representatives of $Z \in H_2(Y,\alpha,\beta)$ and $Z'\in H_2(Y,\alpha',\beta')$ whose interiors $\dot{S}$ and $\dot{S}'$ are transverse and do not intersect near the boundary. We define the *relative intersection pairing* by the following signed count $$Q_\tau(Z,Z'):= \# \left( \dot{S} \cap \dot{S}'\right) - \ell_\tau(S,S').$$ Moreover, $Q_\tau(Z,Z')$ is an integer which depends only on $\alpha, \beta, Z, Z'$ and $\tau$. If $Z=Z'$ then we write $Q_\tau(Z):=Q_\tau(Z,Z)$.
For another collection of trivialization choices $\tau'$, $$Q_\tau(Z,Z') - Q_{\tau'}(Z,Z') = \sum_i m_i m_i' (\tau_i^+ - \tau_i^{+'}) - \sum_i n_j n_j' (\tau_i^- - \tau_i^{-'}).$$ We recall how [@Hrevisit §3.5] permits us to compute the relative intersection pairing using embedded surfaces in $Y$. An admissible representative $S$ of $Z \in H_2(Y,\alpha,\beta)$ is said to be *nice* whenever the projection of $S$ to $Y$ is an immersion and the projection of the interior $\dot{S}$ to $Y$ is an embedding which does not intersect the $\alpha_i$’s or $\beta_j$’s. Lemma 3.9 from [@Hrevisit] establishes that if none of the $\alpha_i$ equal the $\beta_j$ then every class $Z \in H_2(Y,\alpha,\beta)$ admits a nice admissible representative.
If $S$ is a nice admissible representative of $Z$ with associated braids $\zeta_i^+$ and $\zeta_j^-$ then we can define the winding number $$\eta_\tau(S) : = \sum_i \eta_{\tau_i^+}\left(\zeta_i^+\right) -\sum_j \eta_{\tau_j^-}\left(\zeta_j^-\right).$$
Suppose that $S$ is a nice admissible representative of $Z$. Then $$Q_\tau(Z) = -w_\tau(S) - \eta_\tau(S).$$
### The relative adjunction formula
In this section we review the relative adjunction formulas of interest, which are used later in §\[sec:connectors\]. This is taken from [@Hindex §3] and is stated for pseudoholomorphic curves interpolating between orbit sets $\alpha$ and $\beta$ in symplectizations. As explained in [@Hrevisit §4.4] the proof carries over in a straightforward manner to exact symplectic cobordisms.
\[lem:adjunction\] Let $u\in \mathcal{M}^J(\alpha,\beta)$ be somewhere injective, $S$ be a representative of $Z\in H_2(Y,\alpha,\beta)$, and $\tau\in \mathcal{T}(\alpha,\beta)$. Let $N_S$ be the normal bundle to $S$.
1. If $u$ is further assumed to be embedded everywhere then $$\label{relcalc}
c_1^{\tau}(Z) = \chi(S) + c_1^\tau(N_S).$$
2. For general embedded representatives $S$, e.g. ones not necessarily coming from pseuoholomorphic curves, holds mod 2 and $$\label{chernwQ}
c_1^\tau(N_S) = w_\tau(S) + Q_\tau(Z,Z).$$
3. If $u$ is embedded except at possibly finitely many singularities then $$\label{relsingeq}
c_1^\tau(Z) = \chi(u) + Q_\tau(Z) + w_\tau(u) - 2\delta(u),$$ where $\delta(u)$ is a sum of positive integer contributions from each singularity.
If $u$ is a closed pseudoholomorphic curve, then there is no writhe term or trivialization choice, and reduces to the usual adjunction formula $$\langle c_1(TW),[u]\rangle = \chi(u) + [u] \cdot [u] - 2\delta(u).$$
The ECH index
-------------
We are now ready to give the definition of the ECH index.
Let $\alpha=\{(\alpha_i,m_i)\}$ and $\beta=\{(\beta_j,n_j)\}$ be Reeb orbit sets in the same homology class, $\sum_i m_i [\alpha_i]=\sum_j n_j [\beta_j]=\Gamma\in H_1(Y).$ Given $Z \in H_2(Y,\alpha,\beta)$. We define the *ECH index* to be $$I(\alpha,\beta,Z) = c_1^\tau(Z) + Q_\tau(Z) + CZ^I_\tau(\alpha) -CZ^I_\tau(\beta).$$ where $CZ^I$ is the shorthand defined in . When $\alpha$ and $\beta$ are clear from context, we use the notation $I(Z)$.
The Chern class term is linear in the multiplicities of the orbit sets and the relative intersection term is quadratic. The “total Conley-Zehnder" index term $CZ_\tau^I$ typically behaves in a complicated way with respect to the multiplicities. We also have the following general properties of the ECH index, as proven in [@Hindex §3.3].
\[thm:Iproperties\]The ECH index has the following basic properties:
1. [*(Well Defined)*]{} The ECH index $I(Z)$ is independent of the choice of trivialization.
2. \[property:indexamb\] [*(Index Ambiguity Formula)* ]{}If $Z' \in H_2(Y,\alpha,\beta)$ is another relative homology class, then $$I(Z)-I(Z')= \langle Z-Z', c_1(\xi) + 2 \mbox{\em PD}(\Gamma) \rangle.$$
3. [*(Additivity)* ]{} If $\delta$ is another orbit set in the homology class $\Gamma$, and if $W \in H_2(Y,\beta,\delta)$, then $Z+W \in H_2(Y,\alpha,\delta)$ is defined and $$I(Z+W)=I(Z)+I(W).$$
4. [*(Index Parity)* ]{}If $\alpha$ and $\beta$ are generators of the ECH chain complex, then $$(-1)^{I(Z)} = \varepsilon(\alpha)\varepsilon(\beta),$$ where $\varepsilon(\alpha)$ denotes $-1$ to the number of positive hyperbolic orbits in $\alpha$.
To learn more about the wonders of the ECH index see [@Hrevisit §2].
We will also use the notation $I(\alpha,\beta,\mathcal{C})$ and $I(\mathcal{C})$ for $\mathcal{C}=\{ (C_k, d_k) \}\in{{{\mathcal M}}}^J(\alpha,\beta)$ to denote $I(\alpha,\beta,[\pi_YC])$, where $[\cdot]$ denotes equivalence in $H_2(Y,\alpha,\beta)$, as well as $c_\tau(\mathcal{C})$ and $Q_\tau(\mathcal{C})$, following [@Hrevisit Notation 4.7]. In addition, we will occasionally use the notation $Q_\tau(S)$ for $S$ an admissible surface in $[-1,1]\times Y$ to denote $Q_\tau([\pi_YS])$ (recall that we are using the notation $\pi_Y$ to denote both the projections from ${{\mathbb R}}\times Y$ and from $[-1,1]\times Y$ to $Y$).
The ECH index inequality (cf. §\[ECH-ineq\], [@Hrevisit Theorem 4.15]) permits the following results regarding low ECH index curves. A [*trivial cylinder*]{} is a cylinder ${{\mathbb R}}\times \gamma \subset {{\mathbb R}}\times Y$ where $\gamma$ is an embedded Reeb orbit.
\[lowiprop\] Suppose $J$ is generic. Let $\alpha$ and $\beta$ be orbit sets and let $\mathcal{C} \in {{{\mathcal M}}}^J(\alpha,\beta)$ be any $J$-holomorphic current in ${{\mathbb R}}\times Y$, not necessarily somewhere injective. Then:
1. We have $I(\mathcal{C})\geq 0$ with equality if and only if $\mathcal{C}$ is a union of trivial cylinders with multiplicities.
2. If $I(\mathcal{C}) =1$ then $\mathcal{C}= \mathcal{C}_0 \sqcup C_1$, where $I(\mathcal{C}_0)=0$, and $C_1$ is embedded and has ${{\operatorname}{ind}}(C_1)=I(C_1)=1$.
3. If $I(\mathcal{C}) =2$, and if $\alpha$ and $\beta$ are generators of the chain complex $ECC_*(Y,\lambda,\Gamma,J)$, then $\mathcal{C}= \mathcal{C}_0 \sqcup C_2$, where $I(\mathcal{C}_0)=0$, and $C_2$ is embedded and has ${{\operatorname}{ind}}(C_2)=I(C_2)=2$.
The ECH partition conditions and index inequality {#ECH-ineq}
-------------------------------------------------
The ECH partition conditions are a topological type of data associated to the pseudoholomorphic curves (and currents) which can be obtained indirectly from certain ECH index relations. In particular, the covering multiplicities of the Reeb orbits at the ends of the nontrivial components of the pseudoholomorphic curves (and currents) are uniquely determined by the trivial cylinder component information. The genus can be determined by the current’s relative homology class.
\[def:part\] [@Hu2 §3.9] Let $\gamma$ be an embedded Reeb orbit and $m$ a positive integer. We define two partitions of $m$, the *positive partition* $P^+_\gamma(m)$ and the *negative partition* $P^-_\gamma(m)$[^4] as follows.
- If $\gamma$ is positive hyperbolic, then $$P_\gamma^+(m): = P_\gamma^-(m): = (1,...,1).$$
- If $\gamma$ is negative hyperbolic, then $$P_\gamma^+(m): = P_\gamma^-(m): = \left\{ \begin{array}{ll}
(2,...,2) & m \mbox{ even,} \\
(2,...,2,1) & m \mbox{ odd. } \\
\end{array}
\right .$$
- If $\gamma$ is elliptic then the partitions are defined in terms of the quantity $\vartheta \in {{\mathbb R}}\setminus {{\mathbb Z}}$ for which ${{CZ}}^\tau(\gamma^k) = 2\lfloor k \vartheta \rfloor + 1$. We write $$P_\gamma^\pm(m): = P_\vartheta^\pm(m),$$ with the right hand side defined as follows.
Let $\Lambda^+_\vartheta(m)$ denote the highest concave polygonal path in the plane that starts at $(0,0)$, ends at $(m,\lfloor m \vartheta \rfloor)$, stays below the line $y = \vartheta x$, and has corners at lattice points. Then the integers $P^+_\vartheta(m)$ are the horizontal displacements of the segments of the path $\Lambda^+_\vartheta(m)$ between the lattice points.
Likewise, let $\Lambda^-_\vartheta(m)$ denote the lowest convex polygonal path in the plane that starts at $(0,0)$, ends at $(m,\lceil m \vartheta \rceil)$, stays above the line $y = \vartheta x$, and has corners at lattice points. Then the integers $P^-_\vartheta(m)$ are the horizontal displacements of the segments of the path $\Lambda^-_\vartheta(m)$ between the lattice points.
Both $P_\vartheta^\pm(m)$ depend only on the class of $\vartheta$ in ${{\mathbb R}}\setminus {{\mathbb Z}}$. Moreover, $P_\vartheta^+(m) = P_{-\vartheta}^-(m)$.
\[ex:partitions\] If the rotation angle for elliptic orbit $\gamma$ satisfies $\vartheta \in (0,1/m)$ then $$\begin{array}{lcl}
P_\vartheta^+(m) &=& (1,...,1) \\
P_\vartheta^-(m) &=& (m). \\
\end{array}$$ The partitions are quite complex for other $\vartheta$ values, see [@Hu2 Fig. 1].
We end this section with the ECH index inequality [@Hrevisit Theorem 4.15] in symplectic cobordisms. As before we take $\alpha=\{(\alpha_i,m_i)\}$ and $\beta=\{(\beta_j,n_j)\}$ to be Reeb orbit sets in the same homology class. Let $C \in \mathcal{M}^J(\alpha,\beta)$. For each $i$ let $a_i^+$ denote the number of positive ends of $C$ at $\alpha_i$ and let $\{ q_{i,k}^+\}_{k=1}^{a_i^+}$ denote their multiplicities. Thus $\sum_{k=1}^{a_i^+} q_{i,k}^+=m_i$. Likewise, for each $j$ let $b_i^-$ denote the number of negative ends of $C$ at $\beta_j$ and let $\{ q_{j,k}^-\}_{k=1}^{b_j^-}$ denote their multiplicities; we have $\sum_{k=1}^{b_j^-} q_{j,k}^-=n_j$.
\[thm:indexineq\] Suppose $C \in \mathcal{M}^J(\alpha,\beta)$ is somewhere injective. Then $$\mbox{\em ind}(C) \leq I(C) - 2 \delta(C).$$ Equality holds only if $\{ q_{i,k}^+ \} = P_{\alpha_i}^+(m_i)$ for each $i$ and $\{ q_{j,k}^- \} = P_{\beta_j}^-(n_j)$ for each $j$.
The ECH differential and grading
--------------------------------
If $\alpha$ and $\beta$ are orbit sets and $k$ is an integer, define $${{{\mathcal M}}}_k^J(\alpha,\beta) = \{ {{\mathcal C}}\in {{{\mathcal M}}}^J(\alpha, \beta) \ | \ I({{\mathcal C}}) =k\}.$$ If $\alpha$ is a generator of the ECH chain complex we define the differential $\partial$ on $ECC_*(Y,\lambda,\Gamma, J)$ by $$\partial \alpha = \sum_{\beta} \# ({{{\mathcal M}}}_1^J(\alpha,\beta)/{{\mathbb R}}) \beta.$$ The sum is over chain complex generators $\beta$, and ‘$\#$’ denotes the mod 2 count. Here ${{\mathbb R}}$ acts on ${{{\mathcal M}}}_1^J(\alpha,\beta)$ by translation of the ${{\mathbb R}}$-coordinate on ${{\mathbb R}}\times Y$. By Proposition \[lowiprop\] the quotient ${{{\mathcal M}}}_1^J(\alpha,\beta)/{{\mathbb R}}$ is a discrete set. By the arguments in [@Hu2 §5.3] we can conclude that ${{{\mathcal M}}}_1^J(\alpha,\beta)/{{\mathbb R}}$ is finite. Finally, because the differential decreases the symplectic action and since any nondegenerate contact form has only finitely many Reeb orbits with bounded symplectic action, we have that the ECH differential is well-defined.
Proving that $\partial^2=0$ is a substantial undertaking, see [@obg1] and [@obg2]. Moreover, Taubes’ proof in [@taubesechswf1] that the homology of $(ECC_*(Y,\lambda,\Gamma,J),\partial)$ is independent of $J$ and of $\lambda$ up to $c_1(\xi)$ requires Seiberg-Witten theory. In light of this invariance we denote this homology by $ECH_*(Y,\xi,\Gamma)$, and call it the *embedded contact homology*, or ECH, of $(Y,\xi,\Gamma)$.
Since the ECH index depends on a choice of relative second homology class $Z$, for general $(Y,\xi,\Gamma)$ we can only expect a relative ${{\mathbb Z}}_d$ grading, where $d$ is the divisibility of the class $c_1(\xi) + 2 \mbox{\em PD}(\Gamma)$ in $H^2(Y;{{\mathbb Z}})$ mod torsion. This is because of the index ambiguity property of the ECH index, Theorem \[thm:Iproperties\] (\[property:indexamb\]), whereby we set $$I(\alpha,\beta):=[I(\alpha,\beta,Z)]\in{{\mathbb Z}}_d.$$
In the setting of prequantization bundles, we will have $d=0$, because $c_1(\xi) + 2 \mbox{\em PD}(\Gamma)$ is torsion, as we now explain. If $g=0$, then $H^2(Y)\cong H_1(Y)$ is torsion, as shown in §\[subsec:homology\]. If $g>0$, then we will demonstrate in Lemma \[lem:indepZ\] that the divisibility of $c_1(\xi)$ is zero. In §\[subsec:PQBgens\] we explain why we only need to consider $\Gamma$ which are a multiple of the fiber class. The fiber class is torsion as we will show in §\[subsec:homology\]. Thus we can work with a relative ${{\mathbb Z}}$ grading.
We often refine our relative ${{\mathbb Z}}$ grading to an absolute ${{\mathbb Z}}$ grading by setting a chosen generator to have grading zero. In particular, if $\Gamma=[\emptyset]$, we will choose $\emptyset$ to have grading zero.
Filtered ECH
------------
There is a filtration on ECH which enables us to compute it via successive approximations, as explained in §\[subsec:PQBgens\] and §\[subsec:directlimit\]. The *symplectic action* or *length* of an orbit set $\alpha=\{(\alpha_i,m_i)\}$ is $$\mathcal{A}(\alpha):=\sum_im_i\int_{\alpha_i}\lambda.$$ If $J$ is $\lambda$-compatible and there is a $J$-holomorphic current from $\alpha$ to $\beta$, then $\mathcal{A}(\alpha)\geq\mathcal{A}(\beta)$ by Stokes’ theorem, since $d\lambda$ is an area form on such $J$-holomorphic curves. Since $\partial$ counts $J$-holomorphic currents, it decreases symplectic action, i.e., $$\langle\partial\alpha,\beta\rangle\neq0\Rightarrow\mathcal{A}(\alpha)\geq\mathcal{A}(\beta).$$
Let $ECC_*^L(Y,\lambda,\Gamma;J)$ denote the subgroup of $ECC_*(Y,\lambda,\Gamma;J)$ generated by orbit sets of symplectic action less than $L$. Because $\partial$ decreases action, it is a subcomplex. It is shown in [@cc2 Theorem 1.3] that the homology of $ECC_*(Y,\lambda,\Gamma;J)$ is independent of $J$, therefore we denote its homology by $ECH_*^L(Y,\lambda,\Gamma)$, which we call *filtered ECH*.
Given $L<L'$, there is a homomorphism $$\iota^{L,L'}:ECH_*^L(Y,\lambda,\Gamma)\to ECH_*^{L'}(Y,\lambda,\Gamma),$$ induced by the inclusion $ECC_*^L(Y,\lambda,\Gamma;J)\hookrightarrow ECC_*^{L'}(Y,\lambda,\Gamma;J)$ and independent of $J$, as shown in [@cc2 Theorem 1.3]. The $\iota^{L,L'}$ fit together into a direct system $(\{ECC_*^L(Y,\lambda,\Gamma)\}_{L\in{{\mathbb R}}},\iota^{L,L'})$. Because taking direct limits commutes with taking homology, we have $$ECH_*(Y,\lambda,\Gamma)=H_*\left(\lim_{L\to\infty}ECC^L_*(Y,\lambda,\Gamma;J)\right)=\lim_{L\to\infty}ECH^L_*(Y,\lambda,\Gamma).$$
The inclusion maps are compatible with certain cobordism maps as follows. An *exact symplectic cobordism* from $(Y_+,\lambda_+)$ to $(Y_-,\lambda_-)$ is a pair $(X,\lambda)$ where $X$ is a four-manifold with $\partial X=Y_+-Y_-$ and $\lambda$ a one-form on $X$ with $d\lambda$ symplectic and $\lambda|_{Y_\pm}=\lambda_\pm$.
Define $$\label{eqn:ECHdecompH1}
ECH_*(Y,\lambda):=\bigoplus_{\Gamma\in H_1(Y)}ECH_*(Y,\lambda,\Gamma),$$ and define $ECH^L_*(Y,\lambda)$ similarly.
Exact symplectic cobordisms induce maps on filtered ECH. The properties of these cobordism maps will allow us to compute the ECH of prequantization bundles via a nondegenerate perturbation of the contact form, as discussed in §\[sec:ECHI\] and §\[sec:finalcomp\]. For now we state the results from [@cc2] on these cobordism maps which we will need in order to understand §\[sec:ECHI\]-§\[handleslides\]; a more in-depth discussion and detour through Seiberg-Witten theory is postponed to §\[sec:finalcomp\]. We do not need the following two results in their full strength, and so we paraphrase below. In particular, we do not need the full notion of “composition" of exact symplectic cobordisms, so we cite the composition property in the following theorem only in the case which we will need. The fact that if $\varepsilon'<\varepsilon$ then $([\varepsilon',\varepsilon]\times Y,(1+s{\mathfrak{p}}^*H)\lambda)$ (where $s$ is the coordinate on $[\varepsilon',\varepsilon]$) is an exact symplectic cobordism from $(Y,\lambda_\varepsilon)$ to $(Y,\lambda_{\varepsilon'})$ is addressed in the proof of Proposition \[prop:directlimitcomputesfiberhomology\] in §\[subsec:PQBgens\].
\[thm:cc2cobmaps\] Let $\lambda_\pm$ be contact forms on closed, oriented, connected three-manifolds $Y_\pm$, with $(X,\lambda)$ an exact symplectic cobordism from $(Y_+,\lambda_+)$ to $(Y_-,\lambda_-)$. There are maps of ungraded ${{\mathbb Z}}_2$-modules: $$\Phi^L(X,\lambda):ECH_*^L(Y_+,\lambda_+)\to ECH^L_*(Y_-,\lambda_-),$$ for each real number $L$ such that:
(Inclusion) If $L<L'$ then the following diagram commutes: $$\label{eqn:ECHcd}
\xymatrixcolsep{3pc}\xymatrix{
ECH^L(Y,\lambda_+,\Gamma) \ar[r]^{\Phi^L(X,\lambda)} \ar[d]_{\iota^{L,L'}} & ECH^L(Y,\lambda_-,\Gamma) \ar[d]^{\iota^{L,L'}}
\\ECH^{L'}(Y,\lambda_+,\Gamma) \ar[r]_{\Phi^{L'}(X,\lambda)} & ECH^{L'}(Y,\lambda_-,\Gamma)
}$$
(Composition) Given $\varepsilon''<\varepsilon'<\varepsilon$, $$\Phi^L([\varepsilon'',\varepsilon]\times Y,(1+s{\mathfrak{p}}^*H))=\Phi^L([\varepsilon'',\varepsilon']\times Y,(1+s{\mathfrak{p}}^*H))\circ\Phi^L([\varepsilon',\varepsilon]\times Y,(1+s{\mathfrak{p}}^*H)).$$
Furthermore, the maps $\Phi^L(X,\lambda)$ respect the decomposition (\[eqn:ECHdecompH1\]) in the following sense: the image of $ECH_*(Y_+,\lambda_+,\Gamma_+)$ has a nonzero component in $ECH_*(Y_-,\lambda_-,\Gamma_-)$ only if $\Gamma_\pm\in H_1(Y_\pm)$ map to the same class in $H_1(X)$.
In order to understand the impact of these cobordism maps on the chain level in certain well-behaved cobordisms, we will also use the simplification expressed in Lemma \[lem:nicecobmap\] of two technical lemmas and a definition from [@cc2].
\[lem:nicecobmap\] Given a real number $L$, let $\lambda_t$ and $J_t$ be smooth 1-parameter families of contact forms on $Y$ and $\lambda_t$-compatible almost complex structures such that
- The contact forms $\lambda_t$ are of the form $f_t\lambda_0$, where $f:[0,1]\times Y\to{{\mathbb R}}_{>0}$ satisfies $\frac{\partial f}{\partial_t}<0$ everywhere.
- All Reeb orbits of each $\lambda_t$ of length less than $L$ are nondegenerate, and there are no orbit sets of $\lambda_t$ of action exactly $L$ (This condition is referred to in [@cc2] as $\lambda_t$ being “$L$-nondegenerate.")
- Near each Reeb orbit of length less than $L$ the pair $(\lambda_t,J_t)$ satisfies the conditions of [@taubesechswf1 (4-1)]. (This condition is referred to in [@cc2] as $(\lambda_t,J_t)$ being “$L$-flat.")
- For orbit sets of action less than $L$, the ECH differential $\partial$ is well-defined on admissible orbit sets of action less than $L$ and satisfies $\partial^2=0$. (This is a condition on the genericity of $J_t$ described in [@obg2], and referred to in [@cc2] as $J_t$ being “$ECH^L$-generic.")
Then $([-1,0]\times Y,\lambda_{-t})$ is an exact symplectic cobordism from $(Y,\lambda_0)$ to $(Y,\lambda_1)$, and for all $\Gamma\in H_1(Y)$, the cobordism map $\Phi^L([-1,0]\times Y,\lambda_{-t})$ agrees with the map $$ECC^L_*(Y,\lambda_0,\Gamma;J_0)\to ECC^L_*(Y,\lambda_1,\Gamma;J_1),$$ determined by the canonical bijection on generators.
The ECH index for prequantization bundles {#sec:ECHI}
=========================================
In this section we compute the ECH index for certain orbit sets in perturbations of prequantization bundles. The canonical contact form associated to a prequantization bundle is degenerate, so it is not possible to compute its ECH directly. Instead we introduce the perturbation $$\label{eqn:lambdapert}
\lambda_\varepsilon=(1+\varepsilon{\mathfrak{p}}^*H)\lambda,$$ where $H$ is a perfect Morse function on the base $\Sigma_g$. As explained in §\[subsec:pertReeb\], we have:
\[lem:efromL\] Fix a Morse function $H$ such that $|H|_{C^2}<1$.
1. For each $L>0$, there exists $\varepsilon(L)>0$ such that for all $\varepsilon<\varepsilon(L)$, all Reeb orbits with $\mathcal{A}(\gamma) < L$ are nondegenerate and project to critical points of $H$, where $\mathcal{A}$ is computed using $\lambda_\varepsilon$.
2. The action of a Reeb orbit $\gamma_p^k$ of $R_\varepsilon$ over a critical point $p$ of $H$ is proportional to the length of the fiber, namely $$\mathcal{A}(\gamma_p^k) = \int_{\gamma_p^k} \lambda_\varepsilon = 2k \pi (1+\varepsilon {\mathfrak{p}}^*H).$$
By Lemma \[lem:efromL\] (i), it is possible to choose $\varepsilon(L)$ based on $L$ so that the embedded orbits contributing to the generators of $ECC^L_*(Y,\lambda_{\varepsilon(L)},\Gamma;J)$ consist only of fibers above critical points of $H$. In the proof of Lemma \[lem:efromL\] in §\[subsec:pertReeb\] we show that $\varepsilon(L)\sim\frac{1}{L}$. Therefore, by Lemma \[lem:efromL\] (ii), we can choose $L$ large enough so that the generators of $ECC^L_*(Y,\lambda_{\varepsilon(L)}),\Gamma;J)$ include any given orbit set whose embedded Reeb orbits are fibers above critical points.
To capture all these filtered complexes, we prove the following result in §\[subsec:PQBgens\]:
\[prop:directlimitcomputesfiberhomology\] With $Y,\lambda$, and $\varepsilon(L)$ as discussed above, for any $\Gamma$, there is a direct system consisting of all the $ECH^L_*(Y,\lambda_{\varepsilon(L)},\Gamma)$. The direct limit $\lim_{L\to\infty}ECH^L_*(Y,\lambda_{\varepsilon(L)},\Gamma)$ is the homology of the chain complex generated by orbit sets $\{(\alpha_i,m_i)\}$ where the $\alpha_i$ are fibers above critical points of $H$ and $\sum_im_i[\alpha_i]=\Gamma$.
Proposition \[prop:directlimitcomputesfiberhomology\] will allow us to prove Theorem \[thm:mainthm\] in §\[sec:finalcomp\], where we will relate the direct limit to the ECH of the original prequantization bundle. In light of Proposition \[prop:directlimitcomputesfiberhomology\], in §\[subsec:PQBgens\], building on the Conley Zehnder index computations in §\[subsec:CZ\] we prove
\[lem:orbitseh\] The fibers above the index zero and two critical points of $H$ are embedded elliptic orbits, while the fibers above the index one critical points of $H$ are embedded positive hyperbolic orbits.
[For technical reasons]{}, we need to assume $H$ is perfect. We denote the corresponding embedded Reeb orbits by $e_-, e_+$, and $h_i$, respectively, and throughout the rest of the paper consider generators of the form $e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+}$ where $m_\pm,m_i\in{{\mathbb Z}}_\geq0$, denoting the orbit set $$\{(e_-,m_-),(h_1,m_1),\dots,(h_{2g},m_{2g}),(e_+,m_+)\}$$ When specifying a particular orbit set with multiplicative notation, we will follow the convention that $m_\pm, m_i>0$, omitting the term $\gamma^m$ if $m=0$. When using multiplicative notation to denote an unspecified or general orbit set, however, we will allow $m_\pm,m_i=0$, and it will correspond to the orbit set in the usual notation with the pair $(e_\pm,m_\pm)$ or $(h_i,m_i)$ removed.
Given orbit sets $\alpha=e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+}$ and $\beta=e_-^{n_-}h_1^{n_1}\cdots h_{2g}^{n_{2g}}e_+^{n_+}$, let $$d=\frac{M-N}{-e},$$ where $M:=m_-+m_1+\cdots+m_{2g}+m_+$ and similarly $N:=n_-+n_1+\cdots+n_{2g}+n_+$. (Note that $d$ corresponds to the *degree* of any curves counted in $\langle\partial\alpha,\beta\rangle$, as proved in §\[degree\].) We will prove
\[prop:indexcalc\] Let $(Y,\lambda)$ be a prequantization bundle over a surface $\Sigma_g$ with Euler class $e\in{{\mathbb Z}}_{<0}$. The ECH index in $ECH_*^L(Y,\lambda_{\varepsilon(L)},\Gamma)$ satisfies the following formula for any $\Gamma, L$, and orbit sets $\alpha$ and $\beta$. $$\label{eqn:indexformula}
I(\alpha,\beta)=\chi(\Sigma_g)d-d^2e+2dN+m_+-m_--n_++n_-.$$
In particular, note that the ECH index $I(\alpha,\beta)$ depends only on the generators $\alpha$ and $\beta$ and not on a relative homology class $Z\in H_2(Y,\alpha,\beta)$.
We will obtain Proposition \[prop:indexcalc\] in §\[subsec:ECHindexcomp\] by combining the results of Lemma \[lem:indepZ\], §\[subsec:CZ\], §\[subsec:relfirstchern\], and §\[subsec:relselfint\], which prove that the index depends only on the generators, and compute the Conley Zender, relative first Chern class, and relative self intersection number terms, respectively.
Perturbed Reeb dynamics {#subsec:pertReeb}
-----------------------
Let $(Y,\lambda)$ be the prequantization bundle over $(\Sigma_g,\omega)$ with Euler class $e=-\frac{1}{2\pi}[\omega]<0$ and contact structure $\xi=\ker(\lambda)$. Recall that the Reeb orbits of $\lambda$ consist of the $S^1$ fibers of ${\mathfrak{p}}:Y\to\Sigma_g$, all of which have action $2\pi$. We take: $$\label{perturbedform1}
\lambda_\varepsilon=(1+\varepsilon {\mathfrak{p}}^*H)\lambda.$$
A standard computation, cf. [@jo2 Prop. 4.10], yields the following.
The Reeb vector field of $\lambda_\varepsilon$ is given by $$\label{perturbedreeb1}
R_{\varepsilon}=\frac{R}{1+\varepsilon {\mathfrak{p}}^*H} + \frac{\varepsilon \widetilde{X}_H}{{(1+\varepsilon {\mathfrak{p}}^*H)}^{2}},$$ where $X_{H}$ is the Hamiltonian vector field[^5] on $\Sigma$ and $\widetilde{X}_{ H}$ is its horizontal lift.
We now prove Lemma \[lem:efromL\].
To prove (i), note that the horizontal lift $\widetilde{X}_{H}$ is determined by $$dh(q)\widetilde{X}_H(q) = X_{\varepsilon H}(h(q)) \ \ \mbox{ and } \ \ \lambda(\widetilde{X}_H)=0.$$ Thus those orbits which do not project to $p\in \mbox{Crit}(H)$ must project to $X_H$. We have $$\frac{\varepsilon}{(1+\varepsilon)^2} < \frac{\varepsilon}{{(1+\varepsilon {\mathfrak{p}}^*H)}^{2}} < \frac{\varepsilon}{(1-\varepsilon)^2}$$ A Taylor series expansion shows that the $k$-periodic orbits of $X_H$ give rise to orbits of $ \frac{\varepsilon \widetilde{X}_H}{{(1+\varepsilon {\mathfrak{p}}^*H)}^{2}}$ which are $\frac{C}{\varepsilon}$-periodic for some $C$. We note that $C$ and $k$ must be bounded away from 0 since $X_H$ is time autonomous. Nondegeneracy of Reeb orbits $\gamma$ such that $\mathcal{A}(\gamma) <L$ follows from the proof of Theorem 13 in Appendix A of [@ABW].
We obtain (ii) because the period of an orbit of $R_\varepsilon$ over a critical point $p$ of $H$ must be $1+\varepsilon H(p)$ times the period of the orbit of $R$ over $p$ by (\[perturbedreeb1\]).
Homology of prequantization bundles {#subsec:homology}
-----------------------------------
In this subsection we review preliminaries on the homology of prequantization bundles. Let $Y$ be a prequantization bundle over a two-dimensional surface $\Sigma_g$ of negative Euler class $e$. The second page of its Leray-Serre spectral sequence has terms $$E^2_{p,q}=H_p(\Sigma_g;\{H_q(Y_x)\})=H_p(\Sigma_g;{{\mathbb Z}})$$ for $q=0,1$. Since $\partial_2:E^2_{p,q}\to E^2_{p-2,q+1}$, the only differential on the second page which neither starts nor ends at a trivial group is from $E^2_{2,0}=H_2(\Sigma_g;{{\mathbb Z}})$ to $E^2_{0,1}=H_0(\Sigma_g;{{\mathbb Z}})$; this differential sends the element of $E^1_{2,0}$ corresponding to a closed 2-cell in $\Sigma$ to the obstruction to finding a section over $\Sigma_g$, and so the image of $\partial_2$ in $E^2_{0,1}$ is $e{{\mathbb Z}}$. The other groups are unchanged.
Since all higher differentials will either start or end at a trivial group, we obtain $$\label{eqn:HofPQB}
H_*(Y;{{\mathbb Z}})=\begin{cases}{{\mathbb Z}}&*=3
\\{{\mathbb Z}}^{2g}&*=2
\\{{\mathbb Z}}^{2g}\oplus{{\mathbb Z}}_{-e}&*=1
\\{{\mathbb Z}}&*=0
\end{cases}$$
In our computations of the ECH index we will need to understand representatives of the degree one and two homology classes.
\[lem:HYreps\]Let $Y$ be a prequantization bundle over a two-dimensional surface $\Sigma_g$ of negative Euler class $e$.
1. Let $f_p$ denote the fiber over $p\in\Sigma_g$. Its $k$-fold cover represents the class $k \text{ mod}(-e)$ in the ${{\mathbb Z}}_{-e}$ summand of $H_1(Y)$.
2. Each $H_2(Y)$ class is represented by the union of fibers over a representative of an $H_1(\Sigma_g)$ class.
Because $\Sigma_g$ is a CW complex, the Leray-Serre spectral sequence can be constructed using the filtration on $C_*(Y)$ where $F_p(C_*(Y))$ is the subcomplex consisting of singular chains supported in the preimage under ${\mathfrak{p}}$ of the $p$-skeleton of $\Sigma_g$.
To show (i), we first show that $E^2_{0,1}$ is generated by the fiber. Because $E^1_{0,1}$ is generated by the fiber and $E^1_{1,1}$ is generated by 2-chains of $Y$ over the 1-skeleton of $\Sigma_g$, the image of $E^1_{1,1}$ under $\partial_1$ is zero. Therefore $E^2_{0,1}$ is also generated by the fiber.
Secondly, we show that $E^2_{2,0}$ is generated by a section of $Y$ over $\Sigma_g-\{pt\}$. It follows from the definitions that $E^1_{2,0}$ is generated by a section of $Y$ over $\Sigma_g-\{pt\}$, and every such section is in the kernel of $\partial_1$ because its boundary is a 1-chain in $\pi^{-1}(\{pt\})$. Therefore $E^2_{2,0}=E^1_{2,0}$.
The differential $\partial_2$ takes the generator of $E^2_{2,0}$ to $f_p^e$, so (i) is proved.
To show (ii), note that $E^2_{1,1}$ consists of 2-chains over the 1-skeleton of $\Sigma_g$ whose boundaries do not wrap around fibers and which are not the boundary of a 3-chain in the preimage under ${\mathfrak{p}}$ of the 1-skeleton of $\Sigma_g$. Therefore elements of $E^2_{1,1}$ can be represented by preimages under ${\mathfrak{p}}$ of representatives of $H_1(\Sigma_g)$ classes.
In an abuse of notation, we will often refer to elements of the subgroup $\{0\}\times{{\mathbb Z}}_{-e}$ of $H_1(Y)$ simply as elements of ${{\mathbb Z}}_{-e}$.
Trivialization and Conley Zehnder index {#subsec:CZ}
---------------------------------------
We will use the constant trivialization as considered in [@ggm1 §3.1, 4.2] to compute the Conley-Zehnder indices; note that we only have to consider embedded orbits and not multiple covers. For any point $q \in {\mathfrak{p}}^{-1}(p)$, a fixed trivialization of $T_p\Sigma_g$ allows us to trivialize $\xi_q$ because $\xi_q \cong T_p\Sigma$. This trivialization is invariant under the linearized Reeb flow and can be thought of as a *constant trivialization* over the orbit $\gamma_p$ because the linearized Reeb flow, with respect to this trivialization, is the identity map.
Using this constant trivialization, we have the following result regarding the Robbin-Salamon index, see [@vknotes Lem. 3.3], [@jo2 Lem. 4.8].
\[consttrivlem\] Let $(Y,\lambda) \overset{{\mathfrak{p}}}{\rightarrow}(\Sigma_g, \omega)$ be a prequantization bundle of negative Euler class $e$. Then for the constant trivialization $\tau$ along the circle fiber $\gamma = \pi^{-1}(p) $, we obtain $RS_\tau(\gamma)=0$ and $RS_\tau(\gamma^k) =0$, where $RS$ denotes the Robbin-Salamon index of the $k$-fold iterate of the fiber.
We also have the following formula for the Conley-Zehnder indices of iterates of orbits which project to critical points $p$ of $H$. We denote the $k$-fold iterate of an orbit which projects to $p \in \mbox{Crit}(H)$ by $\gamma_p^k$.
\[lempre\] Fix $L>0$ and $H$ a Morse-Smale function on $\Sigma$ which is $C^2$ close to 1. Then there exists $\varepsilon >0$ such that all periodic orbits $\gamma$ of $R_\varepsilon$ with action $\mathcal{A}(\gamma) <L$ are nondegenerate and project to critical points of $H$. The Conley-Zehnder index such a Reeb orbit over $p \in \mbox{\em Crit}(H)$ is given by $$\begin{array}{lcl}
CZ_\tau(\gamma_p^k) &=& {RS}_\tau(\gamma^k) -1 + \mbox{\em index}_pH,\\
&=& \mbox{\em index}_pH -1.\\
\end{array}$$
Detailed definitions and computations of the Conley-Zehnder and Robbin-Salamon index as well as the proofs of the preceding standard computations can be found in [@jo2 §4], [@vknotes].
ECH generators {#subsec:PQBgens}
--------------
We explain why generators of the form $e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+}$ are all we need to consider until §\[sec:finalcomp\]. Our focus through §\[handleslides\] will be to build the foundations necessary to understand the direct limit $\lim_{L\to\infty}ECH^L_*(Y,\lambda_{\varepsilon(L)},\Gamma;J)$. In Theorem \[thm:dlisECH\] we will show that $$\lim_{L\to\infty}ECH^L_*(Y,\lambda_{\varepsilon(L)},\Gamma)=ECH_*(Y,\xi,\Gamma)$$ We will then prove the main theorem in §\[sec:finalcomp\] by relating the direct limit $\lim_{L\to\infty}ECH^L_*(Y,\lambda_{\varepsilon(L)},\Gamma)$, as we understand it via Proposition \[prop:directlimitcomputesfiberhomology\], to the Morse homology of the base, which will require the analysis of §\[sec:modspcs\]-§\[handleslides\]. This section will be devoted to understanding the ECH index of the generators which Proposition \[prop:directlimitcomputesfiberhomology\] tells us are relevant, i.e., those whose embedded orbits are fibers above critical points of $H$.
We require that $H$ be perfect, so that $H$ has exactly as many critical points of index $i$ as its $i^\text{th}$ Betti number. Let $e_+$ denote the orbit whose image is the fiber above the index two critical point of $H$. Similarly, let $e_-$ denote the orbit above the index zero critical point and let $h_1,\dots,h_{2g}$ denote the orbits above the index one critical points.
The notation is derived from the fact that the orbits $e_\pm$ are elliptic, with slightly positive/negative rotation numbers, respectively, in the constant trivialization discussed in §\[subsec:CZ\], and the $h_i$ are positive hyperbolic. This is the content of Lemma \[lem:orbitseh\]. Heuristically, it is true because the linearized return map of an orbit projecting to a critical point $p$ of $H$ approximately agrees with a lift of the linearized flow of $\varepsilon X_H$ on $T_p\Sigma_g$. However, we prove Lemma \[lem:orbitseh\] using the Conley Zehnder index instead.
We next prove Proposition \[prop:directlimitcomputesfiberhomology\].
Given $\varepsilon>\varepsilon'$ there is an exact symplectic cobordism $(X_{\varepsilon,\varepsilon'},\lambda_{\varepsilon,\varepsilon'}):=([\varepsilon',\varepsilon]\times Y,(1+s{\mathfrak{p}}^*H)\lambda)$ from $(Y,\lambda_\varepsilon)$ to $(Y,\lambda_{\varepsilon'})$. (It is symplectic because $d\lambda_{\varepsilon,\varepsilon'}^2$ is a positive multiple of $ds\wedge\lambda\wedge d\lambda$.)
Thus we have cobordism maps $\Phi^L(X_{\varepsilon,\varepsilon'},\lambda_{\varepsilon,\varepsilon'})$ as in Theorem \[thm:cc2cobmaps\], inclusion maps $\iota^{L,L'}$ as in [@cc2 Thm. 1.3], and a commutative diagram $$\label{eqn:trivialcobcd}
\xymatrixcolsep{4pc}\xymatrix{
ECH^L_*(Y,\lambda_{\varepsilon},\Gamma) \ar[r]^{\Phi^L(X_{\varepsilon,\varepsilon'},\lambda_{\varepsilon,\varepsilon'})} \ar[d]_{\iota^{L,L'}} & ECH^L_*(Y,\lambda_{\varepsilon'},\Gamma) \ar[d]^{\iota^{L,L'}}
\\ECH^{L'}_*(Y,\lambda_{\varepsilon},\Gamma) \ar[r]_{\Phi^{L'}(X_{\varepsilon,\varepsilon'},\lambda_{\varepsilon,\varepsilon'})} & ECH^{L'}_*(Y,\lambda_{\varepsilon'},\Gamma)
}$$ by adapting (\[eqn:ECHcd\]) from the Inclusion property of Theorem \[thm:cc2cobmaps\]. (Because $X$ is a product of $Y$ with an interval, the cobordism maps respect the splitting.)
Because if $L<L'$ then $\varepsilon(L)>\varepsilon(L')$, from either path on the commutative diagram (\[eqn:trivialcobcd\]) we get a well-defined map $$\label{eqn:directsystemoverLmaps}
ECH^L_*(Y,\lambda_{\varepsilon(L)},\Gamma)\to ECH^{L'}_*(Y,\lambda_{\varepsilon(L')},\Gamma).$$ For the $ECH^L_*(Y,\lambda_{\varepsilon(L)},\Gamma)$ to form a direct system, it remains to show that the maps (\[eqn:directsystemoverLmaps\]) compose. In the following denote by $\Phi^L(\varepsilon,\varepsilon')$ the cobordism map $\Phi^L(X_{\varepsilon,\varepsilon'},\lambda_{\varepsilon,\varepsilon'})$. It is enough to show that for $L''>L'>L$ and $\varepsilon''<\varepsilon'<\varepsilon$, the composition $$ECH^L_*(Y,\lambda_\varepsilon,\Gamma)\overset{\iota^{L,L'}}{\to}ECH^{L'}(Y,\lambda_\varepsilon,\Gamma)\overset{\Phi^{L'}(\varepsilon',\varepsilon'')\circ\Phi^{L'}(\varepsilon,\varepsilon')}{\longrightarrow}ECH^{L'}(Y,\lambda_{\varepsilon''},\Gamma)\overset{\iota^{L',L''}}{\to}ECH^{L''}(Y,\lambda_{\varepsilon''},\Gamma)$$ equals either $\Phi^{L''}(\varepsilon,\varepsilon'')\circ\iota^{L,L''}$ or $\iota^{L,L''}\circ\Phi^{L}(\varepsilon,\varepsilon'')$. This follows from the four-fold commutative diagram consisting of the versions of (\[eqn:trivialcobcd\]) for $(L,\varepsilon)$ to $(L',\varepsilon')$, $(L,\varepsilon')$ to $(L',\varepsilon'')$, $(L',\varepsilon)$ to $(L'',\varepsilon')$, and $(L',\varepsilon')$ to $(L'',\varepsilon'')$ in concert. In this four-fold commutative diagram, the path across the top and down the right side equals $\Phi^{L''}(\varepsilon,\varepsilon'')\circ\iota^{L,L''}$, by the Composition property of Theorem \[thm:cc2cobmaps\], and similarly the path down the left side and across the bottom equals $\iota^{L,L''}\circ\Phi^{L}(\varepsilon,\varepsilon'')$.
It remains to show that the direct limit is the homology of the chain complex generated by orbit sets whose embedded orbits are fibers above the critical points of $H$ and whose multiplicities can be any element of ${{\mathbb Z}}_{>0}$.
The embedded orbits contributing to the generators of any $ECC^L_*(Y,\lambda_{\varepsilon(L)},\Gamma;J)$ must be orbits over critical points of $H$ by Lemma \[lem:efromL\] (i). And by Lemma \[lem:efromL\] (ii), for any pair $(\gamma,m_\gamma)$ where $\gamma$ is an orbit above a critical point of $H$ and $m_\gamma\in{{\mathbb Z}}_{>0}$, there is some (possibly very large) $L$ for which $m_\gamma\mathcal{A}(\gamma)<L$ when $\mathcal{A}$ is computed using $\lambda_{\varepsilon(L)}$.
To complete the proof we need to know that the maps (\[eqn:directsystemoverLmaps\]) are induced by the obvious inclusion of chain complexes $$ECC^L_*(Y,\lambda_{\varepsilon(L)},\Gamma;J)\to ECC^{L'}_*(Y,\lambda_{\varepsilon(L')},\Gamma;J).$$ Because $\iota^{L,L'}$ is induced by inclusion, it suffices to show that the map $\Phi^L(X,\lambda_{\varepsilon,\varepsilon'})$ is also induced by inclusion. (There is no need to check the cobordism map $\Phi^{L'}(X,\lambda_{\varepsilon,\varepsilon'})$ because the diagram commutes.)
That $\Phi^L(X,\lambda_{\varepsilon,\varepsilon'})$ is induced by inclusion follows if there is a smooth 1-parameter family $\lambda_t$ where $\lambda_t=\lambda_{\varepsilon-(\varepsilon-\varepsilon')t}$ which, for $\lambda_t$-compatible almost complex structures $J_t$, the pairs $(\lambda_t,J_t)$ satisfy the hypotheses of Lemma \[lem:nicecobmap\] for $L$. The first and second bullet points follow immediately from the construction. The fourth bullet point generically holds. The third bullet point can then be accomplished by a deformation as in [@taubesechswf1 Prop. B.1] (see also [@cc2 Lem. 3.6]).
We prove Lemma \[lem:orbitseh\], which classifies the orbits we consider.
By Lemma \[lempre\], we have $$\label{eqn:CZH}
CZ_\tau(\gamma_p^k)=\mbox{index}_pH-1,$$ because in our case $n=1$ and $RS_\tau(\gamma^k)=0$.
Because $\lambda_\varepsilon$ is a small perturbation of $\lambda$, all linearized return maps of Reeb orbits must be close to the identity. Therefore there can be no negative hyperbolic orbits.
Positive hyperbolic orbits have even Conley Zehnder indices, so the $e_\pm$, with Conley Zehnder indices $\pm1$ by (\[eqn:CZH\]), must be elliptic.
Elliptic orbits have odd Conley Zehnder indices, so the $h_i$, which all have Conley Zehnder index zero by (\[eqn:CZH\]), must be positive hyperbolic.
Computation of the ECH index
----------------------------
In this subsection we prove Proposition \[prop:indexcalc\]. We note that the structure of the proof follows [@farris §3]. Let $Y$ be a prequantization bundle over a surface $\Sigma_g$ of negative Euler class $e$, and let $L\in{{\mathbb R}}$ be large. Let $\Gamma$ be a torsion element in the $\{0\}\times{{\mathbb Z}}_{-e}$ subgroup of $H_1(Y)$. In this subsection we prove Proposition \[prop:indexcalc\]. We first introduce some notation which we will use throughout the rest of this section in our computation of the ECH index.
Given generators $\alpha=e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+}$ and $\beta=e_-^{n_-}h_1^{n_1}\cdots h_{2g}^{n_{2g}}e_+^{n_+}$ of $ECC^L_*(Y,\lambda_{\varepsilon(L)},\Gamma)$, let $$M:=m_-+m_1+\cdots+m_{2g}+m_+\text{ and }N:=n_-+n_1+\cdots+n_{2g}+n_+.$$ Because $[\alpha]=[\beta]=\Gamma$, there is some $d\in{{\mathbb Z}}$ so that $$\label{eqn:diffmult-e}
M=N+(-e)d.$$ Throughout the proof of the index formula, which occupies the rest of this section, we will assume $d\geq0$; the $d\leq0$ case is handled similarly, and signs will change appropriately.
\[lem:indepZ\] Given $\alpha$ and $\beta$ as above and $Z\in H_2(Y,\alpha,\beta)$, the ECH index $I(\alpha,\beta,Z)$ does not depend on $Z$.
Let $A\in H_2(Y)$ and $Z\in H_2(Y,\alpha,\beta)$. From the index ambiguity formula, Theorem \[thm:Iproperties\] (\[property:indexamb\]), we have $$I(\alpha,\beta,Z+A)-I(\alpha,\beta,Z)=\langle c_1(\xi)+2PD(\Gamma),A\rangle=c_1(\xi)(A)+2\Gamma\cdot A.$$
Assume $g>0$. Recall from Lemma \[lem:HYreps\] (ii) that $H_2(Y)={{\mathbb Z}}^{2g}$, and if $a_1,b_1,\dots,a_g,b_g$ generate $H_1(\Sigma)$, then the unions of fibers over representatives of $a$ and $b$ will generate $H_2(Y)$. The class $\Gamma$ can be represented by a single fiber, so a representative of $\Gamma$ can be isotoped not to intersect a representative of $A$. Thus $\Gamma\cdot A=0$. Moreover, we have $c_1(\xi)(A)=0$, via $$c_1(\xi)(A)=c_1(T\Sigma_g)({\mathfrak{p}}_*(A))=0,$$ because ${\mathfrak{p}}_*$ will send a representative of $A$ to a representative of a 1-cycle in $\Sigma$.
If $g=0$, then we have $H_2(Y)=0$, and because $H_2(Y,\alpha,\beta)$ is affine over $H_2(Y)$, there is no possibility for index ambiguity.
Therefore $I(\alpha,\beta,Z)$ is independent of $Z$, and will from now on be denoted $I(\alpha,\beta)$. Similarly, we will use $c_\tau(\alpha,\beta)$ and $Q_\tau(\alpha,\beta)$ to denote $c_\tau(Z)$ and $Q_\tau(Z)$.
We will now compute the relative first Chern class and relative self intersection terms in the ECH index. Lemmas \[lempre\] and \[lem:orbitseh\] allow us to compute the Conley Zehnder index term in the final proof of Proposition \[prop:indexcalc\]. Throughout the computation we use the constant trivialization $\tau$ from §\[subsec:CZ\].
### Surfaces in $Y$ {#subsec:surfaces}
Let $\alpha$ and $\beta$ be homologous orbit sets, thus satisfying (\[eqn:diffmult-e\]). Before computing $c_\tau$ and $Q_\tau$, we will define surfaces in $Y$ representing $[\alpha]=[\beta]$ to be used in both the computation of the relative first Chern class in §\[subsec:relfirstchern\] and the computation of relative self intersection number in §\[subsec:relselfint\]. In this section we use the notation $\alpha=\{(\alpha_k,m_k)\}$ and $\beta=\{(\beta_l,n_l)\}$; in particular, the $m_k$ and $n_l$ are not necessarily equal to the multiplicities of hyperbolic orbits.
Let ${\mathfrak{p}}(\alpha)$ and ${\mathfrak{p}}(\beta)$ denote the sets of points $\{{\mathfrak{p}}(\alpha_k)\}$ and $\{{\mathfrak{p}}(\beta_l)\}$, respectively, where the point ${\mathfrak{p}}(\alpha_k)$ appears with multiplicity $m_k$ and ${\mathfrak{p}}(\beta_l)$ appears with multiplicity $n_l$. Choose any subset of ${\mathfrak{p}}(\alpha)$ of total multiplicity $N$ and denote it ${\mathfrak{p}}(\alpha)_\beta$; such a subset exists because we are assuming $d\geq0$ in (\[eqn:diffmult-e\]). Note that the multiplicity of ${\mathfrak{p}}(\alpha_k)$ in ${\mathfrak{p}}(\alpha)_\beta$ does not have to equal $m_k$, though it is at most $m_k$. Denote the set of points in $\Sigma_g$ underlying ${\mathfrak{p}}(\alpha)_\beta$ by $\{{\mathfrak{p}}(\alpha)_\beta\}$.
First we explain how to obtain a surface in $Y$ connecting a set of orbits from $\alpha$ of total multiplicity $N$ with $\beta$. Choose a graph $G_N$ embedded in $\Sigma$ with vertices $\{{\mathfrak{p}}(\alpha)_\beta\}\cup\{{\mathfrak{p}}(\beta)\}$, where the degree of each vertex equals its multiplicity as part of ${\mathfrak{p}}(\alpha)_\beta$ or ${\mathfrak{p}}(\beta)$. Furthermore, we require that the edges of $G_N$ partition $\{{\mathfrak{p}}(\alpha)_\beta\}\cup\{{\mathfrak{p}}(\beta)\}$ into ${\mathfrak{p}}(\alpha)_\beta$ and ${\mathfrak{p}}(\beta)$ in the sense that each edge of $G_N$ can be labeled with a pair in ${\mathfrak{p}}(\alpha)_\beta\times{\mathfrak{p}}(\beta)$ where the edge of $G_N$ connects the underlying pair in $\Sigma_g$, and all points in ${\mathfrak{p}}(\alpha)_\beta\cup{\mathfrak{p}}(\beta)$ are connected in this way. Finally, we require that the edges of $G_N$ intersect only transversely, including at their endpoints (meaning that if $x$ is an endpoint with degree at least two, the one-sided limits of the tangent vectors to those edges form a basis for $T_x\Sigma_g$). In particular, if $x\in{\mathfrak{p}}(\alpha)_\beta\cap{\mathfrak{p}}(\beta)$ then the graph can include transversely intersecting loops from $x$ to $x$. Let $\check{S}_N$ denote the union of the fibers above $G_N$.
Now we explain how to obtain a surface in $Y$ with boundary homologous to the remaining $(-e)d$ orbits in $\alpha$ (counted with multiplicity). Denote ${\mathfrak{p}}(\alpha)-{\mathfrak{p}}(\alpha)_\beta$ by ${\mathfrak{p}}(\alpha)_\alpha$. Divide the points in ${\mathfrak{p}}(\alpha)_\alpha$ into $d$ subsets, each of total multiplicity $-e$. Denote each subset by ${\mathfrak{p}}(\alpha)_\alpha^k$, for $k=1,\dots,d$, and denote the underlying set of points by $\{{\mathfrak{p}}(\alpha)_\alpha^k\}$. Let $\text{mult}_k(x)$ denote the multiplicity of $x$ as an element of ${\mathfrak{p}}(\alpha)_\alpha^k$. For each $k$ choose a section $\check{S}_k$ of $Y$ over $\Sigma-\{{\mathfrak{p}}(\alpha)_\alpha^k\}$ whose boundary forms a $\text{mult}_k(x)$-fold cover of the fiber over $x$, for each $x\in\{{\mathfrak{p}}(\alpha)_\alpha^k\}$.
For $z\not\in\{{\mathfrak{p}}(\alpha)_\alpha^k\}$, denote the point of $\check{S}_k$ above $z$ by $\check{S}_k(z)$.
### Relative first Chern class {#subsec:relfirstchern}
We compute the relative first Chern class $c_\tau(\alpha,\beta)$ of orbit sets $\alpha$ and $\beta$.
\[lem:relfirstCherncalc\] Given orbit sets $\alpha$ and $\beta$ satisfying (\[eqn:diffmult-e\]), their relative first Chern class $c_\tau(\alpha,\beta)$ satisfies the following formula: $$\label{eqn:relfirstChern}
c_\tau(\alpha,\beta)=\chi(\Sigma)d.$$
We will use the surfaces $\check{S}_N$ and $\check{S}_k$ of §\[subsec:surfaces\]. It is immediate from the definition of $c_\tau$ that if $S$ and $S'$ are two admissible surfaces, then $$c_\tau(S\cup S')=c_\tau(S)\cup c_\tau(S').$$ Therefore, we have $$\label{eqn:ctausum}
c_\tau(\alpha,\beta)=c_\tau(\check{S}_N\cup\check{S}_1\cdots\cup\check{S}_d)=c_\tau(\check{S}_N)+\sum_{k=1}^dc_\tau(\check{S}_k)$$
Since $\xi={\mathfrak{p}}^*T\Sigma$, the first Chern class of $\xi$ is ${\mathfrak{p}}^*c_1(T\Sigma)$. Since ${\mathfrak{p}}(\check{S}_N)=G_N$ represents zero in $H_2(\Sigma;{{\mathbb Z}})$, $$\label{eqn:ctauSN}
c_\tau(\check{S}_N)=0.$$
Since $[{\mathfrak{p}}(\check{S}_k)]=[\Sigma]$ in $H_2(\Sigma;{{\mathbb Z}})$, $$\label{eqn:ctauSk}
c_\tau(\check{S}_k)=\chi(\Sigma).$$
Combining equations (\[eqn:ctausum\]), (\[eqn:ctauSN\]), and (\[eqn:ctauSk\]) yields the desired result.
### Relative self intersection number {#subsec:relselfint}
We compute the relative self intersection number $Q_\tau(\alpha,\beta)$ of orbit sets $\alpha$ and $\beta$.
\[lem:relselfint\] Given orbit sets $\alpha$ and $\beta$ satisfying (\[eqn:diffmult-e\]), their relative self-intersection number $Q_\tau(\alpha,\beta)$ satisfies the following formula: $$\label{eqn:relselfint}
Q_\tau(\alpha,\beta)=-ed^2+2dN.$$
We will first lift the surfaces $\check{S}_N$ and $\check{S}_k$ in $Y$ from §\[subsec:surfaces\] to admissible surfaces in $[-1,1]\times Y$ to use in our computation. To lift $\check{S}_N$ to an admissible surface $S_N\subset[-1,1]\times Y$, parameterize the edges of $G_N$ by $[-1,1]$ from ${\mathfrak{p}}(\beta)$ to ${\mathfrak{p}}(\alpha)_\beta$ so that they do not intersect as parameterized curves. Denote these parameterizations by $g_i$. The non-intersecting requirement means that if $g_1,g_2$ parameterize two edges of $G_N$ which intersect in $\Sigma$, then we have $g_1(t_1)=g_2(t_2)$ only if $t_1\neq t_2$. Let $S_N$ be the surface $${\bigcup}_{i=1}^N(t,{\mathfrak{p}}^{-1}(g_i(t))).$$
To construct an admissible surface with boundary on the remaining $(-e)d$ components of $\alpha$, we will define a family of lifts for each $\check{S}_k$ to an admissible surface $S_k\subset[-1,1]\times Y$. The lifts are isotopic, so the relative self-intersection number will not depend on our choice within the family. We will need this flexibility in order to guarantee transverse intersections.
Choose a disc neighborhood $\mathbb{D}^2_x$ for each $x\in\{{\mathfrak{p}}(\alpha)_\alpha\}$ which do not pairwise intersect, and parameterize each $\mathbb{D}^2_x$ with radial function $0\leq r_x\leq2$. For any choice of functions $\epsilon,l:\{1,\dots,d\}\to[0,2)$ with $0\leq\epsilon(k)<l(k)<2$ and $\delta:\{1,\dots,d\}\to{{\mathbb R}}_{>0}$, let $f_k:\Sigma_g\to{{\mathbb R}}$ be a smooth function for which $$f_k(z)=\begin{cases}
\delta(k)r_x&0\leq r_x\leq\frac{\epsilon(k)}{\delta(k)}
\\l(k)&l(k)\leq r_x\leq2
\\l(k)&\text{ outside $\bigcup_{x\in{\mathfrak{p}}(\alpha)_\alpha^k}\mathbb{D}^2_x$}
\end{cases} \text{ and }f'_k>0\text{ for }\frac{\epsilon(k)}{\delta(k)}\leq r_x<l(k)$$ For each $k=1,\dots,d$, we define $S_k\subset[-1,1]\times Y$ to be the surface $$S_k:=(1-f_k(z),\check{S}_k(z)).$$ Heuristically, $S_k$ lifts to $[-1,1]$ near $1$ by the negative of each radial direction $r_x$ times $\delta(k)$, until the $[-1,1]$ coordinate reaches $\epsilon(k)$. After some smooth interpolation depending within the $\mathbb{D}^2_x$ discs, the rest of $S_k$ simply equals $\{1-l(k)\}\times\left(\check{S}_k-\bigcup_{x\in{\mathfrak{p}}(\alpha)_\alpha^k}\mathbb{D}^2_x\right)$.
Expanding [@Hrevisit (3.11)], we have $$\label{eqn:Qtausum}
Q_\tau(S_N\cup S_1\cup\cdots\cup S_d)=Q_\tau(S_N)+2\sum_{k=1}^dQ_\tau(S_N,S_k)+Q_\tau(S_1\cup\cdots\cup S_d).$$ We will compute each term separately.
First, we have $$\label{eqn:QtauSN}
Q_\tau(S_N)=0,$$ because the graph $G_N$ has self-intersection zero as a parameterized graph. That is, any intersections between the edges of $G_N$, including self-intersections, can occur away from $
{\mathfrak{p}}(\alpha)_\beta$ and ${\mathfrak{p}}(\beta)$, and the parameterizations can be adjusted so as to avoid intersection in $[-1,1]\times Y$. In particular, the self-linking of the braids $S_N\cap\{1-\epsilon\}\times Y$ is zero because $G_N$ can be isotoped so that its edges do not intersect near ${\mathfrak{p}}(\alpha)_\beta$ and ${\mathfrak{p}}(\beta)$, even as non-parameterized curves.[^6]
Second, we compute $Q_\tau(S_N,S_k)$. We can choose the parameterizations $g(t)$ of the edges of $G_N$ so that when $t=1-l(k)$, the point $g(t)$ is outside all disks $\mathbb{D}^2_x$, the derivative $g'(1-l(k))\neq0$, and if $g(t)$ has an end at $x$, that when $t\geq1-\frac{\epsilon(x)}{\delta(x)}$, the parameterization $g(t)\equiv x$.
The points $(1-l(k),\check{S}_k(z))\in(1-l(k),{\mathfrak{p}}^{-1}(g(1-l(k))))$ will then be the only points of intersection between $\check{S}_k$ and the edges of $G_N$. Each contributes to the count of intersections with sign $+1$ because, in the oriented local basis $\{\partial_s,R,\partial_1,\partial_2\}$ for ${{\mathbb R}}_s\times Y$, where $\{\partial_1,\partial_2\}$ is an oriented basis for $\Sigma$ and $\partial_1$ equals the tangent vector to the edge in question (i.e., $\partial_1=g'(1-l(k))$), an oriented basis for $TS_N\oplus TS_k$ at their point of intersection is $$\{(1,0,1,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)\}.$$ Therefore $$\label{eqn:QtauSNSk}
Q_\tau(S_N,S_k)=N.$$
Finally we consider the self-intersection of the union of $S_k$s. We have $$Q_\tau(S_1\cup\cdots\cup S_d)=\sum_{k=1}^dQ_\tau(S_k)+\sum_{k_1\neq k_2}Q_\tau(S_{k_1},S_{k_2})$$ We will show that $Q_\tau(S_{k_1},S_{k_2})$ does not depend on the $k_i$ (even if $k_1=k_2$). Therefore, because $$d+2{d\choose2}=d+\frac{2d!}{2(d-2)!}=d+d(d-1)=d^2,$$ we will get $$Q_\tau(S_1\cup\cdots\cup S_d)=d^2Q_\tau(S_1).$$
To compute $Q_\tau(S_{k_1},S_{k_2})$ for any $k_1,k_2$, let $\delta_i,\epsilon_i,l_i$ denote $\delta(k_i)$, etc. Choose $\delta_1>\delta_2$ and $\frac{\epsilon_1}{\delta_1}>l_2$.
Because $\frac{\epsilon_1}{\delta_1}>l_2$, all intersections between $S_{k_1}$ and $S_{k_2}$ must occur at points whose projection to $\Sigma$ lies within the disk neighborhoods of ${\mathfrak{p}}(\alpha)_\alpha^{k_1}$.
Assume $x\in\{{\mathfrak{p}}(\alpha)_\alpha^{k_1}\}$ but $x\not\in\{{\mathfrak{p}}(\alpha)_\alpha^{k_2}\}$. In the local product coordinates $\partial_s,R,\partial_r,\partial_\theta$ determined by the section $\check{S}_{k_2}$, the intersection $S_{k_1}\cap(\{1-l_2\}\times Y)$ consists of a $(1,\text{mult}_{k_1}(x))$ torus knot about the fiber over $x$ in the $s=1-l_2$ level of $[-1,1]$ with $r_x=\frac{l_2}{\delta_1}$, and $S_{k_2}$ consists of the zero section of ${\mathfrak{p}}$, so is parameterized by $(1-l_2,0,r,\theta)$. In particular, by $T_{1,\text{mult}_{k_1}(x)}$ we are referring to $\partial_\theta$ as the meridional coordinate and $R$ as the longitudinal coordinate.
Since in oriented bases for $TS_{k_1}$ and $TS_{k_2}$ near any intersection in the $s=1-l_2$ slice only the first basis vector for $TS_{k_1}$ will have any $\partial_s$ component, and it will be positive, the intersection number in $[-1,1]\times Y$ will agree with the intersection number of the projections to $Y$ in $Y$. These projections will consist of the $T_{1,\text{mult}_{k_1}(x)}$ torus knot in the $r=\frac{l_2}{\delta_1}$ torus and the disk obtained by projecting off the Reeb direction.
Similarly, only the first basis vector for $TS_{k_2}$ will have any $\partial_r$ component, and it will be positive. Therefore the intersection number of the projections to $Y$ will agree with the intersection number in the $r_x=\frac{l_2}{\delta_1}$ torus of the $T_{1,\text{mult}_{k_1}(x)}$ torus knot and the meridian, parameterized by $\theta$. Their intersection number can easily be computed via a matrix: $$\label{eqn:nomult2}
\begin{vmatrix}\text{mult}_{k_1}(x)&0\\1&1\end{vmatrix}=\text{mult}_{k_1}(x).$$
Now assume $x\in\{{\mathfrak{p}}(\alpha)_\alpha^{k_1}\}\cap\{{\mathfrak{p}}(\alpha)_\alpha^{k_2}\}$. Let $0<\epsilon<\epsilon_2$. Using local coordinates $s$, the Reeb direction coming from ${\mathfrak{p}}^{-1}(x)$, and polar coordinates $r_x,\theta$ on the base, the intersections $S_{k_i}\cap(\{1-\epsilon\}\times Y)$ consist of $T_{1,\text{mult}_{k_i}(x)}$ torus knots in the tori $r_x=\frac{\epsilon}{\delta_i}$, respectively. Because $\delta_1>\delta_2$, the $T_{1,\text{mult}_{k_1}(x)}$ torus knot lies on a torus nested “inside" the torus of the $T_{1,\text{mult}_{k_2}(x)}$ torus knot, where “inside" refers to the component of $Y-T^2$ containing ${\mathfrak{p}}^{-1}(x)$. From knot diagrams of the image in ${{\mathbb R}}^3$ under the diffeomorphism of §\[s:writhe\] (see also the version in coordinates defined between Definitions 2.7 and 2.8 in [@Hrevisit]), it is immediate that $$\label{eqn:mult12}
\ell_\tau(\pi_Y(S_{k_1}\cap(\{1-\epsilon\}\times Y)),\pi_Y(S_{k_2}\cap(\{1-\epsilon\}\times Y)))=\text{mult}_{k_1}(x).$$
Equations (\[eqn:nomult2\]) and (\[eqn:mult12\]) show that each $x\in\{{\mathfrak{p}}(\alpha)_\alpha^{k_1}\}$ contributes to $Q_\tau(S_{k_1},S_{k_2})$ according to its multiplicity. Since there are no other intersections or boundary components, we obtain $$\label{eqn:QtauSk}
Q_\tau(S_{k_1},S_{k_2})=-e.$$
Combining equations (\[eqn:Qtausum\]), (\[eqn:QtauSN\]), (\[eqn:QtauSNSk\]), and (\[eqn:QtauSk\]) yields the desired result.
### Proof of the ECH index formula {#subsec:ECHindexcomp}
Combining Lemmas \[lem:orbitseh\], \[lempre\] tells us that $$CZ^I_\tau(\alpha)-CZ^I_\tau(\beta)=m_+-m_--(n_+-n_-).$$ Adding (\[eqn:relfirstChern\]) and (\[eqn:relselfint\]) proves the result.
Checking that our formula satisfies the additivity property of Theorem \[thm:Iproperties\] (iii) is straightforward. Checking that our formula satisfies the index parity property of Theorem \[thm:Iproperties\] (iv) requires relating the sums $m_1+\cdots+m_{2g}$ and $n_1+\cdots+n_{2g}$ to $m_\pm$ and $n_\pm$ via the formula $M=N+(-e)d$ defining $d$.
The many flavors of $J$ {#sec:modspcs}
=======================
In this section we work towards proving that $\partial^{L}$ only counts cylinders which are the union of fibers over Morse flow lines in $\Sigma$. One can count cylinders with a generic fiberwise $S^1$-invariant almost complex structure $\mathfrak{J} := \mathfrak{p}^*j_{\Sigma_g}$, the $S^1$-invariant lift of $j_{\Sigma_g}$, but unfortunately we cannot use $\mathfrak{J}$ for higher genus curves because it cannot be independently perturbed at the intersection points of $\pi_Y C$ with a given $S^1$-orbit by an $S^1$-invariant perturbation; see §\[degree\]. Generically, there will always exist a regular $J \in {\mathscr{J}}(Y,\lambda)$ for moduli spaces of nonzero genus curves, but we cannot assume that this $J$ is $S^1$-invariant.
To resolve this issue, we employ Farris’ strategy [@farris §6] of using a family of $S^1$-invariant domain dependent almost complex structures, ${\mathbb{J}}$, for higher genus curves, which was modeled on Cieliebak and Mohnke’s approach for genus zero pseudoholomorphic curves in [@CM]. To an ($S^1$-invariant) domain dependent almost complex structure ${\mathbb{J}}\in {\mathscr{J}}_{{{\dot{\Sigma}}}}^{S^1}$ and a map $C: ({{\dot{\Sigma}}}, j) \to ({{\mathbb R}}\times Y, {\mathbb{J}})$ where $g({{\dot{\Sigma}}}) >0$, we associate the $(0,1)$-form $$\overline{\partial}_{j,{\mathbb{J}}} : = \tfrac{1}{2}\left( dC + {\mathbb{J}}(z,C)\circ dC \circ j\right),$$ which at the point $z\in {{\dot{\Sigma}}}$ is given by $$\overline{\partial}_{j,{\mathbb{J}}}C(z) := \tfrac{1}{2} \left( dC(z) + {\mathbb{J}}(z,C(z))\circ dC(z) \circ j ( z) \right).$$ We say that $C$ is *${\mathbb{J}}$-holomorphic* whenever $\overline{\partial}_{j,{\mathbb{J}}}C=0$. There are two new phenomena to be accounted for in the case of higher genus ${\mathbb{J}}$-holomorphic curves. The first is that higher genus Riemann surfaces have have finite nontrivial symmetry groups, so the moduli space ${\mathscr{M}_{g,n}}$ is an orbifold, and therefore the moduli spaces of ${\mathbb{J}}$-holomorphic curves are also orbifolds. The second is that, even when using domain dependent almost complex structures, a nodal curve with a constant component of positive genus cannot be perturbed away to achieve transversality. However in dimension 4, we will show how Farris’s index considerations obstruct the latter configurations from arising.
Our scheme for obtaining regularity will be that if $z, z' \in {{\dot{\Sigma}}}$ map under $C$ to the same $S^1$ orbit in $Y$, then we will perturb ${\mathbb{J}}$ independently at $z$ and $z'$ while preserving ${\mathbb{J}}$’s $S^1$-invariance. We will exploit this construction to prove the existence of regular $S^1$-invariant domain dependent almost complex structures in §\[sec:ddacs\]-\[sec:reg\]. Moreoever, we show that for a generic choice of $S^1$-invariant domain dependent almost complex structure ${\mathbb{J}}$, the moduli spaces of curves of nonzero genus with ECH index 1 are empty. In §\[handleslides\] we will consider a one parameter family of domain dependent almost complex structures to relate curve counts defined with a domain dependent ($S^1$-invariant) ${\mathbb{J}}$ and a generic $\lambda$-compatible almost complex structure $J$, permitting us to conclude that the only contributions to an appropriately filtered ECH differential are from cylinders which project to Morse flow lines.
Degree of a completed projected curve {#degree}
-------------------------------------
We first review the notion of a degree of a completed projection for a pseudoholomorphic curve in the symplectization of a prequantization bundle.
\[def:degree\] If we compose the $J$-holomorphic curve $$C: {{\dot{\Sigma}}}\to {{\mathbb R}}\times Y,$$ with the projection $${\mathfrak{p}}: Y \to \Sigma_g,$$ then we obtain a map $${\mathfrak{p}}\circ \pi_Y C: {{\dot{\Sigma}}}\to \Sigma_g,$$ which has a well-defined non-negative *degree* because ${\mathfrak{p}}\circ \pi_Y C$ extends to a map of closed surfaces. We define the *degree* of $C$, denoted ${\operatorname}{deg}(C)$, to be the degree of this map.
One should not confuse degree with *multiplicity*. Recall that in Definition \[covered\] that the multiplicity of a pseudoholomorphic curve $C$ is given by degree of the holomorphic branched covering map between the domain of $C$ and the domain of the underlying somewhere injective curve. The multiplicity of a somewhere injective curve is always 1.
We can relate the degree of the completed projection map ${\mathfrak{p}}\circ \pi_Y C$ to the number of positive and negative ends via the $d\lambda_\varepsilon$-energy and Stokes’ Theorem as follows. First we note the following.
\[acrem\] The action of a Reeb orbit $\gamma_p^k$ of $R_\varepsilon$ over a critical point $p$ of $H$ is proportional to the length of the fiber, namely $$\mathcal{A}(\gamma_p^k) = \int_{\gamma_p^k} \lambda_\varepsilon = 2k \pi (1+\varepsilon {\mathfrak{p}}^*H),$$ because ${\mathfrak{p}}^*H$ is constant on critical points $p$ of $H$.
\[prop-degree\] For all $\lambda_\varepsilon$, we have the following relation between the degree ${\operatorname}{deg}(C)$ of a curve $C\in\mathcal{M}^J(\alpha,\beta)$ and the total multiplicity of the Reeb orbits at the positive and negative ends: $$M-N = |e|{\operatorname}{deg}(C)$$
Note that equality of the total homology classes of $\alpha$ and $\beta$ forces $$M-N = 0 \mbox{ mod } |e|$$
Denote by $H_\pm$ the values of $H$ at ${\mathfrak{p}}(e_\pm)$, respectively, and denote by $H_i$ the value of $H$ at ${\mathfrak{p}}(h_i)$.
Recall that the $d\lambda_\varepsilon$-energy (equivalently, contact area) $A(C)$ of a $J$-holomorphic curve $C$ is given by $$A(C) := \int_{{{\dot{\Sigma}}}} (\pi_YC)^*(d\lambda_\varepsilon).$$ Stokes’ Theorem yields $$\begin{aligned}
A(C)&= \int_{{{\dot{\Sigma}}}} (\pi_YC)^*(d\lambda_\varepsilon)
\\& = \int_{\partial(\pi_YC_*[{{\dot{\Sigma}}}])}\lambda_\varepsilon
\\&=2\pi(M-N+\varepsilon((m_--n_-)H_-+(m_1-n_1)H_1+\cdots+(m_{2g}-n_{2g})H_{2g}+(m_+-n_+)H_+))\end{aligned}$$
On the other hand, we have $$d\lambda_\varepsilon=\varepsilon{\mathfrak{p}}^*dH\wedge\lambda+(1+\varepsilon{\mathfrak{p}}^*H)d\lambda$$ where ${\mathfrak{p}}^*\omega=d\lambda$ and $\omega[\Sigma_g]=2\pi|e|$. Therefore $$\begin{aligned}
A(C)&=\varepsilon\int_{{{\dot{\Sigma}}}}(\pi_YC)^*({\mathfrak{p}}^*dH\wedge\lambda)+\int_{{{\dot{\Sigma}}}}(1+\varepsilon H\circ{\mathfrak{p}}\circ\pi_YC)(\pi_YC)^*({\mathfrak{p}}^*\omega)
\\&=\varepsilon\int_{{{\dot{\Sigma}}}}(\pi_YC)^*({\mathfrak{p}}^*dH\wedge\lambda)+\omega[({\mathfrak{p}}\circ\pi_YC)_*{{{\dot{\Sigma}}}}]\left(1+\varepsilon\int_{\Sigma_g}H\right)
\\&=\varepsilon\int_{{{\dot{\Sigma}}}}(\pi_YC)^*({\mathfrak{p}}^*dH\wedge\lambda)+2\pi|e|{\operatorname}{deg}(C) \text{ because $\int_{\Sigma_g}H=0$}\end{aligned}$$ It remains to show that $$\label{eqn:dHlambda}
\int_{{{\dot{\Sigma}}}}(\pi_YC)^*({\mathfrak{p}}^*dH\wedge\lambda)=2\pi((m_--n_-)H_-+(m_1-n_1)H_1+\cdots+(m_{2g}-n_{2g})H_{2g}+(m_+-n_+)H_+)$$ from which we obtain the conclusion by setting the two values computed for $A(C)$ equal to one another.
(\[eqn:dHlambda\]) also follows from Stokes’ theorem. Again because $\int_{\Sigma_g}H=0$, we have $$\begin{aligned}
\int_{{{\dot{\Sigma}}}}(\pi_YC)^*({\mathfrak{p}}^*dH\wedge\lambda)&=\int_{{{\dot{\Sigma}}}}(\pi_YC)^*({\mathfrak{p}}^*dH\wedge\lambda)+\omega[({\mathfrak{p}}\circ\pi_YC)_*{{{\dot{\Sigma}}}}]\int_{\Sigma_g}H
\\&=\int_{{{\dot{\Sigma}}}}(\pi_YC)^*({\mathfrak{p}}^*dH\wedge\lambda)+\int_{(\pi_YC)_*[{{\dot{\Sigma}}}]}{\mathfrak{p}}^*(H\omega)
\\&=\int_{{{\dot{\Sigma}}}}(\pi_YC)^*(d(H\circ{\mathfrak{p}})\wedge\lambda+(H\circ{\mathfrak{p}})d\lambda)
\\&=\int_{{{\dot{\Sigma}}}}(\pi_YC)^*d((H\circ{\mathfrak{p}})\lambda)
\\&=\int_{\partial(\pi_YC)_*[{{\dot{\Sigma}}}])}(H\circ{\mathfrak{p}})\lambda\end{aligned}$$ which equals the right hand side of (\[eqn:dHlambda\]).
As a consequence of Proposition \[prop-degree\], the degree of any two curves in $\mathcal{M}^J(\alpha,\beta)$ must be equal, and therefore we make the following definition.
\[degree-gen\] The *degree* of a pair of ECH generators $(\alpha,\beta)$, denoted ${\operatorname}{deg}(\alpha,\beta)$, is $${\operatorname}{deg}(\alpha,\beta):=\frac{M-N}{|e|}$$
A curve $C$ which contributes nontrivially to the ECH differential is degree zero if and only if it is a cylinder, as we now explain.
\[lem:d0g0\] Let $C\in\mathcal{M}^J(\alpha,\beta)$ be a $J$-holomorphic curve with domain $({{\dot{\Sigma}}},j)$ and with $I(C)=1$. Then ${\operatorname}{deg}(C)=0$ if and only if ${{\dot{\Sigma}}}$ the union of cylinders.
If ${{\dot{\Sigma}}}$ is a cylinder and ${\operatorname}{deg}(C)>0$, then the completion of ${\mathfrak{p}}\circ\pi_YC$ has $S^2$ as its domain. The Riemann-Hurwitz formula (\[eqn:RH\]) tells us $$\label{eqn:RHcontra}
2={\operatorname}{deg}(C)\chi(\Sigma_g)-\sum_{p\in{{\dot{\Sigma}}}}(e(p)-1)$$ Since we are assuming $g\geq1$, the right hand side is at most zero, therefore (\[eqn:RHcontra\]) is a contradiction. Therefore, if ${{\dot{\Sigma}}}$ is a cylinder, then $C$ has degree zero.
For the opposite implication, assume ${\operatorname}{deg}(C)=0$. Because $I(C)=1$, the curve $C$ is embedded and has ${\operatorname}{ind}(C)=1$, by Proposition \[lowiprop\]. By the ECH index inequality, Theorem \[ECH-ineq\], the positive and negative partitions of the ends of $C$ must equal the positive and negative partitions defined in Definition \[def:part\].
By Example \[ex:partitions\] and the analogous result for $\vartheta$ slightly smaller than zero, i.e. $$\begin{array}{lcl}
P_\vartheta^+(m) &=& (m) \\
P_\vartheta^-(m) &=& (1,\dots,1) \\
\end{array}$$ and the fact that in admissible orbit sets hyperbolic orbits have multiplicity at most one, we find that ${{\dot{\Sigma}}}$ has exactly $1+m_1+\cdots+m_{2g}+m_+$ positive ends and exactly $n_-+n_1+\cdots+n_{2g}+1$ negative ends. Moreover, $$CZ_\tau^{ind}(C)=(m_+-1)-(1-n_-)$$ Therefore, $$\begin{aligned}
1&={\operatorname}{ind}(C)\nonumber
\\&=-2+2g({{\dot{\Sigma}}})+M-m_-+1+N-n_++1+0+m_+-1-1+n_-\text{ by (\ref{eqn:relfirstChern})}\nonumber
\\&=-2+2g({{\dot{\Sigma}}})+2M-m_--n_++0+m_++n_-\text{ because $M=N$}\label{eqn:1ind}\end{aligned}$$
Note that $I(C)=1$ and $\frac{M-N}{|e|}={\operatorname}{deg}(C)=0$, along with the index formula (\[eqn:indexformula\]), imply $$\label{eqn:d0I1}
1=I(C)=m_+-m_--n_++n_-$$
Combining (\[eqn:1ind\]) and (\[eqn:d0I1\]) gives $$0=-2+2g({{\dot{\Sigma}}})+2M \Leftrightarrow 1=g({{\dot{\Sigma}}})+M$$
Since $M>0$, we must have $M=N=1$ and $g({{\dot{\Sigma}}})=0$, therefore ${{\dot{\Sigma}}}$ must consist of a union of cylinders.
In particular we note that fiberwise $S^1$-invariant ${\mathfrak{J}}$-holomorphic cylinders have degree 0. We have the following correspondence between ${\mathfrak{J}}$-holomorphic cylinders asymptotic to Reeb orbits which project to critical points of $H$ and downward gradient flow lines of $H$, which will be key later in relating the filtered ECH differential to the Morse differential on the base.
\[cylinder-to-morse\] For suitable orientation choices, if $p$ and $q$ are critical points of $H$, then there is an orientation-preserving bijection between the moduli space of ${\mathfrak{J}}$-holomorphic cylinders ${{{\mathcal M}}}^{{\mathfrak{J}}}(\gamma_p^k,\gamma_q^k)$ and the moduli space ${{{\mathcal M}}}^{\mbox{\tiny Morse}}(p,q)$ of downward gradient flow lines of $H$ from $p$ to $q$, modulo reparametrization. Furthermore, each of the holomorphic cylinders is a $k$-fold cover which is cut out transversely.
Complete details on this correspondence are found in [@jo2 §5.1] and [@moreno §3.5.1,§6.1]; that the multiple covers are cut out transversely requires automatic transversality, cf. [@dc §4.1-4.2], [@Wtrans]. In ECH, we will only have somewhere injective curves, because the ECH chain complex does not admit hyperbolic orbits with multiplicity greater than one as generators.
If a $J$-holomorphic curve $C: {{\dot{\Sigma}}}\to {{\mathbb R}}\times Y$ has [degree]{} $d=0,1$ then can prove directly or appeal to automatic transversality that a regular $S^1$-invariant $J$ exists. If $d \geq 1$, the projection $\pi_Y C$ has intersection number $d$ with a given $S^1$-orbit, and hence has at least $d$ intersections, which are counted with multiplicity, since $C$ could be a nontrivial branched covering of its image. The complex structure ${\mathfrak{J}}$ cannot be independently perturbed at these $d \geq 2$ points by an $S^1$-invariant perturbation. By Lemma \[lem:d0g0\], a curve $C$ which contributes nontrivially to the ECH differential is degree zero if and only if it the union of cylinders.
Domain dependent almost complex structures {#sec:ddacs}
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Let ${\mathscr{J}}(Y,\lambda)$ denote the set of $\lambda$-compatible almost complex structures and let ${\mathscr{J}}^{S^1}(Y,\lambda) \subset {\mathscr{J}}(Y,\lambda)$ denote the subset of $S^1$-invariant $\lambda$-compatible almost complex structures. Since ${\mathfrak{J}}\in {\mathscr{J}}^{S^1}(Y,\lambda)$ is always obtained from ${\mathfrak{p}}^*j_{\Sigma_g}$, there is a correspondence between the $S^1$-invariant complex structures on $\xi$ and the complex structures on $T\Sigma_g$. Fix a “generic" ${\mathfrak{J}}_0 \in {\mathscr{J}}^{S^1}(Y,\lambda) $, let $\{ N_\gamma\}$ be a disjoint union of tubular neighborhoods associated to the set of Reeb orbits $\{ \gamma \}$, and set $N = \sqcup_\gamma N_\gamma$. We define $$\label{JNs}
{\mathscr{J}}_N : = \{ J \in {\mathscr{J}}(Y,\lambda) \ | \ J_q = ({\mathfrak{J}}_0)_q, \mbox{ for } q\in N \},$$ to be the subset of $\lambda$-compatible almost complex structures which agree with ${\mathfrak{J}}_0$ on $N$, and let ${\mathscr{J}}^{S^1}_N \subset {\mathscr{J}}_N $ consist of the $S^1$ invariant elements of ${\mathscr{J}}_N$. The elements of ${\mathscr{J}}_N $ are in correspondence with complex structures on $T\Sigma_g$ which agree with a fixed ${\mathfrak{J}}_0 = {\mathfrak{p}}^*j_0$ on $N$. We first note that we have regularity for ECH index 1 curves for generic $\lambda$-compatible $J$ satisfying the constraint $J|_N = {\mathfrak{J}}_0$.
\[rem:Js\] Let $\alpha$ and $\beta$ be nondegenerate admissible orbit sets in the same homology class with $I(\alpha,\beta)=1$. If $J \in {\mathscr{J}}_N$ is generic then ${{{\mathcal M}}}^{J}(\alpha, \beta)$ is cut out transversely.
Regularity follows from the subclaim in the proof of [@Hindex Lemma 9.12]. We have that in the absence of trivial cylinders, there is a nonempty set $U \subset C$ away from a neighborhood of of the periodic orbits with action $\leq \mathcal{A}(\alpha)$, such that for each $x \in U$, $\pi_Y^{-1}(\pi_Y(x))= \{x \}$, $C$ is nonsingular, and that a certain projection of the derivative of $\overline{\partial}_{j,J}(C)$ with respect to $J$ is surjective on $U$. The proof actually shows that $U$ is an open dense subset of $C$, and because the intersection with $C^{-1}({{\mathbb R}}\times (Y \setminus N))$ contains a nonempty open set, the result holds.
When ${{\dot{\Sigma}}}\neq {{\mathbb R}}\times S^1$, it is not possible to achieve regularity via a generic choice of $S^1$-invariant ${\mathfrak{J}}$. Instead, we will use an $S^1$-invariant domain dependent family ${\mathbb{J}}$ of $\lambda$-compatible almost complex structures. To define such a ${\mathbb{J}}$, we consider a certain class of functions on the domain ${{\dot{\Sigma}}}$ that are independent of reparameterization, meaning that these functions are to be defined on isomorphism classes of punctured Riemann surfaces, e.g. elements of ${\mathscr{M}_{g,n}}$, where we view the punctures $\mathcal{S}$ of ${{\dot{\Sigma}}}$ as the $n$ marked points and $g = g({{\dot{\Sigma}}})$.
### Preliminaries relating to ${\mathscr{M}_{g,n}}$
To ensure that we have well-defined, nontrivial functions on ${{\dot{\Sigma}}}$ (from which we will construct domain dependent almost complex structures), we need ${{\dot{\Sigma}}}$ to be *stable*, meaning that $\chi({{\dot{\Sigma}}}) < 0$, e.g. $2g+n \geq 3$, where $n$ is the number of punctures of ${{\dot{\Sigma}}}$. If ${{\dot{\Sigma}}}$ is stable then ${\mathscr{M}_{g,n}}$ is an orbifold with $$\mbox{dim}({\mathscr{M}_{g,n}}) = 6g - g + 2n,$$ and the associated automorphism group $${{\operatorname}{Aut}}({{\dot{\Sigma}}}, j):=\{ \varphi \in \mbox{Diff}_+(\Sigma, \mathcal{S}) \ | \ \varphi^* j =j \}$$ is finite for any $j \in {\mathscr{J}}(\Sigma)$. Here ${\mathscr{J}}(\Sigma)$ is the set of smooth complex structures on $\Sigma$ that induce the given orientation and $\mbox{Diff}_+(\Sigma, \mathcal{S})$ is the group of orientation preserving diffeomorphisms on $\Sigma$ that fix the set of punctures $\mathcal{S}$. Our class of domain dependent almost complex structures on ${{\dot{\Sigma}}}$ must respect the orbifold structure of ${\mathscr{M}_{g,n}}$, meaning they must be invariant with respect to the finite symmetry groups of the orbifold points of ${\mathscr{M}_{g,n}}$. While the derivative of such an invariant function (giving rise to ${\mathbb{J}}$) will have nontrivial kernel at an orbifold point of ${\mathscr{M}_{g,n}}$, the set of orbifold points has positive complex codimension, meaning that there is sufficient flexibility in the normal direction.
\[unstable-cases\] Following [@Wtrans §3.1], and because our pseudoholomorphic curves must all have at least one puncture, we conclude that the nonstable cases are ${{\dot{\Sigma}}}= {{\mathbb R}}\times S^1$ and ${{\dot{\Sigma}}}= {{\mathbb C}}$. We previously showed that we can use a domain independent $S^1$-invariant ${\mathfrak{J}}$ to count pseudoholomorphic cylinders, so it remains to consider when ${{\dot{\Sigma}}}= {{\mathbb C}}$. Since any pseudoholomorphic map $C: {{\mathbb C}}\to {{\mathbb R}}\times Y$ must be asymptotic to a Reeb orbit $\gamma$, we can consider its *projected completion* to the base $(\Sigma_g, \omega)$ of the prequantization bundle $$\overline{C}: S^2 \to \Sigma_g,$$ which is null homotopic for $g>0$. Since $\overline{C}$ is null homotopic when $g>0$, if we consider a sufficiently small perturbation, $\overline{C}$ must be close to constant, which means that $C$ is “concentrated" near its limiting Reeb orbit, and thus cannot bound $\overline{C}$, because otherwise we would obtain a contradiction to the fact that far away fibers of ${\mathfrak{p}}: (Y,\lambda) \to (\Sigma_g, \omega)$ are linked. For this reason, in §\[subsec:pfmainthm\] we split the proof of Theorem \[thm:mainthm\] into two cases, depending on whether the genus of the base of the prequantization bundle $\Sigma_g$ is zero or positive. When $g=0$, we show that the differential vanishes for grading reasons, which permits us to use a generic $\lambda$-compatible $J$ without appealing to the results of §\[sec:modspcs\]-\[handleslides\].
In order to vary a domain dependent ${\mathbb{J}}$ and take limits of sequences of ${\mathbb{J}}$-holomorphic curves in the sense of [@BEHWZ], we must actually work with the Deligne-Mumford compactification $\overline{{\mathscr{M}_{g,n}}}$, a compact and metrizable topological space containing ${\mathscr{M}_{g,n}}$ as an open subset, which consists of connected stable nodal Riemann surfaces with $n$ marked points, (presupposing that $2g+n \geq 3$).
Recall that an element of ${\mathscr{M}_{g,n}}$ is *stable* whenever $2g + n \geq 3$. A *stable nodal Riemann surface* is an element $({{\dot{\Sigma}}}, j) \in \overline{{\mathscr{M}_{g,n}}}$, which is itself a disjoint union of elements $({{\dot{\Sigma}}}_i,j_i) \in {\mathscr{M}}_{g_i, n_i+m_i}$, where ${{\dot{\Sigma}}}_i$ is a stable curve whose $n_i+m_i$ marked points consist of a subset $n_i$ of the marked points of ${{\dot{\Sigma}}}$ (hence $\sum n_i = n$), with the induced ordering, and $m_i$ nodes. Every node $z \in {{\dot{\Sigma}}}_i$ is paired with another node $z' \in {{\dot{\Sigma}}}_i'$, with the stipulation that $i'\neq i$ for at least one of the nodes of each $C_i$. We thus obtain a connected singular surface by gluing $z$ to $z'$ for every pair $\{ z, z' \}$ of nodes.
Any sequence of curves $\{{{\dot{\Sigma}}}(k)\} \in {\mathscr{M}_{g,n}}\subset \overline{{\mathscr{M}_{g,n}}}$ has a subsequence whose limit is a nodal curve ${{\dot{\Sigma}}}\in \overline{{\mathscr{M}_{g,n}}}$. Furthermore, if $z_i(k) \in {{\dot{\Sigma}}}(k)$ is a marked point, passing to a subsequence means $z_i(k)$ converges to some marked point $z\in{{\dot{\Sigma}}}$, hence $z \in {{\dot{\Sigma}}}_i$, where ${{\dot{\Sigma}}}_i \in {\mathscr{M}}_{g_i, n_i+m_i}$ for some ${{\dot{\Sigma}}}_i \subset {{\dot{\Sigma}}}$. We recall the following result regarding the topology of the nodal limit, noting that further details can be found in [@wendl-sft §9.3.3] and [@ss92].
\[lem:mgn\] If ${{\dot{\Sigma}}}\in \overline{{\mathscr{M}_{g,n}}}$ then for each component ${{\dot{\Sigma}}}_i \in {\mathscr{M}}_{g_i,n_i+m_i}$ of ${{\dot{\Sigma}}}$ we have that $g_i \leq g$ and if $g_i=g$ then $n_i+m_i < n$.
The nodal limit ${{\dot{\Sigma}}}$ is obtained topologically from any smooth sequence $\{ {{\dot{\Sigma}}}(k) \}$ via the following types of degenerations. The first degeneration is that $\ell$ marked points in some component ${{\dot{\Sigma}}}_j$ can collide and form a bubble attached to ${{\dot{\Sigma}}}_j$. The genus of ${{\dot{\Sigma}}}_j$ does not change, but it loses $\ell$ marked points and gains a node where the bubble arises. Thus, the total number of marked and nodal points on ${{\dot{\Sigma}}}_j$ decreases by $\ell-1$. The bubble itself is a genus 0 component with $\ell$ marked points and one node. If the original smooth curve ${{\dot{\Sigma}}}(k_0)$ had genus 0, then, then it must have had more than $\ell$ marked points. Thus each new component resulting from iterative bubbling has either genus 0 or $g$.
The second kind of degeneration comes from letting the complex structure on the curve degenerate. Topologically, this results in a simple closed curve on some component, i.e. the vanishing cycle, being crushed to a point. If the vanishing cycle is a non-separating curve, it reduces the genus of a component by 1 without creating new components. If the vanishing cycle is a separating curve, it breaks a component into two pieces, whose genera sum to the genus of the original component. The case where one component has genus 0 and the other has full genus is topologically identical to bubbling, cf. [@jhol].
Lemma \[lem:mgn\] induces an ordering on pairs $(g,n)$ wherein $(g',n') < (g,n)$ whenever $g' < g$ or $g' =g$ and $n' < n$. The boundary $\partial \overline{{\mathscr{M}_{g,n}}}$ is a stratified space, and each stratum containing a nodal curve ${{\dot{\Sigma}}}$ is the product over the ${\mathscr{M}}_{g',n'}$ for each component ${{\dot{\Sigma}}}_i$ of ${{\dot{\Sigma}}}$ with ${{\dot{\Sigma}}}_i \in {\mathscr{M}}_{g',n'}$. Moreover, we have that if ${\mathscr{M}}_{g',n'}$ is a factor in a stratum of ${\mathscr{M}_{g,n}}$, then $(g',n') < (g,n)$ and we can distinguish one of the components of a given ${{\dot{\Sigma}}}\in \overline{{\mathscr{M}_{g,n}}}$ whenever it contains the $n^{th}$ marked point, which we will always denote as $z_0$. This prescribes an inductive means of coherently defining functions on all strata of $\overline{{\mathscr{M}_{g,n}}}$ simultaneously, though before we get into this we need to briefly review [@CM §3].
### Coherent maps
In the following discussion, we consider the Deligne-Mumford space $\overline{{\mathscr{M}}_{g,n+1}}$ for $2g+n \geq 3$, with $n+1$-marked points $z_0,...,z_{n}$. We will see momentarily that the point $z_0$ plays a special role, as it serves as the variable for holomorphic maps and is key in the proof of Theorem \[thm:ddacs\].
First consider the case when $g=0$. We call a decomposition $\mathbf{I}=(I_0,...,I_\ell)$ of $\{0,...,n \}$ *stable* whenever $I_0 = \{ 0\}$ and $|\mathbf{I}| := \ell+1 \geq 3$. We will always order the $I_j$ such that the integers $i_j:=\min \{ i \ | \ i \in I_j \}$ satisfy $$0 = i_0 < i_1 < ... < i_\ell.$$ Denote by ${\mathscr{M}}_\mathbf{I} \subset \overline{{\mathscr{M}}_{0,n+1}} $ the union over stable trees that give rise to the stable decomposition $\mathbf{I}$. The ${\mathscr{M}}_{{\mathbf{I}}}$ are submanifolds of $\overline{{\mathscr{M}}_{0,n+1}} $ with $$\overline{{\mathscr{M}}_{0,n+1}} = \cup_{{{\mathbf{I}}}}{\mathscr{M}}_{{\mathbf{I}}}$$ and the closure of ${\mathscr{M}}_{{\mathbf{J}}}$ is a union of certain strata ${\mathscr{M}}_{{\mathbf{I}}}$ with $|{{\mathbf{I}}}| \leq |{{\mathbf{J}}}|$. The above ordering of the $I_j$ determines a projection $$p_{{\mathbf{I}}}: {\mathscr{M}}_{{\mathbf{I}}}\to {\mathscr{M}}_{|{{\mathbf{I}}}|},$$ sending a stable curve to the special points on the component $S_{\alpha0}$.
[@CM Definition 1.3] Let $Z$ be a Banach space and $n \geq3$. We call a continuous map $J_{0,n+1}: \overline{{\mathscr{M}}_{g,n+1}} \to Z$ *coherent* if it satisfies the following two conditions:
(a) $J_{0,n+1} \equiv 0$ in a neighborhood of those ${\mathscr{M}}_{\mathbf{I}}$ with $|\mathbf{I}|=3$;
(b) For every stable decomposition $\mathbf{I}$ with $|\mathbf{I}|\geq4$, there exists a smooth map $J_{{\mathbf{I}}}: {\mathscr{M}}_{0,|{{\mathbf{I}}}|} \to Z$ such that $$J|_{{\mathscr{M}}_{{\mathbf{I}}}} = J_{{\mathbf{I}}}\circ p_{{\mathbf{I}}}: {\mathscr{M}}_{{\mathbf{I}}}\to Z.$$ More generally, let $Z^* \subset Z$ be an open neighborhood of 0 and let $\mathcal{I}$ be a collection of stable decompositions. Then we call a continuous map $J: \cup_{{{\mathbf{I}}}\in \mathcal{I}}{\mathscr{M}}_{{\mathbf{I}}}\to Z^*$ coherent if it satisfies (a) and (b) and in addition:
(c) The image of $J$ is contractible in $Z^*$.
The space of coherent maps from $\overline{{\mathscr{M}}_{0,n+1}}$ to $Z$ is equipped with the $C^0$-topology on $\overline{{\mathscr{M}}_{0,n+1}}$ and the $C^\infty$-topology on each ${\mathscr{M}}_{{\mathbf{I}}}$ via the projection $p_{{\mathbf{I}}}$.
For $g>0$, we deploy Lemma \[lem:mgn\], obtaining an inductive means of coherently defining functions on all strata of $\overline{{\mathscr{M}_{g,n}}}$ simultaneously. In particular, assume for all $(g',n') < (g,n)$ we have defined continuous maps $$J_{g',n'}: {\mathscr{M}}_{g',n'} \to Z$$ Each element of $\partial \overline{{\mathscr{M}_{g,n}}}$ is a nodal curve ${{\dot{\Sigma}}}$ with $n$ marked points . If $p_n$ lies on ${{\dot{\Sigma}}}' \in {\mathscr{M}}_{g',n'}$, then by Lemma \[lem:mgn\], $(g',n') < (g,n)$, and by hypothesis there is a function $J_{g',n'}$ defined on ${\mathscr{M}}_{g',n'}$. We can define $$J_{g,n}({{\dot{\Sigma}}}) : = J_{g',n'}({{\dot{\Sigma}}}').$$ The collection $\{J_{g',n'}\}_{(g',n') < (g,n)}$ thus determines $\{ J_{g',n'}\}_{\overline{{\mathscr{M}_{g,n}}}}$ and we can extend $J_{g,n}$ to the interior ${\mathscr{M}_{g,n}}$. We can continue this procedure, defining $J_{g,N}$ on ${\mathscr{M}}_{g,N}$ for all $N>n$, then $f_{g+1}, n$ for all $n$, and so on. This inductive procedure provides the definition for our continuous maps with $g>0$ to be *coherent*.
[Before we can use this class of to define domain dependent almost complex structures, we need to review a few details regarding Banach manifolds of (parametrized) almost complex structures.]{}
### Banach manifolds of almost complex structures
For a symplectic vector space $(V, \omega)$, denote by ${\mathscr{J}}(V,\omega)$ the space of $\omega$-tamed almost complex structures. The space ${\mathscr{J}}(V,\omega)$ is a manifold with tangent space $$T_{J_0}{\mathscr{J}}(V,\omega) := \{ \mathcal{Y} \in {\operatorname}{End}(V) \ | \ J_0 {\mathcal{Y}}J_0 = {\mathcal{Y}}\}.$$ If $V$ is equipped with a Euclidean metric $g$ then ${\operatorname}{trace}({\mathcal{Y}}^t{\mathcal{Y}})$ defines a Riemannian metric on ${\mathscr{J}}(V,\omega)$. The exponential map of this metric defines embeddings from the open ball of radius $\rho(g,J)>0$ which continuously depend on $g$ and $J$: $$\exp_J:T_J {\mathscr{J}}(V,\omega) \supset B(0, \rho(g,J)) \hookrightarrow {\mathscr{J}}(V,\omega).$$ We review the construction of the *Floer $C^\varepsilon$-space* [@floer], which circumvents the issue that naturally arising spaces of smooth functions are not Banach spaces; see also [@wendl-sft Appendix B]. For a vector bundle $E \to X$ over a closed manifold $X$, we denote the space of Floer’s $C^\varepsilon$-sections in $E$ by $$C^\varepsilon(X,E):= \left \{ s \in \Omega(X,E) \ \bigg \vert \ \sum_{i=1}^\infty \varepsilon_i || s||_{C^i} < \infty \right \}.$$ Here $\varepsilon = (\varepsilon_i)_{i \in {{\mathbb N}}}$ is a fixed sequence of positive numbers and $||\cdot||_{C^i}$ is the $C^i$-norm with respect to some connection on $E$. By [@floer Lemma 5.1], if the $\varepsilon_i$ converge sufficiently fast to zero, then $C^{\varepsilon}(X,E)$ is a Banach space consisting of smooth sections and containing sections with support in arbitrarily small sets in $X$.
Next let $(X,\omega)$ be the symplectization of a closed contact manifold $Y$ (or exact symplectic cobordism). Fix a $\lambda$-compatible almost complex structure $J_0$ on $(X,\omega)$, e.g. a smooth section in the bundle ${\mathscr{J}}(TX,\omega) \to X$ with fibers ${\mathscr{J}}(T_xX,\omega_x)$. Let $g$ be the canonical Riemannian metric on $X$ defined via $\omega$ and $J$. Let $T_{J_0}{\mathscr{J}}(TX,\omega) \to X$ be the vector bundle with fibers $T_{J_0(x)}{\mathscr{J}}(T_xX,\omega_x)$ and set $$\begin{array}{rcl}
T_{J_0}{\mathscr{J}}^\varepsilon & : =& C^\varepsilon(X, T_{J_0}{\mathscr{J}}(TX,\omega) ) \\
{\mathscr{J}}^\varepsilon &:=& {\mathscr{J}}^\varepsilon(X, \omega) \ := \ \exp_{J_0}(B), \\
\end{array}$$ where $B:= \{ {\mathcal{Y}}\in T_{J_0}{\mathscr{J}}^\varepsilon \ | \ {\mathcal{Y}}(x) \in B(0, \rho(g(x),J_0(x))) \}.$ Thus ${\mathscr{J}}^\varepsilon$ is the space of $\lambda$-compatible almost complex structures of $(X,\omega)$ that are of class $C^\varepsilon$ over $J_0$ via $\exp_{J_0}$. We can regard ${\mathscr{J}}^\varepsilon$ as a Banach manifold with a single chart $\exp_{J_0}$.
Next we consider spaces of parametrized complex structures. A *complex structure on $(X, \omega)$ parametrized by a manifold $P$* is a smooth section in the pullback bundle ${\mathscr{J}}(TX, \omega) \to P \times X$. Fix $J_0$ as above and let $T_{J_0}{\mathscr{J}}_P(TX, \omega) \to P \times X $ be the vector bundle with fibers $T_{J_0(p,q)}{\mathscr{J}}(T_xX, \omega_x)$ and set $$\begin{array}{rcl}
T_{J_0}{\mathscr{J}}_P^\varepsilon & := & C^\varepsilon(P \times X, T_{J_0}{\mathscr{J}}_P(TX,\omega) ) \\
{\mathscr{J}}^\varepsilon_P &:=& {\mathscr{J}}_P^\varepsilon(X, \omega) \ := \ \exp_{J_0}(B_P), \\
\end{array}$$ where $B_P:= \{ {\mathcal{Y}}\in T_{J_0}{\mathscr{J}}^\varepsilon_P \ | \ {\mathcal{Y}}(p, x) \in B(0, \rho(g(x),J_0(x))) \}$. We may think of $J \in {\mathscr{J}}^\varepsilon_P$ as a map $P \to {\mathscr{J}}^\varepsilon$. For an open subset $U\subset P$, we denote by $T_J{\mathscr{J}}^\varepsilon_U \subset T_J{\mathscr{J}}^\varepsilon_P$ the subspace of those section having compact support in $U$.
We will be interested in the spaces of domain dependent almost complex structures $${\mathscr{J}}^\varepsilon_{{{\dot{\Sigma}}}}:={\mathscr{J}}^\varepsilon_{\overline{{\mathscr{M}_{g,n}}}} \mbox{ and } {\mathscr{J}}_{{{\dot{\Sigma}}}}:={\mathscr{J}}_{\overline{{\mathscr{M}_{g,n}}}}$$ parametrized by the Deligne-Mumford space $\overline{{\mathscr{M}_{g,n}}}$.
A *domain dependent almost complex structure* ${\mathbb{J}}$ is a coherent collection of $C^\ell$, $\ell >0$ maps $${\mathbb{J}}= \{ J_{g,n}: \overline{{\mathscr{M}_{g,n}}} \to {\mathscr{J}}_N \},$$ (recall ${\mathscr{J}}_N$ was defined in ) that additionally satisfy the following condition. If given a sequence $\{{{\dot{\Sigma}}}(k)\} \in \overline{{\mathscr{M}_{g,n}}}$ converging to ${{\dot{\Sigma}}}_\infty \in \partial \overline{{\mathscr{M}_{g,n}}}$ and ${{\dot{\Sigma}}}^n_\infty \in {\mathscr{M}}_{g',n'}$ is the component of ${{\dot{\Sigma}}}_\infty$ containing the $n^{th}$ marked point, then $$\lim_{k \to \infty}J_{g,n}({{\dot{\Sigma}}}(k)) = J_{g,n}({{\dot{\Sigma}}}_\infty) =J_{g,n}({{\dot{\Sigma}}}^n_\infty).$$ [We denote the set of all such $C^\ell$ domain dependent almost complex structures by ${\mathscr{J}}_{{\dot{\Sigma}}}^\ell$.]{} We call a domain dependent almost complex structure ${\mathbb{J}}$ *generic* if for every $(g,n)$, the extension of $J_{g,n}$ from the boundary, where the values are determined by $J_{g',n'}$, to the interior of ${\mathscr{M}_{g,n}}$ is a generic $C^\ell$ map.
The extension to nodal maps follows from [@CM §5]. The previous discussion guarantees that ${\mathscr{J}}_{{\dot{\Sigma}}}^\varepsilon$ is a Banach manifold. When it is understood that we should be using the Floer $C^\varepsilon$-space we will drop $\varepsilon$ from the notation, and use ${\mathscr{J}}_{{\dot{\Sigma}}}$.
\[ddacsintpos\] Since the target of our collection of functions on ${{\dot{\Sigma}}}$ is ${\mathscr{J}}_N$, we have that if ${\mathbb{J}}\in {\mathscr{J}}_{{\dot{\Sigma}}}$ and if $$C: ({{\dot{\Sigma}}}, j) \to ({{\mathbb R}}\times Y, {\mathbb{J}})$$ is a $(j,{\mathbb{J}})$-holomorphic curve, then $C \vert_{C^{-1}({{\mathbb R}}\times N)}$ is $(j,{\mathfrak{J}}_0)$-holomorphic. Since ${\mathfrak{J}}_0$ is domain independent, the subset $C({{\dot{\Sigma}}}) \cap {{\mathbb R}}\times N \subset C({{\dot{\Sigma}}})$ satisfies intersection positivity, which will be important in the proof of Proposition \[notopbottomconn\].
Before we can conclude that this algorithm for constructing domain dependent almost complex structures is well-defined, it remains to discuss two technicalities, the first being that ${\mathscr{M}_{g,n}}$ is an orbifold, while the other concerns the special role of the “last" marked point $z_0$. We elucidate these points in the following remarks.
A neighborhood of a point in an $k$-dimensional orbifold is modeled on the quotient of ${{\mathbb R}}^k$ by the linear action of some finite group $G$, and a $C^\ell$ function on an orbifold in a neighborhood modeled on ${{\mathbb R}}^k/G$ is a $C^\ell$ function on ${{\mathbb R}}^k$ which is invariant under the group action $G$. For $g>1$, the locus of points on ${\mathscr{M}_{g,n}}$ without automorphisms (e.g. the action of $G$ on ${{\mathbb R}}^k$ is nontrivial) has real codimension at least two. Thus, a generic curve of genus $g >1$ has no nontrivial automorphisms. For $g=0$, every stable curve has a trivial automorphism group. For $g=1$ and $n=1$, $\mbox{dim}_{{\mathbb R}}{\mathscr{M}}_{g,1} =2$, and a generic elliptic curve has an involution and isolated points in ${\mathscr{M}}_{g,1}$ have additional automorphisms, hence functions on ${\mathscr{M}}_{g,1}$ have no constraints at generic points and respect the additional symmetries at the isolated points admitting the extra automorphisms.
[The derivative of a $G$-invariant function always has nontrivial kernel, but on any tangent space $T_z{\mathscr{M}}_{g, n+1}$, there is a subspace of real dimension at least two on which the derivative has no constraints. Hence there exists a map from a neighborhood of any point in ${\mathscr{M}}_{g, n+1}$ to a neighborhood of any point $x$ in a manifold $X$ with $\dim_{{\mathbb R}}X \geq 2,$ sending a two dimensional subspace of the unconstrained subspace to any two dimensional subspace of $T_xX$. If we fix $j$ and the marked points $z_1,...,z_n$ on ${\mathscr{M}}_{g, n+1}$, we may view this as a map $T_{z_{0}}{{\dot{\Sigma}}}\to T_xX$.]{}
Recall that there is a forgetful map $$\pi: \overline{{\mathscr{M}}_{g,n+1}} \to \overline{{\mathscr{M}_{g,n}}}$$ which forgets the special marked point $z_0$ and collapses any resulting unstable components, which are necessarily of genus zero. The fiber over ${{\dot{\Sigma}}}\in \overline{{\mathscr{M}_{g,n}}}$ is itself isomorphic to ${{\dot{\Sigma}}}$. To see why this holds over the marked and nodal points, note the following. The fiber above the $k^{th}$ marked point $z_k$ is a single nodal curve which has a genus zero component containing the marked points $z_k$ and $z_0$ as well as a node, which is glued to $z_k\in {{\dot{\Sigma}}}$. This component collapses when the marked point $z_0$ is removed. The fiber above a node resulting from gluing $z' \in {{\dot{\Sigma}}}'$ to $z''\in {{\dot{\Sigma}}}''$ is a single curve which has a genus zero component containing two nodes and the marked point $z_0$, attached by the first node to ${{\dot{\Sigma}}}'$ at $z'$ and to ${{\dot{\Sigma}}}''$ at $z''$ by the second node. This genus zero component similarly collapses when the marked point is removed. Thus, a point of $\overline{{\mathscr{M}}_{g,n+1}}$ is equivalent to a pair $({{\dot{\Sigma}}},z)$, where ${{\dot{\Sigma}}}\in \overline{{\mathscr{M}_{g,n}}}$ and $z\in {{\dot{\Sigma}}}$.
If ${{\dot{\Sigma}}}\in\overline{{\mathscr{M}_{g,n}}}$, we can delete the first $n$ marked points to obtain an $n$-times punctured Riemann surface with one marked point $z_0$. Fix the $n$ marked points corresponding to the $n$ punctures and let ${\mathbb{J}}:=\{ J_{g',n'}\}$ be a domain dependent almost complex structure. By restricting $J_{g,n+1}$ to ${{\dot{\Sigma}}}\cong \pi^{-1}({{\dot{\Sigma}}}) \subset {\mathscr{M}}_{g,n+1}$, we obtain a family of almost complex structures on $\xi$ parametrized by ${{\dot{\Sigma}}}$, which we denote by ${\mathbb{J}}_{{\dot{\Sigma}}}$. Rather than writing ${\mathbb{J}}_{{\dot{\Sigma}}}$, we will work under the assumption that in the Cauchy-Riemann equation below, the domain of ${\mathbb{J}}$ is restricted to $\pi^{-1}({{\dot{\Sigma}}})$, where $({{\dot{\Sigma}}}, j) \in {\mathscr{M}_{g,n}}$. Returning to the perspective of ${{\dot{\Sigma}}}$ as a $n$-times punctured Riemann surface, a map $C: ({{\dot{\Sigma}}}, j) \to ({{\mathbb R}}\times Y, {\mathbb{J}})$ can be associated with the $(0,1)$-form $$\overline{\partial}_{j,{\mathbb{J}}} : = \tfrac{1}{2}\left( dC + {\mathbb{J}}(z,C)\circ dC \circ j\right),$$ which at the point $z\in {\operatorname}{int}({{\dot{\Sigma}}})$ is given by $$\overline{\partial}_{j,{\mathbb{J}}}C(z) := \tfrac{1}{2} \left( dC(z) + {\mathbb{J}}(z,C(z))\circ dC(z) \circ j ( z) \right).$$ We say that $C$ is *${\mathbb{J}}$-holomorphic* whenever $\overline{\partial}_{j,{\mathbb{J}}}C=0$.
Regularity for generic $S^1$-invariant domain dependent ${\mathbb{J}}$ {#sec:reg}
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In this section, we prove that a generic $S^1$-invariant domain dependent almost complex structure ${\mathbb{J}}$ is regular. We note that the weaker statement that a generic ${\mathbb{J}}\in {\mathscr{J}}_{{\dot{\Sigma}}}$ is regular follows similarly.
\[thm:ddacs\] Let $\alpha$ and $\beta$ be nondegenerate orbit sets with ${\operatorname}{deg}(\alpha,\beta) >0$. If ${\mathbb{J}}\in {\mathscr{J}}_{{\dot{\Sigma}}}^{S^1}$ is generic and ${{\dot{\Sigma}}}$ does not include ${{\mathbb C}}$ or a union of cylinders, then any nonconstant holomorphic curve $C \in {{{\mathcal M}}}^{{\mathbb{J}}}(\alpha, \beta)$ is regular, meaning that the linearization $D\overline{\partial}_{{\mathbb{J}}}(C)$ is surjective and a neighborhood of $C \in {{{\mathcal M}}}^{{\mathbb{J}}}(\alpha, \beta)$ naturally admits the structure of a smoooth orbifold of dimension given by the Fredholm index ${{\operatorname}{ind}}(C)$, whose isotropy group at $C$ is $${\operatorname}{Aut}(C) := \{ \varphi \in {\operatorname}{Aut}({{\dot{\Sigma}}}, j) \ | \ C = C \circ \varphi \}$$ and there is a natural isomorphism $$T_C {{{\mathcal M}}}^{{\mathbb{J}}}(\alpha, \beta) = \ker D\overline{\partial}_{{\mathbb{J}}}(j, C) / \mathfrak{aut}({{\dot{\Sigma}}}, j).$$
At an orbifold point $C \in {{{\mathcal M}}}^{{\mathbb{J}}}(\alpha, \beta)$, $C$ has a nontrivial automorphism group with respect to which $C$ is invariant, so $C$ factors through the branched covering ${{\dot{\Sigma}}}\to \frac{{{\dot{\Sigma}}}}{{\operatorname}{Aut}({{\dot{\Sigma}}}, j)}$. Additional multiple covers may arise which do not come from automorphisms of the domain, but the use of domain-dependent almost complex structures permits us to perturb away the multiple covers of the latter type by choosing different perturbations at the different points in $C^{-1}(C(q))$. However, multiple covers coming from automorphisms of the domain remain because the functions from which we defined domain dependent almost complex structures are invariant with respect to the orbifold symmetry groups of ${\mathscr{M}_{g,n}}$. Since the subset of orbifold points of $\overline{{\mathscr{M}_{g,n}}}$ has real codimension at least 2 in $\overline{{\mathscr{M}_{g,n}}}$, hence we can conclude that the subset of ${\mathbb{J}}$-holomorphic curves in the moduli space whose domains are orbifolds also has real codimension at least 2. Thus a generic ${\mathbb{J}}$-holomorphic curve is not an orbifold point in its moduli space, and a generic path of ${\mathbb{J}}$-holomorphic curves avoids the locus of orbifold points.
Before giving the proof, we provide the corollary, which demonstrates that positive degree curves do not contribute to $ECH$ index 1 moduli spaces.
\[nogenus\] Let $\alpha$ and $\beta$ be nondegenerate admissible orbit sets and ${\mathbb{J}}\in {\mathscr{J}}_{{\dot{\Sigma}}}^{S^1}$ be generic. If ${\operatorname}{deg}(\alpha,\beta) >0$ and $I(\alpha,\beta)=1$ then ${{{\mathcal M}}}^{{\mathbb{J}}}(\alpha,\beta) = \emptyset$.
Given a generic ${\mathbb{J}}\in {\mathscr{J}}_{{\dot{\Sigma}}}^{S^1}$ consider a ${\mathbb{J}}$-holomorphic curve $C: {{\dot{\Sigma}}}\to {{\mathbb R}}\times Y$ with ${\operatorname}{deg}(\alpha,\beta) >0$. Take ${\mathbb{J}}' \in {\mathscr{J}}_{{\dot{\Sigma}}}$ to be generic and sufficiently close to ${\mathbb{J}}$ then ${{\operatorname}{ind}}(C_{\mathbb{J}}) = {{\operatorname}{ind}}(C_{{\mathbb{J}}'})$. Moroever, if we take $J \in {\mathscr{J}}_{reg}(Y,\lambda)$ to be sufficiently close to ${\mathbb{J}}'$ then ${{\operatorname}{ind}}(C_{{\mathbb{J}}'}) = {{\operatorname}{ind}}(C_{J})$. By the definition of degree, the domain cannot be a union of cylinders, so $S^1$ acts locally freely on ${{{\mathcal M}}}^{J}(\alpha,\beta) $. Regularity of $J$ guarantees that Fredholm index 1 curves do not exist unless they are fixed by the $S^1$-action. Since ${{\mathbb R}}$ acts freely on ${{{\mathcal M}}}^{J}(\alpha,\beta)$ and, because these actions commute, we have that $\dim {{{\mathcal M}}}^{J}(\alpha, \beta) \geq 2$ whenever ${{{\mathcal M}}}^{J}(\alpha, \beta) \neq \emptyset$. Since Theorem \[thm:ddacs\] guarantees that ${{{\mathcal M}}}^{{\mathbb{J}}}(\alpha, \beta)$ is cut out transversely and has dimension equal to the Fredholm index, we have that $\dim {{{\mathcal M}}}^{{\mathbb{J}}}(\alpha, \beta) \geq 2$ whenever ${{{\mathcal M}}}^{{\mathbb{J}}}(\alpha, \beta) \neq \emptyset$. We now obtain a contradiction because if $I(\alpha,\beta)=1$ then ${\operatorname}{ind}(C_J) \leq 1$, by the ECH index inequality, Theorem \[thm:indexineq\]: ${{\operatorname}{ind}}(C_{{\mathbb{J}}}) = {{\operatorname}{ind}}(C_J) \leq I(C_J) = I(\alpha,\beta) =1$.
Prior to proving our main regularity result, we provide some definitions and construct the universal moduli space, mostly following [@CM §4-5] and [@wendl-sft §7.2]. Assuming $kp>2$, let $${\mathscr{B}}^{k,p,\delta} :={W^{k,p,\delta}}({{\dot{\Sigma}}},{{\mathbb R}}\times Y; \alpha,\beta) \subset C^0({{\dot{\Sigma}}},{{\mathbb R}}\times Y)$$ be the usual smooth, separable, and metrizable Banach manifold of [exponentially weighted Sobolev spaces of maps which are asymptotically cylindrical curves to the orbit sets $\alpha$ and $\beta$ at the ends]{}. The tangent space to ${\mathscr{B}}^{k,p,\delta}$ at $C\in {\mathscr{B}}^{k,p,\delta}$ can be written as $$T_C{\mathscr{B}}^{k,p,\delta} = {W^{k,p,\delta}}(C^*({{\mathbb R}}\times Y) ) \oplus V_{\mathcal{S}},$$ where $V_{\mathcal{S}}\subset \Gamma(C^*({{\mathbb R}}\times Y))$ is a non-canonical choice of a $2|{\mathcal{S}}|$-dimensional vector space of smooth sections asymptotic at the punctures to constant linear combinations of the vector fields spanning the canonical trivialization of the first factor in $T({{\mathbb R}}\times Y) = \epsilon \oplus \xi$. The space $V_{\mathcal{S}}$ appears due to the fact that two distinct elements of ${\mathscr{B}}^{k,p,\delta}$ are generally asymptotic to collections of trivial cylinders that differ from each other by $|{\mathcal{S}}|$ pairs of constant shifts $(a,b) \in {{\mathbb R}}\times S^1$.
Fix ${\mathbb{J}}\in {\mathscr{J}}_{{\dot{\Sigma}}}^{\varepsilon}$. The nonlinear Cauchy-Riemann operator is then defined as a smooth section $$\overline{\partial}_{j,{\mathbb{J}}}: {\mathscr{B}}^{k,p,\delta} \to {\mathscr{E}}^{k-1,p,\delta}; C \mapsto TC + {\mathbb{J}}\circ TC \circ j$$ of a Banach space bundle $${\mathscr{E}}^{k-1,p,\delta} \to {\mathscr{B}}^{k,p,\delta}$$ with fibers $${\mathscr{E}}^{k-1,p,\delta}_C = W^{k-1,p,\delta}(\overline{{\operatorname}{Hom}}_{{\mathbb C}}(T{{\dot{\Sigma}}}, C^*({{\mathbb R}}\times Y))).$$ The zero set of $\overline{\partial}_{j,{\mathbb{J}}}$ is the set of all maps $C \in {\mathscr{B}}^{k,p,\delta} $ that are ${\mathbb{J}}$-holomorphic.
More generally, the *universal Cauchy-Riemann operator* is the section $$\overline{\partial}: {\mathscr{B}}^{k,p,\delta} \times {\mathscr{J}}_{{\dot{\Sigma}}}^{\varepsilon} \to {\mathscr{E}}^{k-1,p,\delta}; (C, {\mathbb{J}}) \mapsto \overline{\partial}_{j,{\mathbb{J}}}C$$ of a Banach space bundle $${\mathscr{E}}^{k-1,p,\delta} \to {\mathscr{B}}^{k,p,\delta} \times {\mathscr{J}}_{{\dot{\Sigma}}}^{\varepsilon}$$ with fibers $${\mathscr{E}}^{k-1,p,\delta}_C = W^{k-1,p,\delta}(\overline{{\operatorname}{Hom}}_{{\mathbb C}}(T{{\dot{\Sigma}}}, C^*({{\mathbb R}}\times Y))).$$ The zero section gives rise to the *universal moduli space*: $${{{\mathcal M}}}({\mathscr{J}}_{{{\dot{\Sigma}}}}^{\varepsilon}):= \{ (C,{\mathbb{J}}) \ | \ {\mathbb{J}}\in {\mathscr{J}}_{{{\dot{\Sigma}}}}^{\varepsilon}, \ \overline{\partial}_{j,{\mathbb{J}}}C=0 \}.$$ Arguments similar to [@wendl-sft Lemma 7.15] demonstrate that the universal moduli space ${{{\mathcal M}}}({\mathscr{J}}_{{{\dot{\Sigma}}}}^{\varepsilon})$ is a smooth separable Banach manifold, and the projection ${{{\mathcal M}}}({\mathscr{J}}_{{{\dot{\Sigma}}}}^{\varepsilon}) \to {\mathscr{J}}_{{{\dot{\Sigma}}}}^{\varepsilon}; (C,{\mathbb{J}}) \mapsto {\mathbb{J}}$ is smooth.
For any $C \in \overline{\partial}_{j,{\mathbb{J}}}^{-1}(0)$, the linearization $$D\overline{\partial}_{j,{\mathbb{J}}}: T_{\mathbb{J}}{\mathscr{J}}_{{\dot{\Sigma}}}^{\varepsilon} \times T_C{\mathscr{B}}^{k,p,\delta} \to {\mathscr{E}}^{k-1,p,\delta}_C$$ defines a bounded linear operator $$D: W^{k,p,\delta}(C^*T({{\mathbb R}}\times Y)) \oplus T_{\mathbb{J}}{\mathscr{J}}_{{\dot{\Sigma}}}^{\varepsilon} \oplus V_{\mathcal{S}}\to W^{k-1,p,\delta}(\overline{{\operatorname}{Hom}}_{{\mathbb C}}(T{{\dot{\Sigma}}}, C^*({{\mathbb R}}\times Y)))$$ Since $V_{\mathcal{S}}$ is finite dimensional, $D$ will be Fredholm if and only if its restriction to the first two factors is Fredholm; denote this restriction by $${\mathbf{D}}:= {\mathbf{D}}_C + {\mathbf{D}}_{\mathbb{J}}: W^{k,p,\delta}(C^*T({{\mathbb R}}\times Y)) \oplus T_{\mathbb{J}}{\mathscr{J}}_{{\dot{\Sigma}}}^{\varepsilon} \to W^{k-1,p,\delta}(\overline{{\operatorname}{Hom}}_{{\mathbb C}}(T{{\dot{\Sigma}}}, C^*({{\mathbb R}}\times Y)))$$
We will view the $n$ punctures of the domain ${{\dot{\Sigma}}}$ of $C$ as fixed, with $j$ varying on ${\operatorname}{int}({{\dot{\Sigma}}})$, so that the tangent space to ${\mathscr{M}}_{g,n+1}$ at a point $(\Sigma, j, z_0,z_1,...,z_n)$ is $T_j{\mathscr{J}}({{\dot{\Sigma}}}) \oplus T_{z_0}{{\dot{\Sigma}}}$. If $V = (a, A) \in T_{\mathbb{J}}({\mathscr{J}}_{{\dot{\Sigma}}}^{S^1, \varepsilon})$, where $A: T_j({{\dot{\Sigma}}}) \to T_{\mathbb{J}}{\mathscr{J}}_{{\dot{\Sigma}}}^{S^1, \varepsilon}$ and $a \in \overline{{\operatorname}{End}}_j(T{{\dot{\Sigma}}})$, then $${\mathbf{D}}_{\mathbb{J}}(V)=A \circ du \circ j_{{\dot{\Sigma}}}+ {\mathbb{J}}\circ du \circ a.$$
We begin by recalling a few observations in the proof of [@farris Theorem 6.2.1]. [Since $\deg(\alpha,\beta)>0$, the domain ${{\dot{\Sigma}}}$ cannot solely consist of a union of cylinders.]{} The ECH index is additive and positive for pseudoholomorphic curves which are not themselves cylinders, so there is a unique noncylindrical component ${{\dot{\Sigma}}}'$ of ${{\dot{\Sigma}}}$. Trivial cylinders are always cut out transversely [@wendl-sft Proposition 8.2], as are somewhere injective cylinders [@wendl-sft §7-8]. In light of Remark \[unstable-cases\] and without loss of generality, we may prove the theorem in the case when ${{\dot{\Sigma}}}= {{\dot{\Sigma}}}'$ and ${{\dot{\Sigma}}}$ is stable. Since $C$ is not a trivial cylinder, $C^{-1}({{\mathbb R}}\times (Y \setminus N))$ contains a nonempty open set of ${{\dot{\Sigma}}}$.
Next we show that $C$ cannot be a nodal curve with a constant component of positive genus, which crucially relies on ${\operatorname}{dim}({{\mathbb R}}\times Y) = 4$, noting this is in part why [@CM] restricts to genus 0 curves. Suppose to the contrary that $C$ is the union of a nodal curve $C_1$ with a constant component of positive genus $C_2$ and that ${{\operatorname}{ind}}(C)=1$, then $${{\operatorname}{ind}}(C_1) + {{\operatorname}{ind}}(C)=1.$$ Since $C|_{{{\dot{\Sigma}}}_2}$ is constant, the restricted pullback $C^*(T({{\mathbb R}}\times Y)|_{{{\dot{\Sigma}}}_2}$ is trivial, thus $$c_1(C^*(T({{\mathbb R}}\times Y))= c_1(C^*(T({{\mathbb R}}\times Y))|_{{{\dot{\Sigma}}}_1} + c_1(C^*(T({{\mathbb R}}\times Y))|_{{{\dot{\Sigma}}}_2} = c_1(C^*(T({{\mathbb R}}\times Y))|_{{{\dot{\Sigma}}}_1}.$$ Let $$c_1|_{{{\dot{\Sigma}}}_1}=c_1(C^*(T({{\mathbb R}}\times Y))|_{{{\dot{\Sigma}}}_1}.$$ Because $C$ maps ${{\dot{\Sigma}}}_2$ to a constant, all the punctures must lie on ${{\dot{\Sigma}}}_1$. Thus the Fredholm index contribution of the Conley-Zehnder indices of the orbits asymptotic to the ends of ${{\dot{\Sigma}}}$ and ${{\dot{\Sigma}}}_1$ must agree. Denote this contribution by $CZ_\tau^{ind}(C_1)$. By hypothesis, we have that $g({{\dot{\Sigma}}}_2) > 0$, $g({{\dot{\Sigma}}}_1) < g({{\dot{\Sigma}}})$, and $\chi({{\dot{\Sigma}}}_1) > \chi({{\dot{\Sigma}}})$. Thus $$\begin{array}{r c l c l}
1&= &{\operatorname}{ind}(C) &= & -\chi(C) + c_1|_{{{\dot{\Sigma}}}_1} + CZ_\tau^{ind}(C_1), \\
&&{\operatorname}{ind}(C_1)& = & -\chi(C_1) + c_1|_{{{\dot{\Sigma}}}_1} + CZ_\tau^{ind}(C_1), \\
\end{array}$$ hence ${\operatorname}{ind}(C_1)<{\operatorname}{ind}(C) =1$. Therefore ${\operatorname}{ind}(C_2) = 1-{\operatorname}{ind}(C_1) > 0$. Since we assumed that ${\mathbb{J}}$ is a generic $S^1$-invariant domain dependent almost complex structure, we have that all of its restrictions to $\partial \overline{{\mathscr{M}_{g,n}}}$ are generic, which determine the almost complex structure on ${{\dot{\Sigma}}}_1$ and ${{\dot{\Sigma}}}_2$. [However, for generic almost complex structures, positive index curves of positive genus do not exist.]{} Thus $C$ is not constant on a component of positive genus. Since constant components of genus 0 can be eliminated by reparametrization, we can assume without loss of generality that $C$ is not constant on any component of ${{\dot{\Sigma}}}$, hence the zeros of $dC$ are isolated. Note that the above argument also holds if ${{\operatorname}{ind}}(C)>1$. The remainder of the argument is similar to that of [@CM Lemmas 4.1, 5.4, 5.6], [@jhol Proposition 3.4.2], [@wendl-sft §8], so we only sketch the argument.
Let $C: {{\dot{\Sigma}}}\to {{\mathbb R}}\times Y$ be a ${\mathbb{J}}$-holomorphic map. The set of regular points $z$ of ${{\dot{\Sigma}}}$ such that $\pi_Y C(z)$ is a regular value of $\pi_Y \circ C$ form an open dense subset of ${{\dot{\Sigma}}}$. If we intersect the set of regular points with the set of points $z \in {{\dot{\Sigma}}}$ where ${\operatorname}{im}(dC_z) = \xi_{\pi_YC(z)}$, it remains open and dense because the projection $\pi_Y$ is already open and dense by the nonintegrability of $\xi$. Denote the further intersection of these sets with $C^{-1}({{\mathbb R}}\times (Y\setminus N))$ by $U$. Note that $U$ contains a nonempty open set.
After fixing $(C,{\mathbb{J}}) \in {{{\mathcal M}}}({\mathscr{J}}_{{{\dot{\Sigma}}}}^{S^1})$,[^7] we want to show that the linearization ${\mathbf{D}}= {\mathbf{D}}_C + {\mathbf{D}}_{\mathbb{J}}$ is surjective. Since ${\mathbf{D}}_C$ is Fredholm, ${\mathbf{D}}$ has closed range and hence surjectivity is equivalent to the triviality of the annihilator of ${\operatorname}{Im}({\mathbf{D}})$. We prove the result for $k=1$, noting that $k>1$ follows by elliptic regularity, cf. [@jhol Theorem C.2.3]. When $k=1$, we have that the dual space of any space of sections of class $L^{p,q}$ can be identified with sections of class $L^{q,-\delta}$ for $\frac{1}{p} + \frac{1}{q} =1$ [@wendl-sft Remark 7.7]. Using the nondegenerate 2-form $d(e^r\lambda)$ on ${{\mathbb R}}\times Y$ we can use it to define a nondegenerate $L^2$-pairing $$\langle \cdot, \cdot \rangle = L^{p,\delta} \times L^{q,-\delta}.$$ Moreover $\left( L^{p,\delta }\right)^* \cong L^{q,\delta}$, so we can consider the formal adjoint ${\mathbf{D}}_C^*$ of ${\mathbf{D}}_C$. Let $\eta \in {\operatorname}{coker}({\mathbf{D}})$, then the splitting and dualization yield that orthogonality to ${\operatorname}{Im}({\mathbf{D}})$ amounts to the equations $$\label{eqs}
\begin{array}{cc}
\langle {\mathbf{D}}_C(\zeta),\eta\rangle =0 \\
\langle {\mathbf{D}}_{\mathbb{J}}(V),\eta \rangle =0 \\
\end{array}$$ for all $\zeta \in T_C {\mathscr{B}}^{k,p,\delta}$ and $V \in T_{\mathbb{J}}({\mathscr{J}}_{{{\dot{\Sigma}}}}^{S^1})$. By the first equation, $\eta \in {\operatorname}{ker}({\mathbf{D}}_C^*)$. By elliptic regularity, $\eta$ is smooth. The second equation implies that if $\eta$ vanishes on an open set, then $\eta \equiv 0$ by unique continuation; cf. [@jhol Lemma 3.4.7].
The remainder of the argument is similar to the original proof by Farris. Assume that $\eta_z \neq 0$ for some $z \in U \subset {{\dot{\Sigma}}}$. This implies that $$\eta_z \in \overline{{\operatorname}{Hom}}_{{\mathbb{J}}(z,C(z))}(T_z{{\dot{\Sigma}}}, T_{C(z)}T ({{\mathbb R}}\times Y)) \mbox{ and } dC_z \circ j_z \in \overline{{\operatorname}{Hom}}_{{\mathbb{J}}(z,C(z))}(T_z{{\dot{\Sigma}}}, T_{C(z)}T ({{\mathbb R}}\times Y))$$ are injective maps. Thus given any $0\neq v \in T_z{{\dot{\Sigma}}}$ we have that $$\eta_z(v)\neq 0,\ \ \ dC_z \circ j_z(v) \neq 0.$$ Next we find some $A_z \in \overline{{\operatorname}{End}}_{{\mathbb{J}}(z,C(z))}(T ({{\mathbb R}}\times Y), d\lambda_{C(z)})$ such that $$A_z ( dC_z \circ j_z(v)) = \eta_z.$$ On the set $U$, we have that $\xi_{C(z)}$ and ${\operatorname}{im}(dC_z)$ are distinct complex subspaces which span $T_{C(z)}({{\mathbb R}}\times Y)$. Hence the codomain of ${\mathbf{D}}$, admits the following splitting: $$\overline{{\operatorname}{Hom}}_{{\mathbb{J}}(z,C(z))}(T_z{{\dot{\Sigma}}}, T_{C(z)}T ({{\mathbb R}}\times Y)) = \overline{{\operatorname}{Hom}}_{{\mathbb{J}}(z,C(z))}(T_z{{\dot{\Sigma}}}, \xi_{C(z)}) \oplus \overline{{\operatorname}{End}}_{{\mathbb{J}}(z,C(z))}(T_z{{\dot{\Sigma}}}).$$ We split $\eta_z$ accordingly: $$\eta_z = \eta_{{\dot{\Sigma}}}+ \eta_\xi,$$ where $$\begin{array}{lcl}
\eta_\xi & =& \eta_{\xi_{C(z)}}, \\
\eta_{{\dot{\Sigma}}}& = & \eta_{T_z{{\dot{\Sigma}}}}. \\
\end{array}$$ Since $({\mathbb{J}}\circ dC)_z$ is injective, for any given $\nu_z \in T_z{{\dot{\Sigma}}}$, we can choose $a_z \in \overline{{\operatorname}{End}}_{j(z)}(T_z{{\dot{\Sigma}}})$ so that $$({\mathbb{J}}\circ du \circ a)_z(\nu_z) = \eta_{{\dot{\Sigma}}}.$$ Next, we consider the $\xi$-component. Since $\overline{{\operatorname}{End}}_{{\mathbb{J}}(z,C(z))}(\xi_{C(z)}) = {T_{{\mathbb{J}}(z,C(z))} {\mathscr{J}}_{{\dot{\Sigma}}}^{S^1}}$ is complex one dimensional, and for any given $v_z,w_z \in \xi_z$, there is an element $B_z \in \overline{{\operatorname}{End}}_{{\mathbb{J}}(z,C(z))}(\xi_{C(z)})$ sending $v_z$ to $w_z$. Hence we take $B_z:T_z {{\dot{\Sigma}}}\to T_{{\mathbb{J}}(z,C(z))}{\mathscr{J}}_{{\dot{\Sigma}}}^{S^1}$ sending $(du \circ j)_z(v_z)$ to $\eta_\xi$. Thus $$A_z := (a_z, B_z): (\nu_z,v_z) \mapsto (\eta_{{\dot{\Sigma}}}, \eta_\xi),$$ as desired.
We now need to suitably extend $A_z$ to an $A \in T_{{\mathbb{J}}(z,C(z))} {\mathscr{J}}_{{\dot{\Sigma}}}^{S^1}$. When $J_{g,n}$ is restricted to ${{\dot{\Sigma}}}\in {\mathscr{M}}_{g,n+1}$, $V \in T_{\mathbb{J}}({\mathscr{J}}_{{{\dot{\Sigma}}}}^{S^1})$ depends on the special marked point $z_0 \in {{\dot{\Sigma}}}$ and ${\mathfrak{p}}(\pi_YC(z)) \in \Sigma_g$. In order to extend $A$ to all of $T_{{\mathbb{J}}} {\mathscr{J}}_{{\dot{\Sigma}}}^{S^1}$, we must let it vary with the complex structure $j$ on ${{\dot{\Sigma}}}$. The domain of $V(p,q)$ is ${\mathscr{M}}_{g,n+1} \times \Sigma_g$.
Define a smooth cutoff function $\kappa: \Sigma_g \to {{\mathbb R}}$ which is nonnegative, takes the value one at ${\mathfrak{p}}(\pi_YC(z))$, and the value zero outside some open neighborhood of ${\mathfrak{p}}(\pi_YC(z_0))$ which does not contain any critical points of the perfect Morse function used to define $\lambda_\varepsilon$. Let $\nu: {\mathscr{M}}_{g,n+1} \to {{\mathbb R}}$ be a smooth nonnegative function which is one at $({{\dot{\Sigma}}}, j, z_0,z_1,...,z_n)$ and zero outside an appropriately small open neighborhood of this point. The neighborhoods of the $z_i$ should not intersect each other, and the neighborhood $U'$ of the special marked point $z_0$ should not contain any preimages of ${\mathfrak{p}}(\pi_YC(z_0))$ besides $z_0$ itself. Note that these preimages are finite in number, as otherwise they would accumulate, forcing $C$ to be globally constant.
Choose an arbitrary smooth extension $A'$ of $A_{z_0}$, shrinking neighborhoods as necessary to ensure that $$\langle A(j, z_0, z_1, ... , z_n, q) \circ dC_{z_0} \circ j_{z_0}, \eta_{z_0} \rangle >0$$ whenever $q \in {\operatorname}{supp}(\kappa)$ and $(j,z_0,...,z_n) \in {\operatorname}{supp}(\nu)$. We define $$A(j,z_0,...,z_n,q) : = \kappa(q) \nu(j, z_0,...,z_n) A'(j, z_0,...,z_n,q).$$ [Since $\langle {\mathbf{D}}_{\mathbb{J}}(A)_z, \eta_z \rangle >0$, obtain a contradiction to the assumption that $\eta \in {\operatorname}{coker}({\mathbf{D}})$. Thus $\eta \equiv 0$ and ${\mathbf{D}}$ is surjective as claimed.]{} It follows from surjectivity of $D\overline{\partial}_{{\mathbb{J}}}(C)$ that ${{{\mathcal M}}}^{{\mathbb{J}}}(\alpha, \beta)$ naturally admits the structure of a smooth orbifold of dimension given by the Fredholm index by way of a virtual repeat of [@Wtrans Theorem 0].
Classification of ECH connectors {#sec:connectors}
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In this section we carry out some index calculations which allow us to classify *connectors* ${\mathscr{C}}$, which are defined to be branched and unbranched covers of a union of trivial cylinders $\sqcup_i \gamma_i \times {{\mathbb R}}$. We will also use intersection theory to show, under sufficient genericity assumptions, that certain sequences of holomorphic curves cannot converge to a building which has a connector at the top most or bottom most level. Subsequently in §\[handleslides\], we use these classification results to invoke the obstruction bundle gluing theorems [@obg1; @obg2] and prove that the appearance of ECH handleslides does not impact the homology.
Buildings and connectors
------------------------
As in §\[sec:freddie\], if $C$ is a $J$-holomorphic curve with positive ends at Reeb orbits $\alpha_1,\ldots,\alpha_k$ and negative ends at Reeb orbits $\beta_1,\ldots,\beta_l$, then the Fredholm index of $C$ is given by the formula $$\label{eqn:ind}
{\operatorname}{ind}(C)=-\chi(C) + 2c_\tau(C) + \sum_{i=1}^k{{\operatorname}{CZ}}_\tau(\alpha_i) - \sum_{j=1}^l{{\operatorname}{CZ}}_\tau(\beta_j).$$ Here $\chi(C)$ denotes the Euler characteristic of the domain of $C$, so if $C$ is irreducible of genus $g$ then $$\label{eqn:chi}
\chi(C) = 2-2g-k-l.$$
Next we recall the definition of a pseudoholomorphic building from [@BEHWZ]; see also [@wendl-sft §9.4]. In our setting all the curves and their limits are non-nodal and unmarked.
\[building\] For our purposes, a *holomorphic building* is an $m$-tuple $(u_1,\ldots,u_m)$, for some positive integer $m$, of (possibly disconnected) $J$-holomorphic curves $u_i$ in ${{\mathbb R}}\times Y$, called [*levels*]{}. Although our notation does not indicate this, the building also includes, for each $i\in\{1,\ldots,m-1\}$, a bijection between the negative ends of $u_i$ and the positive ends of $u_{i+1}$, such that paired ends are at the same Reeb orbit[^8]. If $m>1$ then we assume that for each $i$, at least one component of $u_i$ is not a trivial cylinder[^9]. A [*positive end*]{} of the building $(u_1,\ldots,u_m)$ is a positive end of $u_1$, and a [*negative end*]{} of $(u_1,\ldots,u_m)$ is a negative end of $u_m$. The [*genus*]{} of the building $(u_1,\ldots,u_m)$ is the genus of the Riemann surface obtained by gluing together negative ends of the domain of $u_i$ and positive ends of the domain of $u_{i+1}$ by the given bijections (when this glued Riemann surface is connected).
We define the *Fredholm index of a holomorphic building* by $${\operatorname}{ind}(u_1,\ldots,u_m) = \sum_{i=1}^m{\operatorname}{ind}(u_i).$$
We recall the Riemann-Hurwitz theorem, in part to fix notation.
\[RH\] Let $\varphi:\widetilde{\dot{\Sigma}} \to {{\dot{\Sigma}}}$ be a compact $k$-fold cover of the punctured Riemann surface ${{\dot{\Sigma}}}$. Then $$\label{eqn:RH}
\chi(\widetilde{\dot{\Sigma}}) = k\chi({{\dot{\Sigma}}}) - \sum_{p \in\widetilde{\dot{\Sigma}}} (e(p)-1),$$ where $e(p)-1$ is the ramification index of $\varphi$ at $p$.
At unbranched points $p$ we have $e(p)-1=0$, thus for any $q \in {{\dot{\Sigma}}}$, $$\sum_{p \in \varphi^{-1}(q)}e(p)=k.$$
The Riemann-Hurwitz theorem provides us with the number of punctures of the cover. The multiplicities of the Reeb orbits at the punctures are determined by the monodromy of the local behavior of a curve near its punctures [@MiWh; @s1], which are in turn governed by the monodromy of the covering.
Low index connectors
--------------------
In this section we investigate the relation between low ECH and Fredholm index connectors ${\mathscr{C}}$ and the configurations of Reeb orbits at the ends of the components of each ${\mathscr{C}}$. Recall that a *connector* ${\mathscr{C}}$ is a branched cover of a union of trivial cylinders, and all or some of the components may be unbranched.
First, we recall that in a symplectization of a contact 3-manifold, all covers of trivial cylinders have non-negative Fredholm index.
\[lem:ht\] Let $C \in {{{\mathcal M}}}^J(\alpha,\beta)$ be a branched or unbranched cover of a trivial cylinder ${{\mathbb R}}\times \gamma$, where $\gamma$ is an embedded Reeb orbit. Then ${\operatorname}{ind}(C)\ge 0$, with equality only if
1. Each component of the domain ${{\dot{\Sigma}}}$ of $C$ has genus 0.
2. If $\gamma$ is hyperbolic, then the covering $C: {{\dot{\Sigma}}}\to {{\mathbb R}}\times \gamma$ has no branch points.
The remainder of this section concerns the proof of the following result.
\[lem:connectortopology\] Let ${\mathscr{C}}:\dot{\Sigma} \to{{\mathbb R}}\times Y$ be a connector, where ${\mathscr{C}}={\bigcup}_iC_i$ and each $C_i$ is connected. The ECH index $I({\mathscr{C}})=0$ and the genus of each component $C_i$ is zero.[Further assuming that the Fredholm index ${\operatorname}{ind}({\mathscr{C}})\in\{0,1\}$]{}, then:
1. If ${{\operatorname}{ind}}({\mathscr{C}})=0$ then ${{\operatorname}{ind}}(C_i)=0$ for all $i$, and either
1. $C_i$ is an unbranched cover of a trivial cylinder.
2. $C_i$ is branched, covers ${{\mathbb R}}\times e_+$, and has a single positive end.
3. $C_i$ branched, covers ${{\mathbb R}}\times e_-$, and has a single negative end.
2. If ${{\operatorname}{ind}}({\mathscr{C}})=1$ then ${\mathscr{C}}=C_0\cup{\bigcup}_iC_i$ where ${{\operatorname}{ind}}(C_0)=1, {{\operatorname}{ind}}(C_i)=0$, and $C_0$ is a branched cover of ${{\mathbb R}}\times h_j$ for some $j\in\{1,\dots,2g\}$ with either one positive end and two negative ends, or two positive ends and one negative end. [Each of the $C_i$ is an unbranched cover of a trivial cylinder.]{}
Let $m_\pm,m_j$ denote the multiplicities of the ends of ${\mathscr{C}}$ at the orbits $e_\pm,h_j$: the multiplicities at the positive and negative ends will be the same because ${\mathscr{C}}$ covers a union of trivial cylinders. In particular, the difference between the total multiplicities at the positive and negative ends of ${\mathscr{C}}$ will be zero. Therefore, from the index formula (\[eqn:indexformula\]), we have $$I({\mathscr{C}})=\chi(\Sigma)\cdot0-0^2e+2\cdot0\cdot\left(m_++\sum_jm_j+m_-\right)+m_+-m_--m_++m_-=0$$ Note that in particular $c_\tau({\mathscr{C}})=0$. These formulas also hold for each component $C_i$ of ${\mathscr{C}}$.
Let $p_\pm(C_i)$ denote the number of positive and negative ends of $C_i$, respectively, and let $g(C_i)$ denote the genus of $C_i$. Recall that the Euler characteristic of a surface with $p$ punctures is $2-2g-p$.
**Case (i)** If ${{\operatorname}{ind}}({\mathscr{C}})=0$, then ${{\operatorname}{ind}}(C_i)=0$ for all $i$ by Lemma \[lem:ht\].
**Case (i.a)** Assume $u(C_i)$ is a branched cover of ${{\mathbb R}}\times h_j$. Because $c_\tau(C_i)=0$ and hyperbolic orbits have Conley-Zehnder index zero, we have $$0={{\operatorname}{ind}}(C_i)=-\chi(C_i)$$ The Euler characteristic of a cylinder is 0, therefore the Riemann-Hurwitz Theorem (\[eqn:RH\]) gives us $$0=\chi(C_i)=-\sum_{p\in\dot\Sigma}(e(p)-1)$$ Because $e(p)\geq1$ for all $p$, each term $e(p)-1\geq0$, so we must have $e(p)=0$ for all $p$. Therefore, $C_i$ is unbranched. Moreover, $$0=2g(C_i)-2+p_+(C_i)+p_-(C_i)\Leftrightarrow 2=2g(C_i)+p_+(C_i)+p_-(C_i).$$ Because $C_i$ is a cover of a cylinder, $p_\pm(C_i)\geq1$. Therefore $g(C_i)=0$, both $p_\pm(C_i)=1$, and $C_i$ unbranched cover of a cylinder. **Case (i.b)** Because the Conley-Zehnder index of a cover of $e_+$ is always 1, we have $$\begin{aligned}
0&={{\operatorname}{ind}}(C_i)\nonumber
\\&=2g(C_i)-2+p_+(C_i)+p_-(C_i)+p_+(C_i)-p_-(C_i),\label{eqn:coveringe+}\end{aligned}$$ hence $$1=g(C_i)+p_+(C_i).$$ Therefore, because $p_+(C_i)\geq1$, we have $g(C_i)=0$ and $p_+(C_i)=1$.
**Case (i.c)** By the same argument as for 1.(b), using the fact that the Conley-Zehnder index of a cover of $e_-$ is always $-1$, we get $g(C_i)=0$ and $p_-(C_i)=1$.
**Case (ii)** [If ${{\operatorname}{ind}}({\mathscr{C}})=1$ then ${{\operatorname}{ind}}(C_i)\leq1$ for all $i$. Because ${{\operatorname}{ind}}(C_i)\geq0$ for all $i$ by Lemma \[lem:ht\], there must be one component $C_0$ with ${{\operatorname}{ind}}(C_0)=1$ and all other $C_i$ have ${{\operatorname}{ind}}(C_i)=0$.]{}
If $C_0$ were a branched cover of ${{\mathbb R}}\times e_+$, then setting the analogue of the right hand side of (\[eqn:coveringe+\]) equal to ${{\operatorname}{ind}}(C_0)$ would imply that $$1=2(g(C_0)-1+p_+(C_0))$$ a contradiction. Similarly $C_0$ being a branched cover of ${{\mathbb R}}\times e_-$ would lead to a contradiction.
Therefore $C_0$ must be a branched cover of ${{\mathbb R}}\times h_j$. In this case, because hyperbolic orbits have Conley-Zehnder index zero, we have $$1=2g(C_0)-2+p_+(C_0)+p_-(C_0)\Leftrightarrow 3=2g(C_0)+p_+(C_0)+p_-(C_0)$$ Because $p_\pm(C_0)\geq1$, this implies $1\geq2g(C_0)$, requiring $g(C_0)=0$. Therefore either $(p_+(C_0),p_-(C_0))=(1,2)$ or $(p_+(C_0),p_-(C_0))=(2,1)$.
Classification of connectors arising in buildings
-------------------------------------------------
In this section we use intersection theory and higher asymptotics of holomorphic curves to rule out connectors from appearing at the top-most and bottom-most level of a building arising as a limit of a (sub)sequence of holomorphic curves defined in terms of a one parameter family of domain dependent almost complex structures, cf. Proposition \[notopbottomconn\]. This result will be key in §\[handleslides\]. We begin by recalling some needed results about the asymptotics of holomorphic curves from [@dc §3.1].
Let $\gamma$ be an embedded Reeb orbit, and let $N$ be a tubular neighborhood of $\gamma$. We can identify $N$ with a disk bundle in the normal bundle to $\gamma$, and also with $\xi|_\gamma$. Let $\zeta$ be a braid in $N$, i.e. a link in $N$ such that that the tubular neighborhood projection restricts to a submersion $\zeta\to\gamma$. Given a trivialization $\tau$ of $\xi|_\gamma$, one can then define the [*writhe*]{} $w_\tau(\zeta)\in{{\mathbb Z}}$. To define this one uses the trivialization $\tau$ to identify $N$ with $S^1\times D^2$, then projects $\zeta$ to an annulus and counts crossings of the projection with (nonstandard) signs; see §\[s:writhe\], [@Hrevisit §2.6], or [@Hu2 §3.3] for details.
Now let $C$ be a $J$-holomorphic curve in ${{\mathbb R}}\times Y$. Suppose that $C$ has a positive end at $\gamma^d$ which is not part of a multiply covered component. Results of Siefring [@s1 Cor. 2.5 and 2.6] show that if $s$ is sufficiently large, then the intersection of this end of $C$ with $\{s\}\times N\subset\{s\}\times Y$ is a braid $\zeta$, whose isotopy class is independent of $s$. We will need bounds on the writhe $w_\tau(\zeta)$, which are provided by the following lemma.
\[lem:positivewrithe\] Let $\gamma$ be an embedded Reeb orbit, let $C$ be a $J$-holomorphic curve in ${{\mathbb R}}\times Y$ with a positive end at $\gamma^d$ which is not part of a trivial cylinder or a multiply covered component, and let $\zeta$ denote the intersection of this end with $\{s\}\times Y$. If $s \gg 0$, then the following hold:
[*(a)*]{} $\zeta$ is the graph in $N$ of a nonvanishing section of $\xi|_{\gamma^d}$. Thus, using the trivialization $\tau$ to write this section as a map $\gamma^d\to{{\mathbb C}}\setminus\{0\}$, it has a well-defined winding number around $0$, which we denote by ${\operatorname}{wind}_\tau(\zeta)$.
[*(b)*]{} ${\operatorname}{wind}_\tau(\zeta) \le {\left\lfloor {\operatorname}{CZ}_\tau(\gamma^d)/2 \right\rfloor}$.
[*(c)*]{} If $J$ is generic, ${\operatorname}{CZ}_\tau(\gamma^d)$ is odd, and ${\operatorname}{ind}(u)\le 2$, then equality holds in (b).
[*(d)*]{} $w_\tau(\zeta) \le (d-1){\operatorname}{wind}_\tau(\zeta)$.
Symmetrically to Lemma \[lem:positivewrithe\], we also have the following:
\[lem:negativewrithe\] Let $\gamma$ be an embedded Reeb orbit, let $C$ be a $J$-holomorphic curve in ${{\mathbb R}}\times Y$ with a negative end at $\gamma^d$ which is not part of a trivial cylinder or multiply covered component, and let $\zeta$ denote the intersection of this end with $\{s\}\times Y$. If $s\ll0$, then the following hold:
[*(a)*]{} $\zeta$ is the graph of a nonvanishing section of $\xi|_{\gamma^d}$, and thus has a well-defined winding number ${\operatorname}{wind}_\tau(\zeta)$.
[*(b)*]{} ${\operatorname}{wind}_\tau(\zeta) \ge {\left\lceil {\operatorname}{CZ}_\tau(\gamma^d)/2 \right\rceil}$.
[*(c)*]{} If $J$ is generic, ${\operatorname}{CZ}_\tau(\gamma^d)$ is odd, and ${\operatorname}{ind}(u)\le 2$, then equality holds in (b).
[*(d)*]{} $w_\tau(\zeta) \ge (d-1){\operatorname}{wind}_\tau(\zeta)$.
\[rem:improved\] Lemma \[lem:positivewrithe\](b),(d) imply that $$w_\tau(\zeta) \le (d-1){\left\lfloor {{\operatorname}{CZ}}_\tau(\gamma^d)/2 \right\rfloor}.$$ [In fact one can improve this to]{} $$\label{eqn:improved}
w_\tau(\zeta) \le (d-1){\left\lfloor {{\operatorname}{CZ}}_\tau(\gamma^d)/2 \right\rfloor} - {\operatorname}{gcd}\left(d,{\left\lfloor {{\operatorname}{CZ}}_\tau(\gamma^d)/2 \right\rfloor}\right)+1,$$ see [@s2]. Recent work of Cristofaro-Gardiner - Hutchings - Zhang obtains equality in in the following situation.
\[lem:writheCZequality\] Let $\gamma$ be an embedded Reeb orbit, let $C$ be a $J$-holomorphic curve in ${{\mathbb R}}\times Y$ with only one positive end at $\gamma^d$, and let $\zeta$ denote the intersection of this end with $\{s\}\times Y$ for $s \gg 0$. Suppose $CZ_\tau(\gamma^d)$ is odd, the index of $u$ is at most 2, and $J$ is generic. Then $\zeta$ is isotopic to the braid given by a regular end and $$w_\tau(\zeta)=(d-1){\left\lfloor {\operatorname}{CZ}(\gamma^d)/2 \right\rfloor}-\gcd\left(d,{\left\lfloor {\operatorname}{CZ}(\gamma^d)/2 \right\rfloor}\right)+1.$$
The definition of a regular end is lengthy, see [@CGHZ Def. 1.3]. It ensures that the topology of the braid near an embedded Reeb orbit is completely determined by the total multiplicity of the orbit and the corresponding partition numbers. However, [@CGHZ Thm. 1.4], guarantees that for generic $J$, every generic curve has regular positive and negative ends. Symmetrically to Lemma \[lem:writheCZequality\] we have the following result for a negative end.
\[lem:writheCZequalityneg\] Let $\gamma$ be an embedded Reeb orbit, let $C$ be a $J$-holomorphic curve in ${{\mathbb R}}\times Y$ with only one negative end at $\gamma^d$, and let $\zeta$ denote the intersection of this end with $\{s\}\times Y$ for $s \ll 0$. Suppose $CZ_\tau(\gamma^d)$ is odd, the index of $u$ is at most 2, and $J$ is generic. Then $\zeta$ is isotopic to the braid given by a regular end and $$w_\tau(\zeta)=(d-1){\left\lceil {\operatorname}{CZ}(\gamma^d)/2 \right\rceil}+\gcd\left(d,{\left\lceil {\operatorname}{CZ}(\gamma^d)/2 \right\rceil}\right)-1.$$
The proof of the main classification result, Proposition \[notopbottomconn\], requires the following direct computation of asymptotic writhes and linking numbers, which uses the preceding lemmas.
\[lem:writhecomputations\] Let $J$ be generic. Let $\zeta_i,\zeta_j$ be connected braids about an embedded Reeb orbit $\gamma$ with multiplicities $d_i, d_j$. If both $\zeta_i, \zeta_j$ arise from either the positive or the negative ends of a curve which covers $\gamma$, then
1. Assuming $\gamma=e_+$:
1. There is only one positive end $\zeta_+$, and $w_\tau(\zeta_+)=1-d_+$.
2. If the $\zeta_i,\zeta_j$ are negative ends, then $w_\tau(\zeta_i)=d_i-1$, $w_\tau(\zeta_j)=d_j-1$, and $$\ell_\tau(\zeta_i,\zeta_j)=\min(d_i,d_j).$$
2. Assuming $\gamma=e_-$:
1. If the $\zeta_i,\zeta_j$ are positive ends, then $w_\tau(\zeta_i)=1-d_i$, $w_\tau(\zeta_j)=1-d_j$, and $$\ell_\tau(\zeta_i,\zeta_j)=-\min(d_i,d_j).$$
2. There is only one negative end $\zeta_-$, and $w_\tau(\zeta_-)=d_--1$.
We proceed casewise.
**Case (i.a)** By Lemma \[lem:connectortopology\] [(i.b)]{}, the end $\zeta_+$ is the only positive end. Therefore Lemma \[lem:writheCZequality\] applies, giving us $$\begin{aligned}
w_\tau(\zeta_+)&=(d_+-1){\left\lfloor {\operatorname}{CZ}(\gamma^{d_+})/2 \right\rfloor}-\gcd\left(d_+,{\left\lfloor {\operatorname}{CZ}(\gamma^{d_+})/2 \right\rfloor}\right)+1
\\&=(d_+-1){\left\lfloor \frac{1}{2} \right\rfloor}-\gcd\left(d_+,{\left\lfloor \frac{1}{2} \right\rfloor}\right)+1
\\&=0-\gcd(d_+,0)+1
\\&=1-d_+.\end{aligned}$$
**Case (i.b)** Firstly, we immediately have ${\operatorname}{wind}_\tau(\zeta_i)=1$ by Lemma \[lem:negativewrithe\] (b,c): $${\operatorname}{wind}_\tau(\zeta_i)={\left\lceil CZ_\tau(\zeta_i^{d_i})/2 \right\rceil}={\left\lceil \frac{1}{2} \right\rceil}=1$$
Therefore $\gcd(d_i,{\operatorname}{wind}_\tau(\zeta_i))=\gcd(d_i,1)=1$, which is a sub-case in the proof of [@Hindex Lemma 6.7]. There the equality $$w_\tau(\zeta_i)=(d_i-1){\operatorname}{wind}_\tau(\zeta_i)$$ is proven by showing that the $\zeta_i$ are isotopic to $(d_i,1)$ torus braids when $\gcd(d_i,{\operatorname}{wind}_\tau(\zeta_i))=1$. Therefore $w_\tau(\zeta_i)=d_i-1$.
For the claim on linking, let $\lambda_i$ denote the smallest eigenvalue of $L_{d_i}$ in the expansion of $\zeta_i$. The proof of [@Hindex Lemma 6.9] is proceeds by considering three cases: without loss of generality, when $\lambda_i<\lambda_j$, when $\lambda_i=\lambda_j$ and the coefficients of the corresponding eigenfunctions are different, and when $\lambda_i=\lambda_j$ and the coefficients of the corresponding eigenfunctions are the same. We are guaranteed by [@obg2 Proposition 3.9] that we are in either of the first two cases, while the proof of [@Hindex Lemma 6.9] gives the equality $$\ell_\tau(\zeta_i,\zeta_j)=\min\{d_i,d_j\}$$ in both of those cases, which is stronger than its general result.
**Case (ii.a)** We immediately have ${\operatorname}{wind}_\tau(\zeta_i)=-1$ by Lemma \[lem:positivewrithe\](b,c): $${\operatorname}{wind}_\tau(\zeta_i)={\left\lfloor CZ_\tau(\zeta_i^{d_i})/2 \right\rfloor}={\left\lfloor -\frac{1}{2} \right\rfloor}=-1.$$
The proof that $w_\tau(\zeta_i)=(d_i-1){\operatorname}{wind}_\tau(\zeta_i)$ and hence that $w_\tau(\zeta_i)=1-d_i$ is a virtual repeat of the proof for negative ends from [@Hindex Lemma 6.7] as in Case (i.b).
For the claim on linking, we can repeat the proof in Case (i.b). Note that [@Hindex Lemma 6.9] only applies to negative ends, but the proof will work using the asymptotic expansion of a positive end from [@hwz1], written in our notation as [@Hu2 Lemma 5.2]. If $\lambda_i<\lambda_j$, or $\lambda_i=\lambda_j$ with corresponding eigenfunctions having different multiplicities in the $\zeta_i$, we know that the braid $\zeta_j$ must be nested inside $\zeta_i$, therefore $$\ell_\tau(\zeta_i,\zeta_j)={\operatorname}{wind}_\tau(\zeta_i)d_j=-d_j.$$ We have $$-d_j=-\min\{d_i,d_j\}$$ because, by pulling back both $\zeta_i$ to covers of $\gamma^{d_id_j}$, we multiply their winding numbers by $d_j$ and $d_i$, respectively, and can apply the analytic perturbation theory of [@hwz2 §3], written in our notation as [@obg1 Lemma 2.11 (a)], to obtain $$d_j{\operatorname}{wind}_\tau(\zeta_i)\geq d_1{\operatorname}{wind}_\tau(\zeta_j)\Leftrightarrow d_j\leq d_i.$$
**Case (ii.b)** By Lemma \[lem:connectortopology\](i.c), the end $\zeta_-$ is the only negative end. Therefore Lemma \[lem:writheCZequalityneg\] applies, giving us $$\begin{aligned}
w_\tau(\zeta_-)&=(d_--1){\left\lceil {\operatorname}{CZ}(\gamma^{d_-})/2 \right\rceil}+\gcd\left(d_1,{\left\lceil {\operatorname}{CZ}(\gamma^{d_-})/2 \right\rceil}\right)-1
\\&=(d_--1){\left\lceil -\frac{1}{2} \right\rceil}+\gcd\left(d_-,{\left\lceil -\frac{1}{2} \right\rceil}\right)-1
\\&=0+\gcd(d_-,0)-1
\\&=d_--1.\end{aligned}$$
Finally, we need the following inequality from intersection theory of holomorphic curves, cf. [@dc §3.2], which is proven similarly to the relative adjunction formula, Lemma \[lem:adjunction\]. As before, let $\gamma$ be an embedded Reeb orbit with tubular neighborhood $N$, and let $\tau$ be a trivialization of $\xi|_\gamma$.
\[lem:adj\] Let $C$ be a $J$-holomorphic curve in $[s_-,s_+] \times N$ with no multiply covered components and with boundary $\zeta_+ - \zeta_-$ where $\zeta_\pm$ is a braid in $\{ s_\pm \} \times N$. Then $$\chi(C) + w_\tau(\zeta_+) - w_\tau(\zeta_-) = 2\delta(C) \geq 0,$$ where $\chi(C)$ denotes the Euler characteristic of the domain of $C$ and $\delta(C)$ is a count of the singularities of $C$ in $Y$ with positive integer weights.
With these preliminaries in place, we are now ready to prove the key classification result which excludes connectors from appearing in the top most or bottom most level of a building arising as a limit in the sense of [@BEHWZ].
\[notopbottomconn\] Let $\{{\mathcal{J}}_t\}_{t\in [0,1]}$ be a generic family of domain dependent almost complex structures and $\alpha$ and $\beta$ be admissible orbit sets with $I(\alpha,\beta)=1$. Let $C(t) \in {{{\mathcal M}}}^{{\mathcal{J}}_t}(\alpha,\beta)$ be a sequence of Fredholm index 1 curves, which, as $t \to 1$, converges in the sense of [@BEHWZ] to a building with $n$ levels given by $C_{i} \in {{{\mathcal M}}}^{{\mathcal{J}}_1}(\gamma_{i-1},\gamma_{i})$, $i=1,...,n$, where $\gamma_0 = \alpha$ and $\gamma_n = \beta$. Then neither the top most level $C_1$ nor the bottom most level $C_n$ are connectors.
We assume that the proposition is false and set up some notation. Suppose to get a contradiction that there exists a sequence of $\{{\mathcal{J}}_t\}$-holomorphic curves $\{ C(t)\} \in {{{\mathcal M}}}^{{\mathcal{J}}_t}(\alpha,\beta)$ which converges in the sense of [@BEHWZ] to a $n$-level building which has either $C_1$ or $C_n$ as a connector. [Recall that $C_i$ is an equivalence class of holomorphic curves in ${{\mathbb R}}\times Y$, where two holomorphic curves are equivalent iff they differ by ${{\mathbb R}}$-translation in ${{\mathbb R}}\times Y$.]{} In the following, we will choose a representative of this equivalence class and still denote it by $C_i$. If necessary, translate the holomorphic curve $C_1$ upward and $C_n$ downward so that Lemmas \[lem:positivewrithe\]-\[lem:writheCZequalityneg\] apply, cf. [@dc §3.3].
Without loss of generality, we can work under the assumption that the connector appears in the top most level, $C_1 \in {{{\mathcal M}}}^{{\mathcal{J}}_1}(\alpha, \alpha)$. Consider an embedded Reeb orbit $\gamma$ appearing in the orbit set $\alpha$. Let $N_\gamma$ be a tubular neighborhood of the Reeb orbit $\gamma$. For some sufficiently large $s_0 \gg 0$ and some $t$ close to 1, the intersection $C(t) \cap ( [s_0, \infty) \times N_\gamma)$ can be identified with the union of components of $C_1$ that cover ${{\mathbb R}}\times \gamma$. Denote both by $C$. Note that as a subset of $C(t)$, $C$ is not a trivial cylinder, but rather an embedding in the complement of a finite number of singular points.
While intersection positivity is not true in general for domain dependent almost complex structures, by Remark \[ddacsintpos\] if $$C: ({{\dot{\Sigma}}}, j) \to ({{\mathbb R}}\times Y, {\mathbb{J}})$$ is a $(j,{\mathbb{J}})$-holomorphic curve, then $C \vert_{C^{-1}({{\mathbb R}}\times N)}$ is $(j,{\mathfrak{J}}_0)$-holomorphic. Since ${\mathfrak{J}}_0$ is domain independent, the subset $C({{\dot{\Sigma}}}) \cap {{\mathbb R}}\times N \subset C({{\dot{\Sigma}}})$ satisfies intersection positivity. Thus we will be in a situation to apply relative adjunction as in Lemma \[lem:adj\] because intersection positivity holds. Moreover, the count of singularities of $C$, satisfies $\delta(C) \geq 0$ with equality if and only if $C$ is embedded.
We will show that if $C$ arises from a nontrivial connector appearing at the top most level, then relative adjunction as in Lemma \[lem:adj\] will imply that $\delta(C) < 0$, a contradiction. Note that a connector cannot be trivial in the sense that it exclusively consists of unbranched components, e.g. trivial cylinders, as explained in [@wendl-sft Remark 9.26].
There are three cases to consider, corresponding to connectors containing components satisfying the conclusions of Lemma \[lem:connectortopology\], (i.b), (i.c), and (ii).
**Case (i.b)** Assume $C$ is a component of a connector covering ${{\mathbb R}}\times e_+$. Then by Lemma \[lem:connectortopology\] we have $g(C)=0$ and by Lemma \[lem:connectortopology\](i.b) $C$ has a single positive end. Let $d_+$ denote the covering multiplicity of this end, and let $d_i$ denote the covering multiplicity of the $i^\text{th}$ negative end of $C$. Because $C$ covers a trivial cylinder, $d_+=\sum_{i=1}^{p_-(C)}d_i$. For $t$ sufficiently close to 1, there is a representative $C$ with the following properties.
1. $C^{-1}([0,\infty) \times Y)$ is an annulus with one puncture, which is mapped by $C$ to $[0,\infty) \times N_{e_+}$
2. $C^{-1}((-\infty, 0] \times Y)$ consists of as many half cylinders $C_i$ as there are $p_-(C)$ negative ends of $C$.
3. $C(C_i)$ is contained in $(-\infty, 0] \times N_{e_+}$ and $C(C_i) \cap (\{ 0 \} \times N)$ is a braid $\zeta_i$ which projects to $e_+$ with degree $d_i$ and has distance at most $\frac{\varepsilon}{p_-(C) +1}$ from $e_+$.
Also let $\zeta_+$ denote the braid corresponding to the positive end of $C$ at $e_+^{d_+}$. It follows that the union $\bigcup_i \zeta_i$ is a braid. We obtain a contradiction: $$\begin{aligned}
2\delta(C)&=2-p_+(C)-p_-(C)+w_\tau(\zeta_+) - {w_\tau\left(\bigcup_i \zeta_i\right)}
\\&=1-p_-(C)+(1-d_+)-\left(\sum_{i=1}^{p_-(C)}(d_i-1)+\sum_{i\neq j}\min(d_i,d_j)\right)\text{ by Lemma \ref{lem:writhecomputations}(i) and (\ref{eqn:writhelinking})}
\\&=2-2d_+-\sum_{i\neq j}\min(d_i,d_j)
\\&\leq -2\end{aligned}$$ (Note that our notation $\sum_{i\neq j}\min(d_i,d_j)$ accounts for the factor of two in (\[eqn:writhelinking\]).) In the inequality we have used the fact that $\sum_{i\neq j}\min(d_i,d_j)\geq2$ whenever there are at least two negative ends, because $d_i\geq1$. There can never be just one negative end lest $C$ be topologically a cylinder and therefore unbranched, by the Riemann-Hurwitz Theorem.
**Case (i.c)** Assume $C$ is a component of a connector covering ${{\mathbb R}}\times e_-$. Then by Lemma \[lem:connectortopology\] we have $g(C)=0$ and by Lemma \[lem:connectortopology\](i.c) $C$ has a single negative end. Let $d_-$ denote the covering multiplicity of this end, and let $d_i$ denote the covering multiplicity of the $i^\text{th}$ positive end of $C$. For $t$ sufficiently close to 1, there is a representative $C$ with the following properties.
1. $C^{-1}((-\infty, 0] \times Y)$ is an annulus with one puncture, which is mapped by $C$ to $(\infty,0] \times N_{e_-}$
2. $C^{-1}([0,\infty) \times Y)$ consists of as many half cylinders $C_i$ as there are $p_+(C)$ positive ends of $C$.
3. $C(C_i)$ is contained in $[0,\infty) \times N_{e_-}$ and $C(C_i) \cap (\{ 0 \} \times N)$ is a braid $\zeta_i$ which projects to $e_-$ with degree $d_i$ and has distance at most $\frac{\varepsilon}{p_+(C) +1}$ from $e_-$.
Also let $\zeta_-$ denote the braid corresponding to the negative end of $C$ at $e_-^{d_-}$. It follows that the union $\bigcup_i \zeta_i$ is a braid. We obtain a contradiction: $$\begin{aligned}
2\delta(C)&=2-p_+(C)-p_-(C) + {w_\tau\left(\bigcup_i \zeta_i\right)} - w_\tau(\zeta_-)
\\&=1-p_+(C)+\left(\sum_{i=1}^{p_+(C)}(1-d_i)-\sum_{i\neq j}\min(d_i,d_j)\right)-(d_--1)\text{ by Lemma \ref{lem:writhecomputations}(ii) and (\ref{eqn:writhelinking})}
\\&=2-2d_--\sum_{i\neq j}\min(d_i,d_j)
\\&\leq-2\end{aligned}$$ As in Case (i.b), we must have $p_+(C)\geq2$, hence $\sum_{i\neq j}\min(d_i,d_j)\geq2$.
**Case (ii)** If a branched component of the connector at the top (respectively, the bottom) covers ${{\mathbb R}}\times h$, where $h$ is hyperbolic, then by Lemma \[lem:connectortopology\](ii), its ends must be asymptotic to $h^2$. Therefore $\alpha$ (respectively, $\beta$) must include the pair $(h,m)$ with $m\geq2$, which contradicts the fact that $\alpha$ (respectively, $\beta$) is an ECH chain complex generator.
From domain dependent ${\mathbb{J}}$ to domain independent $J$ {#handleslides}
==============================================================
In Corollary \[nogenus\], we saw that for a generic $S^1$-invariant domain dependent almost complex structure ${\mathbb{J}}\in {\mathscr{J}}_{{\dot{\Sigma}}}^{S^1}$ that ECH index one moduli spaces of nonzero genus curves are empty. However ECH is defined using a domain independent generic $\lambda$-compatible $J$, so we must prove the analogous result when $J$ is a generic $\lambda$-compatible almost complex structure. In order to do so, we consider a generic one parameter family $\{ {\mathcal{J}}_t\}_{t\in[0,1]}$ of domain dependent almost complex structures interpolating between a generic ${\mathcal{J}}_0 :={\mathbb{J}}\in {\mathscr{J}}_{{\dot{\Sigma}}}^{S^1}$ and a domain independent generic $\lambda$-compatible ${\mathcal{J}}_1:=J \in {\mathscr{J}}_{reg}(Y,\lambda)$ and show that the computation of ECH is not affected.
Overview and sketch of proof
----------------------------
Our main result is the following.
\[ECH-ok\] Let $\alpha$ and $\beta$ be admissible orbit sets with $I(\alpha,\beta)=1$ and ${\operatorname}{deg}(\alpha,\beta) >0.$ For generic paths $\{ J_t\}_{t\in[0,1]}$ connecting ${\mathcal{J}}_0 := {\mathbb{J}}\in {\mathscr{J}}_{{\dot{\Sigma}}}^{S^1}$ and ${\mathcal{J}}_1 := J \in \mathscr{J}_{reg}(Y,\lambda)$, the moduli space $\mathcal{M}_t := \mathcal{M}^{{\mathcal{J}}_t}(\alpha, \beta)$ is cut out transversely save for a discrete number of times $t_0,...,t_\ell \in (0,1)$. At each such $t_i$, the ECH differential can change either by:
1. The creation or destruction of a pair of oppositely signed curves.[^10]
2. An ECH handleslide.
However, in either case, the homology is unaffected.
For each ${\mathcal{J}}_t$ we consider the moduli space ${{{\mathcal M}}}_t(\alpha,\beta)$ of ${\mathcal{J}}_t$-holomorphic curves where $\alpha$ and $\beta$ are admissible orbit sets satisfying $I(\alpha,\beta)=1$ and ${\operatorname}{deg}(\alpha,\beta) >0$. That ${\operatorname}{deg}(\alpha,\beta) >0$ rules out moduli spaces of ${\mathcal{J}}_t$-holmorphic cylinders, for which the domain dependent almost complex structures cannot be used, cf. Lemma \[lem:d0g0\]. We have that ${{{\mathcal M}}}_t(\alpha,\beta)$ is cut out transversely save a discrete number of times $t_i\in[0,1]$ and at such a nonregular ${\mathcal{J}}_{t_i}$, the differential can be impacted by either the creation or destruction of a pair of oppositely signed curves or by an “ECH handleslide." In the former case, the signed and mod 2 counts of curves in ${{{\mathcal M}}}_{t_i-\varepsilon}$ and ${{{\mathcal M}}}_{t_i+\varepsilon}$ are the same. The differential can change at an ECH handleslide, at which a sequence of Fredholm and ECH index 1 curves $\{C(t)\}$ breaks into a holomorphic building in the sense of [@BEHWZ] into components consisting of an ECH and Fredholm index 0 curve, an ECH and Fredholm index 1 curve, and some “connectors," which are Fredholm index 0 branched covers of a trivial cylinder $ {{\mathbb R}}\times \gamma$. In §\[sec:connectors\] we previously demonstrated that connectors cannot appear at the top most or bottom most level of the building via intersection theory arguments similar to [@dc §4]. As a result, in §\[sec:ECHok\] we can appeal to the obstruction bundle gluing theorems of [@obg1; @obg2] in conjunction with an inductive argument involving the degree of a completed curve, cf. Definition \[degree\] and Proposition \[prop-degree\], to show that the differential does not actually change at an ECH handleslide.
Proposition \[ECH-ok\] yields the the domain independent analogue of Corollary \[nogenus\].
\[nogenusJ\] Let $\alpha$ and $\beta$ be nondegenerate admissible orbit sets and $J \in {\mathscr{J}}(Y,\lambda)$ be generic. If ${\operatorname}{deg}(\alpha,\beta) >0$ and $I(\alpha,\beta)=1$ then the mod 2 count $\#_{{{\mathbb Z}}_2}{{{\mathcal M}}}^{J}(\alpha,\beta) = 0$. If $\alpha$ and $\beta$ are associated to $\lambda_\varepsilon$ as in Lemma \[lem:efromL\] and $\mathcal{A}(\alpha), \mathcal{A}(\beta) <L(\varepsilon)$ then $\langle \partial^{L(\varepsilon)} \alpha, \beta \rangle =0$.
In §\[sec:handles\] we demonstrate that the classification results for connectors in §\[sec:connectors\] ensure that at an ECH handleslide $t_i$, a sequence of ${\mathcal{J}}_{t}$-holomorphic curves $\{ C_k \ | \
\mbox{ind}(C_k) =1 \}$ breaks into an *ECH handleslide building* $(C_+, {\mathscr{C}}, C_-)$ wherein
(i) The top most curve $C_+$ has either index 1 or index 0
(ii) Connectors ${\mathscr{C}}$ with $\mbox{ind}({\mathscr{C}})=0$ appearing in the middle;
(iii) The bottom most curve $C_-$ has ${{\operatorname}{ind}}(C_-) = 1-\mbox{ind}(C_+)$.
Moreover, the index 0 curve occurring at either the top most or bottom most level cannot contain any connectors.
We define an *ECH handleslide* to be the index 0 curve which is not a connector in an *ECH handleslide building* $(C_+, {\mathscr{C}}, C_-)$, in analogy with Morse theory.
As observed in [@farris §7.1.1], because connectors, the branched covers of trivial cylinders, cannot appear as the top-most or bottom-most level by Proposition \[notopbottomconn\], we can appeal to the obstruction bundle gluing theorems [@obg1; @obg2] to relate the curve counts occurring immediately prior to and following the appearance of an ECH handleslide at time $t_i$. If we assume that the ECH handleslide is $C_-$, then as explained in Remark \[rem:handleslide\] we obtain: $$\label{eq:handle}
{\# \mathcal{M}_{t_i + \epsilon}(\alpha,\beta)= \# \mathcal{M}_{t_i - \epsilon}(\alpha, \beta) + \# G(C_+, C_-) \cdot \# \mathcal{M}_{t_i }(\alpha, \gamma_+),}$$ where $\gamma$ is another (admissible) orbit set such that $I(\alpha,\gamma_+)=1$ with $C_+ \in \mathcal{M}_{t_i }(\alpha, \gamma_+)$. Note that the connector ${\mathscr{C}}\in \mathcal{M}_{t_i }(\gamma_+, \gamma_-)$ and the ECH handleslide curve $C_- \in \mathcal{M}_{t_i }(\gamma_-, \beta)$. As explained in §\[sec:obg\], obstruction bundle gluing gives a combinatorial formula for $\#G(C_+, C_-) \in {{\mathbb Z}}$, based on the negative asymptotic ends of $C_+$, the positive asymptotic ends of $C_-$, and the partitions associated to the ends of the connectors ${\mathscr{C}}$. For each embedded Reeb orbit $\gamma$, the total covering multiplicity of Reeb orbits covering $\gamma$ in the list $\gamma_+$ is the same as the total for $\gamma_-$. (In contrast, for the usual form of Floer theory gluing, one would assume that $\gamma_+=\gamma_-$.) In §\[sec:ECHok\] we complete the proof of Proposition \[ECH-ok\] by way of an inductive argument involving the degree, which precludes the need to explicitly compute $\#G(C_+, C_-) $ as we obtain $\# \mathcal{M}_{t_i}(\alpha,\gamma_+) =0$ for all admissible $\gamma_+$ such that $I(\alpha,\gamma_+)=1$.
Handleslides and bifurcations {#sec:handles}
-----------------------------
An *ECH handleslide building* is a building arising as a limit of $I(C)=1$, ${\operatorname}{ind}(C)=1$ curves in ${{\mathbb R}}\times Y$ as the complex structure varies through domain-dependent almost complex structures. The terminology arises from the fact that such a building might include levels with $I(C)=0$ which do not consist solely of trivial cylinders, in analogy to the Morse index zero gradient trajectories which arise during a handleslide in a generic homotopy of Morse functions. Note that the characterization from Proposition \[lowiprop\] does not applies to moduli spaces defined using domain-dependent almost complex structures.
\[lem:slide\] Fix admissible orbit sets $\alpha$ and $\beta$ with ${\operatorname}{deg}(\alpha,\beta)>0$ and $I(\alpha,\beta)=1$. Let $\{{\mathcal{J}}_t\}_{t\in [0,1]}$ be a one generic parameter family of almost complex structures. Consider the corresponding moduli spaces $\mathcal{M}_t := \mathcal{M}^{{\mathcal{J}}_t}(\alpha, \beta)$; label the times at which $\mathcal{M}_t$ is not cut out transversely by $t_0,...,t_\ell \in (0,1)$. Let $C(t) \in {{{\mathcal M}}}_t(\alpha,\beta)$ with $t \to t_i$. Then after passing to a subsequence, $\{ C(t) \}$ converges in the sense of [@BEHWZ] either to a curve in ${{{\mathcal M}}}_{t_i}(\alpha,\beta)$ or to an *ECH handleslide building* with
1. An index 1 curve at the top most level $C_+$ (or at bottom most level $C_-$);
2. Connectors ${\mathscr{C}}$ with $\mbox{ind}({\mathscr{C}})=0$;
3. An index 0 curve, the ECH handleslide, at the bottom most level $C_-$ (or at the top most level $C_+$).
By the compactness theorem in [@BEHWZ], any sequence of ECH and Fredholm index 1 curves in ${{{\mathcal M}}}^{{\mathcal{J}}_t}(\alpha,\beta)$ has a subsequence which converges to some broken curve as $t \to t_i$. Moreover, the indices of the levels of the broken curve sum to 1. By Proposition \[notopbottomconn\] we cannot have connectors appear at the top most or bottom most level. Moreover by compactness and the conservation of Fredholm index, a Fredholm index one connector cannot appear as a middle level in a handleslide building. If the sequence is close to breaking, cf. Definition \[def:close\], then by Lemma \[lem:ht\] and the definition of $\mathcal{G}_\delta$, one of the following two scenarios occurs:
(i) The top most level of the broken curve contains the index 1 component $C_+$ and some lower level contains the index 0 ECH handleslide $C_-$.
(ii) The bottom most level of the broken curve contains the index 1 component, $C_-$, and the top most level contains the index 0 ECH handleslide, $C_+$.
Moreover, all other components of all levels are index zero branched covers of ${{\mathbb R}}$-invariant cylinders, e.g. connectors. By analogy with condition (d) in the definition of a gluing pair, Definition \[def:gluepair\], any covers of ${{\mathbb R}}$-invariant cylinders in the top and bottom levels of the broken curve must be unbranched.
Finally, we review the possible bifurcations that appear in a generic 1-parameter family $\{ {\mathcal{J}}_t \}_{t\in[0,1]}$:
Fix a nondegenerate contact form $\lambda$. Then for a 1-parameter family $\{ {\mathcal{J}}\}_{t\in[0,1]}$ of $\lambda$-compatible domain dependent almost complex structures with fixed endpoints we may arrange that the only possible bifurcations are:
1. A cancellation of two oppositely signed holomorphic curves.
2. An ECH handleslide.
In the case of Morse theory, the corresponding transversality statement is [@torsion Lemma 2.11(b)]. In the context of Seiberg-Witten Floer homology, Taubes completes this bifurcation analysis at the end of [@taubes-wconj]. Note that cancellation of two oppositely signed curves does not change the differential. The presence of an ECH handleslide does, change the differential, but in §\[sec:ECHok\], we show that it does not have an impact after passing to homology.
Recap of obstruction bundle gluing {#sec:obg}
----------------------------------
In this section we collect the results from [@obg1] that will be used in the proof of Proposition \[ECH-ok\]. We state everything in the context considered in proving $\partial^2=0$, and explain in a subsequent remark the difference and continued applicability in the setting under consideration. In connection with the index calculations for branched covered cylinders over an elliptic embedded Reeb orbit, cf. Lemma \[lem:ht\], we can define a partial order on the associated set of partitions, which will be used in the construction of a gluing pair.
\[def:partial\] Let $\gamma$ be a nondegenerate elliptic embedded Reeb orbit with a fixed irrational rotation number $\vartheta$, cf. §\[cz-sec\]. Writing $\alpha = \left(\gamma^{a_1},...,\gamma^{a_k}\right)$ and $\beta=\left(\gamma^{b_1},...,\gamma^{b_\ell}\right)$, consider $C \in {{{\mathcal M}}}^J(\alpha,\beta)$a branched cover of ${{\mathbb R}}\times \gamma$. We say $$(a_1,...,a_k) \geq_\vartheta (b_1,...,b_\ell)$$ whenever there exists an index zero branched cover of ${{\mathbb R}}\times \gamma \in {{{\mathcal M}}}^J\left(\left(\gamma^{a_1},...,\gamma^{a_k}\right),\left(\gamma^{b_1},...,\gamma^{b_\ell}\right)\right).$
Following [@obg1 §1.3-1.4] we define a gluing pair, prepare for the definition of the count $\#G(C_+, C_-)$, and state the main obstruction bundle gluing theorem.
\[def:gluepair\] A *gluing pair* is a pair of immersed $J$-holomorphic curves $C_+(\alpha,\gamma_+)$ and $C_-(\gamma_-,\beta)$ such that:
(a) ${{\operatorname}{ind}}(C_+)={{\operatorname}{ind}}(C_-)=1$.
(b) $C_+$ and $C_-$ are not multiply covered, except that they may contain unbranched covers of ${{\mathbb R}}$-invariant cylinders.
(c) For each embedded Reeb orbit $\gamma$, the total covering multiplicity of Reeb orbits covering $\gamma$ in the list $\gamma_+$ is the same as the total for $\gamma_-$. (In contrast, for the usual form of Floer theory gluing, one would assume that $\gamma_+=\gamma_-$.)
(d) If $\gamma$ is an elliptic embedded Reeb orbit with rotation angle $\vartheta$, let $m_1',...,m_k'$ denote the covering multiplicities of the ${{\mathbb R}}$-invariant cylinders over $\gamma$ in $C_+$ and let $n_1',...,n_j'$ denote the corresponding multiplicities in $C_-$. Then under the partial order $\geq_\vartheta$ in Definition \[def:partial\], the partition $(m_1',...,m_k')$ is minimal, and the partition $(n_1',...,n_j')$ is maximal.
Let $(C_+,C_-)$ be a gluing pair. The main gluing result of [@obg1; @obg2] computes an integer $\#G(C_+,C_-)$ which, roughly speaking is a signed count of ends of the index two part of the moduli space ${{{\mathcal M}}}^J(\alpha,\beta)/{{\mathbb R}}$ that break into $C_+$ and $C_-$ along with some index zero connectors (branched covers of ${{\mathbb R}}$-invariant cylinders between them). When $C_\pm$ contain covers of ${{\mathbb R}}$-invariant cylinders, there are some subtleties which require the use of condition (d) above in showing that $\#G(C_+,C_-)$ is well-defined.
Before giving the definition of the count $\#G(C_+,C_-)$, we first define a set $\mathcal{G}_\delta(C_+,C_-)$ of index two curves in ${{{\mathcal M}}}^J(\alpha,\beta)$ which, are close to breaking in the above manner. For the following definition, choose an arbitrary product metric on ${{\mathbb R}}\times Y$.
\[def:close\] For $\delta >0$, define $\mathfrak{C}_\delta(C_+,C_-)$ to be the set of immersed (except possibly at finitely many singular points) surfaces in ${{\mathbb R}}\times Y$ that can be decomposed as $\mathbf{C}_- \cup \mathbf{C}_0 \cup \mathbf{C}_+$ such that the following hold:
- There is a real number $R_-$, and a section $\psi_-$ of the normal bundle to $C_-$ with $|\psi_-|<\delta$, such that $\mathbf{C}_-$ is the set $s \mapsto s + R_-$ translate of the $s \leq \frac{1}{\delta}$ portion of the image of $\psi_-$ under the exponential map.
- Similarily, there is a real number $R_+$, and a section $\psi_+$ of the normal bundle to $C_+$ with $|\psi_+|<\delta$, such that $\mathbf{C}_+$ is the set $s \mapsto s + R_+$ translate of the $s \geq -\frac{1}{\delta}$ portion of the image of $\psi_+$ under the exponential map.
- $R_+ - R_- > \frac{2}{\delta}.$
- $\mathbf{C}_0 $ is contained in the union of the radius $\delta$ tubular neighborhoods of the cylinders ${{\mathbb R}}\times \gamma$, where $\gamma$ ranges over the embedded Reeb orbits covered by orbits in $\gamma_\pm$
- $\partial \mathbf{C}_0 = \partial \mathbf{C}_- \sqcup \partial \mathbf{C}_+$, where the positive boundary circles of $\mathbf{C}_-$ agree with the negative boundary circles of $\mathbf{C}_0$, and the positive boundary circles of $\mathbf{C}_0$ agree with the negative boundary circles of $\mathbf{C}_+$.
Let $\mathcal{G}_\delta(C_+,C_-)$ denote the set of index two curves in ${{{\mathcal M}}}^{J}(\alpha,\beta) \cap \mathfrak{C}_\delta(C_+,C_-)$.
To see that this definition works as expected, we have the following lemma. We include the proof, as it will better elucidate why we can invoke the obstruction bundle gluing formalism in the setting under consideration.
\[lem:glue\] Given a gluing pair $(C_+,C_-)$, there exists $\delta_0 >0$ with the following property. Let $\delta \in (0,\delta_0)$ and let $\{C(k) \}_{k=1,2,...}$ be a sequence in $\mathcal{G}_\delta(C_+,C_-)/{{\mathbb R}}$. Then there is a subsequence which converges in the sense of [@BEHWZ] either to a curve in ${{{\mathcal M}}}^{J}(\alpha,\beta)/{{\mathbb R}}$ or to a broken curve in which the top level is $C_+$, the bottom level is $C_-$, and all intermediate levels are unions of index zero branched covers of ${{\mathbb R}}$-invariant cylinders.
By the compactness theorem in [@BEHWZ], any sequence of index 2 curves in ${{{\mathcal M}}}^J(\alpha,\beta)/{{\mathbb R}}$ has a subsequence which converges to some broken curve. Moreover, the indices of the levels of the broken curve sum to 2. If the sequence is in $\mathcal{G}_\delta(C_+,C_-)/{{\mathbb R}}$ with $\delta>0$ sufficiently small then by Lemma \[lem:ht\] and the definition of $\mathcal{G}_\delta$, one of the following two scenarios occurs:
(i) One level of the broken curve contains the index 1 component of $C_+$, and some lower level contains the index 1 component of $C_-$.
(ii) Some level contains two index 1 components or one index 2 component.
Moreover, all other components of all levels are index zero branched covers of ${{\mathbb R}}$-invariant cylinders. By condition (d) in the definition of a gluing pair, any covers of ${{\mathbb R}}$-invariant cylinders in the top and bottom levels of the broken curve must be unbranched. It follows that in Case (i), the top level is $C_+$ and the bottom level is $C_-$, while in Case (ii), there are no other levels.
\[def:U\] Fix coherent orientations and generic $\lambda$-compatible $J$ and let $(C_+,C_-)$ be a gluing pair. If $\delta \in (0,\delta_0)$, then by Lemma \[lem:glue\], one can choose an open set $U \subset {{{\mathcal M}}}^J(\alpha,\beta)/{{\mathbb R}}$ such that:
- $\mathcal{G}_{\delta'}(C_+,C_-)/{{\mathbb R}}\subset U \subset \mathcal{G}_\delta(C_+,C_-)/{{\mathbb R}}$ for some $\delta' \in (0,\delta)$.
- The closure $\overline{U}$ has finitely many boundary points.
Define $\# G(C_+,C_-) \in {{\mathbb Z}}$ to be minus the signed count of boundary points of $\overline{U}$. By Lemma \[lem:glue\], this does not depend on the choice of $\delta$ or $U$.
Note that by Lemma \[lem:ht\], if $\#G(C_+,C_-)\neq 0$ then for each hyperbolic Reeb orbit $\gamma$, the multiplicities of the negative ends of $C_+$ at covers of $\gamma$ agree, up to reordering, with the multiplicities of the positive ends of $C_-$ at covers of $\gamma$. When this is the case, assume that the orderings of the negative ends of $C_+$ and of the positive ends of $C_-$ are such that for each positive hyperbolic orbit $\gamma$, the aforementioned multiplicities appear in the same order for $C_+$ and for $C_-$. With this ordering convention, the statement of the main gluing theorem is as follows:
[*[@obg1 Theorem 1.13]*]{}\[thm:obg\]\[thm:obg1\] Fix coherent orientations. If $J$ is generic and if $(C_+,C_-)$ is a gluing pair then $$\#G(C_+,C_-) = \epsilon(C_+)\epsilon(C_-)\prod_\gamma c_\gamma(C_+,C_-).$$ Here the product is over embedded Reeb orbits $\gamma$ such that $C_+$ has a negative end at a cover of $\gamma$. The integer $c_\gamma(C_+,C_-)$ depends only on $\gamma$ and on the multiplicities of the ${{\mathbb R}}$-invariant and non-${{\mathbb R}}$-invariant negative ends of $C_+$ and positive ends of $C_-$ at covers of $\gamma$.
We omit the discussion of the explicit computation of the gluing coefficient $c_\gamma(C_+,C_-)$, as we will multiply this by zero; the budding obstruction bundle enthusiast can find further details in [@obg1 §1.5-1.6].
We have that $c_\gamma(C_+,C_-)=1$ if and only if the ECH partition conditions are met, cf. §\[ECH-ineq\]. However, we cannot guarantee this in practice, as such an argument typically relies on the ECH index inequality, Theorem \[thm:indexineq\], which does not apply for domain dependent curves, because its proof requires intersection positivity.
As a result of §\[sec:handles\] (and the classification of connectors in §\[sec:connectors\]), we can analogously define a gluing pair for an ECH handleslide building and invoke Theorem \[thm:obg1\]. It remains to explain how this yields .
\[rem:handleslide\] Roughly speaking and along the lines of [@fl], [@torsion §3.3], for $\epsilon$ small, the results of §\[sec:handles\] yield a one-dimensional cobordism: $$\partial \left( \bigcup_{t\in[t_i - \epsilon, t_i + \epsilon]} \mathcal{M}_{t}(\alpha,\beta) \right) = \mathcal{M}_{t_i + \epsilon}(\alpha,\beta) - \mathcal{M}_{t_i + \epsilon}(\alpha,\beta) \mp \mathcal{M}_{t_i}(\alpha,\gamma_+) \bigsqcup_{{\mathscr{C}}\in {{{\mathcal M}}}_{t_i}(\gamma_+,\gamma_-)} \mathcal{M}_{t_i}(\gamma_-,\beta).$$ However, technically speaking we cannot compute the boundary of the compactified moduli space on the left hand side. Instead, we must truncate this moduli space in order to invoke the obstruction bundle gluing theorem and obtain similarly to the proof that the ECH differential squares to zero, cf. [@obg1 Theorem 7.20]. In particular, if $(C_+,C_-)$ is a gluing pair arising from an ECH handleslide in which $C_+ \in \mathcal{M}_{t_i}(\alpha,\gamma_+) $ and $C_- \in \mathcal{M}_{t_i}(\gamma_-,\beta)$, let $V(C_+,C_-) \subset {{{\mathcal M}}}_t(\alpha,\beta)$ for $t\in[t_i - \epsilon, t_i + \epsilon]$ be an open set like the open set $U$ in Definition \[def:U\], but where the curves do not have any asymptotic markings or orderings of the ends. We truncate the interior of the cobordism by removing curves which are close to breaking, $$\overline {{{\mathcal M}}}: = \bigcup_{t\in[t_i - \epsilon, t_i + \epsilon]} \mathcal{M}_{t}(\alpha,\beta) \ \setminus \bigsqcup_{(C_-,C_+)} V(C_-,C_+)$$ By the analogue of [@obg1 Lem. 7.23], namely Lemma \[lem:slide\], we have that $\overline {{{\mathcal M}}}$ is compact. Because the handleslide is isolated, the signed count of truncated boundary points is $$0 = \# \partial \overline{{{{\mathcal M}}}} = - \# \partial \overline{V(C_+,C_-)}.$$ The count $\#G(C_+,C_-) = -\#\partial\overline{U}$ distinguishes curves in $\partial\overline{U}$ that have different asymptotic markings and orderings of the ends, but represent the same element of $\partial\overline{V(C_+,C_-)}$, resulting in .
Proof of Proposition \[ECH-ok\] {#sec:ECHok}
-------------------------------
Similarly to [@farris §7.1.1], we complete the proof of Proposition \[ECH-ok\], by demonstrating that the occurrence of ECH handleslides do not impact the homology.
Number the ECH handleslides $t_0,...,t_k$. We omit the cancellation bifurcations as they do not change the curve counts, and note that one should occur before the first ECH handleslide, as otherwise the moduli space under consideration are empty [by Corollary \[nogenus\]]{}. Without loss of generality we will always assume that $C_+$ is the index 1 curve and $C_-$ is the index 0 ECH handleslide curve in an ECH handleslide building.
By Remark \[rem:handleslide\] we have that at each handleslide $t_i$, $$\label{obg-handleslide}
{\# \mathcal{M}_{t_i + \epsilon}(\alpha,\beta)= \# \mathcal{M}_{t_i - \epsilon}(\alpha, \beta) + \# G(C_+, C_-) \cdot \# \mathcal{M}_{t_i }(\alpha, \gamma_+),}$$ where $\gamma_+$ is another (admissible) orbit set such that $I(\alpha,\gamma_+)=1$ with $C_+ \in \mathcal{M}_{t_i }(\alpha, \gamma_+)$. Note that the connector ${\mathscr{C}}\in \mathcal{M}_{t_i }(\gamma_+, \gamma_-)$ and the ECH handleslide curve $C_- \in \mathcal{M}_{t_i }(\gamma_-, \beta)$.
Since ${\mathcal{J}}_0:={\mathbb{J}}$ is a generic $S^1$ invariant domain dependent almost complex structure, by Corollary \[nogenus\] $${{{\mathcal M}}}_{t_0-\varepsilon}(\alpha,\beta) = \emptyset.$$ If we can show for all possible $\gamma$ that $$\label{desire}
\# {{{\mathcal M}}}_{t_0}(\alpha, \gamma) =0$$ then yields $$\# {{{\mathcal M}}}_{t_0+\varepsilon}(\alpha,\beta) = \# {{{\mathcal M}}}_{t_0-\varepsilon}(\alpha,\beta) + 0 =0.$$ An inductive argument on $k \in {0,...,\ell}$, where each $t_k$ realizes an ECH handleslide, would then complete the proof.
It remains to show ; this will be done by a reductive degree argument. The degree as defined in §\[degree\] of a connector is always zero and degree is additive, hence $${\operatorname}{deg}(C(t)) = {\operatorname}{deg}(C_+) + {\operatorname}{deg}(C_-).$$ Moreoever, if the ECH handleslide curve $C_-$ has degree 0, then it is a union of branched covers of cylinders, at least one of which is not a trivial cylinder, as otherwise $C_-$ is a connector. But a nontrivial cylinder, and hence the union of cylinders including it, has positive Fredholm index. Thus the ECH handleslide curve $C_-$ must have positive degree (and positive genus) by Lemma \[lem:d0g0\]. Hence, if at a handleslide, $\{C(t) \} \in {{{\mathcal M}}}_t(\alpha,\beta)$ converges to an ECH handleslide building $(C_+,{\mathscr{C}}, C_-)$ then $$\label{eq:deg}
{\operatorname}{deg}(C_+) < {\operatorname}{deg}(C(t))$$ Going back to the task at hand, consider ${{{\mathcal M}}}_{t_0}(\alpha,\gamma_+)$, a smooth one dimensional moduli space of Fredholm and ECH index 1 curves. We wish to show that $$\# {{{\mathcal M}}}_{t_0}(\alpha, \gamma_+) =0.$$ This will be accomplished by inductive iteration:
1. Consider another generic one parameter family $\{{\mathcal{J}}'_t\}_{t \in [0,t_0]}$ of domain dependent almost complex structures from ${\mathbb{J}}_0'$ to ${\mathbb{J}}_{t_0}$. Note that before the first handleslide, call it $t_0'$, in this new deformation $${{{\mathcal M}}}_{t_0'-\varepsilon}(\alpha,\gamma_+) = \emptyset.$$
2. Use to relate the curve counts occurring immediately prior to and following the first appearance of an ECH handleslide at $t_0'$ in this new deformation $\{{\mathcal{J}}_t'\}_{t\in [0,t_0]}$: $$\begin{array}{lcl}
\# \mathcal{M}_{t_0' + \epsilon}(\alpha,\gamma_+) &= &\# \mathcal{M}_{t_0' - \epsilon}(\alpha, \gamma_+) + \# G(C_+', C_-') \cdot \# \mathcal{M}_{t_0' }(\alpha, \gamma'_+) \\
&= & \# G(C_+', C_-') \cdot \# \mathcal{M}_{t_0' }(\alpha, \gamma'_+). \\
\end{array}$$
3. Observe the degree reduction for the resulting ECH index 1 component $C_+' \in {{{\mathcal M}}}_{t_0'}(\alpha,\gamma'_+)$ in the handleslide, namely, $${\operatorname}{deg}(C_+')< {\operatorname}{deg}(C_+)$$ because the index 0 ECH handleslide curve must always have positive degree.
We repeat this process until either the resulting index 1 curve $C_+^{(k)}\in{{{\mathcal M}}}_{{t}^{(k)}_0}(\alpha, \gamma^{(k)}_+)$ arising from associated to $\left\{{\mathcal{J}}^{(k)}_t\right\}$ for ${t \in \left[0,t_0^{(k-1)}\right]}$ at the “next" handleslide at time $t_0^{(k)}$ can no longer degenerate via handleslides or is a degree 1 curve. (Note that we can also stop at degree 2, since transversality for degree 1 curves can be achieved by $S^1$-invariant domain dependent almost complex structures). This permits us to conclude that $$\# {{{\mathcal M}}}_{t_0}(\alpha, \gamma_+) =0.$$ for all admissible $\gamma_+$ with $I(\alpha,\gamma_+)=1$ as desired.
Computation of $ECH$ {#sec:finalcomp}
====================
In this section we prove Theorem \[thm:mainthm\].
Before invoking the results in §\[sec:modspcs\]-§\[handleslides\] to prove the first and second conclusions of Theorem \[thm:mainthm\], which relate the chain complex $\lim_{L\to\infty}ECH_*^L(Y,\lambda_{\varepsilon(L)},\Gamma)$ to $\Lambda^*H_*(\Sigma_g,{{\mathbb Z}}_2)$, we must first relate $\lim_{L\to\infty}ECH_*^L(Y,\lambda_{\varepsilon(L)},\Gamma)$ to the ECH of the original prequantization bundle. In §\[subsec:directlimit\], we prove
\[thm:dlisECH\] With $Y,\lambda,\varepsilon(L)$ as in Lemma \[lem:efromL\], for any $\Gamma \in H_1(Y;{{\mathbb Z}})$, $$\lim_{L\to\infty}ECH_*^L(Y,\lambda_{\varepsilon(L)},\Gamma)=ECH_*(Y,\xi,\Gamma).$$
In §\[subsec:pfmainthm\], we will use Theorem \[thm:dlisECH\] to prove Theorem \[thm:mainthm\] by showing the ${{\mathbb Z}}_2$-graded isomorphism of ${{\mathbb Z}}_2$-modules $$\label{eqn:computedl}
\bigoplus_{\Gamma\in H_1(Y)}\lim_{L\to\infty}ECH_*^L(Y,\lambda_{\varepsilon(L)},\Gamma)\cong\Lambda^*H_*(\Sigma_g;{{\mathbb Z}}_2).$$ The idea of the proof is as follows. As a consequence of Lemma \[lem:HYreps\] (i), the groups $ECC^L_*(Y,\lambda_{\varepsilon(L)},\Gamma;J)$ are all zero unless $\Gamma$ is in the ${{\mathbb Z}}_{-e}$ summand of $H_1(Y)$. We abuse notation by using $\Gamma$ to also indicate the corresponding element of ${{\mathbb Z}}_{-e}$. By Proposition \[prop:directlimitcomputesfiberhomology\] we can restrict attention to orbits above critical points of the Morse function $H$, and by the analysis of §\[sec:modspcs\]-§\[handleslides\], the ECH differential “agrees" with the Morse differential in the sense that if a $J$-holomorphic curve count contributing to the ECH differential were nonzero, then it must equal a count of gradient flow lines defining the Morse differential on $\Sigma_g$. However, because $H$ is perfect, all such counts of gradient flow lines are zero. By analyzing the implications of index parity and the fact that hyperbolic orbits can appear with multiplicity at most one, we prove that the chain complexes have zero differential and satisfy $$\label{eqn:dlG}
\lim_{L\to\infty}ECH^L_*(Y,\lambda_{\varepsilon(L)},\Gamma)\cong\bigoplus_{d\in{{\mathbb Z}}_{\geq0}}\Lambda^{\Gamma+(-e)d}H_*(\Sigma_g;{{\mathbb Z}}_2)$$ as ${{\mathbb Z}}_2$-graded ${{\mathbb Z}}_2$-modules, which implies (\[eqn:computedl\]).
We conclude this introductory section by proving the third and fourth conclusions of Theorem \[thm:mainthm\], assuming the first and second.
In §\[subsec:pfmainthm\] we will show that (\[eqn:dlG\]) follows from the natural map taking an admissible orbit set to the wedge product the homology classes of the critical points to which ${\mathfrak{p}}$ sends the orbits. Note that the images of generators with ${\operatorname}{deg}(\alpha,e_-^\Gamma)=d$ lie in $\Lambda^{\Gamma+(-e)d}H_*(\Sigma_g;{{\mathbb Z}}_2)$. The ${{\mathbb Z}}$-valued grading of the image of $e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+}$ under this map is $$|e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+}|_\bullet = m_1+\cdots+m_{2g}+2m_+ = M+m_+-m_- = \Gamma-ed+m_+-m_-,$$ where $d={\operatorname}{deg}(e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+},e_-^\Gamma)=\frac{M-\Gamma}{-e}$. On the other hand, $$\begin{aligned}
I(e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+},e_-^\Gamma)&=-ed^2+(\chi(\Sigma_g)+2\Gamma)d+m_+-m_-+\Gamma
\\&=-ed^2+(\chi(\Sigma_g)+2\Gamma)d+|e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+}|_\bullet+ed.\end{aligned}$$
Direct limits and Seiberg-Witten Floer cohomology {#subsec:directlimit}
-------------------------------------------------
Because there are no Morse-Bott methods for ECH, we must compute $ECH_*(Y,\xi,\Gamma)$ by relating it to $\lim_{L\to\infty}ECH^L_*(Y,\lambda_{\varepsilon(L)},\Gamma)$. Our discussion so far allows us to understand the latter. In this section we prove Theorem \[thm:dlisECH\], obtaining $$\lim_{L\to\infty}ECH^L_*(Y,\lambda_{\varepsilon(L)},\Gamma)=ECH_*(Y,\xi,\Gamma).$$
It is not possible to prove Theorem \[thm:dlisECH\] solely via ECH. This is because we cannot relate the filtered homologies computed for $\varepsilon>0$ to any chain complex when $\varepsilon=0$, since the contact form is degenerate and the filtered homology is not defined for any $L$. However, ECH is isomorphic to Seiberg-Witten Floer cohomology (including at the level of filtrations), and in the latter case the degeneracy of $\lambda$ is not an issue. The key result is the last equation in [@cc2 §3.5], whose role we explain in the proof of Theorem \[thm:dlisECH\] in §\[sssec:pfThm7.1\] below.
In §\[sssec:SWFc\] we review Seiberg-Witten Floer cohomology. In §\[sssec:cfpert\] we describe the perturbation of the Seiberg-Witten equations which allows us to incorporate a contact form into the chain complex. In §\[sssec:HML\] we explain an energy filtration analogous to that in ECH and collect results from [@cc2] about cobordism maps on filtered Seiberg-Witten Floer cohomology necessary for our arguments in §\[sssec:HML\]. Our review follows §[@cc2 §2]. The proof of Theorem \[thm:dlisECH\] is completed in §\[sssec:pfThm7.1\].
### Seiberg-Witten Floer cohomology {#sssec:SWFc}
For a definition of Seiberg-Witten Floer cohomology in our setting, we refer the reader to [@cc2 §2]. In this section we quickly review the necessary notation, following [@cc2 §2].
We consider closed oriented connected three-manifolds $Y$ with Riemannian metrics $g$. Recall that a *spin-c structure* on $Y$ is a rank two Hermitian vector bundle $\mathbb{S}$ on $Y$ together with a *Clifford multiplication* $\text{cl}:TY\to\text{End}(\mathbb{S})$. We denote a spin-c structure by $\mathfrak{s}=(\mathbb{S},\text{cl})$. Sections of $\mathbb{S}$ are called *spinors*. A *spin-c connection* on a spin-c structure $\mathfrak{s}$ is a connection $\mathbb{A}_\mathbb{S}$ on $\mathbb{S}$ which is compatible with the Clifford multiplication, meaning that if $v$ is a vector field on $Y$ and $\psi$ is a spinor, then $$\nabla_{\mathbb{A}_\mathbb{S}}(\text{cl}(v)\psi)=\text{cl}(\nabla v)\psi+\text{cl}(v)\nabla_{\mathbb{A}_\mathbb{S}}\psi,$$ where $\nabla v$ denotes the covariant derivative of $v$ with respect to the Levi-Civita connection of $g$. Such a spin-c connection is equivalent to a Hermitian connection $\mathbb{A}$ on the determinant line bundle $\det(\mathbb{S})$. The *Dirac operator* $D_\mathbb{A}$ of $\mathbb{A}_\mathbb{S}$ is the composition $$\label{eqn:Do}
C^\infty(Y;\mathbb{S})\overset{\nabla_{\mathbb{A}_\mathbb{S}}}{\to}C^\infty(Y;T^*Y\otimes\mathbb{S})\overset{\text{cl}}{\to}C^\infty(Y;\mathbb{S}),$$ where $\text{cl}$ is the composition $T^*Y\overset{g}{\cong}TY\overset{\text{cl}}{\to}\text{End}(\mathbb{S})$.
Let $\eta$ be an exact 2-form on $Y$. Let $\mathbb{A}$ be a connection on $\det(\mathbb{S})$ and $\Psi$ be a spinor. Define a bundle map $\tau:\mathbb{S}\to iT^*Y$ by $\tau(\Psi)(v)=g(\text{cl}(v)\Psi,\Psi)$. The *Seiberg-Witten equations* for $(\mathbb{A},\Psi)$ with perturbation $\eta$ are $$\label{eqn:SWeqns}
D_\mathbb{A}\Psi=0 \text{ and }*F_\mathbb{A}=\tau(\Psi)+i*\eta.$$ Seiberg-Witten Floer cohomology is generated by certain solutions $(\mathbb{A},\Psi)$ to the (\[eqn:SWeqns\]). The *gauge group* $\mathcal{G}:=C^\infty(Y;S^1)$ acts on the set of all pairs $(\mathbb{A},\Psi)$ by $$u\cdot(\mathbb{A};\Psi):=(\mathbb{A}-2u^{-1}du,u\Psi),$$ and if $(\mathbb{A},\Psi)$ is a solution to the Seiberg-Witten equations, so is $u\cdot(\mathbb{A},\Psi)$. Solutions are *(gauge) equivalent* if they are equivalent under the action of $\mathcal{G}$. If $\eta$ is generic then modulo gauge equivalence there are only finitely many solutions with $\Psi\not\equiv0$, and each is cut out transversely. Such solutions are called *irreducible* (those with $\Psi\equiv0$ are *reducible*). We assume $\eta$ is sufficiently generic in this way.
Denote by $\widehat{CM}^*_\text{irr}$ the free ${{\mathbb Z}}_2$-module generated by the irreducible solutions to the Seiberg-Witten equations, modulo gauge equivalence.
We next describe the part of the Seiberg-Witten differential which maps $\widehat{CM}^*_\text{irr}$ to itself. Let $(\mathbb{A}_\pm,\Psi_\pm)$ be solutions to the Seiberg-Witten equations. An *instanton* from $(\mathbb{A}_-,\Psi_-)$ to $(\mathbb{A}_+,\Psi_+)$ is a smooth one-parameter family $(\mathbb{A}(s),\Psi(s))$ parameterized by $s\in{{\mathbb R}}$, for which $$\begin{aligned}
&\frac{\partial}{\partial s}\Psi(s)=-D_{\mathbb{A}(s)}\Psi(s) \nonumber
\\&\frac{\partial}{\partial s}\mathbb{A}(s)=-*F_{\mathbb{A}(s)}+\tau(\Psi(s))+i*\eta
\\&\lim_{s\to\pm\infty}(\mathbb{A}(s),\Psi(s))=(\mathbb{A}_\pm,\Psi_\pm). \nonumber\end{aligned}$$ The gauge group and ${{\mathbb R}}$ both act on the space of instantons.
If $(\mathbb{A}_\pm,\Psi_\pm)$ are irreducible, then the differential coefficient $\langle\partial(\mathbb{A}_+,\Psi_+),(\mathbb{A}_-,\Psi_-)\rangle$ is a count of instantons from $(\mathbb{A}_-,\Psi_-)$ to $(\mathbb{A}_+,\Psi_+)$, modulo the actions of $\mathcal{G}$ and ${{\mathbb R}}$, living in a moduli space of local expected dimension one. This local expected dimension defines a relative ${{\mathbb Z}}/d(c_1(\mathfrak{s}))$ grading on the chain complex, and the differential increases this grading by one. More generally, so long as there is no moduli space of instantons to $(\mathbb{A}_+,\Psi_+)$ from a reducible solution to the Seiberg-Witten equations of local expected dimension one, then $\partial(\mathbb{A}_+,\Psi_+)\in\widehat{CM}^*_\text{irr}$. For a discussion of further abstract perturbations necessary to obtain the transversality required to fully define the differential, see [@cc2 §2.1]; they are not necessary in our arguments.
There is a differential on the chain complex generated over ${{\mathbb Z}}_2$ by all solutions to the Seiberg-Witten equations, modulo gauge equivalence, whose differential extends the differential from $\widehat{CM}^*_\text{irr}$ to itself discussed above. We will not need to discuss this extension further here, because the key to the proof of Theorem \[thm:dlisECH\] is a filtered version of Seiberg-Witten Floer cohomology whose generators are all irreducible, introduced in §\[sssec:HML\]. However, we do mention that the homology of the chain complex including reducible solutions is denoted by $\widehat{HM}^*(Y,\mathfrak{s};g,\eta)$. Because this homology is independent of the choices of $(g,\eta)$, we denote the canonical isomorphism class of all such homologies by $\widehat{HM}^*(Y,\mathfrak{s})$.
By $\mathfrak{s}_{\xi,\Gamma}$ we denote the spin-c structure $\mathfrak{s}_\xi+PD(\Gamma)$ on $Y$, where $\mathfrak{s}_\xi$ is the spin-c structure determined by $\xi$ as in [@cc2 Example 2.1]. Taubes [@taubesechswf1]-[@taubesechswf4] has shown $$\label{eqn:fullECHHMiso}
ECH_*(Y,\lambda,\Gamma;J)\cong\widehat{HM}^{-*}(Y,\mathfrak{s}_{\xi,\Gamma});$$ here we use the notation for ECH emphasizing the roles of $\lambda$ and $J$, although inherent in the result is the fact that both sides do not depend on $\lambda$ or $J$ but only on $(Y,\xi,\Gamma)$.
### The contact form perturbation of the Seiberg-Witten equations {#sssec:cfpert}
If $Y$ has a contact form $\lambda$, let $J$ be an almost complex structure on $\xi$ which extends to a $\lambda$-compatible almost complex structure on ${{\mathbb R}}\times Y$. From $\lambda$ and $J$ we obtain a metric $g$ for which $g(R,R)=1$ and $g(R,\xi)=0$. In particular, on $\xi$, we have $g(v,w)=\frac{1}{2}d\lambda(v,Jw)$. Any spin-c structure $\mathfrak{s}=(\mathbb{S},\text{cl})$ can be canoncially decomposed into eigenbundles of $\text{cl}(\lambda)$, i.e. $\mathbb{S}=E\oplus\xi E$, where $E$ is the $i$ eigenbundle and concatenation denotes the tensor product of line bundles. In this decomposition, a connection $\mathbb{A}$ on $\det(\mathbb{S})=\xi E^2$ can be written $\mathbb{A}=A_\xi+2A$ for some connection $A$ on $E$. Similarly to (\[eqn:Do\]) we can define the Dirac operator $D_A$ of $A$.
Let $r>0$ and $\mu$ be an exact 2-form satisfying the genericity conditions described in [@cc2 §2.2]. Replacing $D_\mathbb{A}$ with $D_A$ and setting $$\eta=-rd\lambda+2\mu,\; \psi=\frac{1}{\sqrt{2r}}\Psi$$ in (\[eqn:SWeqns\]) gives us the perturbed Seiberg-Witten equations for the pair $(A,\psi)$: $$\label{eqn:pSWeqns}
D_A\Psi=0 \text{ and } *F_A=r(\tau(\Psi)-i\lambda)-\frac{1}{2}*F_{A_\xi}+i*\mu.$$
Taking into account abstract perturbations as in [@cc2 §2.2], the chain complex $\widehat{CM}^*(Y,\mathfrak{s};\lambda,J,r)$ is generated by the solutions to (\[eqn:pSWeqns\]), modulo gauge equivalence, and its homology is denoted $\widehat{HM}^*(Y,\mathfrak{s};\lambda,J,r)$. As in the original chain complex, if $(A_\pm,\psi_\pm)$ are irreducible solutions, then $\langle\partial(A_+,\psi_+),(A_-,\psi_-)\rangle$ is a count (modulo the actions of $\mathcal{G}$ and ${{\mathbb R}}$) of solutions to the perturbed instanton equations $$\begin{aligned}
&\frac{\partial}{\partial s}\psi(s)=-D_{A(s)}\psi(s) \nonumber
\\&\frac{\partial}{\partial s}A(s)=-*F_{A(s)}+r(\tau(\psi(s))-i\lambda)-\frac{1}{2}*F_{A_\xi}+i*\mu
\\&\lim_{s\to\pm\infty}(A(s),\psi(s))=(A_\pm,\psi_\pm), \nonumber\end{aligned}$$ which live in a moduli space of local expected dimension one. Again we denote by $\widehat{CM}^*_\text{irr}$ the component of $\widehat{CM}^*(Y,\mathfrak{s};\lambda,J,r)$ generated by irreducible solutions to (\[eqn:pSWeqns\]). Although the notation $\widehat{CM}^*_\text{irr}$ denotes two subcomplexes, it will be clear from context which it denotes.
### The energy filtration on $\widehat{HM}^*$ {#sssec:HML}
Analogous to the action of Reeb orbit sets, the *energy* of a solution $(A,\psi)$ to the perturbed Seiberg-Witten equations (\[eqn:pSWeqns\]) is defined as $$E(A):=i\int_Y\lambda\wedge F_A.$$ In analogy to $ECH^L_*$, for $L>0$, define $\widehat{CM}^*_L$ to be the submodule of $\widehat{CM}^*_\text{irr}$ generated by the irreducible solutions to (\[eqn:pSWeqns\]) with $E(A)<2\pi L$.
Note that the energy of a reducible solution $(A,0)$ to (\[eqn:pSWeqns\]) is a linear increasing function in $r$, so if $r$ is sufficiently large then the condition that elements of $\widehat{CM}^*_L$ be elements of $\widehat{CM}^*_\text{irr}$ is redundant: if $E(A)<2\pi L$ then if $r$ is large enough, the pair $(A,0)$ cannot be a solution to (\[eqn:pSWeqns\]).
We quote a lemma necessary for defining the homology of the submodule $\widehat{CM}^*_L$:
\[lem:CMLsubcx\] Fix $Y,\lambda, J$ as above and $L\in{{\mathbb R}}$. Suppose that $\lambda$ has no orbit set of action exactly $L$. Fix $r$ sufficiently large, and a 2-form $\mu$ so that all irreducible solutions to (\[eqn:pSWeqns\]) are cut out transversely. Then for every $\mathfrak{s}$ and for every sufficiently small generic abstract perturbation, $\widehat{CM}^*_L(Y,\mathfrak{s};\lambda,J,r)$ is a subcomplex of $\widehat{CM}^*(Y,\mathfrak{s};\lambda,J,r)$.
When the hypotheses of Lemma \[lem:CMLsubcx\] apply, we denote the homology of $\widehat{CM}^*_L(Y,\mathfrak{s};\lambda,J,r)$ by $\widehat{HM}^*_L(Y,\lambda,\mathfrak{s})$. In particular, if $r$ is sufficiently large then this homology is independent of $\mu$ and $r$, and it is also independent of $J$, as shown in [@cc2 Cor. 3.5]. We will use the notation $\widehat{HM}^*_L(Y;\lambda,J,r)$ when we wish to emphasize the roles of $J$ and $r$.
Filtered Seiberg-Witten Floer cohomology is isomorphic to ECH:
Suppose that $\lambda$ is $L$-nondegenerate and $J$ is $ECH^L$-generic (see Lemma \[lem:nicecobmap\]). Then for all $\Gamma\in H_1(Y)$, there is a canonical isomorphism of relatively graded ${{\mathbb Z}}_2$-modules $$\label{eqn:filterediso}
\Psi^L:ECH^L_*(Y,\lambda,\Gamma;J)\overset{\cong}{\to}\widehat{HM}^{-*}_L(Y,\lambda,\mathfrak{s}_{\xi,\Gamma}).$$
Analogous to the cobordism maps on $ECH^L_*$, there are cobordism maps on $\widehat{HM}^*_L$. The following is a modified version of [@cc2 Cor. 5.3 (a)] which keeps track of the spin-c structures in our setting. Note that therefore our notation for the cobordism maps on $\widehat{HM}^*_L$ differs slightly from that of [@cc2].
\[lem:HMLcobmapdef\] Let $(X,\lambda)$ be an exact symplectic cobordism from $(Y_+,\lambda_+)$ to $(Y_-,\lambda_-)$ where $\lambda_\pm$ is $L$-nondegenerate. Let $\mathfrak{s}$ be a spin-c structure on $X$ and let $\mathfrak{s}_\pm$ denote its restrictions to $Y_\pm$, respectively. Let $J_\pm$ be $\lambda_\pm$-compatible almost complex structures. Suppose $r$ is sufficiently large. Fix 2-forms $\mu_\pm$ and small abstract perturbations sufficient to define the chain complexes $\widehat{CM}^*(Y_\pm,\mathfrak{s}_\pm;\lambda_\pm,J_\pm,r)$. Then there is a well-defined map $$\label{eqn:HMLcobmap}
\widehat{HM}^*_L(X,\lambda,\mathfrak{s}):\widehat{HM}^*_L(Y_+,\mathfrak{s}_+;\lambda_+,J_+,r)\to\widehat{HM}^*_L(Y_-,\mathfrak{s}_-;\lambda_-,J_-,r),$$ depending only on $X,\mathfrak{s},\lambda,L,r,J_\pm,\mu_\pm$, and the perturbations, such that if $L'<L$ and if $\lambda_\pm$ are also $L'$-nondegenerate, then the diagram $$\label{eqn:HMLcd}
\xymatrixcolsep{5pc}\xymatrix{
\widehat{HM}^*_{L'}(Y_+,\mathfrak{s}_+;\lambda_+,J_+,r) \ar[r]^{\widehat{HM}^*_{L'}(X,\lambda,\mathfrak{s})} \ar[d] & \widehat{HM}^*_{L'}(Y_-,\mathfrak{s}_-;\lambda_-,J_-,r) \ar[d]
\\\widehat{HM}^*_L(Y_+,\mathfrak{s}_+;\lambda_+,J_+,r) \ar[r]_{\widehat{HM}^*_L(X,\lambda,\mathfrak{s})} & \widehat{HM}^*_L(Y_-,\mathfrak{s}_-;\lambda_-,J_-,r)
}$$ commutes, where the vertical arrows are induced by inclusions of chain complexes.
Finally, in order to define direct systems, we will need to compose cobordism maps on $\widehat{HM}^*_L$. For certain cobordisms (e.g. those defining the direct system $\lim_{\varepsilon\to0}\widehat{HM}^{-*}_{L(\varepsilon)}(Y,\lambda_\varepsilon,\mathfrak{s}_{\xi,\Gamma})$ it is enough to use the composition property [@cc2 Lem. 3.4 (b)], but we will need to understand cobordism maps on slightly more complex cobordisms as well. The next lemma is a version of [@cc2 Prop. 5.4] explaining the composition law for $\widehat{HM}^*_L$ in our setting.
In the following lemma, we will consider the following composition. Assume $\varepsilon''<\varepsilon'<\varepsilon$. We consider the exact symplectic cobordism $([\varepsilon'',\varepsilon]\times Y,(1+s{\mathfrak{p}}^*H)\lambda)$ from $(Y,\lambda_\varepsilon)$ to $(Y,\lambda_{\varepsilon''})$. It is the composition of the exact symplectic cobordism $([\varepsilon'',\varepsilon']\times Y,(1+s{\mathfrak{p}}^*H)\lambda)$ from $(Y,\lambda_{\varepsilon'})$ to $(Y,\lambda_{\varepsilon''})$ with the exact symplectic cobordism $([\varepsilon',\varepsilon]\times Y,(1+s{\mathfrak{p}}^*H)\lambda)$ from $(Y,\lambda_\varepsilon)$ to $(Y,\lambda_{\varepsilon'})$ in the sense of [@cc2 §1.5], where $\lambda_\varepsilon, \lambda_{\varepsilon'}$, and $\lambda_{\varepsilon''}$ are $L$-nondegenerate. We also assume $J, J'$, and $J''$ are $\lambda_\varepsilon$-, $\lambda_{\varepsilon'}$-, and $\lambda_{\varepsilon''}$-compatible almost complex structures, respectively. Further, we choose a spin-c structure $\mathfrak{s}''$ on $[\varepsilon'',\varepsilon]\times Y$ which restricts to spin-c structures $\mathfrak{s}'$ and $\mathfrak{s}$ on $[\varepsilon'',\varepsilon']\times Y$ and $[\varepsilon',\varepsilon]\times Y$, respectively, where $\mathfrak{s}'$ restricts to $\mathfrak{s}_2$ on $\{\varepsilon''\}\times Y$, $\mathfrak{s}$ restricts to $\mathfrak{s}_0$ on $\{\varepsilon\}\times Y$, and both $\mathfrak{s}'$ and $\mathfrak{s}$ restrict to $\mathfrak{s}_1$ on $\{\varepsilon'\}\times Y$. Finally we choose abstract perturbations and $r$ large enough to define the chain complexes $\widehat{CM}^*_L$.
\[lem:HMLcomp\] The maps of Lemma \[lem:HMLcobmapdef\] for the above data satisfy $$\widehat{HM}^*_L([\varepsilon'',\varepsilon]\times Y,(1+s{\mathfrak{p}}^*H)\lambda,\mathfrak{s})=\widehat{HM}^*_L([\varepsilon'',\varepsilon']\times Y,(1+s{\mathfrak{p}}^*H)\lambda,\mathfrak{s}^+)\circ\widehat{HM}^*_L([\varepsilon',\varepsilon]\times Y,(1+s{\mathfrak{p}}^*H)\lambda,\mathfrak{s}^-).$$
Note that [@cc2 Prop. 5.4] does not discuss the spin-c structures, but since it is proved with a neck-stretching argument for holomorphic curves whose ends must be homologous, it will preserve spin-c structures in the case considered in Lemma \[lem:HMLcomp\], see [@cc2 Rmk. 1.10].
### $ECH_*$ via $\widehat{HM}^*_L$ {#sssec:pfThm7.1}
In this section we prove Theorem \[thm:dlisECH\] using the machinery from Seiberg-Witten theory reviewed in the previous sections.
Because all $\lambda_\varepsilon$ have the same contact structure $\xi$ as $\lambda$, we have $$\lim_{L\to\infty}ECH^L_*(Y,\lambda_\varepsilon,\Gamma)=ECH_*(Y,\xi,\Gamma).$$ However, if $\varepsilon>\varepsilon(L)$ then we cannot compute $ECH^L_*(Y,\lambda_\varepsilon,\Gamma)$ using our methods, because the chain complex $ECC^L_*(Y,\lambda_\varepsilon,\Gamma;J)$ may contain orbits which do not project to critical points of $H$. If $\varepsilon$ is fixed and only $L$ is sent to $\infty$, then because $\varepsilon(L)\sim\frac{1}{L}$, there will be some $L$ beyond which $\varepsilon>\varepsilon(L)$ and we can no longer compute $ECH^L_*(Y,\lambda_\varepsilon,\Gamma)$.
Instead, we will explain how to obtain $ECH_*(Y,\xi,\Gamma)$ from $\lim_{L\to\infty}ECH^L_*(Y,\lambda_{\varepsilon(L)},\Gamma)$. Let $L(\varepsilon)$ denote the value of $L$ for which $\varepsilon(L)=\varepsilon$. Note that for all $L<L(\varepsilon)$, the generators of $ECH^L_*(Y,\lambda_\varepsilon,\Gamma)$ all project to critical points of $H$. In particular, $$ECH^{L(\varepsilon)}_*(Y,\lambda_\varepsilon,\Gamma)=ECH^L_*(Y,\lambda_{\varepsilon(L)},\Gamma)$$ and therefore $$\lim_{L\to\infty}ECH^L_*(Y,\lambda_{\varepsilon(L)},\Gamma)=\lim_{\varepsilon\to0}ECH^{L(\varepsilon)}_*(Y,\lambda_\varepsilon,\Gamma).$$
To prove Theorem \[thm:dlisECH\] we will prove the following sequence of isomorphisms: $$\begin{aligned}
\lim_{\varepsilon\to0}ECH_*^{L(\varepsilon)}(Y,\lambda_\varepsilon,\Gamma)&\cong\lim_{\varepsilon\to0}\widehat{HM}^{-*}_{L(\varepsilon)}(Y,\lambda_\varepsilon,\mathfrak{s}_{\xi,\Gamma})\label{eqn:ECHeHMe}
\\&\cong\lim_{\varepsilon\to0}\lim_{L\to\infty}\widehat{HM}^{-*}_L(Y,\lambda_\varepsilon,\mathfrak{s}_{\xi,\Gamma})\label{eqn:factoringdleL}
\\&\cong\lim_{\varepsilon\to0}\widehat{HM}^{-*}(Y,\mathfrak{s}_{\xi,\Gamma})\label{eqn:sendLtoinfty}
\\&\cong ECH_*(Y,\xi,\Gamma).\label{eqn:HMtoECH}\end{aligned}$$ Note that the groups $\widehat{HM}^{-*}_L(Y,\lambda_\varepsilon,\mathfrak{s}_{\xi,\Gamma})$ on the right hand side of the second equation (\[eqn:factoringdleL\]) are only defined for $L$ and $\varepsilon$ such that $\lambda_\varepsilon$ has no Reeb orbit sets of action exactly $L$. This includes all $L\leq L(\varepsilon)$ (similarly all $\varepsilon<\varepsilon(L)$); for a given $\varepsilon$, this is still a full measure set of $L$ because for generic perfect $H$, the set of actions of orbits of $X_H$ is discrete.
The direct limit on the right hand side of the first equation (\[eqn:ECHeHMe\]) is defined using either composition in the commutative diagram (\[eqn:HMLcd\]) and the exact symplectic cobordisms $([\varepsilon',\varepsilon]\times Y,(1+s{\mathfrak{p}}^*H)\lambda)$ discussed in the proof of Proposition \[prop:directlimitcomputesfiberhomology\] in §\[subsec:PQBgens\]. That these maps compose properly is derived from the composability of the cobordism maps in Lemma \[lem:HMLcomp\], following the same logic in the proof of Proposition \[prop:directlimitcomputesfiberhomology\] to prove that the maps in the direct system on the left hand side of (\[eqn:ECHeHMe\]) compose. (\[eqn:ECHeHMe\]) follows from the isomorphism (\[eqn:filterediso\]).
The direct limit in $L$ on the right hand side of the second equation (\[eqn:factoringdleL\]) is defined using the maps induced on homology by the inclusion of chain complexes. The direct limit in $\varepsilon$ is defined using the cobordism maps (\[eqn:HMLcobmap\]) and the same exact symplectic cobordisms as in the above paragraph.
To obtain (\[eqn:factoringdleL\]), we first show that the “obvious" map is well-defined: $$F:\lim_{\varepsilon\to0}\widehat{HM}^{-*}_{L(\varepsilon)}(Y,\lambda_\varepsilon,\mathfrak{s}_{\xi,\Gamma})\to\lim_{\varepsilon\to0}\lim_{L\to\infty}\widehat{HM}^{-*}_L(Y,\lambda_\varepsilon,\mathfrak{s}_{\xi,\Gamma}).$$ Note that every element of the direct system on the left hand side appears on the right hand side; the map $F$ is given by sending the equivalence class of $a\in\widehat{HM}^{-*}_{L(\varepsilon)}(Y,\lambda_\varepsilon,\mathfrak{s}_{\xi,\Gamma})$ as a member of the left hand direct limit to its equivalence class as a member of the right hand direct limit. Throughout the proof of (\[eqn:factoringdleL\]) we will use the notation $$\widehat{HM}^{-*}_L(\varepsilon):=\widehat{HM}^{-*}_{L}(Y,\lambda_\varepsilon,\mathfrak{s}_{\xi,\Gamma}).$$
To show that $F$ is well-defined, assume $a\in\widehat{HM}^{-*}_{L(\varepsilon)}(\varepsilon), b\in\widehat{HM}^{-*}_{L(\varepsilon')}(\varepsilon')$, and $a\sim b$ as elements of $\lim_{\epsilon\to0}\widehat{HM}^{-*}_{L(\varepsilon)}(\varepsilon)$. That means there is some $\varepsilon''$ for which the image of $a$ under the map $$\label{eqn:aLHS}
\widehat{HM}^{-*}_{L(\varepsilon)}(\varepsilon)\to\widehat{HM}^{-*}_{L(\varepsilon'')}(\varepsilon'')$$ equals the image of $b$ under the map $$\widehat{HM}^{-*}_{L(\varepsilon')}(\varepsilon')\to\widehat{HM}^{-*}_{L(\varepsilon'')}(\varepsilon'').$$ Call this shared image $c$. On the right hand side, we also have $F(a)\sim F(c)$ under the composition $$\label{eqn:RHScomp}
\widehat{HM}^{-*}_{L(\varepsilon)}(\varepsilon)\to\widehat{HM}^{-*}_{L(\varepsilon'')}(\varepsilon)\to\widehat{HM}^{-*}_{L(\varepsilon'')}(\varepsilon''),$$ where the first map comes from the first direct limit (and is defined because $\varepsilon''<\varepsilon$) and the second from the second, because (\[eqn:RHScomp\]) is precisely one of the compositions defining (\[eqn:aLHS\]) in the commutative diagram (\[eqn:HMLcd\]). Similarly, we have $F(b)\sim F(c)$.
Next we show that $F$ is an injection. Let $a\in\widehat{HM}^{-*}_{L(\varepsilon)}(\varepsilon), b\in\widehat{HM}^{-*}_{L(\varepsilon')}(\varepsilon')$, but we do not assume $a\sim b$ on the left hand side. Assume $F(a)\sim F(b)$ on the right hand side. We want to show $a\sim b$. Because $F(a)\sim F(b)$, there are $L''\geq L(\varepsilon), L(\varepsilon')$ and $\varepsilon''\leq\varepsilon,\varepsilon'$ for which the image of $a$ under $$\widehat{HM}^{-*}_{L(\varepsilon)}(\varepsilon)\to\widehat{HM}^{-*}_{L''}(\varepsilon)\to\widehat{HM}^{-*}_{L''}(\varepsilon'')$$ equals the image of $b$ under $$\widehat{HM}^{-*}_{L(\varepsilon')}(\varepsilon')\to\widehat{HM}^{-*}_{L''}(\varepsilon)\to\widehat{HM}^{-*}_{L''}(\varepsilon'').$$ Call this shared image $d$. Let $\varepsilon'''=\min\{\varepsilon'',\varepsilon(L'')\}$. We have $d=F(c)$, where $c$ is the image of $a$ under $$\widehat{HM}^{-*}_{L(\varepsilon)}(\varepsilon)\to\widehat{HM}^{-*}_{L(\varepsilon''')}(\varepsilon'''),$$ as well as the image of $b$ under $$\widehat{HM}^{-*}_{L(\varepsilon')}(\varepsilon')\to\widehat{HM}^{-*}_{L(\varepsilon''')}(\varepsilon'''),$$ both again by the definition (\[eqn:HMLcd\]) of the maps in the direct limit on the left hand side. Therefore $a\sim b$ on the left hand side.
Finally we show that $F$ is a surjection, essentially because the direct systems on the right hand side of (\[eqn:factoringdleL\]) are very simple. If $d\in\widehat{HM}^{-*}_L(\varepsilon)$ with $L<L(\varepsilon)$, then it is eventually equivalent to some element of $\widehat{HM}^{-*}_{L(\varepsilon)}(\varepsilon)$ because there is an inclusion map sending $\widehat{HM}^{-*}_L(\varepsilon)$ to $\widehat{HM}^{-*}_{L(\varepsilon)}(\varepsilon)$. If $d\in\widehat{HM}^{-*}_L(\varepsilon)$ with $L>L(\varepsilon)$, then it is eventually equivalent to some element of $\widehat{HM}^{-*}_L(\varepsilon(L))$ because there is a cobordism map sending $\widehat{HM}^{-*}_L(\varepsilon)$ to $\widehat{HM}^{-*}_{L}(\varepsilon(L))$.
We have that (\[eqn:sendLtoinfty\]) follows from the last equation of [@cc2 §3.5], which itself follows from [@taubesechswf1 Thm. 4.5]. Note that although the equation in [@cc2 §3.5] is only required to hold for nondegenerate $\lambda$, it is true for all $\lambda$.
We obtain (\[eqn:HMtoECH\]) by the fact that the groups $\widehat{HM}^{-*}(Y,\mathfrak{s}_{\xi,\Gamma})$ on the right hand side of (\[eqn:sendLtoinfty\]) are all equal and independent of $\varepsilon$, together with the isomorphism (\[eqn:fullECHHMiso\]).
Proof of the main theorem {#subsec:pfmainthm}
-------------------------
We split the proof of the first two conclusions of Theorem \[thm:mainthm\] into the case where $g>0$ and the case where $g=0$. This is because our methods for $g>0$ with domain-dependent almost complex structures do not work when $g=0$: see Remark \[unstable-cases\]. Instead, because a perfect Morse function on $S^2$ has only elliptic critical points, the differential vanishes by index parity (Theorem \[thm:Iproperties\]), as explained in §\[sssec:pfg=0\].
### Proof of the main theorem when $g>0$ {#sssec:pfg>0}
In this section we prove the first two conclusions of Theorem \[thm:mainthm\] in the case $g>0$.
By Theorem \[thm:dlisECH\] and the discussion in the introduction to §\[sec:finalcomp\], it is enough to show (\[eqn:dlG\]). By Proposition \[prop:directlimitcomputesfiberhomology\], the direct limit on the left hand side of (\[eqn:dlG\]) is the homology of the chain complex generated by orbit sets of the form $e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+}$ in the class $\Gamma$. We will denote this chain complex by $(C_*,\partial)$.
Recall that we are abusing notation by thinking of $\Gamma$ as an element of ${{\mathbb Z}}_{-e}$ rather than as an element of the ${{\mathbb Z}}_{-e}$ summand of $H_1(Y)$. Therefore $e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+}$ is in the class $\Gamma$ precisely when $M=m_-+m_1+\cdots+m_{2g}+m_+=\Gamma+(-e)d$ for some $m\in{{\mathbb Z}}_{\geq0}$.
We claim that the chain complex $C_*$ splits into the submodules $\Lambda^{\Gamma+(-e)d}H_*(\Sigma_g;{{\mathbb Z}}_2)$ on the right hand side of (\[eqn:dlG\]). Let $E_-$ denote the index zero generator of $H_*(\Sigma_g;{{\mathbb Z}}_2)$, let $H_i$ denote the $i^\text{th}$ index one generator, and let $E_+$ denote the index two generator. If $A$ is a generator of $H_*(\Sigma_g;{{\mathbb Z}}_2)$ let $A^m$ denote the $m$-fold wedge product $A\wedge\cdots\wedge A$, where $m=0$ indicates that there is no factor of $A$ in the wedge product. The ${{\mathbb Z}}_2$ grading on $E_-^{m_-}\wedge H_1^{m_1}\wedge\cdots\wedge H_{2g}^{m_{2g}}\wedge E_+^{m_+}$ is the ${{\mathbb Z}}_2$ equivalence class of $$\begin{aligned}
m_-|E_-|+m_1|H_1|+\cdots+m_{2g}|H_{2g}|+m_+|E_+|&\equiv_2 0+m_1+\cdots+m_{2g}+2m_+ \nonumber
\\&\equiv_2 m_1+\cdots+m_{2g} \nonumber
\\&\equiv_2 I(e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+},e_-^\Gamma) \label{eqn:Imod2}\end{aligned}$$ where (\[eqn:Imod2\]) follows from the Index Parity property of the ECH index, see Theorem \[thm:Iproperties\] (iv). Moreover, the exterior product $\Lambda^{\Gamma+(-e)d}H_*(\Sigma_g;{{\mathbb Z}}_2)$ consists precisely of wedge products of the $E_\pm$ and $H_i$ of total multiplicity $\Gamma+(-e)d$, where for all $A,B$ generators of $H_*(\Sigma_g;{{\mathbb Z}}_2)$, $$B\wedge A=(-1)^{|A|\cdot|B|}A\wedge B.$$ Therefore all elements of the exterior product can be rearranged so that all $E_-$ terms occur first and all $E_+$ terms occur last. However, no $H_i$ term can occur twice, and because we are using ${{\mathbb Z}}_2$ coefficients, $$H_j\wedge H_i=(-1)^{1\cdot1}H_i\wedge H_j=(-1)H_i\wedge H_j=H_i\wedge H_j,$$ hence all $H_i$ terms can be arranged in the order of ascending index. Therefore the map $$e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+}\leftrightarrow E_-^{m_-}\wedge H_1^{m_1}\wedge\cdots\wedge H_{2g}^{m_{2g}}\wedge E_+^{m_+}$$ defines a bijection $$C_*\leftrightarrow \bigoplus_{d\in{{\mathbb Z}}_{\geq0}}\Lambda^{\Gamma+(-e)d}H_*(\Sigma_g;{{\mathbb Z}}_2),$$ which respects the splitting over $d$ in the sense that there is a splitting of $C_*$ over $d$ such that $M=\Gamma+(-e)d$, and respecting the mod two gradings.
Finally, we show that the differential vanishes, so that these submodules are in fact subcomplexes, and (\[eqn:dlG\]) holds. By Corollary \[nogenusJ\] we can assume that all all ECH index one curves contributing to $\partial$ have degree zero and thus are cylinders by Lemma \[lem:d0g0\]. By Proposition \[cylinder-to-morse\], we know that any moduli space of cylinders which could contribute a nonzero coefficient to $\partial$ must have the same mod two count as the space of currents consisting of trivial cylinders together with a single ECH index one cylinder above a gradient flow line of $H$ which contributes to the Morse differential, and that the count must be the same as the corresponding count for the Morse differential. Specifically, if there is a pseudoholomorphic current contributing to $\langle\partial(e_-^{m_-}h_1^{m_1}\cdots h_{2g}^{m_{2g}}e_+^{m_+}), e_-^{m'_-}h_1^{m'_1}\cdots h_{2g}^{m'_{2g}}e_+^{m'_+}\rangle$ then either
- There is some $h_i$ for which $m_i=1, m'_i=0$, and $m'_-=m_-+1, m'_+=m_+$, and for $j\neq i$, $m'_j=m_j$.
- There is some $h_i$ for which $m_i=0, m'_i=1$, and $m'_+=m_+-1, m'_-=m_-$, and for $j\neq i$, $m'_j=m_j$.
In either case, the mod two count of all such pseudoholomorphic currents must equal the count of Morse flow lines from ${\mathfrak{p}}(h_i)$ to ${\mathfrak{p}}(e_-)$ or ${\mathfrak{p}}(e_+)$ to ${\mathfrak{p}}(h_i)$, respectively. Because $H$ is perfect, both of the latter counts are zero.
### Proof of the main theorem when $g=0$ {#sssec:pfg=0}
In this section we prove the first two conclusions of Theorem \[thm:mainthm\] in the case $g=0$. Note that the manifolds in question are the lens spaces $L(-e,1)$.
\[thm:echS2\] Let $(Y,\lambda)$ be the prequantization bundle of Euler number $e\in{{\mathbb Z}}_{<0}$ over $S^2$ with contact structure $\xi$. Let $\Gamma$ be a class in $H_1(Y;{{\mathbb Z}})={{\mathbb Z}}_{-e}$. Then $$\label{eqn:mainthmg=0}
\bigoplus_{\Gamma\in H_1(Y)}ECH_*(Y,\xi,\Gamma)\cong\Lambda^*H_*(S^2;{{\mathbb Z}}_2)$$ as ${{\mathbb Z}}_2$-graded ${{\mathbb Z}}_2$-modules, which proves Theorem \[thm:mainthm\] when $g=0$. Furthermore, $$\label{eqn:ECHg=0}
ECH_*(Y,\xi,\Gamma)=\begin{cases}
{{\mathbb Z}}&\text{ if $*\in2{{\mathbb Z}}_{\geq0}$}
\\0&\text{ else}
\end{cases}$$
By Theorem \[thm:dlisECH\], it is enough to understand each $\lim_{L\to\infty}ECH^L_*(Y,\lambda_{\varepsilon(L)},\Gamma)$, which, by Proposition \[prop:directlimitcomputesfiberhomology\], is the homology of the chain complex generated by orbit sets of the form $e_-^{m_-}e_+^{m_+}$ in the class $\Gamma$. We will denote this chain complex by $(C_*,\partial)$.
Because a perfect Morse function on $S^2$ has only elliptic critical points, index parity (Theorem \[thm:Iproperties\]) tells us that $\partial=0$. Therefore $$\label{eqn:g=0diffl0}
ECH_*(Y,\xi,\Gamma)\cong C_*$$ Recall that we are abusing notation by thinking of $\Gamma$ as an element of ${{\mathbb Z}}_{-e}$ rather than as an element of $H_1(Y)$. The generator $e_-^{m_-}e_+^{m_+}$ is therefore in the class $\Gamma$ if $m_-+m_+=\Gamma+(-e)d$ for some $d\in{{\mathbb Z}}_{\geq0}$.
If $E_-,E_+$ denote the grading zero and two homology classes in $H_*(S^2;{{\mathbb Z}}_2)$, respectively, then $\Lambda^\delta H_*(S^2;{{\mathbb Z}}_2)$ is the group generated over ${{\mathbb Z}}_2$ by terms $E_-^{m_-}E_+^{m_+}$ with $m_-+m_+=\delta$, where $E^m$ denotes the $m$-fold wedge product $E\wedge\cdots\wedge E$. By a simplification of the proof of the analogous fact in §\[sssec:pfg>0\], we obtain $$C_*\cong\bigoplus_{d\in{{\mathbb Z}}_{\geq0}}\Lambda^{\Gamma+(-e)d}H_*(S^2;{{\mathbb Z}}_2)$$ as ${{\mathbb Z}}_2$-graded ${{\mathbb Z}}_2$-modules. Invoking (\[eqn:g=0diffl0\]) and taking the sum over all $\Gamma\in H_1(Y)$ proves (\[eqn:mainthmg=0\]).
To prove the improvement (\[eqn:ECHg=0\]), we will prove that the ECH index is a bijection from the generators of $(C_*,\partial)$ to $2{{\mathbb Z}}_{\geq0}$ which sends $e_-^\Gamma$ to zero. Therefore $$C_*=\begin{cases}
{{\mathbb Z}}&\text{ if $*\in2{{\mathbb Z}}_{\geq0}$}
\\0&\text{ else}
\end{cases}$$ and by (\[eqn:g=0diffl0\]), we obtain (\[eqn:ECHg=0\]).
The remainder of the proof of (\[eqn:ECHg=0\]) therefore consists of proving that the ECH index is a bijection from generators $e_-^{m_-}e_+^{m_+}$ to $2{{\mathbb Z}}_{\geq0}$. Our perspective is similar to that of Choi [@choi], but the contact forms are not the same (by choosing a specific perturbation function $H$ we could make them essentially the same, but do not need to do so).
The index bijection factors through a bijection to a lattice in the fourth quadrant in ${{\mathbb R}}^2$ determined by the vertical axis and the line through the origin of slope $\frac{1}{-e}$. We will first describe the bijection between generators of $C_*$ and this lattice, and then describe the bijection between the lattice and the nonnegative even integers.
The generators of $C_*$ in the class $\Gamma$ are of the form $e_-^{m_-}e_+^{m_+}$, where $m_\pm\in{{\mathbb Z}}_{\geq0}$ and $m_-+m_+\equiv_{-e}\Gamma$. See Figure \[fig:lattice\_fig\_722\]. The union of all such generators over $\Gamma\in{{\mathbb Z}}_{-e}$ are in bijection with the intersection of the lattice spanned by $(1,0)$ and $\left(0,\frac{1}{-e}\right)$ with the fourth quadrant determined by the vertical axis and the line through the origin of slope $\frac{1}{-e}$, where the bijection is given by $$e_-^{m_-}e_+^{m_+}\mapsto\left(m_-,\frac{m_--m_+}{-e}\right).$$ The image of $e_-^{m_-}e_+^{m_+}$ is to the right of the vertical axis, inclusive, because $m_-\geq0$, and is below the line through the origin of slope $\frac{1}{-e}$, inclusive, because $$\frac{m_--m_+}{-e}\leq\frac{m_-}{-e}.$$ The map is a bijection because it has an inverse, which can be computed directly from the formula. Let $V(m_-,m_+)$ denote $\left(m_-,\frac{m_--m_+}{-e}\right)$.
[lattice\_fig\_722]{}(-8,128)[$V(0,0)$]{} (149,111)[$V(2,2)$]{} (88,73)[$V(1,3)$]{} (28,32)[$V(0,4)$]{} (269,193)[$V(4,0)$]{} (209,152)[$V(3,1)$]{}
The lattice splits into $-e$ sublattices, each corresponding to the homology class $\Gamma$. Each is spanned by $\left(1,\frac{2}{-e}\right)$ and $(0,1)$, but they can be differentiated by a translation: they contain the points $\left(0,-\frac{\Gamma}{-e}\right)$, respectively.
Next we explain the bijections between each of these sublattices and the nonnegative even integers. The essential idea is the following. Let $T(m_-,m_+)$ denote the triangle bounded by the axes and the line through $V(m_-,m_+)$ of slope $\frac{2}{-e}$. The relative first Chern class records the approximate height of $T(m_-,m_+)$, the relative self intersection term records approximately twice its area; when this line moves to the right and/or down, both the height and area of $T(m_-,m_+)$ increase, so moving the line to the right and down groups points in the lattice into batches of roughly increasing ECH index. The Conley-Zehnder index differentiates between lattice points on the same line of slope $\frac{2}{-e}$ by increasing the index by two as $m_1$ increases and $m_0$ decreases, and makes the indices of the groups on different lines of slope $\frac{2}{-e}$ match up exactly.
Because these correspondences are only approximate when $\Gamma\neq0$, we will explain them in detail to show that the index is a bijection to $2{{\mathbb Z}}_{\geq0}$.
The index difference between $e_-^{m_-}e_+^{m_+}$ and $e_-^\Gamma$ is $$I(e_-^{m_-}e_+^{m_+},e_-^\Gamma)=\frac{2(m_-+m_+-\Gamma)}{-e}+\frac{(m_-+m_+-\Gamma)^2}{-e}+\frac{2\Gamma(m_-+m_+-\Gamma)}{-e}+m_+-m_-+\Gamma.$$ The first term is the relative first Chern class $c_\tau(e_-^{m_-}e_+^{m_+},e_-^\Gamma)$. The triangle has vertices $(0,0),\left(-\frac{m_-+m_+}{-e},0\right),\left(m_-+m_+,\frac{m_-+m_+}{-e}\right)$, so its height is $$\frac{2(m_-+m_+)}{-e}=c_\tau(e_-^{m_-}e_+^{m_+},e_-^\Gamma)+\frac{2\Gamma}{-e}\Leftrightarrow c_\tau(e_-^{m_-}e_+^{m_+},e_-^\Gamma)=\text{Height}(T(m_-,m_+))-\frac{2\Gamma}{-e}.$$
The second two terms comprise the relative self intersection number $Q_\tau(e_-^{m_-}e_+^{m_+},e_-^\Gamma)$. Twice the area of the triangle is $$2\text{Area}(T(m_-,m_+))=\left(\frac{m_-+m_+}{-e}\right)(m_-+m_+),$$ while $$\begin{aligned}
Q_\tau(e_-^{m_-}e_+^{m_+},e_-^\Gamma)&=\frac{(m_-+m_+-\Gamma)^2}{-e}+\frac{2\Gamma(m_-+m_+-\Gamma)}{-e}
\\&=\frac{1}{-e}\left((m_-+m_+)^2-2\Gamma(m_-+m_+)+\Gamma^2+2\Gamma(m_-+m_+)-2\Gamma^2\right)
\\&=2\text{Area}(T(m_-,m_+))-\frac{2\Gamma^2}{-e}.\end{aligned}$$
Notice that we can split the sublattices into lattices along the lines of slope $\frac{2}{-e}$ through $\left(-\frac{M+\Gamma}{-e},0\right)$, where $M\in{{\mathbb Z}}_{\geq0}$. Over each such line, the Conley-Zehnder term ranges from $-M+\Gamma$ to $M+\Gamma$, where $M=m_-+m_+$, and is strictly increasing in $m_+$. Since there is exactly one generator with each value of $m_+$ between zero and $m_-+m_+$ on this line, no values of $I(e_-^{m_-}e_+^{m_+},e_-^\Gamma)$ are repeated on a given line.
Along each line, the triangle $T(m_-,m_+)$ is constant, therefore its height and area are constant, and thus both $c_\tau(e_-^{m_-}e_+^{m_+},e_-^\Gamma)$ and $Q_\tau(e_-^{m_-}e_+^{m_+},e_-^\Gamma)$ are constant. They are also both increasing in $M$. In order to prove the theorem it therefore suffices to show that the smallest value the index takes on the line corresponding to $M+(-e)$ must be two greater than the largest value the index takes on the line corresponding to $M$.
The smallest value the index takes on the line corresponding to $M+(-e)$ is $I(e_-^{M+(-e)},e_-^\Gamma)$, while while the largest value the index takes on the line corresponding to $M$ is $I(e_+^M,e_-^\Gamma)$. It is a straightforward computation to show that $I(e_-^{M+(-e)},e_-^\Gamma)=I(e_+^M,e_-^\Gamma)+2$.
[CFHWI]{} P. Albers, B. Bramham, and C. Wendl, *On nonseparating contact hypersurfaces in symplectic 4-manifolds.* Algebr. Geom. Topol. 10 (2010), no. 2, 697-737.
F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, E. Zehnder, *Compactness results in symplectic field theory.* [Geometry and Topology]{} Vol. 7, 799-888 (2003).
K. Choi, *Combinatorial embedded contact homology for toric contact manifolds*, arXiv:1608.07988, preprint.
K. Cieliebak and K. Mohnke, [*Symplectic hypersurfaces and transversality in Gromov-Witten theory.*]{} J. Symplectic Geom. 5 (2007), no. 3, 281-356.
V. Colin and K. Honda, *Reeb vector fields and open book decompositions*, J. Eur. Math. Soc. 15 (2013) no. 2, 443-507.
D. Cristofaro-Gardiner, M. Hutchings, and B. Zhang, *Generic higher asymptotics of holomorphic curves and applications*, <https://cpb-us-e1.wpmucdn.com/sites.ucsc.edu/dist/0/521/files/2019/07/index-inequality.pdf>, preprint.
D. Farris, *The embedded contact homology of nontrivial circle bundles over Riemann surfaces*, PhD Thesis UC Berkeley, 2011., 41 pp.
A. Floer, *Morse theory for Lagrangian intersections.* J. Differential Geom. 28 (1988), no. 3, 513-547
A. Floer, *The unregularized gradient flow of the symplectic action.* Comm. Pure Appl. Math. 41(6), (1988), 775-813.
H. Hofer, K. Wysocki and E. Zehnder, *Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics*, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), 337-379.
H. Hofer, K. Wysocki and E. Zehnder, *Properties of pseudo-holomorphic curves in symplectizations. II. Embedding controls and algebraic invariants*, Geom. Funct. Anal. 5 (1995), 270-328.
M. Hutchings, [*Reidemeister torsion in generalized Morse theory*]{}. Forum Math. 14 (2002), no. 2, 209-244
M. Hutchings, [*An index inequality for embedded pseudoholomorphic curves in symplectizations*]{}, J. Eur. Math. Soc. [**4**]{} (2002), 313–361.
M. Hutchings, [*The embedded contact homology index revisited*]{}, New perspectives and challenges in symplectic field theory, 263–297, CRM Proc. Lecture Notes 49, Amer. Math. Soc., 2009.
M. Hutchings, [*Lecture notes on embedded contact homology*]{}, Contact and Symplectic Topology, Bolya Society Mathematical Studies [**26**]{} (2014), 389–484, Springer.
M. Hutchings, *Mean action and the Calabi invariant*. J. Mod. Dyn. 10 (2016), 511-539.
M. Hutchings and J. Nelson, [*Cylindrical contact homology for dynamically convex contact forms in three dimensions*]{}. J. Symplectic Geom. 14 (2016), no. 4, 983–1012.
M. Hutchings and C. H. Taubes, [*Gluing pseudoholomorphic curves along branched covered cylinders I*]{}, J. Symplectic Geom. [**5**]{} (2007), 43–137.
M. Hutchings and C. H. Taubes, [*Gluing pseudoholomorphic curves along branched covered cylinders II*]{}, J. Symplectic Geom. [**7**]{} (2009), 29–133.
M. Hutchings and C. H. Taubes, [*Proof of the Arnold chord conjecture in three dimensions, II*]{}, Geom. Topol. [**17**]{} (2013), 2601–2688.
O. van Koert, [*Simple computations in prequantization bundles*](http://www.math.snu.ac.kr/~okoert/tools/CZ_index_BW_bundle.pdf). Accessed 7/2020.
P. Kronheimer and T. Mrowka, *Monopoles and three-manifolds*, New Mathematical Monographs, 10, Cambridge University Press, (2007).
P. Kronheimer, T. Mrowka, P. Ozsváth and Z. Szabó, *Monopoles and lens space surgeries*, Annals of Mathematics, 165 (2007), no. 2, 457-546.
D. McDuff and D. Salamon, [*Introduction to symplectic topology.*]{} Third edition. Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, 2017.
D. McDuff and D. Salamon, [*$J$-holomorphic curves and symplectic topology.*]{} Second edition. American Mathematical Society Colloquium Publications, 52. American Mathematical Society, Providence, RI, 2012.
M. Micallef and B. White, *The structure of branch points in minimal surfaces and in pseudoholomorphic curves*, [Ann. Math.]{}, **139** (1994), 35-85.
A. Moreno, *Algebraic torsion in higher-dimensional contact manifolds*, arXiv:1711.01562.
T. Mrowka, P. Ozsváth, and B. Yu, *Seiberg-Witten monopoles on Seifert fibered spaces*, Comm. Anal. Geom. 5 (1997), no. 4, 685-791.
J. Nelson, [*Automatic transversality in contact homology I: Regularity*]{}, Abh. Math. Semin. Univ. Hambg. 85 (2015), no. 2, 125-179.
J. Nelson, [*Automatic transversality in contact homology II: filtrations and computations*]{}, Proc. Lon. Math. Soc. Volume 120, Issue 6, 853-917 June 2020.
P. Ozsváth and Z. Szabó, [*Knot Floer homology and integer surgeries*]{}, Algebr. Geom. Topol. 8 (2008), no. 1, 101–153.
A. Ritter, *Floer theory for negative line bundles via Gromov-Witten invariants.* Adv. Math. 262 (2014), 1035-1106.
D. Salamon and E. Zehnder, *Morse theory for periodic solutions of Hamiltonian systems and the Maslov index*, [Comm. Pure Appl. Math.]{} 45 (1992), no 10, 1303-1360.
M. Seppälä and T. Sorvali, [*Geometry of Riemann surfaces and Teichmüller spaces*]{}, North-Holland Mathematics Studies, vol. 169, North-Holland Publishing Co., Amsterdam, 1992.
R. Siefring, [*Relative asymptotic behavior of pseudoholomorphic half-cylinders*]{}, Pure Appl. Math. [**61**]{} (2008).
R. Siefring, [*Intersection theory of punctured pseudoholomorphic curves*]{}, Geom. Topol. [**15**]{} (2011), 2351–2457.
C. H. Taubes, *The Seiberg-Witten equations and the Weinstein conjecture*, Geom. Topol. 11 (2007), 2117-2202.
C. H. Taubes, *Embedded contact homology and Seiberg-Witten Floer cohomology I*, Geom. Topol. 14 (2010), 2497-2581.
C. H. Taubes, *Embedded contact homology and Seiberg-Witten Floer cohomology II*, Geom. Topol. 14 (2010), 2583-2720.
C. H. Taubes, *Embedded contact homology and Seiberg-Witten Floer cohomology III*, Geom. Topol. 14 (2010), 2721-2817.
C. H. Taubes, *Embedded contact homology and Seiberg-Witten Floer cohomology IV*, Geom. Topol. 14 (2010), 2819-2960.
C. B. Thomas, *Almost regular contact manifolds.* J. Differential Geometry 11 (1976), no. 4, 521-533.
M. Weiler, *Mean action of periodic orbits of area-preserving annulus diffeomorphisms*, J. Topol. Anal. Online Ready <https://doi.org/10.1142/S1793525320500363>.
C. Wendl, *Automatic transversality and orbifolds of punctured holomorphic curves in dimension four*, [Comment. Math. Helv.]{} **85** (2010), no. 2, 347-407.
C. Wendl, [*Lectures on Symplectic Field Theory*]{}, to appear in EMS Lectures in Mathematics series, arXiv:1612.01009v2
<span style="font-variant:small-caps;">Jo Nelson\
Rice University</span>\
[*email:* ]{}`[email protected]`\
<span style="font-variant:small-caps;">Morgan Weiler\
Rice University</span>\
[*email:* ]{}`[email protected]`\
[^1]: Partially supported by NSF grant DMS-1810692.
[^2]: Our convention for defining the Hamiltonian vector field is $X_H=\omega(\cdot, dH)$.
[^3]: Because we are using ${{\mathbb Z}}_2$-coefficients, we will not sort through the signs.
[^4]: Previously the papers [@Hindex; @Hrevisit] used the terminology incoming and outgoing partitions.
[^5]: We use the convention $\omega(X_H, \cdot) = dH.$
[^6]: Alternately, one could show that $S_N$ is a “$\tau$-representative" of $[{\mathfrak{p}}_Y(S_N)]$, following [@Hindex], an alternate construction of $Q_\tau$ for which there is no need to consider boundary self-linking.
[^7]: We are now dropping the $\varepsilon$ from the notation, as it is understood we should be working with the Floer $C^\varepsilon$-space.
[^8]: One might also want a holomorphic building to include appropriate gluing data when Reeb orbits are multiply covered, but we will not need this.
[^9]: A [*trivial cylinder*]{} is a $J$-holomorphic cylinder ${{\mathbb R}}\times\gamma$ in ${{\mathbb R}}\times Y$ where $\gamma$ is a Reeb orbit, which is not required to be embedded.
[^10]: Because we are using ${{\mathbb Z}}_2$-coefficients, we will not sort through the signs.
|
---
abstract: 'Calude, Jain, Khoussainov, Li, and Stephan (2017) proposed a quasi-polynomial-time algorithm solving parity games. After this breakthrough result, a few other quasi-polynomial-time algorithms were introduced; none of them is easy to understand. Moreover, it turns out that in practice they operate very slowly. On the other side there is the Zielonka’s recursive algorithm, which is very simple, exponential in the worst case, and the fastest in practice. We combine these two approaches: we propose a small modification of the Zielonka’s algorithm, which ensures that the running time is at most quasi-polynomial. In effect, we obtain a simple algorithm that solves parity games in quasi-polynomial time. We also hope that our algorithm, after further optimizations, can lead to an algorithm that shares the good performance of the Zielonka’s algorithm on typical inputs, while reducing the worst-case complexity on difficult inputs.'
author:
- Paweł Parys
bibliography:
- 'bib.bib'
title: 'Parity Games: Zielonka’s Algorithm in Quasi-Polynomial Time'
---
Introduction
============
The fundamental role of parity games in automata theory, logic, and their applications to verification and synthesis is doubtless, hence it is pointless to elaborate on their importance. Let us only mention that the algorithmic problem of finding the winner in parity games is polynomial-time equivalent to the emptiness problem for nondeterministic automata on infinite trees with parity acceptance conditions, and to the model-checking problem for modal $\mu$-calculus [@EJS01]. It also lies at the heart of algorithmic solutions to the Church’s synthesis problem [@RabinBook]. The impact of parity games reaches relatively far areas of computer science, like Markov decision processes [@FearnleyMDP] and linear programming [@FHZ-simplex].
It is a long-standing open question whether parity games can be solved in polynomial-time. Several results show that they belong to some classes “slightly above” polynomial time. Namely, deciding the winner of parity games was shown to be in $\mathsf{NP}\cap\mathsf{coNP}$ [@EJS01], and in $\mathsf{UP}\cap\mathsf{coUP}$ [@up-co-up], while computing winning strategies is in , , and even in their subclass [@Daskalakis-Papadimitriou]. The same holds for other kinds of games: mean-payoff games [@mean-payoff], discounted games, and simple stochastic games [@stochastic]; parity games, however, are the easiest among them, in the sense that there are polynomial-time reductions from parity games to the other kinds of games [@up-co-up; @mean-payoff], but no reductions in the opposite direction are known.
Describing the algorithmic side of solving parity games, one has to start with the Zielonka’s algorithm [@Zielonka], being an adaptation of an approach proposed by McNaughton to solve Muller games [@McNaughton]. This algorithm consists of a single recursive procedure, being simple and very natural; one may say that it computes who wins the game “directly from the definition”. Its running time is exponential in the worst case [@exponential-lower-bound; @exponential-lower-bound2; @gazda-phd], but on many typical inputs it works much faster. For over two decades researchers were trying to cutback the complexity of solving parity games, which resulted in a series of algorithms, all of which were either exponential [@BCJLM97; @Seidl96; @old-progress-measure; @strategy-improvement; @Schewe-big-steps; @priority-promotion], or mildly subexponential [@randomized-subexponential; @subexponential]. The next era came unexpectedly in 2017 with a breakthrough result of Calude, Jain, Khoussainov, Li, and Stephan [@calude] (see also [@parity-short; @parity-excursion]), who designed an algorithm working in quasi-polynomial time. This invoked a series of quasi-polynomial-time algorithms, which appeared soon after [@small-progress-measure; @Fearnley; @Lehtinen]. These algorithms are quite involved (at least compared to the simple recursive algorithm of Zielonka), and it is not so trivial to understand them.
The four quasi-polynomial-time algorithms [@calude; @small-progress-measure; @Fearnley; @Lehtinen], at first glance being quite different, actually proceed along a similar line (as observed by Bojańczyk and Czerwiński [@separation-toolbox] and Czerwiński et al. [@lower-bound]). Namely, out of all the four algorithms one can extract a construction of a safety automaton (nondeterministic in the case of Lehtinen [@Lehtinen], and deterministic in the other algorithms), which accepts all words encoding plays that are decisively won by one of the players (more precisely: plays consistent with some positional winning strategy), and rejects all words encoding plays in which the player loses (for plays that are won by the player, but not decisively, the automaton can behave arbitrarily). This automaton does not depend at all on the game graph; it depends only on its size. Having an automaton with the above properties, it is not difficult to convert the original parity game into an equivalent safety game (by taking a “product” of the parity game and the automaton), which can be solved easily—and all the four algorithms actually proceed this way, even if it is not stated explicitly than such an automaton is constructed. As shown in Czerwiński et al. [@lower-bound], all automata having the aforementioned properties have to look very similar: their states have to be leaves of some so-called universal tree; particular papers propose different constructions of these trees, and of the resulting automata (of quasi-polynomial size). Moreover, Czerwiński et al. [@lower-bound] show a quasi-polynomial lower bound for the size of such an automaton.
In this paper we propose a novel quasi-polynomial-time algorithm solving parity games. It is obtained by applying a small modification to the Zielonka’s recursive algorithm; this modification guarantees that the worst-case running time of this algorithm, being originally exponential, becomes quasi-polynomial. The simplicity of the Zielonka’s algorithm remains in place; we avoid complicated considerations accompanying all the previous quasi-polynomial-time algorithms. Another point is that our algorithm exploits the structure of parity games in a rather different way from the four previous quasi-polynomial-time algorithms. Indeed, the other algorithms construct automata that are completely independent from a particular game graph given on input— they work in exactly the same way for every game graph of a considered size. The behaviour of our algorithm, in contrast, is highly driven by an analysis of the game graph given on input. In particular, although our algorithm is not faster than quasi-polynomial, it does not fit to the “separator approach” in which a quasi-polynomial lower bound of Czerwiński et al. [@lower-bound] exists.
The running time of our algorithm is quasi-polynomial, and the space complexity is quadratic (more precisely, $O(n\cdot h)$, where $n$ is the number of nodes in the game graph, and $h$ is the maximal priority appearing there).
Let us also mention the practical side of the world. It turns out that parity games are one of the areas where theory does not need to meet practice: the quasi-polynomial-time algorithms, although fastest in theory, are actually the slowest. The most exhaustive comparison of existing algorithms was performed by Tom van Dijk [@oink]. In his Oink tool he has implemented several algorithms, with different optimizations. Then, he has evaluated them on a benchmark of Keiren [@benchmark], containing multiple parity games obtained from model checking and equivalence checking tasks, as well as on different classes of random games. It turns out that the classic recursive algorithm of Zielonka [@Zielonka] performs the best, *ex aequo* with the recent priority promotion algorithm [@priority-promotion]. After that, we have the strategy improvement algorithm [@strategy-improvement; @strategy-improvement-implem], being a few times slower. Far later, we have the small progress measure algorithm [@old-progress-measure]. At the very end, with a lot of timeouts, we have the quasi-polynomial-time algorithm of Fearnley, Jain, Schewe, Stephan, and Wojtczak [@Fearnley]. The other quasi-polynomial-time algorithms were not implemented due to excessive memory usage.
While developing the current algorithm, we hoped that it will share the good performance with the Zielonka’s algorithm, on which it is based. Unfortunately, preliminary experiments have shown that this is not necessarily the case. It turns out that
- on random games our algorithm performs similarly to the slowest algorithms implemented in Oink;
- on crafted game families that are difficult for the Zielonka’s algorithm, our algorithm is indeed faster from it, but not dramatically faster;
- the only think that is optimistic is that on games with a very low number of priorities our algorithm performs similarly to the fastest algorithms.
Because the empirical results of a direct implementation of the algorithm are completely unsatisfactory, we do not include a full description of our experiments. Instead, we leave an efficient implementation for a future work. Beside of the discouraging outcomes, we believe that our idea, via further optimizations, can lead to an algorithm that is both fast in practice and has a good worst-case complexity (see the concluding section for more comments).
Preliminaries
=============
A parity game is played on a *game graph* between two players, called Even or Odd (shortened sometimes to $E$ and $O$). A game graph consists of
- a directed graph $G$, where we require that every node has at least one successor, and where there are no self-loops (i.e., edges from a node to itself);
- a labeling of every node $v$ of $G$ by a positive natural number $\pi(v)$, called its *priority*;
- a partition of nodes of $G$ between nodes owned by Even and nodes owned by Odd.
An infinite path in $G$ is called a *play*, while a finite path in $G$ is called a *partial play*. The game starts in a designated starting node. Then, the player to which the current node belongs, selects a successor of this node, and the game continues there. In effect, after a finite time a partial play is obtained, and at the end, after infinite time, this results in a play. We say that a play $v_1,v_2,\dots$ is winning for Even if $\limsup_{i\to\infty}\pi(v_i)$ is even (i.e., if the maximal priority seen infinitely often is even). Conversely, the play is winning for Odd if $\limsup_{i\to\infty}\pi(v_i)$ is odd.
A *strategy* of player $P\in\{\mbox{Even},\mbox{Odd}\}$ is a function that maps every partial play that ends in a node of $P$ to some its successor. Such a function says how $P$ will play in every situation of the game (depending on the history of that game). When a (partial) play $\pi$ follows a strategy $\sigma$ in every step in which player $P$ is deciding, we say that $\pi$ *agrees* with $\sigma$. A strategy $\sigma$ is *winning* for $P$ from a node $v$ if every play that starts in $v$ and agrees with $\sigma$ is winning for $P$. While saying “player $P$ wins from a node $v$” we usually mean that $P$ has a winning strategy from $v$. Let ${\mathit{Win}}_P(G)$ be the set of nodes of $G$ from which $P$ wins; it is called the *winning region* of $P$. By the Martin’s theorem [@Martin-determinacy] we know that parity games are determined: in every game graph $G$, and for every node $v$ of $G$ either Even wins from $v$, or Odd wins from $v$. In effect, ${\mathit{Win}}_E(G)$ and ${\mathit{Win}}_O(G)$ form a partition of the node set of $G$.
During the analysis, we also consider games with other winning conditions. A *winning condition* is a set of plays. The winning conditions of Even and Odd considered in parity games are denoted and , respectively. Beside of that, for every set $S$ of nodes, let ${\textup{Safety}\xspace}(S)$ be the set of plays that use only nodes from $S$.
A *dominion* for Even is a set $S$ of nodes such that from every $v\in S$ Even wins the game with the condition ${\textup{Limsup\-Even}\xspace}\cap{\textup{Safety}\xspace}(S)$; in other words, from every node of $S$ he can win the parity game without leaving $S$. Likewise, a dominion for Odd is a set $S$ of nodes such that from every $v\in S$ Odd wins the game with the condition ${\textup{Limsup\-Odd}\xspace}\cap{\textup{Safety}\xspace}(S)$. Notice that the whole ${\mathit{Win}}_P(G)$ is a dominion for $P$ (where $P\in\{\mbox{Even},\mbox{Odd}\}$). Indeed, if Even is going to win from some $v\in{\mathit{Win}}_E(G)$, the play cannot leave ${\mathit{Win}}_E(G)$ and enter a node $v'\in{\mathit{Win}}_O(G)$, as then Odd could use his winning strategy from $v'$ and win the whole game; here we use the fact that all suffixes of a play in are also in . For $P=\mbox{Odd}$ the situation is symmetric.
Standard Zielonka’s Algorithm
=============================
Before presenting our algorithm, we recall the standard Zielonka’s algorithm, as a reference.
For a set of nodes $N$ in a game graph $G$, and for a player $P\in\{\mbox{Even},\mbox{Odd}\}$, we define the *attractor* of $N$, denoted ${\textsc{Atr}}_P(G,N)$, to be the set of nodes of $G$ from which $P$ can force to reach a node from $N$. In other words, ${\textsc{Atr}}_P(G,N)$ is the smallest set such that
- $N\subseteq{\textsc{Atr}}_P(G,N)$,
- if $v$ is a node of $P$ and some its successor is in ${\textsc{Atr}}_P(G,N)$, then $v\in{\textsc{Atr}}_P(G,N)$, and
- if $v$ is a node of the opponent of $P$ and all its successors are in ${\textsc{Atr}}_P(G,N)$, then $v\in{\textsc{Atr}}_P(G,N)$.
Clearly ${\textsc{Atr}}_P(G,N)$ can be computed in time proportional to the size of $G$.
$N_h=\{v\in{\mathsf{nodes}}(G)\mid \pi(v)=h\}$; $H=G\setminus{\textsc{Atr}}_E(G,N_h)$; $W_O=\textsc{Solve}_O(H,h-1)$; $G=G\setminus{\textsc{Atr}}_O(G,W_O)$;
Algorithm \[alg:std\] is the standard Zielonka’s algorithm. The procedure $\textsc{Solve}_E(G,h)$ returns ${\mathit{Win}}_E(G)$, the winning region of Even, if $h$ is an even number that is greater or equal than all priorities appearing in $G$. A procedure $\textsc{Solve}_O(G,h)$ is also needed; it is identical to $\textsc{Solve}_E(G,h)$ except that the roles of $E$ and $O$ are swapped; it returns ${\mathit{Win}}_O(G)$, the winning region of Odd. While writing $G\setminus S$, we mean the game obtained by removing from $G$ all nodes in $S$, and all edges leading to nodes in $S$ or starting from nodes in $S$. We use this construct only when $S$ is an attractor; in such a case, if all successors of a node $v$ are removed, then $v$ is also removed (i.e., if all successors of $v$ belong to an attractor, then $v$ belongs to the attractor as well). In effect $G\setminus S$ is a valid game graph (every its node has at least one successor).
We remark that the algorithm is presented in a slightly different way than usually. Namely, we use here a loop, while the usual presentation does not use a loop but rather calls recursively $\textsc{Solve}_E(G\setminus{\textsc{Atr}}_O(G,W_O),h)$ at the end of the procedure. This is only a superficial difference in the presentation, but is useful while modifying the algorithm in the next section.
The algorithm can be understood while looking at Figure \[fig:1\]. Let $h$ be the highest priority used in $G$; assume that it is even. The game graph $G$ can be divided into two parts: ${\mathit{Win}}_E(G)$ and ${\mathit{Win}}_O(G)$. In ${\mathit{Win}}_E(G)$ we can distinguish the attractor of nodes with priority $h$ (denoted $A_E$). Odd either loses inside ${\mathit{Win}}_E(G)\setminus A_E$, or enters $A_E$, which causes that a node with priority $h$ is seen, and then the game continues in some node of ${\mathit{Win}}_E(G)$. The winning region of Odd, ${\mathit{Win}}_O(G)$, can be divided into multiple parts. We have a part $W_O^0$, where Odd can win without seeing a node of priority $h$. Then, we have nodes of priority $h$ from which Even is forced to enter $W_O^0$, and their attractor, denoted $A_1$. Then, we have a part $W_O^1$, where Odd can ensure that the play is either winning for him inside $W_O^1$ or enters $A_1$; in other words, from nodes of $W_O^1$ Odd can win while seeing $h$ at most once. We also have parts $W_O^i$ for larger $i$, and corresponding attractors $A_i$.
While running the algorithm, this partition of $G$ is not known, and has to be discovered. To this end, the algorithm assumes first (in the game $H$) that all nodes of priority $h$ are winning for Even. The first call to $\textsc{Solve}_O(H,h-1)$ returns the set $W_O^0$ of nodes where Odd wins without seeing a node of priority $h$. We then remove them from the game, together with the attractor $A_1$. In the next step, $\textsc{Solve}_O(H,h-1)$ returns the set $W_O^1$, and so on. At the end the whole ${\mathit{Win}}_O(G)$ becomes removed, and the procedure returns ${\mathit{Win}}_E(G)$.
Quasi-Polynomial-Time Algorithm
===============================
We now present a modification to Algorithm \[alg:std\] that results in obtaining quasi-polynomial running time, in the worst case.
The modification can be understood while looking again at Figure \[fig:1\]. The key observation is that, while ${\mathit{Win}}_O(G)$ is of size at most $n$ (where $n$ is the number of nodes in $G$), then most of its parts $W_O^i$ are smaller. Namely, most of them have to be of size at most $\frac{n}{2}$, and only one of them can be larger than $\frac{n}{2}$. We use this observation, and while looking for $W_O^i$, we search for a winning region (for a dominion) of size at most $\frac{n}{2}$. Usually this is enough; only once it is not enough: one $W_O^i$ can be larger than $\frac{n}{2}$ and it will not be found if we only look for a set of size at most $\frac{n}{2}$. But when the algorithm finds no set of size at most $\frac{n}{2}$, we can once search for $W_O^i$ of an arbitrary size. After that, we know that all following sets $W_O^i$ are again of size at most $\frac{n}{2}$. While going recursively, we notice that every $W_O^i$ can be further subdivided in a similar way, while splitting on the priority $h-2$. If $|W_O^i|\leq\frac{n}{2}$, we again have the property that most of the parts of $W_O^i$ are of size at most $\frac{n}{4}$, and only one of them can be larger than $\frac{n}{4}$.
To exploit this observation, in the recursive calls we pass two precision parameters, $p_E$ and $p_O$ (one for every of the players), saying that we search for winning sets of size at most $p_E$ for Even, and at most $p_O$ for Odd. The modified procedure is presented as Algorithm \[alg:new\]. Again, one also needs a procedure $\textsc{Solve}_O$, which is obtained from $\textsc{Solve}_E$ by literally changing every $E$ to $O$ and *vice versa*.
[**return** ]{}$\emptyset$; $N_h=\{v\in{\mathsf{nodes}}(G)\mid \pi(v)=h\}$; $H=G\setminus{\textsc{Atr}}_E(G,N_h)$; $W_O=\textsc{Solve}_O(H,h-1,\lfloor p_O/2\rfloor,p_E)$; $G=G\setminus{\textsc{Atr}}_O(G,W_O)$; $N_h=\{v\in{\mathsf{nodes}}(G)\mid \pi(v)=h\}$; $H=G\setminus{\textsc{Atr}}_E(G,N_h)$; $W_O=\textsc{Solve}_O(H,h-1,p_O,p_E)$; $G=G\setminus{\textsc{Atr}}_O(G,W_O)$; $N_h=\{v\in{\mathsf{nodes}}(G)\mid \pi(v)=h\}$; $H=G\setminus{\textsc{Atr}}_E(G,N_h)$; $W_O=\textsc{Solve}_O(H,h-1,\lfloor p_O/2\rfloor,p_E)$; $G=G\setminus{\textsc{Atr}}_O(G,W_O)$; [**return** ]{}${\mathsf{nodes}}(G)$;
We start the algorithm with $p_E=p_O=n$, where $n$ is the number of nodes in $G$. In the procedure we have now, in a sense, three copies of the previous procedure, corresponding to three stages. In the first stage, in lines 5-10, we look for sets $W_O^i$ of size at most $\lfloor\frac{p_O}{2}\rfloor$. If the returned set is empty, this may mean that the next $W_O^i$ either is empty, or is of size greater than $\lfloor\frac{p_O}{2}\rfloor$. Then, in lines 11-14, we once search for a set $W_O^i$ of size at most $p_O$ (knowing that if it is nonempty, then its size is greater than $\lfloor\frac{p_O}{2}\rfloor$). Finally, in the loop in lines 15-20, we again look for sets $W_O^i$ of size at most $\lfloor\frac{p_O}{2}\rfloor$ (because we have already found a set of size greater than $\lfloor\frac{p_O}{2}\rfloor$, all the remaining sets have size at most $\lfloor\frac{p_O}{2}\rfloor$).
Complexity Analysis
===================
Let us analyze the complexity of our algorithm.
First, we observe that the space complexity is $O(n\cdot h)$, where $n$ is the number of nodes, and $h$ is the maximal priority. Indeed, the depth of the recursion is at most $h$, and on every step we only need to remember some sets of nodes.
We now come to the running time. As it is anyway worse than the running time of the other quasi-polynomial-time algorithms, we do not aim in proving a very tight upper bound; we only prove that the running time is quasi-polynomial.
Let $R(h,l)$ be the number of (nontrivial) executions of the $\textsc{Solve}_E$ and $\textsc{Solve}_O$ procedures performed during one call to $\textsc{Solve}_E(G,h,p_E,p_O)$ with $\lfloor\log p_E\rfloor+\lfloor\log p_O\rfloor=l$, and with $G$ having at most $n$ nodes (where $n$ is fixed). We only count here nontrivial executions, that is, such that do not leave the procedure in line 4. Clearly $R(0,l)=R(h,0)=0$. For $h,l\geq 1$ it holds that $$\begin{aligned}
R(h,l)\leq 1+n\cdot R(h-1,l-1)+R(h-1,l)\,.\label{eq:1}
\end{aligned}$$ Indeed, in $\textsc{Solve}_E$ after every call to $\textsc{Solve}_O$ we remove at least one node from $G$, with the exception of two such calls: the last call in line 8, and the last call ever. In effect, in lines 8 and 18 we have at most $n$ calls to $\textsc{Solve}_O$ with decreased precision (plus, potentially, the $(n+1)$-th call with empty $G$, which is not included in $R(h,l)$), and in line 13 we have one call to $\textsc{Solve}_O$ with full precision. Notice that $\lfloor\log p_O\rfloor$ (hence also $l$) decreases by $1$ in the decreased-precision call.
Using Inequality we now prove by induction that $R(h,l)\leq n^l\cdot\binom{h+l}{l}-1$. For $h=0$ and for $l=0$ the inequality holds. For $h,l\geq 1$ we have that $$\begin{aligned}
R(h,l)&\leq 1+n\cdot R(h-1,l-1)+R(h-1,l)\\
&\leq 1+n\cdot\left(n^{l-1}\cdot\binom{h-1+l-1}{l-1}-1\right)+n^l\cdot\binom{h-1+l}{l}-1\\
&\leq n^l\cdot\left(\binom{h-1+l}{l-1}+\binom{h-1+l}{l}\right)-1\\
&=n^l\cdot\binom{h+l}{l}-1\,.
\end{aligned}$$ In effect, $R(h,l)\leq n^l\cdot(h+l)^l$. Recalling that we start with $l=2\cdot\lfloor\log n\rfloor$, we see that this number is quasi-polynomial in $n$ and $h$. This concludes the proof, since obviously a single execution of the $\textsc{Solve}_E$ procedure (not counting the running time of recursive calls) costs polynomial time.
Correctness
===========
We now justify correctness of the algorithm. This amounts to proving the following lemma.
\[lem:1\] Procedure $\textsc{Solve}_E(G,h,p_E,p_O)$ returns a set $W_E$ such that for every $S\subseteq{\mathsf{nodes}}(G)$,
- if $S$ is a dominion for Even, and $|S|\leq p_E$, then $S\subseteq W_E$, and
- if $S$ is a dominion for Odd, and $|S|\leq p_O$, then $S\cap W_E=\emptyset$.
Notice that in $G$ there may be nodes that do not belong to any dominion smaller than $p_E$ or $p_O$; for such nodes we do not specify whether or not they are contained in $W_E$.
Recall that ${\mathit{Win}}_E(G)$ is a dominion for Even, and ${\mathit{Win}}_O(G)$ is a dominion for Odd. Thus, using Lemma \[lem:1\] we can conclude that for $p_E=p_O=n$ the procedure returns ${\mathit{Win}}_E(G)$, the winning region of Even.
One may wonder why we use dominions in the statement of the lemma, instead of simply saying that if $|{\mathit{Win}}_E(G)|\leq p_E$, then ${\mathit{Win}}_E(G)\subseteq W_E$. Such a simplified statement, however, is not suitable for induction. Indeed, while switching from the game $G$ to the game $H$ (created in lines 7, 12, 17) the winning regions of Even may increase dramatically, because in $H$ Odd is not allowed to visit any node with priority $h$. Nevertheless, the winning region of Even in $G$, and any dominion of Even in $G$, remains a dominion in $H$ (when restricted to nodes of $H$).
Before proving Lemma \[lem:1\], let us observe two facts about dominions. In their statements $P\in\{\mbox{Even},\mbox{Odd}\}$ is one of the players, and $\overline P$ is his opponent.
\[fact:usun-moj\] If $S$ is a dominion for $P$ in a game $G$, and $X$ is a set of nodes of $G$, then $S\setminus{\textsc{Atr}}_P(G,X)$ is a dominion for $P$ in $G\setminus{\textsc{Atr}}_P(G,X)$.
Denote $S'=S\setminus{\textsc{Atr}}_P(G,X)$ and $G'=G\setminus{\textsc{Atr}}_P(G,X)$. By definition, from every node $v\in S$ player $P$ wins with the condition ${\textup{Limsup}\ensuremath{P}\xspace}\cap{\textup{Safety}\xspace}(S)$ in $G$, using some winning strategy. Observe that using the same strategy he wins with the condition ${\textup{Limsup}\ensuremath{P}\xspace}\cap{\textup{Safety}\xspace}(S')$ in $G'$ (assuming that the starting node $v$ is in $S'$). The strategy remains valid in $G'$, because every node $u$ of player $P$ that remains in $G'$ has the same successors in $G'$ as in $G$ (conversely: if some of successors of $u$ belongs to ${\textsc{Atr}}_P(G,X)$, then $u$ also belongs to ${\textsc{Atr}}_P(G,X)$).
\[fact:usun-jego\] If $S$ is a dominion for $P$ in a game $G$, and $X$ is a set of nodes of $G$ such that $S\cap X=\emptyset$, then $S$ is a dominion for $P$ in $G\setminus{\textsc{Atr}}_{\overline P}(G,X)$ (in particular $S\subseteq{\mathsf{nodes}}(G\setminus{\textsc{Atr}}_{\overline P}(G,X))$).
Denote $G'=G\setminus{\textsc{Atr}}_{\overline P}(G,X)$. Suppose that there is some $v\in S\cap{\textsc{Atr}}_{\overline P}(G,X)$. On the one hand, $P$ can guarantee that, while starting from $v$, the play stays in $S$ (by the definition of a dominion); on the other hand, $\overline P$ can force to reach the set $X$ (by the definition of an attractor), which is disjoint from $S$. Thus such a node $v$ could not exist, we have $S\subseteq{\mathsf{nodes}}(G')$.
It remains to observe that from every node $v\in S$ player $P$ wins with the condition ${\textup{Limsup}\ensuremath{P}\xspace}\cap{\textup{Safety}\xspace}(S)$ also in the restricted game $G'$, using the same strategy as in $G$. Indeed, a play in $G$ following this strategy never leaves $S$, and the whole $S$ remains unchanged in $G'$.
We are now ready to prove Lemma \[lem:1\].
We prove the lemma by induction on $h$. Consider some execution of the procedure. By $G^i, N_h^i, H^i, W_O^i$ we denote values of the variables $G, N_h, H, W_O$ just after the $i$-th call to $\textsc{Solve}_O$ in one of the lines 8, 13, 18; in lines 9, 14, 19 we create $G^{i+1}$ out of $G^i$ and $W_O^i$. In particular $G^1$ equals the original game $G$, and at the end we return ${\mathsf{nodes}}(G^{m+1})$, where $m$ is the number of calls to $\textsc{Solve}_O$.
Concentrate on the first item of the lemma: fix an Even’s dominion $S$ in $G$ (i.e., in $G^1$) such that $|S|\leq p_E$. Assume that $S\neq\emptyset$ (for $S=\emptyset$ there is nothing to prove). Notice first that a nonempty dominion has at least two nodes (by assumption there are no self-loops in $G$, hence every play has to visit at least two nodes), thus, because $S\subseteq{\mathsf{nodes}}(G)$ and $|S|\leq p_E$, we have that $G\neq\emptyset$ and $p_E>1$. It means that the procedure does not return in line 4. We thus need to prove that $S\subseteq{\mathsf{nodes}}(G^{m+1})$.
We actually prove that $S$ is a dominion for Even in $G^i$ for every $i\in\{1,\dots,m+1\}$, meaning in particular that $S\subseteq{\mathsf{nodes}}(G^i)$. This is shown by an internal induction on $i$. The base case ($i=1$) holds by assumption. For the induction step, consider some $i\in\{1,\dots,m\}$. By the induction assumption $S$ is a dominion for Even in $G^i$, and we need to prove that it is a dominion for Even in $G^{i+1}$.
Consider $S^i=S\cap{\mathsf{nodes}}(H^i)$. Because $S^i=S\setminus{\textsc{Atr}}_E(G^i,N_h^i)$, by Fact \[fact:usun-moj\] the set $S^i$ is a dominion for Even in $H^i=G^i\setminus{\textsc{Atr}}_E(G^i,N_h^i)$, and obviously $|S^i|\leq|S|\leq p_E$. By the assumption of the external induction (which can be applied to $\textsc{Solve}_O$, by symmetry) it follows that $S^i\cap W_O^i=\emptyset$, so also $S\cap W_O^i=\emptyset$ (because $W_O^i$ contains only nodes of $G^i$, while $S\setminus S^i$ contains no nodes of $G^i$). Thus, by Fact \[fact:usun-jego\] the set $S$ is a dominion for Even in $G^{i+1}=G^i\setminus{\textsc{Atr}}_O(G^i,W_O^i)$. This finishes the proof of the first item.
Now we prove the second item of the lemma. To this end, fix some Odd’s dominion $S$ in $G$ such that $|S|\leq p_O$. If $p_E\leq 1$, we return $W_E=\emptyset$ (line 4), so clearly $S\cap W_E=\emptyset$. The interesting case is when $p_E\geq 2$. Denote $S^i=S\cap{\mathsf{nodes}}(G^i)$ for all $i\in\{1,\dots,m+1\}$; we first prove that $S^i$ is a dominion for Odd in $G^i$. This is shown by induction on $i$. The base case of $i=1$ holds by assumption, because $G^1=G$ and $S^1=S$. For the induction step, assume that $S^i$ is a dominion for Odd in $G^i$, for some $i\in\{1,\dots,m\}$. By definition $G^{i+1}=G^i\setminus{\textsc{Atr}}_O(G^i,W_O^i)$ and $S^{i+1}=S^i\setminus{\textsc{Atr}}_O(G^i,W_O^i)$, so $S^{i+1}$ is a dominion for Odd in $G^{i+1}$ by Fact \[fact:usun-moj\], which finishes the inductive proof.
For $i\in\{1,\dots,m\}$, let $Z^i$ be the set of nodes (in $S^i\setminus N_h^i$) from which Odd wins with the condition ${\textup{Limsup\-Odd}\xspace}\cap{\textup{Safety}\xspace}(S^i\setminus N_h^i)$ in $G^i$ (that is, where Odd can win without seeing priority $h$—the highest even priority). Let us observe that if $S^i\neq\emptyset$ then $Z^i\neq\emptyset$ ($\clubsuit$). Indeed, suppose to the contrary that $Z^i=\emptyset$, and consider an Odd’s strategy that allows him to win with the condition ${\textup{Limsup\-Odd}\xspace}\cap{\textup{Safety}\xspace}(S^i)$ in $G^i$, from some node $v_0\in S^i$. Because $v_0\not\in Z^i$, this strategy in not winning for the condition ${\textup{Limsup\-Odd}\xspace}\cap{\textup{Safety}\xspace}(S^i\setminus N_h^i)$, so Even, while playing against this strategy, can reach a node $v_1$ in $N_h^i$ (as he cannot violate the parity condition nor leave $S^i$). For the same reason, because $v_1\not\in Z^i$, Even can continue and reach a node $v_2$ in $N_h^i$. Repeating this forever, Even gets priority $h$ (which is even and is the highest priority) infinitely many times, contradicting the fact that the strategy was winning for Odd.
Observe also that from nodes of $Z^i$ Odd can actually win with the condition ${\textup{Limsup\-Odd}\xspace}\cap{\textup{Safety}\xspace}(Z^i)$ in $G^i$, using the strategy that allows him to win with the condition ${\textup{Limsup\-Odd}\xspace}\cap{\textup{Safety}\xspace}(S^i\setminus N_h^i)$. Indeed, if a play following this strategy enters some node $v$, then from this node $v$ Odd can still win with the condition ${\textup{Limsup\-Odd}\xspace}\cap{\textup{Safety}\xspace}(S^i\setminus N_h^i)$, which means that these nodes belongs to $Z^i$. It follows that $Z^i$ is a dominion for Odd in $G^i$. Moreover, because $Z^i\cap N_h^i=\emptyset$, from Fact \[fact:usun-jego\] we have that $Z^i$ is a dominion for Odd in $H^i=G^i\setminus{\textsc{Atr}}_E(G^i,N_h^i)$.
Let $k$ be the number of the call to $\textsc{Solve}_O$ that is performed in line 13 (calls number $1,\dots,k-1$ are performed in line 8, and calls number $k+1,\dots,m$ are performed in line 18). Recall that $W_O^i$ is the set returned by a call to $\textsc{Solve}_O(H^i,h-1,p_O^i,p_E)$, where $p_O^k=p_O$, and $p_O^i=\lfloor\frac{p_O}{2}\rfloor$ if $i\neq k$. From the assumption of the external induction, if $|Z^i|\leq\lfloor\frac{p_O}{2}\rfloor$ or if $i=k$ (since $Z^i\subseteq S^i\subseteq S$ and $|S|\leq p_O$, clearly $|Z^i|\leq p_O$), we obtain that $Z^i\subseteq W_O^i$ ($\spadesuit$).
We now prove that $|S^{k+1}|\leq\lfloor\frac{p_O}{2}\rfloor$. This clearly holds if $S^{k-1}=\emptyset$, because $S^{k+1}\subseteq S^k\subseteq S^{k-1}$. Suppose thus that $S^{k-1}\neq\emptyset$. Then $Z^{k-1}\neq\emptyset$, by ($\clubsuit$). On the other hand, $W_O^{k-1}=\emptyset$, because we are just about to leave the loop in lines 5-10 (the $k$-th call to $\textsc{Solve}_O$ is in line 13). By ($\spadesuit$), if $|Z^{k-1}|\leq\lfloor\frac{p_O}{2}\rfloor$, then $Z^{k-1}\subseteq W_O^{k-1}$, which does not hold in our case. Thus $|Z^{k-1}|>\lfloor\frac{p_O}{2}\rfloor$. Because $W_O^{k-1}=\emptyset$, we simply have $G^k=G^{k-1}$, and $S^k=S^{k-1}$, and $Z^k=Z^{k-1}$. Using ($\spadesuit$) for $i=k$, we obtain that $Z^k\subseteq W_O^k$, and because $S^{k+1}=S^k\setminus{\textsc{Atr}}_O(G^k,W_O^k)\subseteq S^k\setminus W_O^k\subseteq S^k\setminus Z^k$ we obtain that $|S^{k+1}|\leq|S^k|-|Z^k|\leq p_O-(\lfloor\frac{p_O}{2}\rfloor+1)\leq\lfloor\frac{p_O}{2}\rfloor$, as initially claimed.
If $k=m$, we have $Z^m\subseteq W_O^m$ by ($\spadesuit$). If $k+1\leq m$, we have $S^m\subseteq S^{k+1}$ (our procedure only removes nodes from the game) and $Z^m\subseteq S^m$, so $|Z^m|\leq\lfloor\frac{p_O}{2}\rfloor$ by the above paragraph, and also $Z^m\subseteq W_O^m$ by ($\spadesuit$). Because after the $m$-th call to $\textsc{Solve}_O$ the procedure ends, we have $W_O^m=\emptyset$, so also $Z^m=\emptyset$, and thus $S^m=\emptyset$ by ($\clubsuit$). We have $S^{m+1}\subseteq S^m$, so $S^{m+1}=S\cap{\mathsf{nodes}}(G^{m+1})=\emptyset$. This is exactly the conclusion of the lemma, since the set returned by the procedure is ${\mathsf{nodes}}(G^{m+1})$.
Conclusions
===========
To the list of the four existing quasi-polynomial-time algorithms solving parity games, we have added a new one. It uses a rather different approach: it analyses recursively the game graph, like the Zielonka’s algorithm.
Notice that the number of recursive calls in our algorithm may be smaller than in the original Zielonka’s algorithm, because of the precision parameters, but it may also be larger. Indeed, while $\textsc{Solve}_E$ in the original Zielonka’s algorithm stops after the first time when a recursive call returns $\emptyset$, in our algorithm the procedure stops after the second time when a recursive call returns $\emptyset$.
The algorithm, as is, turns out not to be very efficient in practice. Beside of that, we believe that it can serve as a good starting point for a more optimized algorithm. Over the years, some optimizations to the Zielonka’s algorithm were proposed. For example, Liu, Duan, and Tian [@LiuDT14] replace the loop guard $W_O=\emptyset$ by $W_O={\textsc{Atr}}_O(G,W_O)$ (which ensures that $W_O$ will be empty in the next iteration of the loop). Verver [@verver] proposes to check whether ${\textsc{Atr}}_E(G,N_h)$ contains all nodes of priority $h-1$, and if so, to extend $N_h$ by nodes of the next highest Even priority (i.e., $h-2$). It seems that these optimizations can be applied to our algorithm as well.
A straightforward optimization is to decrease $p_O$ and $p_E$ to $|G|$ at the beginning of every recursive call.
Another idea is to extend the recursive procedure so that it will return also a Boolean value saying whether the returned set surely equals the whole winning region (i.e., whether the precision parameters have not restricted anything). If while making the recursive call with smaller precision (line 8) the answer is positive, but the returned set $W_O$ is empty, we can immediately stop the procedure, without making the recursive call with the full precision (line 13).
One can also observe that the call to $\textsc{Solve}_O$ in line 13 (with the full precision) gets the same subgame $H$ as the last call to $\textsc{Solve}_O$ in line 8 (with decreased precision). A very rough idea is to make some use of the computations performed by the decreased-precision call during the full-precision call.
We leave implementation and evaluation of the above (and potentially some other) optimizations for a future work.
|
---
abstract: 'Global folds between Banach spaces are obtained from a simple geometric construction: a Fredholm operator $T$ of index zero with one dimensional kernel is perturbed by a compatible nonlinear term $P$. The scheme encapsulates most of the known examples and suggests new ones. Concrete examples rely on the positivity of an eigenfunction. For the standard Nemitskii case $P(u) = f(u)$ (but $P$ might be nonlocal, non-variational), $T$ might be the Laplacian with different boundary conditions, as in the Ambrosetti-Prodi theorem, or the Schrödinger operators associated with the quantum harmonic oscillator or the Hydrogen atom, a spectral fractional Laplacian, a (nonsymmetric) Markov operator. For self-adjoint operators, we use results on the nondegeneracy of the ground state. On Banach spaces, a similar role is played by a recent extension by Zhang of the Krein-Rutman theorem.'
author:
- 'Marta Calanchi, Carlos Tomei and André Zaccur'
title: |
Global folds between Banach spaces\
as perturbations
---
[**Keywords:**]{} Ambrosetti-Prodi theorem, folds, Krein-Rutman theorem, positivity preserving semigroups.
[**MSC-class:**]{} 35J65, 47H15, 47H30, 58K05.
Introduction {#sec:intro}
============
A continuous function $F:X \to Y$ between real Banach spaces $X$ and $Y$ is a [*global fold*]{} if there is a Banach space $W$ and homeomorphisms $\zeta:X \to W \oplus {{\mathbb{R}}}$ and $\xi: Y \to W \oplus{{\mathbb{R}}}$ for which $\tilde{F}(w,t) \, = \, \xi \circ F \circ \zeta^{-1} (w,t) = (w,-|t|)$. The celebrated Ambrosetti-Prodi theorem [@AP], refined and reinterpreted by subsequent work of Micheletti and Manes ([@MM]), Berger and Podolak ([@BP]) and Berger and Church ([@BC]), describes a class of functions given by nonlinear differential operators which are global folds (the result is presented below in this introduction).
More than a theorem, these works present a strategy to identify folds. In a similar vein, Church and Timourian ([@CT1]) introduced other characterizations, leading to new examples and simpler arguments in known situations. Difficulties frequently arise when verifying the necessary hypotheses. The same authors also provided sufficient conditions for a global fold, closer in spirit to Berger and Podolak ([@BP]), that are easier to check.
We take a similar point of view. Start with a Fredholm operator $T: X \to Y$ of index 0, $\dim \ker T = 1$, which by simple linear changes of variable becomes $$T^a: W \oplus {{\mathbb{R}}}\to W \oplus {{\mathbb{R}}}, \quad T^a(w,t) = (w,0) \ ,$$ where $W = {\operatorname{Ran}}T$. For each $w_0 \in W$ fixed, the image under $T^a$ of a vertical line $\{(w_0,t), t \in {{\mathbb{R}}}\}$ is the point $(w_0,0)$. Consider a unimodal function $h^a(w_0,.)$ whose domain splits in two intervals on which it is first strictly increasing and then strictly decreasing, and suppose that $h^a(w_0,t) \to -\infty$ as $|t| \to \infty$. Clearly $$(w,t) \in W \oplus {{\mathbb{R}}}\mapsto (w, h^a(w,t)) \in W \oplus {{\mathbb{R}}}$$ is a global fold. And a homeomorphism on the domain which keeps invariant each horizontal plane, $\Psi^a(w,t) = (F^a_t(w),t)$ preserves the fold structure. As we shall see, the familiar global folds described in [@CT1] are of the form $F^a(w,t) = (F^a_t(w), h^a(F^a_t(w),t))$. This geometric approach leads also to algorithms for the numerical solution of a class of nonlinear differential equations ([@S], [@CT]).
This approach motivates Theorem \[theo:fold\], implicit in [@STZ]. A continuous (resp. Lipschitz, $C^k$) map $F:X \to Y$ admits an [*$LS$-decomposition*]{} ( [@BP], [@CTZ1],[@MST2],[@M]) if there is a (Lipschitz, $C^k$) homeomorphism $\Phi: Y \to X$ and a continuous (Lipschitz, $C^k$) $h^a: W_Y \oplus {{\mathbb{R}}}\to {{\mathbb{R}}}$ such that $F^a = F \circ \Phi $ is a rank one nonlinear perturbation of the identity, $$F^a : Y = (W_Y \oplus {{\mathbb{R}}}) \to W_Y \oplus {{\mathbb{R}}}\, , \quad (z, t) \mapsto (z, h^a(z,t)) \, .$$ Clearly, $F$ is a global fold if and only if $F^a$ is. Singularity theorists say that $F^a$ provides [*adapted coordinates*]{} to $F$.
\[theo:fold\] Suppose $F: X \to Y$ is a continuous map. If $F$ satisfies the hypotheses below, it is either a homeomorphism or a global fold.
[ $F$ admits an $LS$-decomposition.]{}
[ $F$ is proper.]{}
[ No point of $Y$ has three preimages under $F$.]{}
Theorem \[theo:fold\] implies Theorems \[theo:likem\] and \[theo:likeM\] below, from which we obtain more examples of global folds, with simpler arguments. We consider $F(u) = T u - P(u)$, where $T$ is a linear operator and $P$ a nonlinear perturbation. The Ambrosetti-Prodi theorem is the case $T = -\Delta - \lambda_m I$ with Dirichlet boundary conditions on a smooth bounded domain, $\lambda_m$ is its smallest eigenvalue, and $P$ is the operator $P(u) = f(u) - \lambda_m u$ for a strictly convex $C^2$ function $f: {{\mathbb{R}}}\to {{\mathbb{R}}}$ such that $f'({{\mathbb{R}}})$ intersects the spectrum of $T$ only at $\lambda_m$. By taking appropriate translations, we may consider relevant eigenvalues of $T$ to be different from zero.
We sacrifice generality and concentrate on two classes of [*amenable*]{} operators. An eigenvalue $\lambda \in {{\mathbb{C}}}$ of $T: X \to Y$ is [*elementary*]{} if
\(I) $T - \lambda I: X \to Y$ is a Fredholm operator of index zero and $\ker (T- \lambda I)$ is spanned by the eigenvector $\phi$.
\(II) There is no $w \in X$ for which $(T - \lambda I) w = \phi$.
\(III) $\lambda$ is an isolated point of $\sigma(T)$.
Let $H = L^2(M, d\mu)$ for a $\sigma$-finite measure space $(M,\mu)$. A self-adjoint operator $T: D \subset H \to H$ is [*$m$-amenable*]{} if
(m-a) $ \lambda_m = \min \ \sigma(T)$ is an elementary eigenvalue associated with a normalized eigenvector $\phi_m >0$ a.e.
(m-b) For all $t>0$, the operators $e^{-tT}: H \to H$ are positivity improving: for any nonzero $g \ge 0$, $\ e^{-tT} g > 0$ a.e.
Self-adjoint operators with positive ground state $\phi_m$ are the archetypical examples of $m$-amenability. Also, $m$-amenability will still hold for the Jacobians $DF(u) = T - DP(u)$ for [*$m$-compatible*]{} perturbations $P(u): H \to H$, defined in Section \[sec:Laplike\]. And such perturbations lead to folds.
\[theo:likem\] Let $T:D \to H$ be an $m$-amenable operator and $P: D \to H$ and be an $m$-compatible $C^1$ function. Then $F = T - P : D \to H$ is a global fold.
An operator $T: X \to X$ with spectral radius $r(T)$ is [*$M$-amenable*]{} if
(M-a) $r(T)$ is an elementary eigenvalue of $T$ and $\sigma(T) \cap \{ z \in {{\mathbb{C}}}, |z|=r\} = r(T)$.
(M-b) There is a cone $K \subset X$ with interior $\mathring{K} \ne \emptyset$ for which $T(K \setminus \{0\}) \subset \mathring{K}$.
Markov operators are related to $M$-amenability. The usual fact that $\phi_M \in \mathring{K}$ under small $M$-compatible perturbations $P(u)$, defined in Section \[sec:Nusslike\], is proved by an extension of the Krein-Rutman theorem due to Zhang [@Zh].
\[theo:likeM\] Let $T:X \to X$ be an $M$-amenable operator and $P: X \to X$ and be an $M$-compatible $C^1$ function. Then there is $R_T \in {{\mathbb{R}}}$ so that if, for all $u \in X$, we have $\|DP(u) - r(T) \ I \| < R_T$ then $F = T - P : X \to X$ is a global fold.
Amenability of $T$ and of a translation $T + R \ I$, for a large $R$, are essentially equivalent. As the attentive reader will notice along the text, this fact essentially masks the relevance of this feature in the original Ambrosetti-prodi context, when $T$ is the Laplacian on a bounded set and $P(u) = f(u)$ is a Nemitskii operators , where $f:{{\mathbb{R}}}\to {{\mathbb{R}}}$ has bounded derivative.
In the self-adjoint case, the hypotheses needed for our arguments are given by spectral data of $T$. For $M$-amenable operators, however, we are limited to a perturbation result: the nonlinear term $P - r(T) \ I$ (or better, its Jacobian) has to be sufficiently small.
Most arguments are geometric and do not depend on the locality of amenable or compatible operators. We work with positive functions in the Hilbert context and privileged cones in Banach spaces, so as to preserve familiar procedures.
Smoothness is a key point in this text. In Section \[section:general\] we introduce hypotheses yielding (LS) and (PR) for a Lipschitz map $F$ (providing details for arguments in [@TZ], [@CTZ1]), and the proofs of Theorems \[theo:likem\] and \[theo:likeM\] then depend only on the verification of (NT). For Lipschitz compatible perturbations given by maps close to Nemitskii operators, a proof of (NT) is presented in Section \[sub:equi\]. For more general compatible perturbations, we require that the maps $F$ be $C^1$, and consider spectral properties of the Jacobians $DF(u)$ and some averages, the mean Jacobians $MF(u,v)$, defined in Section \[sec:Laplike\]. For $m$-compatible operators, (NT) is obtained in Section \[sec:Laplike\] from the study of the smallest eigenvalue of mean Jacobians. In the $M$-compatible case instead it is the spectral radius which has to be controlled and we prove (NT) in Section \[sec:Nusslike\].
As we shall see among the examples in Section \[section:examples\], some amenable operators are the Laplacian itself (with Dirichlet, Neumann or periodic boundary conditions), the hydrogen atom $T v = - \Delta v - v/r$ in ${{\mathbb{R}}}^3$, the quantum harmonic oscillator $T v (x)= - v''(x) + x^2 v(x)$, spectral fractional Laplacians in bounded domains, integral operators with positive kernels. We consider only perturbations $T-P(u)$ but nonautonomous maps $T-P(x,u)$ may be handled with minor modifications. Section \[section:rev\] comments on the rarity of global folds associated with compatible Nemitskii operators, a fact shown in [@CTZ2] for the Laplacian. Section \[sub:equi\] shows an equivariant version of the Ambrosetti-Prodi theorem: in this case, the associated compatible operator is nonlocal.
For a $C^2$ function $F$, the differentiability of the eigenvalues of the Jacobians permits additional simplifications, as shown in Corollary \[cor:corC\]. The result is then used in Section \[section:hybrid\], which in a sense is an ideal situation. We describe a map from $L^2$ to $L^2$ which restricts appropriately to one from $C^0$ to $C^0$ with the following properties. There is sufficient (spectral) information of the self-adjoint kind to derive (LS) and (PR) on the $L^2$ scenario. We then transplant these facts to an associated map in $C^0$, on which the hypotheses leading to (NT) are especially weak, and end up with global folds in both scenarios.
This text provides the proofs of most of the results stated in [@CTZ1].
Hypotheses (LS) and (PR), fibers and heights {#section:general}
============================================
The first two hypotheses of Theorem \[theo:fold\], (LS) and (PR), admit a unified treatment for both kinds of amenable operators. There are other approaches to (LS) ([@R], [@MST2], also Section \[section:hybrid\]), which are not relevant in Sections \[sec:Laplike\] and \[sec:Nusslike\].
For $m$-admissibile operators $T: D \to H$, the original Banach spaces are the real Hilbert space $Y=H = L^2(M, d\mu)$ and $X = D \subset H$, the domain of self-adjointness of $T$ (with norm $\|u\|_D = \|u\| + \|Tu\|)$. Let $V$ be the subspace spanned by $\phi_m$, the eigenvector associated with the simple eigenvalue $\lambda_m$. Since $ T$ is $m$-admissible, $W_H = {\operatorname{Ran}}(T - \lambda_m \ I)$ is a closed subspace of codimension one. Then $W_D = W_H \cap D$ induce orthogonal decompositions $$D = W_D \oplus V \quad , \quad \quad H = W_H \oplus V$$ and an orthogonal projection $\Pi: H \to W_H$. The restriction $T - \lambda_m \ I: W_D \to W_H$ is an invertible operator.
For $M$-admissible operators $T: X \to X$, $X=Y$ is a real Banach space. Let $V$ be the span of $\phi_M$, and set $W_X= {\operatorname{Ran}}(T - \lambda_M \ I)$, so that again there are decompositions $ X = Y = W_X \oplus V $, a projection $\Pi: X \to W_X$ and an invertible restriction $T - \lambda_M \ I: W_X \to W_X$.
To unify notation, we write the functions of interest as $$F = T - P : W_X \oplus V \to W_Y \oplus V$$ where the spaces are defined in terms of a privileged eigenpair $(\lambda_p, \phi_p)$ ($p$ is $m$ or $M$, depending on the type of amenability). We also write $\Pi: Y \to W_Y$. Define the translations $P_\gamma = P - \gamma I , T_\gamma = T - \gamma I: X \to Y$ and the restriction $T_{\gamma,W} = T_W - \gamma I: W_X \to W_Y$
\[prop:LS\] Let $F= T - P: X \to Y$ for an amenable operator $T: X \to Y$ and assume
1. There exists $\gamma \in {{\mathbb{R}}}$ for which the restriction $T_{\gamma,W}: W_X \to W_Y$ is invertible and the map $\Pi P_\gamma: X \to W_Y$ is a Lipschitz function with a constant $L$ satisfying $ L \| T_{\gamma,W}^{-1} \| < 1$.
Then hypothesis (LS) holds for the map $F: X \to Y$.
The projection $\Pi$ in (HLS) does not appear in arguments when $P$ is a Nemitskii operator. Indeed, a function $f$ whose derivatives takes values close to $\lambda_p$ gives rise to a Nemitskii map $P(u)$ which looks pretty much like a multiple of the identity. Thus, $f$ acts rather homogeneously in all directions of space. The situation for more general maps $P$ is different: in adapted coordinates, as presented in the Introduction, large distortions along vertical lines in the image do not change the global nature of the fold. Such distortions correspond to large values of $P(u)$ along $\phi_m$ and are trivialized by the action of the projection $\Pi$.
The argument follows [@BP]. Write $u = \Pi u + t \phi_p = w + t \phi_p$, for $w \in W_X$. For $t \in {{\mathbb{R}}}$, define the projected restrictions $F_t : W_X \to W_Y$, $$F_t(w) = \Pi F(w + t \phi_p)= \Pi (T-P)(w + t \phi_p)= T_{\gamma,W} w - \Pi P_\gamma(w + t \phi_p) \ .$$ Set $T_{\gamma,W} w = y$ to obtain $$F_t \circ T_{\gamma,W}^{-1}: W_Y \to W_Y, \ F_t (y) = y - \Pi P_\gamma( T_{\gamma,W}^{-1} y + t \phi_p) = y - K_t(y) .$$ We bound variations of $K_t: W_Y \to W_Y$. For $z, {\tilde z} \in W_Y$, $s , {\tilde s} \in {{\mathbb{R}}}$, $$\| K_s(z) - K_{\tilde s}({\tilde z}) \| = \| \Pi P_\gamma(T_{\gamma,W}^{-1}z + s \phi_p) - \Pi P_\gamma(T_{\gamma,W}^{-1}{\tilde z} + {\tilde s} \phi_p) \|$$ $$\le L \| T_{\gamma,W}^{-1}(z - \tilde z)\| + {{\cal{C}}}|s - \tilde s| \le c \| z - \tilde z\| + {{\cal{C}}}|s - \tilde s|$$ for $c < 1$ by (HLS) and ${{\cal{C}}}$ possibly large.
From the Banach contraction theorem, $I - K_t: W_Y \to W_Y$ are (uniform) Lipschitz bijections. We show that the maps $(Id-K_t)^{-1}: W_Y \to W_Y$ are also uniformly Lipschitz: set $z_i=(Id-K_t)^{-1}(y_i),\ i=1,2$ and then $$||z_1-z_2||\le ||y_1-y_2||+||K_t(z_1)-K_t(z_2)||\le ||y_1-y_2|| +c||z_1-z_2||$$ so that $$\displaystyle \|z_1 - z_2 \|\le \frac{1}{1-c} \ \|y_1 - y_2 \| \ .$$
Thus $F_t = (I - K_t) \circ T_{\gamma,W}: W_X \to W_Y$ are also uniformly bilipschitz homeomorphisms. We now show that $$\Phi^{-1} = \Psi=\left(F_t, \ \hbox{Id} \right): Y\to X$$ is a bilipschitz homeomorphism. Clearly, $\Psi$ is a Lipschitz bijection. To handle $\Psi^{-1} = \Phi$, take $v=y+s\phi$ and $\tilde v=\tilde y+\tilde s\phi$ (the letter ${{\cal{C}}}$ represents different constants along the computations): $$||\Phi(v)-\Phi(\tilde v)||_X \le
{{\cal{C}}}\left( ||(F_s)^{-1}(y)-(F_{\tilde s})^{-1}(\tilde y)||_X+||s\phi-\tilde s\phi||_X \right)$$ $$\le {{\cal{C}}}\left( ||(F_s)^{-1}(y)-(F_{\tilde s})^{-1}( y)||_X+||(F_{\tilde s})^{-1}(y)-(F_{\tilde s})^{-1}(\tilde y)||_X+|s-{\tilde s}|_X \right)$$ For the second term use the Lipschitz bound for $F_{\tilde s}^{-1}$. For the first, we prove $ || F_s^{-1}(y) - F_{\tilde s}^{-1}(y)|| \le {{\cal{C}}}| s - {\tilde s} | $. As before, set $w = F_s^{-1}(y), {\tilde w} = F_{\tilde s}^{-1}(y)$ and $$z = T_{\gamma,W} \circ F_s^{-1}(y)= (1 - K_s)^{-1}(y) \ , \ {\tilde z} = T_{\gamma,W} \circ F_{\tilde s}^{-1}(y)= (1 - K_{\tilde s})^{-1} (y) \ .$$ The iterations yielding $z$ and $\tilde z$, $$z_0=0, \; z_{j+1}=K_s(z_j)+y \ \ {\rm and}\ \ {\tilde z}_0=0, \; {\tilde z}_{j+1}=K_{\tilde s}({\tilde z}_j)+y$$ imply the estimates $$||z_{j+1} - {\tilde z}_{j+1}|| = ||K_s(z_j) - K_{\tilde s}({\tilde z}_j))||
\le c \ ||z_j - {\tilde z}_j|| + {{\cal{C}}}\ |s - {\tilde s} | \ ,$$ and for $j\to +\infty$, $$|| F_s^{-1}(y) - F_{\tilde s}^{-1}(y)|| = || w - {\tilde w}|| \le ||T_{\gamma,W}^{-1}||
|| z - {\tilde z} || \le {{\cal{C}}}\frac{ \ ||T_{\gamma,W}^{-1}||}{1-c} \ | s - {\tilde s}| \ .$$ Adding up, $$||\Phi(v)-\Phi(\tilde v)||_X \le {{\cal{C}}}\frac{ \ ||T_{\gamma,W}^{-1}||}{1-c} \ | s - {\tilde s}| + {{\cal{C}}}\|y - {\tilde y}\|_X + ||s\phi-\tilde s\phi||_X \ ,$$ completing the proof that $\Psi: X \to Y$ is a bilipschitz homeomorphism. Hypothesis (LS) is clear from the following diagram, for $F^a=F \circ \Phi$: $$\begin{array}{ccl}
{D = W_X \oplus V}&
\stackrel{{\scriptstyle F}}{\longrightarrow}&
{H = W_Y \oplus V} \\
{\scriptstyle \Psi=( F_t, Id)}\searrow &
&
\nearrow{\scriptstyle F^a=F \circ \Phi=(Id, h^a)}\\
&{W_Y \oplus V}
& \\
\end{array}$$
For a fixed $z \in W_Y$, the inverse of a vertical line $\{(z, s), s \in {{\mathbb{R}}}\}$ under $\Psi$ is a [*fiber*]{} $\{ u(z,t) = w(z,t) + t \phi_p , \ t \in {{\mathbb{R}}}\}$. The [*height function*]{} is $$h: D= W_X \oplus V \to {{\mathbb{R}}}, \ h(u) = \langle \phi_p, F(u) \rangle = \langle \phi_p,\ Tu - P(u) \rangle\ .$$
In the $M$-amenable case, the dual operator $T^\ast: X^\ast \to X^\ast$ also has a simple eigenvalue $\lambda_M$, associated with the eigenvector $\phi_M^\ast$ given by the functional which is zero on $W_X$ and normalized so as $\langle \phi_M^\ast, \phi_M \rangle = 1$ (notice the coupling between $X$ and $X^\ast$). To unify notation, set $\lambda_p^\ast = \phi_m$ or $\phi_M^\ast$.
\[prop:proper\] For $T:X \to Y$ amenable and $F= T - P: X \to Y$, assume
1. There exist $\lambda_-, \lambda_+, c_-, c_+ \in {{\mathbb{R}}}$ with $\lambda_- < \lambda_p < \lambda_+ $ for which $$\forall \ u \in X \ , \quad
\ \langle \phi_p^\ast, P(u) \rangle \, \ge \, \lambda_- \, \langle \phi_p^\ast , u \rangle \ + c_- \, , \ \lambda_+ \, \langle \phi_p^\ast , u \rangle \ + c_+ \, .$$
and (HLS). Then $F: X \to Y$ satisfies hypothesis (PR).
From (HPR), height functions go to $-\infty$ along fibers. Indeed, $$h(u(z,t))
= \langle \phi_p, T w(z,t) + tT \phi_p\rangle - \langle \phi_p, P(u(z,t))\rangle$$ $$= t \lambda_p - \langle \phi_p, P(u(z,t))\rangle \ \le \ (\lambda_p - \lambda_-) \ t - c_- \ , \ (\lambda_p - \lambda_+) \ t - c_+ ,$$ and the limits follow ($t \to -\infty$ from the first bound, $t \to \infty$ from the second), together with their uniformity in $z$. Since the homeomorphism $\Psi: D \to H$ preserves the horizontal component, $\lim_{|t| \to \infty} h^a(z,t) = - \infty$.
The properness of $F: D \to H$ is equivalent to that of $F^a: W_Y \oplus V \to W_Y \oplus V$, which we prove. Take $(z_n, y_n) =(z_n, h^a(z_n, t_n) ) \to (z_\infty, y_\infty)$. If $|t_n| \to \infty$, then $h^a(z_\infty, t_n) \to -\infty$, contradicting the uniform convergence at $z_\infty$.
[**A sketch of proof of Theorem \[theo:fold\]**]{}
By hypothesis (LS), Theorem \[theo:fold\] reduces to the statement that the proper map $$F^a: W_Y \oplus {{\mathbb{R}}}\to W_Y \oplus {{\mathbb{R}}}\, , \quad (z, t) \mapsto (z, h^a(z,t))$$ is either a homeomorphism or a global fold. But, by hypothesis (NT), this is exactly Proposition 9 in [@STZ].
$m$-amenable operators {#sec:Laplike}
======================
More general self-adjoint operators giving rise to global folds were presented before ([@B]). Here we consider $m$-amenable operators $T: D \to H$: more concretely, the Banach spaces $X$ and $Y$ from the previous sections are $Y = H = L^2(M, d\mu)$ for a $\sigma$-finite measure space $(M,\mu)$ and $X= D \subset H$ is the domain of self-adjointness of $T$, equipped with the norm $\| u \|_D = \| u \|_H + \| T u \|_H$. Euclidean space $D = H = {{\mathbb{R}}}^n$ is the case when $\mu$ is a finite collection of deltas.
The spectrum of $T: D \subset H \to H$ will be denoted by $\sigma(T)$. Following [@RS], we say that a function of $u \in H$ is [*positive*]{} if $u \ge 0 \ a.e.$ and $u \ne 0$. A bounded operator $A: H \to H$ is [*positivity preserving*]{} if $Au$ is positive for all positive $u$.
A $C^1$ function $P: H \to H$ is [*$m$-compatible*]{} with an $m$-amenable operator $T:D \to H$ if it satisfies the following properties. Let $\mu = \min \sigma(T) \setminus \{ \lambda_m\}$.
1. There are $a, b \in {{\mathbb{R}}}$, $a < \lambda_m < b < \mu$, such that, for $u \in D$, $DP(u): H \to H$ is a bounded, symmetric operator with $\sigma(DP(u)) \subset [a,b]$.
2. There exist $\lambda_-, \lambda_+, c_-, c_+ \in {{\mathbb{R}}}$ with $\lambda_- < \lambda_m < \lambda_+ $ for which $$\forall \ u \in D \ , \quad
\ \langle \phi_m, P(u) \rangle \, \ge \, \lambda_- \, \langle \phi_m , u \rangle \ + c_- \, , \ \lambda_+ \, \langle \phi_m , u \rangle \ + c_+ \, .$$
3. For every $t >0, u \in D$, $e^{tDP(u)}: H \to H$ is positive preserving.
4. For $u, v \in D$, if $v- u > 0 \ a.e.$ then $\inf \sigma(DF(v)) < \inf \sigma(DF(u))$.
As we shall see in Proposition \[prop:mamenom\], the $m$-amenability of $T$ together with (m-LS) above implies the $m$-amenability of any Jacobian $DF(u): D \to H$. In particular, there is a continuous function $\lambda_m: D \to {{\mathbb{R}}}$ which associates to $u \in D$ the smallest (elementary) eigenvalue of $DF(u)$ (and then $\lambda_m(DF(u)) = \inf \sigma(DF(u))$).
Proof of Theorem \[theo:likem\] {#sec:NT}
--------------------------------
We must verify hypotheses (LS), (PR) and (NT) of Theorem \[theo:fold\] for $m$-amenable operators and $m$-compatible $C^1$ perturbations.
\[prop:mca\] Let $T:H \to D$ be $m$-amenable, and $P:H \to H$ be a $C^1$ map for which (m-LS) and (m-PR) hold. Then $F$ satisfies (HLS) and (HPR).
We show that (m-LS) implies (HLS) for $\gamma = (a+b)/2$. For $P_\gamma = P - \gamma I$ and $u(\tau) = \tau u + (1 - \tau) \tilde u $, $$P_\gamma(u) - P_\gamma(\tilde u)
= \int_0^1 DP_\gamma(u(\tau)) \ \big( u - \tilde u \big) \ d\tau$$ and then, since $\| \Pi \| = 1$, $$\| \Pi P_\gamma(u) - \Pi P_\gamma(\tilde u) \|
= \| \int_0^1 \Pi DP_\gamma(u(\tau)) \ \big( u - \tilde u \big) \ d\tau \|
\le (b - \gamma) \| u - \tilde u \| \ .$$
Thus $L = b - \gamma$ and since $ \| T_{\gamma,W}^{-1} \| = \| (T_W - \gamma I)^{-1} \| \ge (\mu - \gamma)^{-1}$, (HLS) follows. Clearly (m-PR) implies (HPR).
In [@STZ], (NT) was verified for weaker differentiability conditions, but less general perturbations $P$: the argument was given for Nemitskii operators which are only Lipschitz, and is presented in slightly modified form in Section \[sub:equi\]. We need an extension of the argument.
By (m-LS), for all $ u \in D$, the operators $DP(u): H \to H$ are bounded, and by the Kato-Rellich theorem, the Jacobians $DF(u) = T - DP(u): D \to H$ are self-adjoint. For $u,v \in D$, define the [*mean Jacobian*]{} $$MF(u,v) = \int_0^1 DF( u + t(v-u)) dt = T - \int_0^1 DP( u + t(v-u)) dt \ ,$$ so that $MF = T - MP(u,v)$, for a symmetric, bounded operator $MP(u,v)$. Notice that $MF(u,u) = DF(u)$ and $MF(u,v) = MF(v,u)$: to simplify statements, we treat Jacobians as special cases of mean Jacobians.
Lemma A below is exercise 91 of Chapter XIII from [@RS], a result by Faris [@F]. Lemma B is Theorem XIII.44 of [@RS]. A bounded self-adjoint operator $A: H \to H$ is [*positivity improving*]{} if for any nonzero $f \ge 0$, $Af > 0$ a.e.
\[lemma:Faris\] Let $T: D \to H$ be a self-adjoint operator for which $e^{-tT}, \ t >0$, is positivity improving. Let $-V: H \to H$ be a positivity preserving bounded operator. Then, for $t>0$, $e^{-t(T+V)}$ is positivity improving.
\[lemma:44\] Let $T: D \to H$ be a self-adjoint operator that is bounded from below. Suppose that $e^{-tT}$ is positivity preserving for all $t>0$ and that $E = \min \sigma(T)$ is an eigenvalue. Then the following are equivalent.
(a) $E$ is a simple eigenvalue with a strictly positive eigenvector.
(b) For all $ t > 0 $, $e^{-tT}$ is positivity improving.
\[prop:mamenom\] For $u, v \in D$, if $0 \in \sigma(MF(u,v))$ then $0 = \min \sigma(MF(u,v))$. The operators $MF(u,v): D \to H$ and $MF(u,v): D \to H$ are $m$-amenable.
Clearly (m-LS) holds for $DP(u)$ and $MP(u,v)$. By the Weyl inequalities, (m-LS) implies that the eigenvalues in $\sigma(T) \setminus \{ \lambda_m\}$ are not perturbed enough to reach 0, and thus only the smallest eigenvalue of $MF(u,v)$ can be zero. We now prove $m$-amenability of $S = MF(u,v)= T - MP(u,v)$. For $t >0$, $e^{-tT}$ is positivity improving by (m-a) and $ e^{t MP(u,v)}$ is positivity preserving by (m-NT1). By Lemma A, $e^{-tS}$ is positivity improving, so that (m-b) holds for $S$. By Lemma \[lemma:44\], condition (m-a) for $S$ also holds.
We are ready for the proof of (NT). Suppose by contradiction that there is $g \in H$ with three preimages $u_1, u_2, u_3 \in D$, $ F(u_1)= F(u_2) = F(u_3) = g $, so that $$\label{0} F(u_2)-F(u_1)=0=F(u_3)-F(u_2).$$ The $u_i$’s belong to the same fiber (the inverse under $F$ of the vertical line through $g$), so write $ u_i = t_i \phi_m + w_i, i=1,2,3$ for $t_1 < t_2 < t_3$ and $w_i \in W_D = \langle \phi_m \rangle^\perp$.
From the fundamental theorem of calculus, $$F(u_{i+1})-F(u_i)= \int_0^1DF(t u_{i+1} +(1-t)u_i) \ dt \ (u_{i+1}-u_i)$$ $$=MF(u_{i+1},u_i)(u_{i+1}-u_i) =0 \ , \quad i = 1, 2 \ .$$ Thus $u_{i+1} - u_i \in \ker MF(u_{i+1}, u_i)$ and form the proposition above, the functions $u_{i+1} - u_i$ have a well defined sign. More, $$\langle u_{i+1} - u_i \ , \ \phi_m \rangle \ = \ \langle (t_{i+1} \phi_m + w_{i+1}) - (t_i \phi_m + w_i) \ , \ \phi_m \rangle$$ $$= \ \langle (t_{i+1} - t_i) \ \phi_m \ , \ \phi_m \rangle \ = \ t_{i+1} - t_i > 0$$ and since $\phi_m >0$ we must have $u_3 > u_2 > u_1 $ a.e.
The operators $MF(u_{i+1},u_i), i=1,2$ both have a zero eigenvalue, at the bottom of their spectra, by Proposition \[prop:mamenom\]. As $ u_1 + t (u_2 - u_1) < u_2 + t(u_3 - u_2)$ for each $t \in [0,1]$,thus the integrand defining $MF(u_2,u_1)$ is smaller than that of $MF(u_3,u_2)$. Since the smallest eigenvalue $\lambda_m(MF(u,v))$ is the minimal value of the associated quadratic form (and it is achieved!), zero cannot be a common eigenvalue. The proof of (NT) is complete.
Finally, $F:D \to H$ cannot be a homeomorphism, a possibility left by Theorem \[theo:likem\]: at each fiber, $F^a$ behaves like $x \to - x^2$, by the proof of (HPR).
$M$-amenable operators {#sec:Nusslike}
======================
Let $X$ be a real Banach space, ${\cal B}(X)$ be the space of bounded linear operators from $X$ to $X$ equipped with the usual sup norm. Among the many almost equivalent definitions of the [*essential spectrum*]{} $\sigma_e(T)$ in [@EE], we use the fourth one, $$\sigma_e(T) = \{ \lambda \in {{\mathbb{C}}}\ : \ T - \lambda \ I: X \to X \ \hbox{is not a Fredholm operator of index zero} \} \ .$$ Denote the spectral radius of $T$ by $r(T)$ and the essential spectral radius by $r_e(T)$. The points of $\sigma(T)$ outside of the circle $\{ |z| \le r_e(T) \}$ are isolated eigenvalues of finite (algebraic) multiplicity. For the dual operator $T^\ast: X^\ast \to X^\ast$, $r(T^\ast) = r(T)$ and $r_e(T^\ast)= r_e(T)$, from the characterization in terms of Fredholm theory.
In this text, a [*cone*]{} $K \subset X$ is a closed set with nonempty interior $\mathring{K}$ for which $$0 \in K \ , \quad K + K \subset K \ , \quad t \ge 0 \Rightarrow tK \subset K , \quad v, -v \in K \Rightarrow v = 0 \ .$$ The [*dual cone*]{} $ K^{\ast} = \{\ell \in X^{\ast}: \ell(x) \ge 0 \ \forall x \in K \} $ then also has nonempty interior. An operator $T \in {\cal B}(X)$ is [*positive*]{} (resp. [*strongly positive*]{}) with respect to $K$ if $T K \subset K$ (resp. $T(K \setminus \{0\}) \subset \mathring{K}$). Clearly, if $T: X \to X$ is positive (resp. strongly positive) with respect to $K$, then $T^\ast: X^\ast \to X^\ast$ is positive (resp. strongly positive) with respect to $K^\ast$.
It is easy to see that if $\lambda \in {{\mathbb{R}}}$ is an elementary eigenvalue of $T$, then it is also an elementary eigenvalue of $T^\ast: Y^\ast \to X^\ast$.
We will use Zhang’s extension of Nussbaum’s theorem ([@Zh], [@N]).
\[theo:zhang\] Let $K \subset X$ be a cone and $T \in {\cal B}(X)$ be a strongly positive operator with respect to $K$ with $r(T) > r_e(T)$. Then $\lambda_M = r(T)$ is an elementary eigenvalue of $T$ with an associated eigenvector $\phi_M \in \mathring{K}$. Any other eigenvalue $\lambda$ of $T$ satisfies $|\lambda| < r(T)$. Moreover, $\lambda_M$ is an elementary eigenvalue of $T^\ast \in {\cal B}(X^\ast)$, associated with an eigenvector $\phi_M^\ast \in \mathring{K}^\ast$, normalized so that $\phi_M^\ast(\phi_M) = 1$.
From Theorem \[theo:zhang\], the $M$-amenable operators $T$ and $T^\ast$ have a common eigenvalue $\lambda_M = r(T)$ and eigenvectors $\phi_M \in \mathring{K}$ and $\phi_M^\ast \in \mathring{K}^\ast$. Also, the restriction $T_{\lambda_M,W} = T - \lambda_M \ I: W_X \to W_X$ is an invertible operator.
Since $\sigma(T)$ is bounded and $r(T) = \lambda_M$ is an elementary eigenvalue, the upper semi-continuity of the spectrum implies the existence of an open ball $B_{R(T)} \subset {\cal B}(X)$ centered at the origin with radius $R(T)$ such that the operator $T+S:X \to X$ also has an elementary eigenvalue of largest modulus $r(T+S) = \lambda_M(T+S)$ for $S \in B_{R(T)} $. Again, the largest eigenvalue $\lambda_M: B_{R(T)} \to {{\mathbb{R}}}$ is analytic. To avoid confusion, we denote by $r = r(T) = \lambda_M(T)$ the eigenvalue of largest modulus of $T$ and reserve the notation $\lambda_M$ for the eigenvalue as a function.
As in Section \[sec:NT\], for $u,v \in X$, we consider the Jacobian $DF(u): X \to X$ and the mean Jacobian $MF(u,v): X \to X$, $ MF(u,v) = T - MP(u,v)$, for a bounded operator $MP(u,v): X \to X$.
Let $K$ be the cone associated with $T:X \to X$ and write $P = r + P_r$. A $C^1$ function $P: X \to X$ is [*$M$-compatible*]{} with $T$ if it satisfies the properties below. Hypotheses (HLS) and (HPR) have been rewritten for $P_r$ instead of $P$.
1. The maps $\Pi P_r(u): X \to W_X$ are Lipschitz with a common constant $L$ for which $ L \| T_{r,W}^{-1} \| < 1$.
2. There exist $\lambda_-, \lambda_+, c_-, c_+ \in {{\mathbb{R}}}$ with $\lambda_- < r < \lambda_+$ such that $$\forall \ u \in X , \quad \langle \phi_M^\ast, P_r(u) \rangle \, \ge \, (\lambda_- - r) \, \langle \phi_M^\ast , u \rangle \ + c_- \, , \ (\lambda_+ - r) \, \langle \phi_M^\ast , u \rangle \ + c_+ \, .$$
3. For $u \in X$, $-DP(u)$ is positive with respect to $K$.
4. For $v - u \in \mathring{K}$, $\ \lambda_M(DF(v)+ r) < \lambda_M(DF(u)+ r)$.
Proof of Theorem \[theo:likeM\] {#sub:NC}
--------------------------------
We derive the counterpart of Propositions \[prop:mca\] and \[prop:mamenom\].
\[prop:mamenoM\] There is $R_T >0$ such that, if for all $u \in X , \| DP_r(u) \| < R_T$, then (M-LS) holds. Also, for $u,v \in X$, if $0 \in \sigma(MF(u,v))$ then the spectral radius $\lambda_M(MF(u,v)+ r)$ is $r = r(T)$. The operators $DF(u), MF(u,v): D \to H$ are $M$-amenable with respect to the same cone $K$ associated with $T$.
To prove (M-LS), take $\gamma= r$ in the argument of Section \[section:general\]. A small $R$ implies a small Lipschitz constant for $P_r(u)$: an appropriate $R$ implies (M-LS), since $T_{r,W}$ is fixed. It also implies small changes in $DF$, $MF$ and their spectra, so that (M-a) and the statement for $MF(u,v)$ follows. The proof of (M-b) is trivial from (M-NT1). As in Proposition \[prop:mca\], (M-PR) implies (HPR).
Thus, an appropriate choice of $R_T$ implies (HLS) and (HPR) for the mean Jacobians. We follow Section \[sec:NT\] and prove (NT). Three points $u_i = t_i \phi_M + w_i, i=1,2,3$ with the same image imply $$MF(u_{i+1},u_i)(u_{i+1}-u_i) =0 \ , \ i = 1, 2 \ .$$ From the $M$-amenability of $MF$ proven in Proposition \[prop:mamenoM\] and Theorem \[theo:zhang\], $$u_2 - u_1, \ u_3 - u_2 \in \mathring{K} \quad \hbox{for} \quad t_1 < t_2 < t_3 \ .$$
From (M-NT1), the spectral radius of $MF(u_2,u_1)+r$ and $MF(u_3,u_2)+r$ is $r = \lambda_M$. From (M-NT2), $$r \ = \ \lambda_M(MF(u_2,u_1)+r) < \lambda_M(DF(u_2) + r) < \lambda_M( MF(u_3,u_2)+r) \ = r \ ,$$ and we are done: as in Section \[sec:Laplike\], $F:X \to X$ cannot be a homeomorphism.
The presence of $R_T$ in the statement of Theorem \[theo:likeM\] is the price of considering operators which are not self-adjoint. Some improvements are possible in special cases. The main result in [@STZ] is about a Berestycki-Nirenberg-Varadhan operator $T: W^{2,n}(\Omega) \to L^n(\Omega)$, for a bounded set $\Omega \subset {{\mathbb{R}}}^n$ with smooth boundary [@BNV]. It is not self-adjoint, but admits an eigenvalue $\lambda_m$ of smallest real part. Given $a <\lambda_m$, there is $b(a) > \lambda_m$ such that, for any Lipschitz strictly convex function $f: {{\mathbb{R}}}\to {{\mathbb{R}}}$ with $f'({{\mathbb{R}}}) = (a,b(a))$, the map $F(u) = Tu - P(u): W^{2,n}(\Omega) \to L^n(\Omega)$ is a global fold. The additional structure implies the positivity of ground states from a maximal principle.
Section \[section:hybrid\] presents a scenario in which spectral estimates are available for a map between Banach spaces.
Examples, variations, remarks {#section:five}
=============================
Some amenable operators {#section:examples}
-----------------------
The identification of operators $T$ generating positivity preserving semigroups (hypothesis (m-b)) is by itself a field of mathematics. Arguments in the spirit of Bochner’s theorem characterizing distributions of positive type and the Levy-Khintchine formula (Appendix 2 to Section XIII.12, [@RS], vol.IV) lead to a wealth of examples of such operators. If an operator $T_0$ satisfies (m-b), few weak hypotheses suffice to obtain (m-b) for $T = T_0 + V$, from the Lie-Trotter formula.
We list a few examples, chosen in order to indicate different techniques.
\[prop:m-amenable\] The following operators are $m$-amenable.
- $-\Delta$ for Dirichlet, Neumann, periodic or mixed boundary conditions on bounded smooth domains.
- The Schrödinger operator for the hydrogen atom in ${{\mathbb{R}}}^3$, $T v = - \Delta v - v/r$.
- The quantum oscillator in ${{\mathbb{R}}}$, $Tv (x) = - v''(x) + x^2 v(x)$.
- Fractional powers $T^s, s \in (0,1)$ of positive $m$-amenable operators.
- Spectral fractional Laplacians on bounded smooth domains.
Hypothesis (m-a) is standard for all examples, we check (m-b). For (I), see Sections 8.1 and 8.2 of [@Ar]. For (II), take $T_0 = -\Delta$ in its usual domain $H^2({{\mathbb{R}}}^3)$. For the potential $V = - 1/r$, and define $T = T_0 + V$. We prove (m-b) for $T$ using a perturbation argument, Theorem XIII.45, vol. IV of [@RS]. Define the bounded truncations $V_n$ which coincide with $V$ for $|x| > 1/n$ and are zero otherwise. Set $q_n = V - V_n$. Both $T_0$ and $T$ are bounded from below. Comparing quadratic forms, $$T \le T_0 + V_n \quad \hbox{and} \quad T_0 \le T - V_n \ ,$$ so that $T_0 + V_n$ and $T - V_n$ are uniformly bounded from below. We are left with showing that $T_0 + V_n \to T$ and $T - V_n \to T_0$ in the strong resolvent sense. By Theorem VIII.25, vol. I of [@RS] it suffices to show that, for a given $u \in H^2$, $$q_n \ u \to 0 \quad \hbox{in} \ L^2({{\mathbb{R}}}^3) \ , \quad \hbox{i.e.} \quad \lim_{\epsilon \to 0} \ \int_{|x| \le \epsilon} \frac{u^2(x)}{|x|^2} \ dx \ = \ 0 \ ,$$ which is true, since $H^2({{\mathbb{R}}}^3)$ consists of bounded, continuous functions. The proof of (III) is similar. For (IV), use the arguments with Laplace transforms in Section IX.11 of [@Yo] (see also [@Tay]). Finally, (V) is a special case of (IV).
What about $M$-amenable operators? Any operator satisfying the hypothesis of Theorem \[theo:zhang\] would do, for example.
Nemitskii operators revisited {#section:rev}
-----------------------------
In the standard Ambrosetti-Prodi theorem, the associated compatible perturbation is a Nemitskii map $P(u) = f(u)$ for a strictly convex function $f: {{\mathbb{R}}}\to {{\mathbb{R}}}$.
Our original motivation for considering more general compatible perturbations was the realization ([@CTZ2]) that convexity was essentially a [*necessary*]{} condition to obtain global folds for $F(u)= - \Delta u - f(u)$. It turns out that this property carries through to other amenable operators, once a technical hypothesis on the behavior of eigenvectors detailed in Theorem 1 of [@CTZ2] is satisfied.
Nemitskii maps between Hölder spaces are smooth, but are usually only Lipschitz between Hilbert spaces, so that Jacobians and their eigenvalues are not available. This difficulty led to the arguments for (LS) presented in Section \[section:general\] and those for (NT) illustrated in Section \[sub:equi\] for a nonlocal variation.
The convexity of $f$ suggests a more stringent hypothesis on $P$ leading to a simpler proof of (NT) in the $m$-amenable scenario. The strict convexity of $f$ gives $$x < y < z \quad \Rightarrow \quad \frac{f(y) - f(x)}{y-x} \ < \ \frac{f(z) - f(y)}{z-y}$$ and we are led to the following alternative.
1. If $u < v < w$, then $\langle w - v , P(v) - P(u) \rangle \ < \ \langle v - u , P(w) - P(v) \rangle \ . $
To prove (NT) assuming (m-NT3) instead of (m-NT2), suppose again, by contradiction, that $ F(u)= F(v) = F(w) = g $ with $u < v < w$ and $$T (v - u) - (P(v) - P(u) ) = 0 = T (w - v) - (P(w) - P(v)) \ .$$ Take inner products, $$\langle w - v \ ,\ T (v - u) - (P(v) - P(u) \rangle = 0 = \langle v - u \ ,\
T (w - v) - (P(w) - P(v)) \rangle$$ and use the self-adjointness of $T$ to derive the contradiction $$\langle w - v \ ,\ P(v) - P(u) \rangle \ = \ \langle v - u \ ,\ P(w) - P(v) \rangle \ .$$
An equivariant case: nonlocal Nemitskii operators {#sub:equi}
-------------------------------------------------
Take a closed subgroup $G \subset SO(n,{{\mathbb{R}}})$ of rigid motions and a bounded, smooth, domain $\Omega \subset {{\mathbb{R}}}^n$ which is invariant under the natural action of $G$. Thus $G$ might be $SO(n,{{\mathbb{R}}})$ acting on the unit ball, or $G= \{ I, -I\}$ and $\Omega$ an even set, $x \in \Omega \Leftrightarrow -x \in \Omega$. Denote by $H_G$ the subspace of $G$-invariant functions and by $\pi:H\to H_G$ the associated orthogonal projection. An explicit formula for $\pi$ is obtained by taking averages: for the normalized Haar measure $\mu$ on $G$, $$\pi u (x) = \int_G u \circ g (x) \ d\mu(g) \ ,$$ so that $\pi$ is positive preserving. Take as an example the Laplacian on $\Omega$ (with Dirichlet, Neumann, periodic b.c.) $T = - \Delta: D \to H = L^2(\Omega)$. Its first eigenvalue $\lambda_m$ is simple and the associated eigenfunction $\phi_m >0$ is $G$-invariant (take averages and use the simplicity of $\lambda_m$).
Let $f:\mathbb R\to\mathbb R$ be a Lipschitz strictly convex function such that $$\overline{f'({{\mathbb{R}}})} \cap \sigma(-\Delta) = \{ \lambda_m\} \ .$$ Set $ P: H \to H, P=f\circ\pi$. Then $F = T - P: D \to L^2(\Omega)$ is a global fold.
Since $f$ is Lipschitz, $f'$ exists a.e. Write the hypothesis in terms of the usual Lipschitz quotients: for $\lambda_- = \inf f', \lambda_+ = \sup f'$, $$\lambda_- \ (x - y) \ < \ f(x) - f(y) \ < \ \lambda_+ \ (x - y ) \ , \quad \forall \ x, y \in {{\mathbb{R}}}\ .$$
We use Theorem \[theo:fold\]. The verification of (LS) imitates Section \[section:general\], since a Lipschitz map $f$ yields a Nemitskii operator $N_f: H \to H$ which is also Lipschitz. We consider (PR). By hypothesis, $$\lambda_- < \lambda_m < \lambda_+ < \min \ (\sigma(T) \setminus \{\lambda_m\}) \ .$$ From the convexity of $f$, there are constants $c_-, c_+ \in {{\mathbb{R}}}$ for which $$\ \langle \phi_m, f(u) \rangle \, \ge \, \lambda_- \, \langle \phi_m , u \rangle \ + c_- \, , \ \lambda_+ \, \langle \phi_m , u \rangle \ + c_+$$ and, since $\pi \phi_m = \phi_m$, we obtain (m-PR), $$\ \langle \phi_m, P(u) \rangle \ = \ \langle \phi_m, f(\pi u) \rangle \, \ge \, \lambda_- \, \langle \phi_m , u \rangle \ + c_- \, , \ \lambda_+ \, \langle \phi_m , u \rangle \ + c_+ \ .$$
We prove (NT) following the argument in [@STZ]. Suppose by contradiction that there exist three distinct functions $u_i$ for which
$$Tu_i - P(u_i) = T u_i - f(\pi u_i) = z \ , \quad i = 1, 2, 3 .$$ Then, for $i = 1, 2$, $$T ( u_{i+1} - u_i) - (P(u_{i+1}) - P(u_i)) = 0 \quad \hbox{or} \quad \big( T - V_{i+1,i} \big) ( u_{i+1} - u_i) = 0$$ for the real, bounded potential $$V_{i+1,i}(x):=
\left\{ \begin{array}{ccc} \displaystyle\frac{f(\pi u_{i+1})-f(\pi u_i)}{u_{i+1}-u_i} & , & \hbox{for} \ x\in \Omega \ : \ u_i(x) \neq u_{i+1}(x) . \medskip \\
a & , & \hbox{for} \ x\in \Omega \ : \ u_i(x) = u_{i+1}(x) .
\end{array}
\right .$$ From the hypothesis, the only eigenvalue associated with a potential $V_{i+1,i}$ which can be 0 is the smallest. Thus the ground states $u_2 - u_1$ and $u_3 - u_2$ have a definite sign and are $G$-invariant. Moreover, since the three preimages $u_i$ lie in the same fiber, i.e., $u_i = w_i + t_i \phi_m$, we may suppose without loss that $t_1 < t_2 < t_3$ and thus, by taking inner products with $\phi_m >0$, $$u_1 < u_2 < u_3 \quad \hbox{and thus} \quad \pi u_1 < \pi u_2 < \pi u_3 \quad \hbox{in} \quad \Omega \ .$$ From the strict convexity of $f$, and the equivariance of $u_{i+1} - u_i$, $$V_{2,1} \ = \ \frac{f(\pi u_2) - f(\pi u_1)}{\pi u_2 - \pi u_1} < \ \frac{f(\pi u_3) - f(\pi u_2)}{\pi u_3 - \pi u_2} \ = \ V_{3,2}$$ and thus zero cannot be the smallest eigenvalue of $V_{2,1}$ and $V_{3,2}$. Again, as in Section \[section:general\], $F$ cannot be a homeomorphism.
The case $G= \{e\}$ is the usual Ambrosetti-Prodi theorem.
A geometric corollary {#sec:geomcor}
---------------------
We present a result with a strong geometric content. For real Banach spaces $X \subset Y$, let $F: X \to Y$ be a $C^2$ map satisfying (LS). As in Section \[section:general\], the $C^2$ diffeomorphism $\Phi: Y \to X$ takes vertical lines $$\{ z + t \phi_p, \ t \in {{\mathbb{R}}}\} \subset W_Y \oplus {{\mathbb{R}}}$$ to fibers in $X$, on which one may prescribe a uniform orientation, the one corresponding to going up along vertical lines in $Y$. Let $\phi(u)$ be the normal tangent vector to the fiber at $u \in X$ which is compatible with this orientation.
The [*critical set*]{} $\cal C$ of $F$ consists of [*critical points*]{}, where $DF(u_c): X \to Y$ is not an linear isomorphism. We assume the following hypothesis.
- For all $u_c \in {\cal C}$, $\lambda(u_c) = 0$ is an elementary eigenvalue of $DF(u_c): X \to Y$.
From hypothesis (LS), at a critical point $u_c \in {\cal C}$, $\phi(u_c)$ generates $\ker DF(u_c)$: simply check the statement in adapted coordinates, i.e., for $$F^a : X = (W_X \oplus {{\mathbb{R}}}) \to W_X \oplus {{\mathbb{R}}}\, , \quad (z, t) \mapsto (z, h^a(z,t)) \ .$$
Moreover, $\lambda(u_c) = 0$ is also an elementary eigenvalue of $DF(u_c)^\ast: Y^\ast \to X^\ast$, associated with the eigenvector $\phi^\ast(u_c) \in Y^\ast$ given by the functional which is zero on $W_Y= {\operatorname{Ran}}DF(u_c)$ and normalized so that $\langle \phi^\ast(u_c), \phi(u_c) \rangle = 1$ (the normalization is possible because 0 is an elementary eigenvalue).
From the Appendix, there is a (unique) $C^1$ function $\lambda : U \to {{\mathbb{R}}}$ from an open neighborhood $U \subset X$ of $u_c$ with $\lambda(u_c) = 0$ and $\lambda(u)$ is an eigenvalue of $DF(u)$. We also require that $0$ is a regular value of $\lambda$:
- At a critical point $u_c \in X$, $D \lambda(u_c): X \to {{\mathbb{R}}}$ is nonzero.
By the inverse function theorem, $\cal C$ is a codimension 1 manifold of $X$. The corollary below uses the notation of the previous paragraphs. Recall that $\phi^\ast_p$ is the normalized eigenfunction of $T^\ast: Y^\ast \to X^\ast$ associated with $\lambda_p$.
\[cor:corC\] Let $F: X \subset Y \to Y$ be a proper $C^2$ function satisfying (LS), (PR), (C1), (C2) and (C3) below.
- For each $u_c \in {\cal C}$ with $\phi(u_c) \ne 0$ and $DF(u_c) \phi(u_c) = 0$, $$\langle \ \phi^\ast_p \ , \ [ D (DF(u_c)) \ \phi(u_c)] \ \phi(u_c) \ \rangle \ <0 \ .$$
Then $F: X \to Y$ is either a homeomorphism or a global fold.
The expression $[ D (DF(u)) \ \phi(u)]$ is the directional derivative of the Jacobian $DF(u)$ along $\phi(u)$ and the inner product in (C3) is the directional derivative of the zero eigenvalue of $DF(u)$ along $\phi(u_c)$. In words, the eigenvalue of $DF(u)$ which is zero at $u_c$ always decreases when going up along a fiber. In [@CTZ2], the authors show that the quotient $\lambda(u)/h(u)$ extends to a strictly positive continuous function: said differently, at a critical point $u_c$ the growth properties of the height function along a fiber are equivalent to those of the eigenvalue of $DF(u)$ near zero.
The sign in hypothesis (C3) is reversible. Our choice above is compatible with the behavior of $F$ along fibers used throughout the text.
Since $F: X \to Y$ is a $C^2$ function, hypothesis (C1) and Lemma \[prop:abstract\] in the Appendix imply that, near a point $u_c \in {\cal C}$ the eigenvalue of $DF(u): X \to Y$ admits a (unique) $C^1$ continuation. We follow this eigenvalue as we move up along a fiber. Take the directional derivative along the tangent vector $\phi(u_c)$ to the fiber (which also spans $\ker DF(u_c)$) and use the chain rule $u \mapsto DF(u) \mapsto \lambda(DF(u))$ combined with the formula in the lemma, $$D \lambda(DF(u_c))\cdot \phi(u_c) = \langle \ \phi^\ast \ , \ [ D (DF(u_c)) \ \phi(u_c)] \ \phi(u_c) \ \rangle \ ,$$ which is [*negative*]{}, according to (C3). Said differently, as we move up along the fiber, whenever we pass by a point $u_c$ for which $\lambda(u_c) =0$, it must decrease. Thus there can be no two such points in a fiber. And hence there are no points in $Y$ with three preimages, since they would have to belong to the same fiber. Hypothesis (PR) completes the argument: $F$ is proper, but is not a homeomorphism.
A hybrid: transplanting fibers {#section:hybrid}
------------------------------
We present a variant of the extension by Church and Timourian ([@CT1]) of a global fold described by Mandhyan ([@M2]). We treat Nemitskii maps for simplicity.
For a compact set $\Omega \subset {{\mathbb{R}}}^n$, let $H= L^2(\Omega)$, $X= C^0(\Omega)$ with the usual norms. We require some hypotheses for $ G_H: H \to H, \ G_H(u) = Eu - g(u)$ and its restriction $G_X: X \to X $.
1. The linear operator $E: H \to X$ is bounded, defines a symmetric operator $E_H: H \to H$ and a (bounded) restriction $E_X: X \to X$.
2. $\| E_H \| = \lambda_M$ is an elementary eigenvalue of $E_H$ with an eigenvector $\phi_M > 0 $ a.e. Assume $0 < \mu = \max \sigma(E_H) \setminus \{ \lambda_M\} < \lambda_M$.
3. $E_X:X \to X$ is a positivity preserving operator.
4. $g: {{\mathbb{R}}}\to {{\mathbb{R}}}$ is a strictly convex $C^2$ function with $g'({{\mathbb{R}}}) = (a,b)$ satisfying $$\mu < a < \lambda_M < b .$$
5. For each critical point $u_c \in X$ of $G_X$, there is $R = R(u_c) > 0 $ for which $$R \ = \ r( R\ I + DG_X(u_c)) > r_e( R \ I + DG_X(u_c)) \ .$$
\[theo:hybrid\] Under these hypotheses for $E$ and $g$, the maps $G_H: H \to H$ and its restriction $G_X : X \to X$ are global folds.
The Ambrosetti-Prodi map $u \mapsto - \Delta u - f(u)$ may suggest the inversion of the Laplacian, yielding maps of the form $u \mapsto - E f(u)$. A map like $G_X$ comes up if we go one step beyond: since $f' $ is bounded away from zero (and in the original Ambrosetti-Prodi theorem may be either strictly convex or strictly concave), use $g = f^{-1}$ to invert the nonlinear part $u \mapsto f(u)$ (below, we take $g$ convex to minimize bifurcations). This phrasing is closer in spirit to the $M$-amenable scenario, in which the nonlinearity interacts with the largest eigenvalue.
We deduce Theorem \[theo:hybrid\] again from Theorem \[theo:fold\], but complications arise. The proof of (LS) for $G_H:H \to H$ is easy. We then use Theorem \[theo:zhang\] in order to obtain (LS) for $G_X:X \to X$ and juxtapose information from both approaches.
The proof breaks in steps.
0\. The eigenvector $\phi_M$ lies in $X$.
Indeed, $E_H \phi_M = \lambda_M \phi_M$, with $\lambda_M \ne 0$ and ${\operatorname{Ran}}E_H \subset X$ by (E1).
1\. (HLS) and (HPR) hold for $G_H:H \to H$.
Just mimic Section \[sec:Nusslike\]. Again, split $H = W_H \oplus V$, where $\phi_M$ spans $V$ and $W_H = V^\perp$. Fibers, the inverses of vertical lines $V_z = \{(z,t), t \in {{\mathbb{R}}}\}$, are Lipschitz curves parameterized by height $t \in {{\mathbb{R}}}$, $u(t) = w(z,t) + t \phi_M, w(t) \in W_H$.
2\. If $G_H(u) = x$ for $x \in X$, then $u \in X$.
Set $g^{-1} = f$ (recall that $g$ is strictly increasing): both Nemitskii operators $u \mapsto g(u)$ and $w \mapsto f(w)$ take $H$ to $H$ and $X$ to $X$. Write $G_H(u) = E u - g(u) = E f(w) - w$, so that $G_H(u) = x$ implies that $w = Ef(w) - x \in X$, since $E \ H \subset X$, by (E1). But then $u = f(w) \in X$.
Since $\phi_M \in X$, a vertical line $V_x$ through a point $x \in X$ is completely in $X$. From the previous step, their inverses, which are fibers in $H$, are necessarily in $X$. To show that (HLS) and (HPR) hold for $G_X:X \to X$, we prove that each fiber $\alpha_u = G^{-1}_H (V_x)\subset X$ contained in $X$ is indeed a $C^2$ curve in $X$.
3\. Let $u_c \in X$ be a critical point of $G_X$. The eigenvalue 0 of $DG_X(u_c): X \to X$ is elementary and there is a normal positive vector $\phi_M(u_c)$ spanning $\ker DG_X(u_c)$.
This follows from (G2) and Theorem \[theo:zhang\] applied to $R \ I + DG(u)$ for a large $R >0$. The kernel vector $\phi_M(u_c)$ is not in $W_X = V^\perp$, since $\phi_M(u_c), \phi_M >0$.
Split $X = W_X \oplus V$, where again $W_X = V^\perp \subset X$ and $\Pi:X \to W_X$ is $\Pi v = v - \langle \phi_M, \ v \rangle \ \phi_M$.
4\. The projected restrictions $\Pi (G_t)_X: W_X \to W_X \ , \ w \mapsto \Pi G(w + t \phi_M)$ are local diffeomorphisms for each $t \in {{\mathbb{R}}}$.
We prove the invertibility of $\Pi D(G_t)_X(w): W_X \to W_X$ for every $w \in W_X$. The images of the maps $D(G_t)_X(w): W_X \to W_X$ are closed, either because for $u = w + t \phi_M$ $DG_X(u): X \to X$ is an isomorphism or because $0$ is an elementary eigenvalue of $DG_X(u)$. Moreover, the extensions $D(G_t)_H(w): W_H \to W_H$ never contain $\phi_M$ in their range, from the argument which proves (LS) for $G_H: H \to H$, so that the restriction of $\Pi$ to ${\operatorname{Ran}}D(G_t)_X$ is also an isomorphism.
5\. Near every $u \in X$, the set $\alpha_u = G^{-1}_H (V_x)\subset X$ is an arc near $u$.
An [*arc*]{} is the image of a $C^1$ function $\gamma: (-1,1) \to X$ admitting a continuous, injective extension to $[-1,1]$. At a regular point $u \in X$, $DG(u): X \to Y$ is invertible: from the inverse function theorem, the inversion of small segments of vertical lines through $G(u)$ consists of an arc. Now take a critical point $u_c \in X$ of $G_X$. Simply stack the local diffeomorphisms obtained in the previous step: again, we obtain the required arc.
We completed the proof of the next step: fibers have been transplanted.
6\. (HLS) and (HPR) hold for $G_X:X \to X$.
We are left with proving (NT), both for $G_X: X \to X$ and $G_H: H \to H$. Here instead we start with $G_X$. Since $G_X$ is smooth, we may use Corollary \[cor:corC\]. Its hypotheses are verified with systematic use of Theorem \[theo:zhang\] to the Jacobians $DG_X(u_c): X \to X$ (or better, $R I + DG_X(u_c)$ for large $R$). At a critical point $u_c$ the eigenvalue $0$ is elementary and the associated eigenvector $\phi(u_c)$ is positive (as well as the dual eigenvector $\phi^\ast(u_c)$, which may be interpreted as $\phi(u_c)$ itself by the self-adjointness of $DG_H: H \to H$).
From Lemma \[prop:abstract\] in the Appendix, the zero eigenvalue at $u_c$ is $C^1$ and $$D\lambda(u_c) S = \langle \phi(u_c), \ [ D (DF(u_c)) \ \phi(u_c)] \ \phi(u_c) \ \rangle = \langle \phi(u_c), \ -g''(u_c) \phi(u_c)^2 \ \rangle \ <0$$ so that from Corollary \[cor:corC\], we have (NT) for $G_X$.
Finally, if there is a fiber of $G_H$ with three preimages, then a nearby fiber completely in $X$ also admits three preimages, which we have just shown to be impossible. We leave the density arguments to the reader.
We finish indicating some hypotheses leading to (G2). We need some easy facts from spectral theory. Fix $u \in X$.
I. The eigenvalues of $DG_X(u)$ and $DG_H(u)$ are the same.
Clearly eigenvalues of $DG_X(u)$ are eigenvalues of $DG_H(u)$. If $DG_H(u) v = \lambda v$ then $E_Hv - g'(u)v = \lambda v$ and again, since $g'(u) \in X$, we have $v \in X$.
II\. $\sigma(DG_X(u)) \cup \{0\} \ \subseteq \ \sigma(DG_H(u)) \cup \{0\} $.
Let $\nu \ne 0$ belong to the resolvent of $DG_H(u)$. If $DG_X(u) - \nu I: X \to X$ is not injective, there is an eigenvector in $X$ and then $\nu \in \sigma(DG_H(u))$. Also, for $x \in X$, the equation $DG_H(u) v - \nu v = x$ has a solution $v \in H$: mimic the argument above to show that $v \in X$. Thus $DG_X(u) - \nu I: X \to X$ is an isomorphism.
III\. For a continuous function $f: {{\mathbb{R}}}\to {{\mathbb{R}}}$ with $\overline{f({{\mathbb{R}}})} = [a,b]$, consider the multiplication operators $M_X: X \to X$ and $M_H: H \to H$, $u \mapsto f u$. Then $$\sigma(M_X) = \sigma_e(M_X) = \sigma(M_H) = \sigma_e(M_H) = [a,b] \ .$$
This is a simple computation related to solving $f u = \lambda u$.
By hypothesis, $\lambda_M$ is an elementary eigenvalue of $E_H$. Standard perturbation arguments then imply that $0 \in \sigma(DG_H(u_c))$ is an elementary eigenvalue and the rest of the spectrum of $DG_H(u_c)$ is to its left. From (I), 0 is a common eigenvalue of $DG_H(u_c)$ and $DG_X(u_c)$, and, from (II), $\sigma(DG_X(u_c)) \ \subset \ \sigma(DG_H(u_c)) $, so that $\sigma(DG_X(u_c))$ is real and negative. Thus, for large $R$, $$R \ = \ r( R\ I + DG_H(u_c))= \ r( R\ I + DG_X(u_c)) \ .$$
If $E_X$ is compact, Weyl’s lemma implies (for weaker hypotheses, see [@LD]) $$\sigma_e( R \ I + DG_X(u_c)) = \sigma_e( R \ I - g'(u_c) + E_X) = \sigma_e( R \ I - g'(u_c)) = [R-b,R-a] \ ,$$ by (III), and then (G2) holds.
Another simple hypothesis leading to (G2) is $$\sigma(DG_X(u_c)) \setminus \{0\} \subset {{\mathbb{C}}}_- = \{z \in {{\mathbb{C}}}\ , \Re z < 0 \} \ .$$ Indeed, for each $z \in {{\mathbb{C}}}_-$, there is $R(z)$ for which $| R(z) + z | < R$. Such $R(z)$ may be taken to be the same for all $z \in \sigma(DG_X(u_c)) \setminus \{0\}$ by compactness and (G2) follows, since $R$ is an eigenvalue of $R\ I + DG_X(u_c)$ not in its essential spectrum.
[**Appendix: Smoothness of simple eigenvalues**]{}
We use a simplified version of Proposition 79.15 from [@Z]. For real Banach spaces $X \subset Y$, ${\cal B}(X,Y)$ is the Banach space of bounded linear maps from $X$ to $Y$ with the operator norm. Let $T_0 \in {\cal B}(X,Y)$ and $T_0^\ast \in {\cal B}(Y^\ast,X^\ast)$ have an elementary eigenvalue $\lambda_0 \in {{\mathbb{R}}}$ (defined in the Introduction) associated with the eigenvector $\phi_0 \in X$ of $T_0$ and $\phi_0^\ast$ of $T^\ast_0: Y^\ast \to X^\ast$, so that $\langle \phi_m^\ast, \phi_m \rangle = 1$.
\[prop:abstract\] Suppose $T_0 \in B(X,Y)$ as above. Then there are open neighborhoods $U_0 \subset {\cal B}(X,Y)$ of $T_0$ and $\Lambda_0 \subset {{\mathbb{R}}}$ of $\lambda_0$ for which every operator $T \in U_0$ has a unique eigenvalue $\lambda(T) \in \Lambda_0$. This eigenvalue is elementary and defines an analytic map $T \mapsto \lambda(T)$ with $\lambda(T_0) = \lambda_0$. Appropriately normalized eigenvectors $\phi(T), \phi^\ast(T), T \in U_0$ are also smooth. Finally, along any direction $S \in B(X,Y)$, $$D \lambda(T) \ . \ S \ = \ \langle \ \phi^\ast(T) \ , \ S \ \phi(T) \ \rangle \ .$$
Acknowledgements
----------------
The first author acknowledges support from GNAMPA. The second and third authors from CAPES, CNPq and FAPERJ.
[\[10\]]{} .
Marta Calanchi, Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133 Milano, Italia
Carlos Tomei and André Zaccur, Departamento de Matemática, PUC-Rio, R. Mq. de S. Vicente 225, Rio de Janeiro, RJ 22453-900, Brazil
[email protected] [email protected] [email protected]
|
---
abstract: 'Let $(R,{\mathfrak{m}})\to (S,{\mathfrak{n}})$ be a flat local extension of local rings. Lech conjectured around 1960 that there should be a general inequality $e(R)\leq e(S)$ on the Hilbert-Samuel multiplicities [@Lechnoteonmultiplicity]. This conjecture is known when the base ring $R$ has dimension less than or equal to two [@Lechnoteonmultiplicity], and remains open in higher dimensions. In this paper, we prove Lech’s conjecture in dimension three when $R$ has equal characteristic. In higher dimension, our method yields substantial partial estimate: $e(R)\leq (d!/2^d)\cdot e(S)$ where $d=\dim R\geq 4$, in equal characteristic.'
address: |
Department of Mathematics\
University of Utah\
Salt Lake City\
Utah 84112
author:
- Linquan Ma
bibliography:
- 'CommonBib.bib'
title: 'Lech’s conjecture in dimension three'
---
[^1]
[*Dedicated to Professor Craig Huneke on the occasion of his 65th birthday*]{}
Introduction
============
Around 1960, Lech made the following remarkable conjecture on the Hilbert-Samuel multiplicities [@Lechnoteonmultiplicity]:
\[Lech’s Conjecture\] Let $(R,{\mathfrak{m}})\rightarrow (S,{\mathfrak{n}})$ be a flat local extension of local rings. Then $e(R)\leq e(S)$.
We note that the Hilbert-Samuel multiplicity is a classical invariant that measures the singularity of $R$. Morally speaking, the larger the multiplicity, the worse the singularity. It is very natural to expect that if $(R,{\mathfrak{m}})\to (S,{\mathfrak{n}})$ is a flat local extension, then $R$ cannot have a worse singularity than $S$. Hence, Lech’s conjecture seems quite natural and interesting. However, the conjecture has now stood for over fifty years and remains open in most cases, with the best partial results still those proved in Lech’s original two papers [@Lechnoteonmultiplicity],[@Lechinequalitiesofflatcouples]. There the conjecture was proved in the following cases:
1. $\dim R \leq 2$ [@Lechnoteonmultiplicity];
2. $S/{\mathfrak{m}}S$ is a complete intersection [@Lechnoteonmultiplicity], [@Lechinequalitiesofflatcouples].
Lech’s conjecture has caught great interests to commutative algebraists, and some partial positive answers have been obtained. For example, it follows from results of [@HerzogUlrichBacklinLinearMCMoverStrictcompleteintersections] that Lech’s conjecture holds when the base ring $R$ is a [*strict complete intersection*]{}: the associated graded ring $\operatorname{gr}_{\mathfrak{m}}R$ is a complete intersection. The conjecture was also proved when $R$ is a three-dimensional $\mathbb{N}$-graded $K$-algebra generated over $K$ by one forms for $K$ a perfect field of characteristic $p>0$ [@HanesThesis]. Moreover, partial results were obtained when we put various conditions on the closed fibre $S/{\mathfrak{m}}S$ [@HerzogBLechHironakainequalities], [@HerzogBKodairaSpencermap]. We refer to [@HanesThesis], [@HanesLengthapproximationsforIndependentlygeneratedideals], [@HerzogBKodairaSpencermap] and [@MaThesis] for other related results on Lech’s conjecture.
However, despite these partial results, to the best of our knowledge Lech’s conjecture remains open as long as $\dim R\geq 3$. Our main theorem in this paper settles Lech’s conjecture in dimension three in equal characteristic.
\[theorem–main theorem\] Let $(R,{\mathfrak{m}})\rightarrow (S,{\mathfrak{n}})$ be a flat local extension between local rings of equal characteristic. If $\dim R=3$, then $e(R)\leq e(S)$.
In [@Lechnoteonmultiplicity], Lech proved that, in general, we have $e(R)\leq d!\cdot e(S)$ for $(R,{\mathfrak{m}})\to (S,{\mathfrak{n}})$ flat local extension with $d=\dim R$. Our main technical result [*greatly*]{} generalizes this inequality in equal characteristic, and from which Theorem \[theorem–main theorem\] follows immediately.
\[theorem–main technical theorem\] Let $(R,{\mathfrak{m}})\rightarrow (S,{\mathfrak{n}})$ be a flat local extension between local rings of equal characteristic. If $\dim R=d$, then we have $$e(R)\leq \max\{1,\frac{d!}{2^d}\}\cdot e(S).$$
This paper is organized as follows: in Section 2 we start with general preliminaries on multiplicities, and we recall some important tools that will be employed. In Section 3 we prove a technical lemma on the structure of flat local homomorphism, Lemma \[lemma–factoring maps with c.i. closed fibres\], which is a key ingredient in later proofs. In Section 4 we prove two results on the behavior of Hilbert-Samuel multiplicities under faithfully flat extensions of local rings of characteristic $p>0$: Theorem \[theorem–inequalities on multiplicities in terms of difference of embdim\] and Theorem \[theorem–inequalities on multiplicities when the difference of embdim is large\]. Theorem \[theorem–main technical theorem\] in characteristic $p>0$ then follows immediately by combining them. Finally in Section 5, we use reduction to characteristic $p>0$ to obtain Theorem \[theorem–main technical theorem\] in characteristic $0$.
Acknowledgement {#acknowledgement .unnumbered}
---------------
First of all, it is my great pleasure to thank Mel Hochster for introducing Lech’s conjecture to me and for many enjoyable discussions. In fact, the Gorenstein case of Theorem \[theorem–main theorem\] in characteristic $p>0$ appeared in the last section of my doctorial thesis [@MaThesis] written under the direction of Mel. In addition, Mel also explained to me the ideas in the reduction to characteristic $p>0$ process in Section 5.
I would like to thank Craig Huneke and Bernd Ulrich for answering my questions, for their extremely helpful comments, and for their encouragements. In particular, in discussion with Craig Huneke, we established Lemma \[lemma–choose generically CM sop\] which eventually leads to the current proof of Theorem \[theorem–inequalities on multiplicities in terms of difference of embdim\], and following the comments of Bernd Ulrich, our arguments in Section 3 and Section 4 are largely simplified and shortened. I also thank Kevin Tucker and Wenliang Zhang for their comments on Theorem \[theorem–limit in terms of local chern characters\] and related results.
The main result in dimension three in characteristic $p>0$ was first announced at the Midwest Commutative Algebra Conference at Purdue University in August 2015. At that time, our method only works in dimension three and we need some extra mild assumptions on the residue field of $R$. Since then we have largely improved our techniques to obtain Theorem \[theorem–main technical theorem\] and thus the general version of Theorem \[theorem–main theorem\].
Preliminaries on multiplicities
===============================
Throughout this paper, $(R,{\mathfrak{m}})$ will always be a Noetherian local ring. In most cases, we will work with rings of equal characteristic, i.e., $R$ contains a field. We use $\nu_R(M)$ to denote the minimal number of generators of a module $M$ over $R$. If $M$ has finite length as an $R$-module, we use $l_R(M)$ to denote its length over $R$. We will drop the subscript and write $\nu(M)$, $l(M)$ when $R$ is clear from the context. We will use $\operatorname{edim}R$ to denote the embedding dimension of $R$, which is the same as $\nu_R({\mathfrak{m}})$.
Hilbert-Samuel multiplicity
---------------------------
For an ${\mathfrak{m}}$-primary ideal $I$ of $R$ and a finitely generated $R$-module $M$, the [*Hilbert-Samuel multiplicity*]{} of $M$ with respect to $I$ can be defined as: $$e(I, M)=\lim_{t\to\infty}d!\cdot\frac{l_R(M/I^tM)}{t^d}$$ where $d=\dim R$. Of great importance is the case that $M=R$ and $I={\mathfrak{m}}$, where we just write $e(R)$ for the multiplicity $e({\mathfrak{m}}, R)$.
The Hilbert-Samuel multiplicity is a classical invariant that measures the singularity of $R$ (and of $M$). In general, $e(I,M)$ is always an integer and it is positive if and only if $\dim M=d$. The multiplicity $e(I, -)$ is additive on short exact sequences: if $M$ has a finite filtration by $\{M_i\}$, then $e(I, M)=\sum_ie(I, M_i)$. A rather non-trivial result is the following localization formula, which appeared in [@Nagata62], and also follows from more general results on Hilbert functions [@Lechinequalitiesofflatcouples], [@BennettOnthecharacteristicfunctionsofalocalRing].
\[theorem–localization theorem on multiplicities\] If $P$ is a prime ideal of an excellent local ring $R$ (e.g., a complete local ring $R$) such that $\dim R/P+\height P=\dim R$, then we have $e(R_P)\leq e(R)$.
The above discussion allows us to prove some reductions on Lech’s conjecture. The following result is well known to experts and we include a short proof here for completeness.
\[lemma–reduction\] Let $(R,{\mathfrak{m}})\to (S,{\mathfrak{n}})$ be a flat local map with $\dim R=d$. In order to prove Lech’s conjecture that $e(R)\leq e(S)$, or more generally, to prove $e(R)\leq C\cdot e(S)$ for certain constant $C$ depending only on $d$, it suffices to prove the case where $\dim S=\dim R=d$, $R$ and $S$ are both complete, $R$ is a domain, and $S$ has algebraically closed residue field.
As completion preserves flatness and the multiplicities, we can replace $R$ and $S$ by their completions to assume both $R$ and $S$ are complete. We can choose a minimal prime $Q$ of ${\mathfrak{m}}S$ such that $\dim S/Q=\dim S/{\mathfrak{m}}S$. Since $R\to S$ is faithfully flat, $\dim S=\dim S/{\mathfrak{m}}S+\dim R$ and hence $$\dim S\geq \dim S/Q+\height Q\geq \dim S/{\mathfrak{m}}S+\dim R=\dim S.$$ Now by Theorem \[theorem–localization theorem on multiplicities\], we have $e(S_Q)\leq e(S)$. Thus, if $e(R)\leq e(S_Q)$ (resp., $e(R)\leq C\cdot e(S_Q)$) then $e(R)\leq e(S)$ (resp., $e(R)\leq C\cdot e(S)$) as well. It follows that we may replace $S$ by $S_Q$ and assume $\dim S=\dim R=d$. We may lose completeness, but we can complete again. Next, we can give a filtration of $R$ by prime cyclic modules $R/P_i$, $1\leq i\leq h$. We then have $e(R)=\sum_{\dim R/P_i=d}e(R/P_i)$. After tensoring with $S$ we get a corresponding filtration of $S$ by modules $S/P_iS$, and $e(S)=\sum_{\dim S/P_iS=d}e(S/P_iS)$. Since $S$ is faithfully flat over $R$, the values of $i$ such that $\dim R/P_i=d$ are precisely those such that $\dim S/P_iS=d$. Therefore, by considering each $R/P_i\to S/P_iS$ with $\dim R/P_i=\dim R=d$, we reduce to the case that $\dim S=\dim R=d$, $R$ and $S$ are both complete, and $R$ is a domain. Finally, we can take a flat local extension $(S,{\mathfrak{n}}) \to (S',{\mathfrak{n}}')$ such that ${\mathfrak{n}}'={\mathfrak{n}}S'$ and $S'/{\mathfrak{n}}'$ is the algebraic closure of $S/{\mathfrak{n}}$ (such $S'$ always exists: it is a suitable [*gonflement*]{} of $S$; see [@BourbakiBookChapter89]).[^2] Replacing $S$ by $S'$ and completing $S'$ if necessary, we get the desired reduction.
We do not know whether Lech’s conjecture can be reduced to the case that $R$ has algebraically closed (or perfect) residue field in general: we can only do this when $R$ has equal characteristic $0$ (see the proof of Lemma \[lemma–reduction to module finite\] and Remark \[remark–reduction to module finite not clear in general\] for further discussion on this). It is also not clear to us whether we can assume $S$ is a domain, or even reduced.
If two ${\mathfrak{m}}$-primary ideals $I$ and $J$ in $R$ have the same integral closure, then $e(I, M)=e(J, M)$. This is quite useful because it reduces the computation of multiplicities to the case where $I$ is a parameter ideal: when $R/{\mathfrak{m}}$ is an infinite field, every ${\mathfrak{m}}$-primary ideal $I$ is integral over an ideal generated by a system of parameters $\underline{x}=x_1,\dots,x_d$, which is called a [*minimal reduction*]{} of $I$. In particular, we have $e(I, M)=e(\underline{x}, M)$. The latter one can be computed using the Euler characteristic of the Koszul complex on $\underline{x}$: $$e(\underline{x}, M)=\chi(\underline{x}, M)=\chi(K_\bullet(\underline{x}, M))=\sum_{i=0}^d(-1)^il_R(H_i(\underline{x}, M)).$$ We will use this formula repeatedly throughout the paper. Let us also mention that one defines the higher Euler characteristic by $\chi_j(\underline{x}, M)=\sum_{i=j}^d(-1)^{j-i}l_R(H_i(\underline{x}, M))$.
Frobenius map and the Hilbert-Kunz multiplicity
-----------------------------------------------
In this subsection we always assume $(R,{\mathfrak{m}})$ has equal characteristic $p>0$. We will use $R^{(e)}$ to denote the target ring of the $e$-th Frobenius map $F^e$: $R\rightarrow R$. If $M$ is an $R$-module, we will use $M^{(e)}$ to denote the corresponding module over $R^{(e)}$. We shall let $F^e_R(-)$ denote the Peskine-Szpiro’s Frobenius functor from $R$-modules to $R$-modules (we will write $F^e(-)$ if $R$ is clear from the context). In detail, $F^e_R(M)$ is given by base change to $R^{(e)}$ and then identifying $R^{(e)}$ with $R$.
We say $R$ is [*$F$-finite*]{} if $R^{(1)}$ (or equivalently, every $R^{(e)}$) is finitely generated as an $R$-module. It is well known that, when $(R,{\mathfrak{m}})$ is complete, $R$ is $F$-finite if and only if its residue field $K=R/{\mathfrak{m}}$ is $F$-finite, i.e., $[K:K^p]<\infty$. We denote $\alpha(R)=\log_p[K:K^p]$. In particular, if $(R,{\mathfrak{m}})$ has perfect residue field, then $\alpha(R)=0$. If $M$ is an $R$-module of finite length, then $l_R(M^{(e)})=p^{e\cdot\alpha(R)}l_R(M)$. A result of Kunz [@KunzOnNoetherianRingsOfCharP Proposition 2.3] shows that, when $R$ is $F$-finite, $\alpha(R_P)=\alpha(R)+\dim R/P$. For an ${\mathfrak{m}}$-primary ideal $I$ of $R$ and a finitely generated $R$-module $M$, it was shown by Monsky [@MonskyTheHilbertKunzfunction] that the following limit $$e_{HK}(I, M)=\lim_{e\to\infty}\frac{l_R(M/I^{[p^e]}M)}{p^{ed}}$$ exists, and this is called the [*Hilbert-Kunz multiplicity*]{} of $M$ with respect to $I$. When $I={\mathfrak{m}}$ and $M=R$, we simplify our notation and set $e_{HK}(R)$ to be $e_{HK}({\mathfrak{m}}, R)$. It is straightforward to see that, when $(R,{\mathfrak{m}})$ is $F$-finite, we have $$e_{HK}(I,R)=\lim_{e\to\infty}\frac{l_{R}(F^e(R/I))}{p^{ed}}=\lim_{e\to\infty}\frac{l_R(R^{(e)}/IR^{(e)})}{p^{e(d+\alpha(R))}}.$$ It is worth to pointing out that if $(R,{\mathfrak{m}})$ is a local ring of dimension $d$ and $\underline{x}=x_1,\dots,x_d$ is system of parameters of $R$, then $e_{HK}(\underline{x}, R)=e(\underline{x}, R)$. However, computations of Hilbert-Kunz multiplicities of arbitrary ${\mathfrak{m}}$-primary ideals have proved quite difficult. In general, we only know that if $I$ is an ${\mathfrak{m}}$-primary ideal, then the multiplicities $e_{HK}(I, R)$ and $e(I, R)$ are related by the inequalities: $$\frac{e(I, R)}{d!}\leq e_{HK}(I, R)\leq e(I, R),$$ and this inequality is the best possible in general [@WatanabeYoshidaHilbertKunzmultiplicity]. It is perhaps also worth remarking that, although the Hilbert-Kunz multiplicity is in general hard to study, the analog statement of Lech’s conjecture for Hilbert-Kunz multiplicity is known to be true. To be more precise, if $(R,{\mathfrak{m}})\rightarrow (S,{\mathfrak{n}})$ is a flat local extension between local rings of characteristic $p>0$, then $e_{HK}(R)\leq e_{HK}(S)$ [@KunzOnNoetherianRingsOfCharP], [@HanesThesis]. This immediately gives $$e(R)\leq d!\cdot e_{HK}(R)\leq d!\cdot e_{HK}(S)\leq d!\cdot e(S)$$ where $d=\dim R$. Thus we quickly recovered Lech’s result $e(R)\leq d!\cdot e(S)$ in characteristic $p>0$ (and hence in equal characteristic, see Section 5). Our main result, Theorem \[theorem–main technical theorem\], improved the constant $d!$ to $\max\{1, (d!/2^d)\}$.
Local Chern characters and Dutta multiplicity
---------------------------------------------
Local Chern characters were first introduced in [@BaumFultonMacPhersonRiamannRochforsingularvarieties], and a good description of them can be found in [@FultonIntersectiontheory Chapter 18]. In this paper we will need their connections with Dutta multiplicities as developed in [@RobertsIntersectionTheoremsMSRI], [@RobertsMultiplicitiesandChernclassinLocalAlgebra], and further extended in [@KuranoNumericalequivalenceonChowgroupsoflocalrings]. Below we present an outline of this theory for local rings, which will be suffices for our applications in Section 4.
Let $(R,{\mathfrak{m}})$ be a complete local ring (of arbitrary characteristic) of dimension $d$. Let $G_\bullet$ be a bounded complex of finite free $R$-modules. Let $Z$ be the support of $G_\bullet$: this is the closed subset of $\Spec R$ consisting of those primes $P$ for which the localization of $G_\bullet$ at $P$ is not exact. We denote the [*local Chern character*]{} of $G_\bullet$ by $\operatorname{ch}(G_\bullet)$. This is a sum of components: $$\operatorname{ch}(G_\bullet)=\operatorname{ch}_0(G_\bullet)+\operatorname{ch}_1(G_\bullet)+\cdots+\operatorname{ch}_d(G_\bullet).$$ For each integer $i$, $\operatorname{ch}_i(G_\bullet)$ defines, for each integer $k$, a homomorphism of $\mathbb{Q}$-modules from $A_k(X)_\mathbb{Q}$ to $A_{k-i}(Z)_\mathbb{Q}$, where $A_k(X)_\mathbb{Q}$ is the $k$-th component of the Chow group with rational coefficients. These operators satisfy a lot of properties. We refer to [@RobertsIntersectionTheoremsMSRI Page 422] for a good list, [@RobertsMultiplicitiesandChernclassinLocalAlgebra] and [@FultonIntersectiontheory] for more details and proofs. Of great importance to us here is the local Riemann-Roch formula (see [@RobertsMultiplicitiesandChernclassinLocalAlgebra Section 12.6] or [@FultonIntersectiontheory Example 18.3.12] for more general versions). This formula states that for every finitely generated $R$-module $M$ there is an element $\tau(M)$ in $A_*(\Spec R)$: $$\tau(M)=[M]_d+[M]_{d-1}+\cdots+[M]_0$$ such that for every $G_\bullet$ with homology of finite length (i.e., $H_i(G_\bullet)$ is supported only at ${\mathfrak{m}}$ for every $i$), we have $$\chi(G_\bullet\otimes M)=\operatorname{ch}_d(G_\bullet)[M]_d+\operatorname{ch}_{d-1}(G_\bullet)[M]_{d-1}+\cdots+\operatorname{ch}_0(G_\bullet)[M]_0.$$ In particular, if $G_\bullet$ is a bounded complex (of finite free $R$-modules) with homology of finite length, then $$\label{equation--chi in terms of local RR}
\sum_i (-1)^il_R(H_i(G_\bullet))=\chi(G_\bullet)=\sum_{j=0}^d\operatorname{ch}_j(G_\bullet)[R]_j.$$
One crucial and ingenious observation of Roberts (see [@RobertsIntersectionTheoremsMSRI] or [@RobertsMultiplicitiesandChernclassinLocalAlgebra Theorem 12.7.1]) is that, when $R$ has characteristic $p>0$, we have $$\label{equation--dutta multiplicity in terms of chern characters}
\chi_\infty(G_\bullet):=\lim_{e\to\infty}\sum_{i}(-1)^i\frac{l_R(H_i(F^e(G_\bullet)))}{p^{ed}}=\operatorname{ch}_d(G_\bullet)[R]_d,$$ where $\chi_\infty(G_\bullet)$ is called the [*Dutta multiplicity*]{} of the complex $G_\bullet$, which was first introduced and studied by Dutta [@DuttaFrobeniusandMultiplicities]. This is one of the key ingredients in the solution of the new intersection theorem in mixed characteristic.
Comparing (\[equation–chi in terms of local RR\]) and (\[equation–dutta multiplicity in terms of chern characters\]), we see that $\chi(G_\bullet)$ and $\chi_\infty(G_\bullet)$ differs by $\sum_{j=0}^{d-1}\operatorname{ch}_j(G_\bullet)[R]_j$. We say $R$ is a [*numerically Roberts ring*]{} if $\chi(G_\bullet)=\chi_{\infty}(G_\bullet)$ for any such $G_\bullet$. This notion was formally introduced and studied intensively in [@KuranoNumericalequivalenceonChowgroupsoflocalrings].[^3] Let us point out that any two-dimensional equidimensional complete local ring is numerically Roberts by [@KuranoNumericalequivalenceonChowgroupsoflocalrings Example 6.6]. In higher dimension, it is well known that complete intersections are always numerically Roberts [@DuttaFrobeniusandMultiplicities] (see [@KuranoNumericalequivalenceonChowgroupsoflocalrings Remark 6.9]). This would be our main applications. However, we also mention that there exist Cohen-Macaulay rings of dimension three and Gorenstein rings of dimension five in characteristic $p>0$ that are not numerically Roberts [@RobertsIntersectionTheoremsMSRI], [@SinghandMillerIntersectionmultiplicitiesoverGorensteinrings].
Dutta multiplicities have deep connections with the Hilbert-Kunz multiplicities. We want to sketch this relation briefly. We recall the following important result, which first appeared in the main theorem of [@RobertsIntersectionTheoremsMSRI]. A stronger form of this theorem was also obtained in [@HochsterHunekePhantomhomology Theorem 6.2] using tight closure. Both ideas of the proofs can be traced back to [@DuttaFrobeniusandMultiplicities].
\[theorem–vanishing of higher Koszul\] Let $(R,{\mathfrak{m}})$ be a complete local ring of characteristic $p>0$. Let $G_\bullet$ be a bounded complex of finite free $R$-modules of length $d=\dim R$ with homology of finite length. Then for every $i\geq 1$ we have $$\lim_{e\to\infty}\frac{l_R(H_i(F^e(G_\bullet)))}{p^{ed}}=0.$$
Now, suppose we have a bounded complex $G_\bullet$ of length exactly $d=\dim R$, with homology of finite length. It then follows from Theorem \[theorem–vanishing of higher Koszul\] that $$\chi_\infty(G_\bullet)=\lim_{e\to\infty}\frac{l_R(H_0(F^e(G_\bullet)))}{p^{ed}}.$$ Therefore, if we also have $H_0(G_\bullet)=R/I$, then $\chi_\infty(G_\bullet)=e_{HK}(I, R)$. In particular, if $(R,{\mathfrak{m}})$ is a numerically Roberts ring and $I$ is an ${\mathfrak{m}}$-primary ideal of $R$ of finite projective dimension (this implies $R$ is Cohen-Macaulay by the new intersection theorem [@RobertsIntersectionTheoremsMSRI]), then $$\lim_{e\to\infty}\frac{l(R/I^{[p^e]})}{p^{e\cdot \dim R}}=l(R/I).$$ We will use similar ideas repeatedly in Section 4. We also refer to [@KuranoNumericalequivalenceonChowgroupsoflocalrings Section 6] for more general results of this type.
Structure of flat local maps
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Our goal in this section is to prove Lemma \[lemma–factoring maps with c.i. closed fibres\], which will be used in Section 4. We first recall the main theorem from [@AvramovFoxbyHerzogStructureoflocalmap], which can be viewed as a natural generalization of Cohen’s structure theorem for complete local rings.
\[theorem–Cohen factorization\] A local homomorphism $(R,{\mathfrak{m}})\rightarrow (S,{\mathfrak{n}})$ with $S$ complete can be factored as $(R,{\mathfrak{m}})\rightarrow (T,{\mathfrak{n}}_T)\rightarrow (S,{\mathfrak{n}})$ such that $(R,{\mathfrak{m}})\rightarrow (T,{\mathfrak{n}}_T)$ is flat local with $T/{\mathfrak{m}}T$ regular, $(T,{\mathfrak{n}}_T)$ is complete, and $(T,{\mathfrak{n}}_T)\rightarrow (S,{\mathfrak{n}})$ is surjective.
Moreover, if $S$ has finite flat dimension over $R$ (e.g., $S$ is flat over $R$), then $S$ has finite projective dimension over $T$.
A [*Cohen-factorization*]{} as in Theorem \[theorem–Cohen factorization\] is called [*minimal*]{} if, with $S=T/J$, we have $J\subseteq {\mathfrak{m}}T+{\mathfrak{n}}_T^2$. This can be reduced from any Cohen-factorization by killing part of a regular system of parameters on $T/{\mathfrak{m}}T$ that is contained in $J$ but not contained in ${\mathfrak{n}}_T^2(T/{\mathfrak{m}}T)$ [@AvramovFoxbyHerzogStructureoflocalmap Proposition 1.5]. However in this paper, to get the desired inequalities on multiplicities, we need to factor the map $(R,{\mathfrak{m}})\to (T, {\mathfrak{n}}_T) \to (S,{\mathfrak{n}})=T/J$ such that $\operatorname{pd}_TS<\infty$ and $J\subseteq {\mathfrak{n}}_T^2$. We cannot always achieve this while keeping $T/{\mathfrak{m}}T$ regular. Our key observation here is that, at least for flat local extensions of rings of the same dimension, we can achieve this at the expense of letting $T/{\mathfrak{m}}T$ be a complete intersection. To establish such a factorization we first recall two classical and crucial results.
\[theorem–flat map with regular fibre\] Let $(R,{\mathfrak{m}})\rightarrow (T,{\mathfrak{n}}_T)$ be a flat local map. If $x_1,\dots,x_t$ is a regular sequence in $T/{\mathfrak{m}}T$, then it is a regular sequence on $T/IT$ for every $I\subseteq R$, and $T/(x_1,\dots,x_t)T$ is faithfully flat over $R$.
\[lemma–killing linear term preserves finite projective dimension\] Let $(T,{\mathfrak{n}})$ be a local ring and $M$ be a finitely generated $T$-module such that $\operatorname{pd}_TM<\infty$. If $x\in \Ann_RM$ such that $x\notin {\mathfrak{m}}^2\cup(\cup_{P\in\operatorname{Ass}(T)}P)$, then $\operatorname{pd}_{T/xT}M<\infty$.
It should be pointed out that Lech proved in [@Lechinequalitiesofflatcouples] that for every flat local extension $(R,{\mathfrak{m}})\to (S,{\mathfrak{n}})$, we always have $\operatorname{edim}R-\dim R\leq \operatorname{edim}S-\dim S$. In particular, we have $\operatorname{edim}S-\operatorname{edim}R\geq 0$ (note that in general, $\operatorname{edim}S-\operatorname{edim}R$ is not the same as $\operatorname{edim}(S/{\mathfrak{m}}S)$). Below we give a short proof of a more general fact regarding local maps of finite flat dimension. This also answers a question in [@AvramovFoxbyHalperinDecentandascent Remark 6.3].
\[theorem–regularity defect of flat local extension\] Let $(R,{\mathfrak{m}})\rightarrow (S,{\mathfrak{n}})$ be a local map such that $S$ has finite flat dimension over $R$ (e.g., $S$ is flat over $R$), then we have $$\operatorname{edim}R-\dim R\leq \operatorname{edim}S-\dim S.$$ In particular, if $\dim S\geq \dim R$ (e.g., $S$ is flat over $R$), then $\operatorname{edim}S-\operatorname{edim}R\geq 0$.
We may assume both $R$ and $S$ are complete. By Theorem \[theorem–Cohen factorization\], we have $$(R,{\mathfrak{m}})\rightarrow (T,{\mathfrak{n}}_T) \rightarrow (S,{\mathfrak{n}})$$ such that $R\rightarrow T$ is faithfully flat with $T/{\mathfrak{m}}T$ regular, and $S=T/J$ with $\operatorname{pd}_TS<\infty$. It is easy to see that $\operatorname{edim}R-\dim R=\operatorname{edim}T-\dim T$, so it suffices to prove $\operatorname{edim}T-\dim T\leq \operatorname{edim}S-\dim S$ when $\operatorname{pd}_TS<\infty$. We claim that: $$\label{equation--inequality of embedding dimension and depth}
\operatorname{edim}T\leq \operatorname{edim}S+\operatorname{depth}_JT.$$
This is easy if $J\subseteq {\mathfrak{n}}_T ^2$, because in this case we have $\operatorname{edim}S=\operatorname{edim}T$, so (\[equation–inequality of embedding dimension and depth\]) holds trivially. Now assume $0\neq J\nsubseteq {\mathfrak{n}}_T ^2$. Since $\operatorname{pd}_TS<\infty$, we know that $J$ contains a nonzerodivisor of $T$ [@Eisenbud95 Corollary 20.13]. Thus by prime avoidance, there exists $x\in J$ such that $x\notin {\mathfrak{n}}_T ^2 \cup(\cup_{P\in\operatorname{Ass}T} P)$. Let $\overline{T}=T/xT$ and $\overline{J}=J\overline{T}$. We still have $S=\overline{T}/\overline{J}$ with $\operatorname{pd}_{\overline{T}}S<\infty$ by Lemma \[lemma–killing linear term preserves finite projective dimension\]. Moreover, $\operatorname{edim}T$ drops by one while $\operatorname{edim}S$ stays the same. But we can do this process at most $\operatorname{depth}_JT$ times (we either end up with $J=0$ or we stop at some point with $J\subseteq {\mathfrak{n}}_T^2$), thus (\[equation–inequality of embedding dimension and depth\]) follows.
Since we always have $\operatorname{depth}_JT \leq \dim T-\dim T/J=\dim T- \dim S$. Combining this with (\[equation–inequality of embedding dimension and depth\]) we thus get $\operatorname{edim}T-\dim T\leq \operatorname{edim}S-\dim S$. This finishes the proof.
Recall that for a local ring $T$ and an ideal $J\subseteq T$, we always have $$\label{equation--inequality on depth, height and projdim}
\operatorname{depth}_JT\leq \height{J}\leq \dim T-\dim (T/J)\leq \operatorname{pd}_T(T/J).$$ The first two inequalities are trivial, while the third inequality is a consequence of the celebrated new intersection theorem [@RobertsIntersectionTheoremsMSRI]. An ideal $J\subseteq T$ is called [*perfect*]{} if $\operatorname{pd}_T(T/J)=\operatorname{depth}_JT$, and hence for perfect ideals all the inequalities in (\[equation–inequality on depth, height and projdim\]) are equalities.
We next prove the following:
\[lemma–perfect ideal\] Let $(R,{\mathfrak{m}})\to (S,{\mathfrak{n}})$ be a flat local map between complete local rings of the same dimension. Suppose we have a factorization $$(R,{\mathfrak{m}})\to (T,{\mathfrak{n}}_T)\to (S,{\mathfrak{n}})$$ such that $R\to T$ is flat local, and $S=T/J$ with $\operatorname{pd}_TS<\infty$. Then $J$ is a perfect ideal in $T$.
Since $R\to S$ is flat local with $\dim R=\dim S$, we know that ${\mathfrak{m}}S$ is ${\mathfrak{n}}$-primary and $\operatorname{depth}R=\operatorname{depth}S$. By the Auslander-Buchsbaum formula, we have $$\operatorname{pd}_TS+\operatorname{depth}S=\operatorname{depth}T=\operatorname{depth}R+\operatorname{depth}T/{\mathfrak{m}}T.$$ Therefore we get $\operatorname{pd}_TS=\operatorname{depth}T/{\mathfrak{m}}T$. Since ${\mathfrak{m}}S$ is ${\mathfrak{n}}$-primary, ${\mathfrak{m}}T+J$ is ${\mathfrak{n}}_T$-primary. Thus we know that $J\cdot(T/{\mathfrak{m}}T)=(J+{\mathfrak{m}}T)/{\mathfrak{m}}T$ is ${\mathfrak{n}}_T$-primary in $T/{\mathfrak{m}}T$. Hence we can pick $y_1,\dots, y_n \in J$ such that $y_1,\dots,y_n$ form a regular sequence on $T/{\mathfrak{m}}T$ with $n=\operatorname{depth}T/{\mathfrak{m}}T$. By Theorem \[theorem–flat map with regular fibre\], $y_1,\dots, y_n$ is a regular sequence on $T$. This implies $$\operatorname{depth}_JT\geq n=\operatorname{depth}T/{\mathfrak{m}}T=\operatorname{pd}_TS.$$ Since the other direction always holds by (\[equation–inequality on depth, height and projdim\]), we have $\operatorname{pd}_TS=\operatorname{depth}_JT$ and thus $J$ is perfect.
We are ready to state and prove our lemma on factoring flat local maps that will be used in Section 4. This lemma is also of independent interest.
\[lemma–factoring maps with c.i. closed fibres\] Let $(R,{\mathfrak{m}})\to (S,{\mathfrak{n}})$ be a flat local map between complete local rings of the same dimension. Suppose $\operatorname{edim}S-\operatorname{edim}R=c$. Then this map can be factored as $$(R,{\mathfrak{m}})\rightarrow (T,{\mathfrak{n}}_{T})\rightarrow (S,{\mathfrak{n}})=T/J$$ such that the following are satisfied:
1. $(R,{\mathfrak{m}})\rightarrow (T,{\mathfrak{n}}_{T})$ is flat local with $(T,{\mathfrak{n}}_T)$ complete and $T/{\mathfrak{m}}T$ a complete intersection;
2. $J$ is a perfect ideal and $\operatorname{pd}_TS=c$;
3. $J\subseteq {\mathfrak{n}}_T^2$.
We first note that $c\geq 0$ by Theorem \[theorem–regularity defect of flat local extension\]. Now we let: $$(R,{\mathfrak{m}})\to (T',{\mathfrak{n}}_{T'})\to (S,{\mathfrak{n}})$$ be a Cohen-factorization as in Theorem \[theorem–Cohen factorization\], where $R\to T'$ is flat local with $T'/{\mathfrak{m}}T'$ regular, and $S=T'/J'$ with $\operatorname{pd}_{T'}S<\infty$. Suppose $\dim T'/{\mathfrak{m}}T'=\operatorname{depth}T'/{\mathfrak{m}}T'=b$. Since $S$ is a quotient of $T'$, $\operatorname{edim}S \leq \operatorname{edim}T'$, and we have $$b=\operatorname{edim}T'-\operatorname{edim}R\geq\operatorname{edim}S-\operatorname{edim}R=c \geq 0.$$
If $b=c$ then $\operatorname{edim}S=\operatorname{edim}T'$ and hence $J\subseteq {\mathfrak{n}}_{T'}^2$. Thus we simply set $T=T'$ and $J=J'$ and one can check that (1)–(3) are all satisfied by Theorem \[theorem–Cohen factorization\] and Lemma \[lemma–perfect ideal\].
Now suppose $b>c$, we claim that there exists $y_1,\dots,y_{b-c}\in J'$ satisfying the following conditions:
1. The image of $y_1,\dots,y_{b-c}$ in $T'/{\mathfrak{m}}T'$ form a regular sequence on $T'/{\mathfrak{m}}T'$;
2. The image of $y_1,\dots,y_{b-c}$ in ${\mathfrak{n}}_{T'}/{\mathfrak{n}}_{T'}^2$ form part of a basis for ${\mathfrak{n}}_{T'}/{\mathfrak{n}}_{T'}^2$.
We construct these elements inductively: suppose we already find $y_1,\dots,y_j$, $0\leq j<b-c$ ($j=0$ is the initial case). Let $\overline{T}=T'/(y_1,\dots,y_j)T'$ and $\overline{J}=J'\overline{T}$. Since $y_1,\dots,y_j\in J'$, we still have $S=\overline{T}/\overline{J}$. Because $\operatorname{edim}\overline{T}=\operatorname{edim}T'-j>\operatorname{edim}T'-(b-c)=\operatorname{edim}S$, $\overline{J}\nsubseteq{\mathfrak{n}}_{\overline{T}}^2$. Because $(R,{\mathfrak{m}})\to (S,{\mathfrak{n}})$ is flat local with $\dim R=\dim S$, ${\mathfrak{m}}S$ is ${\mathfrak{n}}$-primary, which implies ${\mathfrak{m}}\overline{T}+\overline{J}$ is ${\mathfrak{n}}_{\overline{T}}$-primary since $S=\overline{T}/\overline{J}$. But by the inductive hypothesis $\overline{T}/{\mathfrak{m}}\overline{T}$ is a complete intersection of dimension $b-j>0$. In particular, every associated prime $P$ of $\overline{T}/{\mathfrak{m}}\overline{T}$ has $\dim \overline{T}/P=b-j>0$. This implies ${\mathfrak{m}}\overline{T}+\overline{J}\nsubseteq P$ and hence $\overline{J}\nsubseteq P$ for every associated prime $P$ of $\overline{T}/{\mathfrak{m}}\overline{T}$. Now by prime avoidance, we have: $$\overline{J}\nsubseteq {\mathfrak{n}}_{\overline{T}}^2\cup (\cup_{P\in\operatorname{Ass}\overline{T}/{\mathfrak{m}}\overline{T}}P).$$ Therefore we can pick $y_{j+1}\in \overline{J}$ such that $y_{j+1}\notin \cup_{P\in\operatorname{Ass}\overline{T}/{\mathfrak{m}}\overline{T}}P$ and $y_{j+1}\notin {\mathfrak{n}}_{\overline{T}}^2$. But this precisely means $y_1,\dots,y_{j+1}$ satisfies (a) and (b). This finishes the proof of our claim.
We set $T=T'/(y_1,\dots,y_{b-c})$ and we claim that $T$ satisfies (1)–(3). It follows directly from condition (a) and Theorem \[theorem–flat map with regular fibre\] that $R\to T$ is flat local with $T$ complete and $T/{\mathfrak{m}}T$ a complete intersection, and thus $T$ satisfies (1). Clearly we have $S=T/J$ where $J=J'T$. Since $y_1,\dots,y_{b-c}$ is a regular sequence on $T'/{\mathfrak{m}}T'$ by (a), Theorem \[theorem–flat map with regular fibre\] tells us that $y_1,\dots,y_{b-c}$ is a regular sequence in $T'$. This together with (b) implies $\operatorname{pd}_{T}S<\infty$ by Lemma \[lemma–killing linear term preserves finite projective dimension\]. Hence by Lemma \[lemma–perfect ideal\], $J$ is a perfect ideal of $T$. By the Auslander-Buchsbaum formula, $$\operatorname{pd}_{T}S=\operatorname{depth}T-\operatorname{depth}S=\operatorname{depth}T-\operatorname{depth}R.$$ But since $T$ is obtained from $T'$ by killing a regular sequence of length $b-c$, we have $$\operatorname{pd}_{T}S=\operatorname{depth}T'-\operatorname{depth}R-(b-c)=\operatorname{depth}T'/{\mathfrak{m}}T'-(b-c)=c,$$ which verified (2). Finally, because $y_1,\dots,y_{b-c}$ is part of a basis for ${\mathfrak{n}}_{T'}/{\mathfrak{n}}_{T'}^2$ by (b), $$\operatorname{edim}T=\operatorname{edim}T'-(b-c)=\operatorname{edim}R+c=\operatorname{edim}S,$$ which implies $J\subseteq{\mathfrak{n}}_{T}^2$. Therefore we have verified (3) and hence finished the proof of the lemma.
Lemma \[lemma–factoring maps with c.i. closed fibres\] immediately implies the following corollary, which was originally proved in [@Lechinequalitiesofflatcouples] using different methods.
\[corollary–difference of embedding dimension <=1\] Let $(R,{\mathfrak{m}})\to (S,{\mathfrak{n}})$ be a flat local map between complete local rings of the same dimension. If $\operatorname{edim}S-\operatorname{edim}R\leq 1$, then $S/{\mathfrak{m}}S$ is a complete intersection.
We factor this map as in Lemma \[lemma–factoring maps with c.i. closed fibres\]. By Lemma \[lemma–factoring maps with c.i. closed fibres\] (2), $\operatorname{pd}_TS=\operatorname{pd}_T(T/J)\leq 1$. So $J$ is either $0$ or a principal ideal generated by a nonzerodivisor $y$ in $T$. In the latter case we claim that $y$ must be a nonzerodivisor on $T/{\mathfrak{m}}T$ also. This is because $$\dim (T/{\mathfrak{m}}T)/y(T/{\mathfrak{m}}T)=\dim S/{\mathfrak{m}}S=0,$$ while $T/{\mathfrak{m}}T$ is a complete intersection with $$\dim T/{\mathfrak{m}}T=\dim T-\dim R=\dim T-\dim S=1.$$ Therefore in either case, $S/{\mathfrak{m}}S$ is a ($0$-dimensional) complete intersection.
We end this section by proving a lemma on the behavior of Hilbert-Kunz multiplicities under flat local map with regular closed fiber. We believe this result (and perhaps more general results) are well known to experts. But we could not find a reference that can cover the generality we need, so we give the proof.
\[lemma–Appendix\] Let $(R,{\mathfrak{m}})\to (T,{\mathfrak{n}}_T)$ be a flat local extension between complete local rings of characteristic $p>0$ such that $T/{\mathfrak{m}}T$ is regular. Then
1. $e_{HK}(T)=e_{HK}(R);$
2. $e_{HK}({\mathfrak{n}}_T^2, T)=e_{HK}({\mathfrak{m}}^2, R)+ (\operatorname{edim}T-\operatorname{edim}R)\cdot e_{HK}(R).$
Let $\dim T/{\mathfrak{m}}T=\operatorname{edim}T-\operatorname{edim}R=n$ and let $x_1,\dots,x_n$ be a regular system of parameters in $T/{\mathfrak{m}}T$. We set $(T_0,{\mathfrak{n}}_0)=R[[x_1,\dots,x_n]]$. Note that $(T_0,{\mathfrak{n}}_0)\to (T, {\mathfrak{n}}_T)$ is a flat local map such that the closed fiber $T/{\mathfrak{n}}_0T$ is a field, i.e., ${\mathfrak{n}}_0T={\mathfrak{n}}_T$. Hence it is clear that $$e_{HK}(T)=e_{HK}(T_0)=e_{HK}(R) \text{ and } e_{HK}({\mathfrak{n}}_T^2, T)=e_{HK}({\mathfrak{n}}_0^2, T_0).$$ It thus remains to show that $$e_{HK}({\mathfrak{n}}_0^2, T_0)=e_{HK}({\mathfrak{m}}^2, R)+ (\operatorname{edim}T_0-\operatorname{edim}R)\cdot e_{HK}(R).$$ Since $T_0$ is a power series over $R$ with $n=\operatorname{edim}T_0-\operatorname{edim}R$ variables, by an easy induction it suffices to prove that $$e_{HK}(({\mathfrak{m}}+x)^2, R[[x]])=e_{HK}({\mathfrak{m}}^2, R)+e_{HK}(R).$$ Now we observe that $$l\left(\frac{R[[x]]}{(({\mathfrak{m}}+x)^2)^{[p^e]}}\right)=l\left(\frac{R[[x]]}{({\mathfrak{m}}^2)^{[p^e]}+{\mathfrak{m}}^{[p^e]}x^{p^e}+(x^2)^{[p^e]}}\right)=p^e\cdot l(R/({\mathfrak{m}}^2)^{[p^e]})+p^e\cdot l(R/{\mathfrak{m}}^{[p^e]}).$$ Dividing both sides by $p^{e(\dim R+1)}$ and taking limit we find that $e_{HK}(({\mathfrak{m}}+x)^2, R[[x]])=e_{HK}({\mathfrak{m}}^2, R)+e_{HK}(R)$, as desired.
The main result in characteristic $p>0$
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In this section we will prove Theorem \[theorem–main technical theorem\] in equal characteristic $p>0$. Our proof heavily uses Hilbert-Kunz theory. We will give two inequalities on the multiplicities from which Theorem \[theorem–main technical theorem\] follows immediately. Throughout this section, we assume all rings have equal characteristic $p>0$.
An inequality on multiplicities in terms of $\operatorname{edim}S-\operatorname{edim}R$
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We begin by proving the following lemma that is a consequence of Theorem \[theorem–vanishing of higher Koszul\].
\[lemma–vanishing of higher Koszul\] Let $(T,{\mathfrak{n}}_T)$ be a complete local ring of characteristic $p>0$ with $T/{\mathfrak{n}}_T$ a perfect field. Let $J$ be a perfect ideal of $T$ and $(S,{\mathfrak{n}})=T/J$. Then for every system of parameters $\underline{x}=x_1,\dots,x_d$ of $S$, we have $$\lim_{e\to\infty}\frac{l_S(H_i(\underline{x}, T^{(e)}\otimes S))}{p^{e\cdot\dim T}}=0$$ for every $i\geq 1$.
Since $J$ is a perfect ideal, we have $\operatorname{pd}_TS=\dim T-\dim S$. Let $G_\bullet$ be a minimal free resolution of $S$ over $T$ and let $K_\bullet(\underline{x}, T)$ be the Koszul complex on $\underline{x}$. We have $$H_i(\underline{x}, T^{(e)}\otimes S)=H_i(K_\bullet(\underline{x}, T)\otimes T^{(e)}\otimes S)=H_i(T^{(e)}\otimes K_\bullet(\underline{x}, T)\otimes G_\bullet).$$ Next we note that $K_\bullet(\underline{x}, T)\otimes G_\bullet$ is a complex of free $T$-modules of length $\dim S+\operatorname{pd}_TS=\dim T$, with homology of finite length. Hence by Theorem \[theorem–vanishing of higher Koszul\], $$\lim_{e\to\infty}\frac{l_T\left(H_i(F_T^e(K_\bullet(\underline{x}, T)\otimes G_\bullet))\right)}{p^{e\cdot\dim T}}=0$$ for every $i\geq 1$. But since $T/{\mathfrak{n}}_T$ is a perfect field, we have $$l_S(H_i(\underline{x}, T^{(e)}\otimes S))=l_T(H_i(T^{(e)}\otimes K_\bullet(\underline{x}, T)\otimes G_\bullet))=l_T\left(H_i(F_T^e(K_\bullet(\underline{x}, T)\otimes G_\bullet)\right). \qedhere$$
Before we proceed, we emphasize that if $(T,{\mathfrak{n}}_T)$ is a complete local ring of characteristic $p>0$ such that $T/{\mathfrak{n}}_T$ is a perfect field, then $T^{(e)}$ is a finitely generated $T$-module (i.e., $T$ and hence all localizations of $T$ are $F$-finite). Because a complete local ring is $F$-finite if and only if its residue field is $F$-finite. Therefore $T^{(e)}\otimes_TS$ is a finitely generated $S$-module for every map $T\to S$, and in particular, we can talk about the multiplicities of $T^{(e)}\otimes S$.
\[lemma–multiplicity for perfect ideal generically numerically Roberts\] Let $(T,{\mathfrak{n}}_T)$ be a complete local ring of characteristic $p>0$ with $T/{\mathfrak{n}}_T$ a perfect field. Let $J$ be a perfect ideal of $T$ and $S=T/J$. Then for every system of parameters $\underline{x}=x_1,\dots,x_d$ of $S$, we have $$\label{equation--equality on multiplicities and the crucial limit}
e_{HK}(J+\underline{x}, T)=\lim_{e\to\infty}\frac{1}{p^{e\cdot \dim T}}\cdot e(\underline{x}, T^{(e)}\otimes_TS)=\sum_Pe(\underline{x}, T/P)e_{HK}(J, T_P)$$ where the sum is taken over all minimal prime $P$ of $J$ such that $\dim T/P=\dim S$.
Moreover, if $T_P$ is a numerically Roberts ring (e.g., $T_P$ is a complete intersection) for every such $P$, then the above is also equal to $e(\underline{x}, S)$.
By the Koszul characterization of multiplicity, we know that $$\lim_{e\to\infty}\frac{1}{p^{e\cdot \dim T}}\cdot e(\underline{x}, T^{(e)}\otimes_TS)=\lim_{e\to\infty}\sum_i(-1)^i\cdot \frac{l_S\left(H_i(\underline{x}, T^{(e)}\otimes S)\right)}{p^{e\cdot\dim T}}.$$ Since $J$ is a perfect ideal by Lemma \[lemma–perfect ideal\], Lemma \[lemma–vanishing of higher Koszul\] tells us all the higher terms on the right hand side vanish, hence the above is equal to $$\lim_{e\to\infty}\frac{l_S\left(H_0(\underline{x}, T^{(e)}\otimes S)\right)}{p^{e\cdot\dim T}}=\lim_{e\to\infty}\frac{l_T\left(F^e_T(T/((\underline{x})+J))\right)}{p^{e\cdot\dim T}}=e_{HK}(J+(\underline{x}), T).$$ This proves the first equality in (\[equation–equality on multiplicities and the crucial limit\]).
Next, suppose $P$ is a minimal prime of $J$ such that $\dim T/P=\dim S$. Because $T/{\mathfrak{n}}_T$ is perfect, we have $$\alpha(T_{P})=\alpha(T)+\dim T/P=\dim T/P.$$ Since $J$ is a perfect ideal, we have $\operatorname{pd}_TS=\height J=\dim T-\dim S$, thus by our choice of $P$, $$\dim T\geq \dim T/P+\dim T_{P}\geq \dim S+\height J=\dim T.$$ Hence we have $\dim T_{P}+\alpha(P)=\dim T$ for every minimal prime $P$ of $J$ such that $\dim T/P=\dim S$. Now by the associativity formula for multiplicities [@HunekeSwansonIntegralClosure Theorem 11.2.4], we have $$\begin{aligned}
\frac{1}{p^{e\cdot\dim T}}\cdot e(\underline{x}, T^{(e)}\otimes_TS)&=& \frac{1}{p^{e\cdot\dim T}}\cdot\sum_Pl_{T_{P}}\left((T^{(e)}\otimes_T S)_{P}\right)\cdot e(\underline{x}, T/P)\\
&=&\frac{1}{p^{e\cdot\dim T}}\cdot\sum_Pl_{T_{P}}\left((T_{P})^{(e)}/J(T_{P})^{(e)}\right)\cdot e(\underline{x}, T/P)\\
&=&\frac{1}{p^{e\cdot\dim T}}\cdot p^{e\cdot\alpha(T_{P})}\sum_Pl_{T_{P}}(T_{P}/J^{[p^e]}T_{P})\cdot e(\underline{x}, T/P)\\
&=&\sum_P\frac{1}{p^{e\cdot\dim T_{P}}}l_{T_{P}}(T_{P}/J^{[p^e]}T_{P})\cdot e(\underline{x}, T/P).\end{aligned}$$ Now we take the limit as $e\to\infty$ and by the definition of Hilbert-Kunz multiplicity, we have $$\lim_{e\to\infty}\frac{1}{p^{e\cdot \dim T}}\cdot e(\underline{x}, T^{(e)}\otimes_TS)=\sum_Pe(\underline{x}, T/P)e_{HK}(J, T_P).$$ This proves the second equality in (\[equation–equality on multiplicities and the crucial limit\]).
Finally, if $T_{P}$ is a numerically Roberts ring, then we know that $e_{HK}(J, T_P)=l_{T_{P}}(T_{P}/JT_{P})$ because $JT_P$ is a $PT_P$-primary ideal of finite projective dimension. So we have: $$\sum_Pe(\underline{x}, T/P)e_{HK}(J, T_P)=\sum_Pe(\underline{x}, T/P)l_{T_{P}}(T_{P}/JT_{P})=e(\underline{x}, T/J)=e(\underline{x}, S). \qedhere$$
It is worth to mention that the last assertion of Lemma \[lemma–multiplicity for perfect ideal generically numerically Roberts\] is [*false*]{} in general if we do not assume $T_P$ is numerically Roberts. In [@RobertsIntersectionTheoremsMSRI], based on earlier work of [@DuttaHochsterMcLaughlinModulesoffiniteprojectivedimension], Roberts constructed an example of a three-dimensional Cohen-Macaulay ring $(T,{\mathfrak{n}}_T)$ and an ${\mathfrak{n}}_T$-primary ideal $J$ of finite projective dimension such that $$e_{HK}(J, T)=\lim_{e\to\infty}\frac{l_T(T/J^{[p^e]})}{p^{e\cdot\dim T}}\neq l_T(T/J).$$ Therefore we can set $S=T/J$ (and $\underline{x}$ to be the empty system of parameters, i.e., the zero ideal) in Lemma \[lemma–multiplicity for perfect ideal generically numerically Roberts\] to see that the desired equality fails: for an Artinian local ring, the multiplicity of any ideal is equal to the length of the ring. Of course, the problem is that our $T$ is not numerically Roberts.
In order to apply Lemma \[lemma–multiplicity for perfect ideal generically numerically Roberts\] in our setting, we need the following celebrated result on the localization problem of Grothendieck, see [@TabaaCompleteintersectionhomomorphism] or [@AvramovFoxbyGrothendiecklocalizationproblem Theorem 4.1]:
\[theorem–flat map with complete intersection closed fibers\] Let $\varphi$: $(R,{\mathfrak{m}})\to (T,{\mathfrak{n}}_T)$ be a flat local map with $R$ complete. If the closed fiber $T/{\mathfrak{m}}T$ is a complete intersection, then all fibers of $\varphi$ are complete intersections.
We are ready to prove the following:
\[lemma–multiplicity in terms of the limit\] Let $(R,{\mathfrak{m}})\to (S,{\mathfrak{n}})$ be a flat local map between complete local rings of characteristic $p>0$ with $\dim R=\dim S$. Suppose we have a factorization $$(R,{\mathfrak{m}})\to (T,{\mathfrak{n}}_T)\to (S,{\mathfrak{n}})=T/J$$ such that $R\to T$ is flat local with $T/{\mathfrak{m}}T$ a complete intersection and $\operatorname{pd}_TS<\infty$ (for example, we can take a Cohen-factorization, or any factorization as in Lemma \[lemma–factoring maps with c.i. closed fibres\]). If $R$ is a domain and $S$ has perfect residue field, then $$e_{HK}(J+\underline{x}, T)=\lim_{e\to\infty}\frac{1}{p^{e\cdot \dim T}}\cdot e(\underline{x}, T^{(e)}\otimes_TS)=e(\underline{x}, S)$$ for every system of parameters $\underline{x}$ of $S$.
By Lemma \[lemma–perfect ideal\], we know $J$ is a perfect ideal. Let $P_1,\dots,P_m$ be the minimal primes of $J$. Since $R\to S=T/J$ is faithfully flat, we know $P_i\cap R=0$ because we assumed $R$ is a domain. Since $T/{\mathfrak{m}}T$ is a complete intersection, by Theorem \[theorem–flat map with complete intersection closed fibers\], $T_{P_i}/(P_i\cap R)T_{P_i}=T_{P_i}$ is a complete intersection and hence numerically Roberts. Therefore all the hypotheses of Lemma \[lemma–multiplicity for perfect ideal generically numerically Roberts\] are satisfied and the result follows.
If we choose $\underline{x}$ to be a minimal reduction of ${\mathfrak{n}}$ in Lemma \[lemma–multiplicity in terms of the limit\], then we see immediately that $e(S)=e(\underline{x}, S)$ is equal to the Hilbert-Kunz multiplicity $e_{HK}(J+\underline{x}, T)$. However, in general, the dimension of $T$ is large and we don’t have control on the ideal $J+(\underline{x})$ either. So $e_{HK}(J+(\underline{x}), T)$ will not give us a good estimate of $e(S)$ in terms of the Hilbert-Kunz multiplicity of $R$. Motivated by Roberts’s work [@RobertsIntersectionTheoremsMSRI], [@RobertsMultiplicitiesandChernclassinLocalAlgebra], in the next theorem we analyze $\lim_{e\to\infty}\frac{1}{p^{e\cdot \dim T}}\cdot e(\underline{x}, T^{(e)}\otimes_TS)$ in more details using local Chern characters. As a consequence we will see that for certain carefully chosen $\underline{x}=x_1,\dots, x_d$, we will have $$e(S)=e(\underline{x}, S)=e_{HK}(J, T/(\underline{x})T).$$ It turns out that this is crucial to get our desired estimate on $e(S)$ once we choose our factorization as in Lemma \[lemma–factoring maps with c.i. closed fibres\].
During the preparation of this paper, we were aware of a beautiful result [@SmirnovEquimultiplicityinHilbertKunztheory Corollary 4.11]. Based on this result and our Lemma \[lemma–multiplicity for perfect ideal generically numerically Roberts\], we are also able to give a completely elementary proof of the next theorem which could avoid the use of local Chern characters. Since we feel both proofs reveal some nature about the limit of multiplicities, we decide to keep both proofs here.
\[theorem–limit in terms of local chern characters\] Let $(T,{\mathfrak{n}}_T)$ be a complete local ring of characteristic $p>0$ with $T/{\mathfrak{n}}_T$ a perfect field. Let $J\subseteq T$ be a perfect ideal and $S=T/J$. Set $n=\height J$ and $d=\dim T/J$ (thus $\dim T=n+d$), and let $\underline{x}=x_1,\dots,x_d$ be a system of parameters of $S$ that is also part of a system of parameters of $T$. Then we have $$\label{equation--interchanging two limits}
\lim_{e\to\infty}\frac{1}{p^{e\cdot \dim T}}\cdot e(\underline{x}, T^{(e)}\otimes_TS)=\sum_Pe_{HK}(J, T/P)e(\underline{x}, T_P)$$ where the sum is taken over all minimal primes $P$ of $(x_1,\dots,x_d)$ of dimension $n$.
In particular, if $T_P$ is Cohen-Macaulay for every such $P$, then we have $$\lim_{e\to\infty}\frac{1}{p^{e\cdot \dim T}}\cdot e(\underline{x}, T^{(e)}\otimes_TS)=e_{HK}(J, T/(\underline{x})T).$$
We first note that, if $T_P$ is Cohen-Macaulay, then $e(\underline{x}, T_P)=l_{T_P}(T_P/(\underline{x})T_P)$. Therefore the second assertion follows immediately from the first one by the associativity formula for Hilbert-Kunz multiplicities. Thus below we aim to prove (\[equation–interchanging two limits\]).
[***First proof of (\[equation–interchanging two limits\])***]{}: Fix a minimal free resolution $G_\bullet$ of $T/J$ over $T$. Let $K_\bullet=K_\bullet(\underline{x}, T)$ be the Koszul complex with respect to $\underline{x}=x_1,\dots,x_d$. By the Koszul characterization of multiplicity, we have $$\begin{aligned}
\frac{1}{p^{e\cdot \dim T}}\cdot e(\underline{x}, T^{(e)}\otimes_TS)&=&\frac{1}{p^{e\cdot \dim T}}\cdot\chi(T^{(e)}\otimes K_\bullet\otimes G_\bullet)\\
&=&\frac{1}{p^{e(n+d)}}\sum_{j=0}^{n+d}\operatorname{ch}_j(F^e_T(K_\bullet\otimes G_\bullet))[T]_j\\
&=&\frac{1}{p^{e(n+d)}}\sum_{j=0}^{n+d}\operatorname{ch}_j(K_\bullet\otimes G_\bullet)[F^e_*T]_j\\
&=&\frac{1}{p^{e(n+d)}}\sum_{j=0}^{n+d}p^{ej}\operatorname{ch}_j(K_\bullet\otimes G_\bullet)[T]_j\end{aligned}$$ where the equality on the second line is by the local Riemann-Roch formula and the equality on the third line is by the projection formula (note that we also used repeatedly here that $T/{\mathfrak{n}}_T$ is perfect).
Next we observe that, when $e\to\infty$, the only term in $\frac{1}{p^{e(n+d)}}\sum_{j=0}^{n+d}p^{ej}\operatorname{ch}_j(K_\bullet\otimes G_\bullet)[T]_j$ that can survive is the top term, i.e., when $j=n+d$. Hence we have: $$\lim_{e\to\infty}\frac{1}{p^{e\cdot \dim T}}\cdot e(\underline{x}, T^{(e)}\otimes_TS)=\operatorname{ch}_{n+d}(K_\bullet\otimes G_\bullet)[T]_{n+d}=\sum_j\operatorname{ch}_{n+d-j}(G_\bullet)\operatorname{ch}_{j}(K_\bullet)[T]_{n+d}$$ Since $\operatorname{ch}(K_\bullet)\eta=\operatorname{ch}_d(K_\bullet)\eta=(x_1)\cap(x_2)\cap\cdots\cap(x_d)\cap \eta$ for every cycle $\eta$ (for example, see [@RobertsMultiplicitiesandChernclassinLocalAlgebra Corollary 12.3.2]) and the computation of intersection with divisors can be explicitly expressed using Koszul homologies [@RobertsMultiplicitiesandChernclassinLocalAlgebra Proposition 5.2.11], we have $$\begin{aligned}
\label{equation--intersection with divisors}
\lim_{e\to\infty}\frac{1}{p^{e\cdot \dim T}}\cdot e(\underline{x}, T^{(e)}\otimes_TS)&=&\operatorname{ch}_n(G_\bullet)\operatorname{ch}_d(K_\bullet)[T]_{n+d}\\
&=&\operatorname{ch}_n(G_\bullet)((x_1)\cap\cdots\cap(x_d)\cap[T]_{n+d})\notag\\
&=&\operatorname{ch}_n(G_\bullet)\sum_j (-1)^j[H_j(\underline{x}, T)]_n\notag\\
&=&\operatorname{ch}_n(G_\bullet)\sum_P \chi(\underline{x}, T_P)[T/P]_n\notag\\
&=&\sum_Pe(\underline{x}, T_P)(\operatorname{ch}_n(G_\bullet)[T/P]_n)\notag\end{aligned}$$ where the sum in the last two lines is taken over all minimal primes $P$ of $(x_1,\dots,x_d)$ of dimension $n$.
Finally, by [@RobertsMultiplicitiesandChernclassinLocalAlgebra Theorem 12.7.1] (or we can run the argument in the beginning of our proof), we have $\operatorname{ch}_n(G_\bullet)[T/P]_n=\chi_\infty(\overline{G}_\bullet)$ where $\overline{G}_\bullet$ denote the the complex $G_\bullet\otimes_TT/P$. At this point, we observe that $\overline{G}_\bullet$ is a complex of finite free $\overline{T}=T/P$-modules with finite length homology, and its length as a complex is exactly $n=\dim T/P$ (remember that $J$ is a perfect ideal of height $n$ and $G_\bullet$ is a minimal free resolution of $T/J$ over $T$). Hence by Theorem \[theorem–vanishing of higher Koszul\], we have $$\operatorname{ch}_n(G_\bullet)[T/P]_n=\chi_\infty(\overline{G}_\bullet)=\lim_{e\to\infty}\frac{l\left(H_0(F^e_{\overline{T}}(\overline{G}_\bullet))\right)}{p^{en}}
=\lim_{e\to\infty}\frac{l\left(\overline{T}/J^{[p^e]}\overline{T}\right)}{p^{e\cdot\dim \overline{T}}}=e_{HK}(J, T/P).$$ Now it is clear that (\[equation–interchanging two limits\]) follows from (\[equation–intersection with divisors\]).
[***Second proof of (\[equation–interchanging two limits\])***]{}: We will prove that $$\label{equation--eHK(J+x) is the sum}
e_{HK}(J+\underline{x}, T)=\sum_Pe_{HK}(J, T/P)e(\underline{x}, T_P).$$ This will establish (\[equation–interchanging two limits\]) by (\[equation–equality on multiplicities and the crucial limit\]) in our Lemma \[lemma–multiplicity for perfect ideal generically numerically Roberts\]. To see (\[equation–eHK(J+x) is the sum\]), note that by [@SmirnovEquimultiplicityinHilbertKunztheory Corollary 4.11] (applied to $I=(\underline{x})$ and $M=T$), we always have: $$\label{equation--chains}
\sum_Pe_{HK}(J, T/P)e(\underline{x}, T_P)=\sum_Pe_{HK}(J, T/P)e_{HK}(\underline{x}, T_P)=\lim_{e\to\infty}\frac{1}{p^{en}}e_{HK}(J^{[p^e]}+\underline{x}, T).$$ Next we recall that if $M$ is a finitely generated $T$-module of finite projective dimension, then $\Tor_i^T(M, T^{(e)})=0$ for all $i\geq 1$ (see [@PeskineSzpiroDimensionProjective Théorème (I.7)]). This implies that if $G_\bullet$ is a finite free resolution of $T/J$, then $F_T^{e}(G_\bullet)$ is a finite free resolution of $F_T^e(T/J)\cong T/J^{[p^e]}$. In particular, $J^{[p^e]}$ is a perfect ideal in $T$ for every $e$. Therefore we can apply (\[equation–equality on multiplicities and the crucial limit\]) in Lemma \[lemma–multiplicity for perfect ideal generically numerically Roberts\] to $J^{[p^e]}$ for every $e$ to obtain: $$e_{HK}(J^{[p^e]}+\underline{x}, T)=\sum_Qe(\underline{x}, T/Q)e_{HK}(J^{[p^e]}, T_Q)$$ where the sum is taken over all minimal primes $Q$ of $J$ such that $\dim T/Q=d$. Since $\height J=n$, we have $$\sum_Qe(\underline{x}, T/Q)e_{HK}(J^{[p^e]}, T_Q)=p^{en}\sum_Qe(\underline{x}, T/Q)e_{HK}(J, T/Q)=p^{en}e_{HK}(J+\underline{x}, T)$$ where the second equality is by (\[equation–equality on multiplicities and the crucial limit\]) again. Therefore, the sequence $$\{\frac{1}{p^{en}}e_{HK}(J^{[p^e]}+\underline{x}, T)\}$$ is a constant sequence! Now (\[equation–eHK(J+x) is the sum\]) follows immediately from (\[equation–chains\]).
In the second proof of Theorem \[theorem–limit in terms of local chern characters\], we crucially used [@SmirnovEquimultiplicityinHilbertKunztheory Corollary 4.11], which in turn follows from some delicate “uniform convergence" results in that paper (in the spirit of [@TuckerFsignatureExists]). Let us also point out that, once we combined (\[equation–equality on multiplicities and the crucial limit\]) in Lemma \[lemma–multiplicity for perfect ideal generically numerically Roberts\] with [@SmirnovEquimultiplicityinHilbertKunztheory Theorem 5.17], we find that perfect ideals in complete (unmixed) local rings [*satisfy colon capturing*]{} of tight closure in the sense of [@SmirnovEquimultiplicityinHilbertKunztheory Definition 5.15]. We suspect that some version of this should be known in the literature, as tight closure of ideals of finite projective dimension (and more generally, of finite phantom projective dimension) has been studied intensively [@HochsterHunekePhantomhomology], [@Aberbachfinitephantomprojectivedimension]. However we have not been able to find a precise reference at the moment.
We need one more lemma on the choice of system of parameters. We recall that a sequence of elements $y_1,\dots,y_n \in {\mathfrak{m}}$ of $(R,{\mathfrak{m}})$ is called a [*filter regular sequence*]{} of $R$ if $y_i$ is not contained in any associated prime of $R/(y_1,\dots,y_{i-1})$ except ${\mathfrak{m}}$ for every $i$. Standard prime avoidance shows that filter regular sequence (of any length) always exists.
\[lemma–choose generically CM sop\] Let $(T,{\mathfrak{n}}_T)$ be a complete local ring with $T/{\mathfrak{n}}_T$ an infinite field. Let $J\subseteq T$ be an ideal and $S=T/J$. Suppose $\height J=n\geq 1$ and $\dim S=d$. Then there exists $x_1,\dots,x_d\in {\mathfrak{n}}_T$ such that
1. $x_1,\dots,x_d$ is part of a system of parameters of $T$;
2. $x_1,\dots,x_d$ is a minimal reduction of ${\mathfrak{n}}_TS$ in $S$;
3. $T_P$ is Cohen-Macaulay for every minimal prime $P$ of $(x_1,\dots,x_d)$.
We first notice that condition (3) is satisfied if $x_1,\dots,x_d$ is a filter regular sequence on $T/I$, where $I$ is the defining ideal of the non-Cohen-Macaulay locus of $T$. The reason is as follows. Since $\height I\geq 1$ and $\dim T\geq n+d\geq 1+d$, $I+(x_1,\dots,x_d)$ has height at least $\min\{\height I+d, \dim T\}\geq d+1$. Now suppose $P$ is a minimal prime of $(x_1,\dots,x_d)$. If $T_P$ is not Cohen-Macaulay, then $I\subseteq P$ and thus $I+(x_1,\dots,x_d)\subseteq P$. But then we have $$d\geq \height P\geq \height(I+(x_1,\dots,x_d))\geq d+1$$ which is a contradiction.
The remaining argument is standard. We can pick $x_1,\dots,x_d$ inductively and thus it suffices to construct $x_1$. First of all we want $x_1$ not contained in any minimal prime of $T$, and $x_1$ not contained in any associated prime of $T/I$ except possibly ${\mathfrak{n}}_T$. There are only finitely many primes that we need to avoid. Call these $Q_1,\dots,Q_m$. Next we note that ${\mathfrak{n}}_T/{\mathfrak{n}}_T^2$ is a finite dimensional vector space over an infinite field $T/{\mathfrak{n}}_T$. It is clear that $\{(Q_i+{\mathfrak{n}}_T^2)/{\mathfrak{n}}_T^2\}_{i=1}^{m}$, as well as the degree one elements of each minimal prime of $$\operatorname{gr}_{\mathfrak{n}}S\cong \frac{T}{{\mathfrak{n}}_T}\oplus \frac{{\mathfrak{n}}_T}{{\mathfrak{n}}_T^2+J}\oplus\frac{{\mathfrak{n}}_T^2+J}{{\mathfrak{n}}_T^3+J}\oplus\cdots$$ form a finite set of finite dimensional subspaces of ${\mathfrak{n}}_T/{\mathfrak{n}}_T^2$. Since none of these subspaces is equal to the whole ${\mathfrak{n}}_T/{\mathfrak{n}}_T^2$ and $T/{\mathfrak{n}}_T$ is infinite, we can pick $x_1\in{\mathfrak{n}}_T/{\mathfrak{n}}_T^2$ that is not contained in all these subspaces. But this is precisely saying that $x_1$ is part of a system of parameter on $T$, is part of a minimal reduction of $S=T/J$, and is part of a filter regular sequence on $T/I$. Therefore we are done by the discussion above.
We are now ready to prove our first inequality on multiplicities under flat local extensions between local rings.
\[theorem–inequalities on multiplicities in terms of difference of embdim\] Let $(R,{\mathfrak{m}})\to (S,{\mathfrak{n}})$ be a flat local map between complete local rings of characteristic $p>0$ with $\dim R=\dim S=d$. Suppose $R$ is a domain and $S/{\mathfrak{n}}$ is algebraically closed. If $\operatorname{edim}S-\operatorname{edim}R=c$, then we have $$e(R)\leq \frac{c!}{2^c}e(S).$$
We first note that $c\geq 0$ by Theorem \[theorem–regularity defect of flat local extension\]. Applying Lemma \[lemma–factoring maps with c.i. closed fibres\], we obtain $$(R,{\mathfrak{m}})\to (T,{\mathfrak{n}}_T)\to (S,{\mathfrak{n}})=T/J$$ such that $T/{\mathfrak{m}}T$ is a complete intersection, $J\subseteq {\mathfrak{n}}_T^2$ is a perfect ideal of $T$, and $\operatorname{pd}_TS=c$. If $c=0$, then $S/{\mathfrak{m}}S$ is a complete intersection by Corollary \[corollary–difference of embedding dimension <=1\]. Hence $e(R)\leq e(S)$ follows from [@Lechinequalitiesofflatcouples] (see [@HerzogBLechHironakainequalities Theorem 1]). We thus assume $c\geq 1$, which implies $\height J=\operatorname{pd}_TS\geq 1$. Since $S/{\mathfrak{n}}=T/{\mathfrak{n}}_T$ is algebraically closed and $\height J\geq 1$, by Lemma \[lemma–choose generically CM sop\] we can pick a minimal reduction $\underline{x}=x_1,\dots,x_d$ of ${\mathfrak{n}}$ such that $x_1,\dots,x_d$ is part of a system of parameters of $T$ and $T_P$ is Cohen-Macaulay for every minimal prime of $(x_1,\dots,x_d)$. Now the hypotheses of Lemma \[lemma–multiplicity in terms of the limit\] and Theorem \[theorem–limit in terms of local chern characters\] are both satisfied. Applying them we obtain $$\label{equation--interchanging two ideals e(S)=e_HK(J)}
e(S)=e(\underline{x}, S)=\lim_{e\to\infty}\frac{1}{p^{e\cdot \dim T}}\cdot e(\underline{x}, T^{(e)}\otimes_TS)=e_{HK}(J, T/(\underline{x})T).$$ Because $J$ is perfect, we have $$\dim T/(\underline{x})T=\dim T-d=\operatorname{pd}_TS=c.$$ Since $J\subseteq {\mathfrak{n}}_T^2$, we have $$\label{equation--intermediate comparison of e_HK and e}
e_{HK}(J, T/(\underline{x})T)\geq e_{HK}({\mathfrak{n}}_T^2, T/(\underline{x})T)\geq \frac{1}{c!}e({\mathfrak{n}}_T^2, T/(\underline{x})T)=\frac{2^c}{c!}e(T/(\underline{x})T).$$ Finally, it is well known that $e(T/(\underline{x})T)\geq e(T)$ when $\underline{x}$ is part of a system of parameters of $T$ [@SinghEffectofpermissibleblowup Corollary 4]. But $(R,{\mathfrak{m}})\to (T,{\mathfrak{n}}_T)$ is a flat local map with $T/{\mathfrak{m}}T$ a complete intersection, so $e(R)\leq e(T)$ follows from [@Lechinequalitiesofflatcouples] (see [@HerzogBLechHironakainequalities Theorem 1]). Therefore putting (\[equation–interchanging two ideals e(S)=e\_HK(J)\]) and (\[equation–intermediate comparison of e\_HK and e\]) together we get $$e(S)\geq \frac{2^c}{c!}e(T/(\underline{x})T)\geq\frac{2^c}{c!}e(T)\geq \frac{2^c}{c!}e(R).$$ This finishes the proof of our theorem.
An inequality on multiplicities when $\operatorname{edim}S-\operatorname{edim}R$ is large
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In this subsection we prove another inequality on the behavior of multiplicities under flat local extension. It gives a control on $e(S)$ when $\operatorname{edim}S-\operatorname{edim}R$ is large. We begin with some lemmas.
\[lemma–estimate on e\_S involving higher Euler character\] Let $(S,{\mathfrak{n}})$ be a local ring of dimension $d$ and $N$ be a finitely generated $S$-module. Then for every system of parameters $\underline{x}=x_1,\dots, x_d$ of $S$, we have $$e(\underline{x}, N)\geq \nu_S({\mathfrak{n}}N)+(1-d)\nu_S(N)-\chi_1(\underline{x}, N).$$
By the Koszul characterization of multiplicity, we have $$\begin{aligned}
e(\underline{x}, N)&=&\sum_{i=0}^d(-1)^il_S(H_i(\underline{x}, N))\\
&=&l_S\left(\frac{N}{(\underline{x})N}\right)-\chi_1(\underline{x}, N)\\
&=&l_S\left(\frac{N}{{\mathfrak{n}}(\underline{x})N}\right)-l_S\left(\frac{(\underline{x})N}{{\mathfrak{n}}(\underline{x})N}\right)-\chi_1(\underline{x}, N)\\
&\geq& l_S(N/{\mathfrak{n}}^2N)-d\cdot\nu_S(N)-\chi_1(\underline{x}, N)\\
&=& \nu_S({\mathfrak{n}}N)+(1-d)\nu_S(N)-\chi_1(\underline{x}, N)\end{aligned}$$ where the only $\geq$ is because ${\mathfrak{n}}(\underline{x})\subseteq {\mathfrak{n}}^2$ and we have a natural surjection $$\left(\frac{N}{{\mathfrak{n}}N}\right)^{\oplus d}\twoheadrightarrow \frac{(\underline{x})N}{{\mathfrak{n}}(\underline{x})N}.$$ This finishes the proof of the lemma.
The next lemma is [@HanesThesis Proposition 4.2.3]. We give a proof for completeness. We note that it was assumed that $S$ is a flat local extension of $R$ in [@HanesThesis]. However, this condition is unnecessary in the proof.
\[lemma–Hanes’ lemma\] Let $(R,{\mathfrak{m}})\rightarrow (S,{\mathfrak{n}})$ be a local map and let $M$ be a finitely generated module over $R$. Then: $$\nu_S ({\mathfrak{n}}M')\geq\nu_R({\mathfrak{m}}M)+(\operatorname{edim}S-\operatorname{edim}R)\cdot\nu_R(M)$$ where $M'=S\otimes_RM$.
The conclusion is obviously true if $M\cong R^n$: in this case we trivially have an equality. Therefore by induction, it suffices to prove the statement for $N=M/Ry$ where $y\in {\mathfrak{m}}M$, assuming that it is true for $M$. We note that $N'=S\otimes_RN=M'/Sy$ and since $y\in {\mathfrak{m}}M$, we have $\nu_R(M)=\nu_R(N)$.
If $y\in {\mathfrak{m}}^2 M$, then the image of $y$ is in ${\mathfrak{m}}^2 M'\subseteq {\mathfrak{n}}^2 M'$, we have $\nu_R({\mathfrak{m}}M)=\nu_R({\mathfrak{m}}N)$ and $\nu_S({\mathfrak{n}}M')=\nu_S({\mathfrak{n}}N')$. So all the terms do not change when we pass from $M$ to $N$ and hence the conclusion holds for $N$. If $y\in {\mathfrak{m}}M-{\mathfrak{m}}^2M$, then $\nu_R({\mathfrak{m}}N)=\nu_R({\mathfrak{m}}M)-1$. Because the image of $y$ is in ${\mathfrak{m}}M'\subseteq {\mathfrak{n}}M'$, $\nu_S({\mathfrak{n}}N')\geq \nu_S({\mathfrak{n}}M')-1$. Since $\nu_R(N)=\nu_R(M)$, the remaining terms do not change, and we see that the inequality continues to hold for $N$.
We are now ready to prove our second inequality on multiplicities under flat local extensions. Our strategy of the proof is inspired by [@HanesThesis Proposition 4.3.4].[^4] Our main new ingredients here are Lemma \[lemma–vanishing of higher Koszul\], Lemma \[lemma–multiplicity in terms of the limit\], and Lemma \[lemma–estimate on e\_S involving higher Euler character\], which will drop the additional hypothesis in Hanes’s argument.
\[theorem–inequalities on multiplicities when the difference of embdim is large\] Let $(R,{\mathfrak{m}})\to (S,{\mathfrak{n}})$ be a flat local map between complete local rings of characteristic $p>0$ with $\dim R=\dim S=d$. Suppose $R$ is a domain and $S/{\mathfrak{n}}$ is algebraically closed. If $\operatorname{edim}S-\operatorname{edim}R=c\geq d$, then we have $$e(R)\leq \frac{d!}{2^d+c-d}e(S)\leq \frac{d!}{2^d}e(S).$$
The second inequality is trivial because we assumed $c\geq d$. Let $$(R,{\mathfrak{m}})\to (T,{\mathfrak{n}}_T)\to (S,{\mathfrak{n}})=T/J$$ be a Cohen-factorization as in Theorem \[theorem–Cohen factorization\]. We fix $\underline{x}=x_1,\dots,x_d$ a minimal reduction of ${\mathfrak{n}}$. We apply Lemma \[lemma–estimate on e\_S involving higher Euler character\] to $\underline{x}$ and $N=T^{(e)}\otimes_TS$ (note that $T^{(e)}\otimes_TS$ is a finitely generated $S$-module because $T$ is complete with algebraically closed residue field) to get: $$\label{equation--multiplicity after applying the first lemma}
e(\underline{x}, T^{(e)}\otimes S) \geq \nu_S({\mathfrak{n}}(T^{(e)}\otimes S))+(1-d)\cdot\nu_S(T^{(e)}\otimes S)-\chi_1(\underline{x}, T^{(e)}\otimes S).$$ Next we apply Lemma \[lemma–Hanes’ lemma\] to $R=T$ and $M=T^{(e)}$ to get: $$\label{equation--after applying Hanes' lemma}
\nu_S({\mathfrak{n}}(T^{(e)}\otimes S))\geq \nu_T({\mathfrak{n}}_T T^{(e)})+(\operatorname{edim}S-\operatorname{edim}T)\cdot\nu_T(T^{(e)}).$$ Combining (\[equation–multiplicity after applying the first lemma\]) and (\[equation–after applying Hanes’ lemma\]) and noticing that $\nu_S(T^{(e)}\otimes S)=\nu_T(T^{(e)})$ because $S$ is a quotient of $T$, we have: $$\label{equation--main estimates before dividing p^ed}
e(\underline{x}, T^{(e)}\otimes S)\geq \nu_T({\mathfrak{n}}_T T^{(e)})+(\operatorname{edim}S-\operatorname{edim}T+1-d)\cdot\nu_T(T^{(e)})-\chi_1(\underline{x}, T^{(e)}\otimes S).$$ Now we observe that when $e\to\infty$, we have: $$e(\underline{x}, T^{(e)}\otimes S) \rightarrow e(\underline{x},S)\cdot p^{e\cdot\dim T} \text{ by Lemma \ref{lemma--multiplicity in terms of the limit}},$$ $$\nu(T^{(e)})=l_T(\frac{T^{(e)}}{{\mathfrak{n}}_T T^{(e)}})\rightarrow e_{HK}(T)\cdot p^{e\cdot\dim T},$$ $$\nu({\mathfrak{n}}_T T^{(e)})=l_T(\frac{{\mathfrak{n}}_T T^{(e)}}{{\mathfrak{n}}_T^2T^{(e)}})=l_T(\frac{T^{(e)}}{{\mathfrak{n}}_T^2T^{(e)}})-l_T(\frac{T^{(e)}}{{\mathfrak{n}}_T T^{(e)}})\rightarrow (e_{HK}({\mathfrak{n}}_T^2,T)-e_{HK}(T))\cdot p^{e\cdot\dim T},$$ where the last two follow from the definition of the Hilbert-Kunz multiplicities and the fact that $T/{\mathfrak{n}}_T$ is algebraically closed. More importantly, since we know that $J$ is a perfect ideal of $T$ by Lemma \[lemma–perfect ideal\], Lemma \[lemma–vanishing of higher Koszul\] then implies $$\chi_1(\underline{x}, T^{(e)}\otimes S)=\sum_{i=1}^d(-1)^{i-1}l_S(H_i(\underline{x}, T^{(e)}\otimes S))=o(p^{e\cdot \dim T}).$$ Hence after we divide (\[equation–main estimates before dividing p\^ed\]) by $p^{e\cdot\dim T}$ and let $e\to\infty$, we get: $$\begin{aligned}
\label{equation--main estimates after dividing p^ed}
e(\underline{x}, S)&\geq&(e_{HK}({\mathfrak{n}}_T^2,T)-e_{HK}(T))+(\operatorname{edim}S-\operatorname{edim}T+1-d)\cdot e_{HK}(T)\\
&=&e_{HK}({\mathfrak{n}}_T^2,T)+(\operatorname{edim}S-\operatorname{edim}T-d)\cdot e_{HK}(T)\end{aligned}$$ Finally, we apply Lemma \[lemma–Appendix\] to obtain $$\begin{aligned}
e(\underline{x}, S)&\geq&e_{HK}({\mathfrak{m}}^2, R)+(\operatorname{edim}T-\operatorname{edim}R)\cdot e_{HK}(R)+ (\operatorname{edim}S-\operatorname{edim}T-d)\cdot e_{HK}(R) \\
&=&e_{HK}({\mathfrak{m}}^2, R)+(\operatorname{edim}S-\operatorname{edim}R-d)\cdot e_{HK}(R)\\
&\geq& \frac{1}{d!}e({\mathfrak{m}}^2, R)+\frac{1}{d!}(c-d)\cdot e(R) \\
&=&\frac{2^d+c-d}{d!}\cdot e(R)\end{aligned}$$ where we use our assumption that $\operatorname{edim}S-\operatorname{edim}R=c\geq d$. Since $\underline{x}$ is a minimal reduction of ${\mathfrak{n}}$, $e(S)=e(\underline{x}, S)$ and the above estimate immediately shows that $$e(R)\leq\frac{d!}{2^d+c-d}e(S).$$ This finishes the proof.
Proof of Theorem \[theorem–main technical theorem\] in equal characteristic $p>0$
---------------------------------------------------------------------------------
For the reader’s convenience we restate our main theorem in characteristic $p>0$.
\[theorem–main theorem in characteristic p>0\] Let $(R,{\mathfrak{m}})\rightarrow (S,{\mathfrak{n}})$ be a flat local extension between local rings of equal characteristic $p>0$. If $\dim R=d$, then we have $$e(R)\leq \max\{1,\frac{d!}{2^d}\}\cdot e(S).$$ In particular, if $\dim R=3$, then $e(R)\leq e(S)$.
By Lemma \[lemma–reduction\], we may assume $\dim R=\dim S=d$, $R$ and $S$ are both complete, $R$ is a domain, and $S$ has algebraically closed residue field. Now the hypotheses of Theorem \[theorem–inequalities on multiplicities in terms of difference of embdim\] and Theorem \[theorem–inequalities on multiplicities when the difference of embdim is large\] are both satisfied. By Theorem \[theorem–regularity defect of flat local extension\], $c=\operatorname{edim}S-\operatorname{edim}S\geq 0$, thus Theorem \[theorem–inequalities on multiplicities in terms of difference of embdim\] and Theorem \[theorem–inequalities on multiplicities when the difference of embdim is large\] together tell us that $$e(R)\leq \max\{\frac{c!}{2^c}| 0\leq c\leq d\}\cdot e(S)=\max\{1, \frac{d!}{2^d}\}\cdot e(S).$$ This finishes the proof.
Reduction to characteristic $p>0$
=================================
In this section we will use reduction to characteristic $p>0$ to obtain Theorem \[theorem–main technical theorem\] in characteristic $0$. The process of reducing Lech’s conjecture in characteristic $0$ to characteristic $p>0$ is known at least to Mel Hochster, and most of the arguments are standard. Therefore we will omit the technical details in our presentation and direct the reader to the references (such as [@HochsterNonnegativityfollowingGabber] or [@DuttaAtheoremonSmoothness]) when necessary.
We should point out that, however, there are at least two nontrivial points when passing to characteristic $p>0$. First, we need to reduce Lech’s conjecture to the case that $(R,{\mathfrak{m}})\to (S,{\mathfrak{n}})$ is a [*module-finite*]{} extension. We can only do this in characteristic $0$ in general. Second, we need to apply Artin approximation and reduction to characteristic $p>0$ while [*preserving the multiplicities of $R$ and $S$*]{}. This will be done by choosing minimal reductions of ${\mathfrak{m}}$ and ${\mathfrak{n}}$ and keeping track of the length of all the Koszul homology modules. We begin with the following lemma.
\[lemma–reduction to module finite\] Let $(R,{\mathfrak{m}})\to (S,{\mathfrak{n}})$ be a flat local map with $\dim R=d$. Suppose $R$ has equal characteristic $0$. In order to prove Lech’s conjecture that $e(R)\leq e(S)$, or more generally, to prove $e(R)\leq C\cdot e(S)$ for certain constant $C$ depending only on $d$, we may assume $R\to S$ is a finite free extension.
By Lemma \[lemma–reduction\], we may assume $R$ and $S$ are both complete, $\dim R=\dim S=d$, and $S$ has algebraically closed residue field. Let $K$ be the coefficient field of $R$. Since we are in characteristic $0$, $K$ is contained in the coefficient field $L$ of $S$. Thus the two maps $R\to S$ and $L\to S$ agree on $K$. Hence we have the following natural maps: $$R\to R\widehat{\otimes}_KL\to S.$$
Let $R'=R\widehat{\otimes}_KL$ and ${\mathfrak{m}}'={\mathfrak{m}}R'$. Note that $(R',{\mathfrak{m}}')$ is still a complete local ring of dimension $d$, and we have $e(R)=e(R')$. Moreover, since $R'$ is flat over $R$ with ${\mathfrak{m}}'={\mathfrak{m}}R'$, $\Tor^{R'}_1(R'/{\mathfrak{m}}', S)=\Tor^R_1(R/{\mathfrak{m}}, S)=0$. So by the local criterion of flatness, $R'\to S$ is still a flat local map. Now we can replace $R$ by $R'$ to assume $R\to S$ is a flat local map between complete local rings (of equal characteristic $0$) of the same dimension and same residue field. But it is well known that such an extension must be module-finite. Therefore we reduce to the case that $R\to S$ is a finite free extension.
\[remark–reduction to module finite not clear in general\] To the best of our knowledge, the above “reduction to module-finite case" is not clear in characteristic $p>0$ (or in mixed characteristic) in general. In characteristic $p>0$, one might hope to use the same method as in Lemma \[lemma–reduction to module finite\]. But the subtle point here is that the coefficient field of $R$ might not be contained in a coefficient field of $S$ and thus we cannot construct the desired $R'$ as in Lemma \[lemma–reduction to module finite\].
We now start to prove Theorem \[theorem–main technical theorem\] in characteristic $0$. There are two steps.
Reduction to local rings essentially of finite type over a field
----------------------------------------------------------------
Suppose we have a counter-example $(R,{\mathfrak{m}})\to (S,{\mathfrak{n}})$ to Theorem \[theorem–main technical theorem\]. By Lemma \[lemma–reduction to module finite\], we can assume that $(R,{\mathfrak{m}})\to (S,{\mathfrak{n}})$ is a finite free extension between complete local rings of dimension $d$, and $R$, $S$ have the same coefficient field $K=\overline{K}$ (of characteristic $0$). By Cohen’s structure theorem we can fix $(A,{\mathfrak{m}}_A)=K[[x_1,\dots,x_d]]\to R$ a module-finite extension. Now we think of this counter-example as a pair of finitely generated $A$-modules $R$, $S$ such that:
(1) $R$ and $S$ both have an algebra structure, they are both local of dimension $d$, with residue field $K$, and $S$ is finite free as an $R$-module;
(2) $e(R)=\alpha$, $e(S)=\beta$, such that $\alpha>\max\{1,\frac{d!}{2^d}\}\beta$.
The idea is to use Artin approximation [@ArtinApproximation] to replace this example by an example with the same properties but constructed over $B=(K[x_1,\dots,x_d]_{(x_1,\dots,x_d)})^h$, i.e., the Henselization of the ring $K[x_1,\dots,x_d]_{(x_1,\dots,x_d)}$. Since the Henselization is a direct limit of pointed étale extensions of $K[x_1,\dots,x_d]_{(x_1,\dots,x_d)}$, it follows immediately that the original counter-example descends to $R\to S$, both essentially of finite type over $K$.
We think of $R$ generated over $A$ by $\theta_0=1, \theta_1,\dots, \theta_n$ and $S$ generated over $R$ by $\eta_0=1,\eta_1,\dots,\eta_r$, think of $S$ generated over $A$ by $\xi_{ij}$ (corresponding to $\theta_i\eta_j$). As an $A$-module, $R$ (resp., $S$) can be represented as the cokernel of a finite matrix $(a_{ij})$ (resp., $(b_{ij})$). To descend to the Henselization we want to think of $a_{ij}$ and $b_{ij}$ as solutions in $A$ of a finite system of polynomial equations over $B$. There will also be a lot of additional auxiliary elements involved in these equations, i.e., the system of polynomial equations will involve many variables besides those corresponds to $a_{ij}$ and $b_{ij}$. The idea is to construct a large family of equations satisfied by $a_{ij}$, $b_{ij}$, and those auxiliary variables such that when we take a new solution in $B$, congruent to the original solution modulo a certain high power of ${\mathfrak{m}}_A$, we can use this solution to get a counter-example over $B$. Therefore, the key point here is to express (1) and (2) [*equationally*]{}.
The fact that $R$ and $S$ have an algebra structure can be expressed using equations is well known and can be found in many references (for example, see [@Smithtightclosureofparameterideals]). We can keep track of the dimensions of $R$ and $S$ using equations—this follows from [@HochsterNonnegativityfollowingGabber Page 12, (9)]. We can use equations to characterize certain element of $R$ winds up in ${\mathfrak{m}}_A R$, this is done in [@Smithtightclosureofparameterideals]. Using this, we can then describe $R$ being a local ring with residue field $K$ equationally (and similarly for $S$) by first keeping track of the lengths of $R/{\mathfrak{m}}_A R$ and $R/({\mathfrak{m}}_A+(\theta_1,\dots, \theta_n))R$—which follows from [@HochsterNonnegativityfollowingGabber Page 12, (7)] or [@DuttaAtheoremonSmoothness Lemma 3.2], and then keeping track of the property that all generators of a fixed power of $(\theta_1,\dots, \theta_n)$ lies in ${\mathfrak{m}}_A R$: this forces $(\theta_1,\dots, \theta_n)$ to be the only maximal ideal in the Artinian ring $R/{\mathfrak{m}}_AR$ and thus $R$ is local with unique maximal ideal ${\mathfrak{m}}_A+(\theta_1,\dots, \theta_n)$, whose residue field is $K$ (since we keep track that it has length $1$). The fact that $S$ is finite free over $R$ can be expressed using equations of $(a_{ij})$ and $(b_{ij})$: because we pick $\xi_{ij}=\theta_i\eta_j$ and so all $A$-relations of $\xi_{ij}$ are coming from $A$-relations of $\theta_i$. This shows that everything in (1) can be traced using equations.
Next we explain why (2) can be expressed equationally. This follows from the following more general results:
(i) We can keep track of a sequence of elements $\underline{y}=y_1,\dots,y_d\in R$ (resp., $\underline{z}=z_1,\dots,z_d\in S$) such that $(y_1,\dots,y_d)$ is a minimal reduction of ${\mathfrak{m}}$ (resp., $(z_1,\dots,z_d)$ is a minimal reduction of ${\mathfrak{n}}$) using equations.
(ii) We can keep track of the Euler characteristic $\chi(\underline{y}, R)$ (resp., $\chi(\underline{z}, S)$).
To see (i), we set $y_i=r_{i1}\theta_1+\cdots+r_{in}\theta_n$ where $r_{ij}$ are (solutions) in $A$. To make sure that $(y_1,\dots,y_d)$ is a minimal reduction of ${\mathfrak{m}}$, note that we can express ${\mathfrak{m}}^N=(y_1,\dots,y_d){\mathfrak{m}}^{N-1}$ for some fixed $N$ using equations: this is clear since ${\mathfrak{m}}^N=(y_1,\dots,y_d){\mathfrak{m}}^{N-1}$ amounts to say that for every $k_1+\cdots+k_n=N$, we have $$\label{equation--describing a minimal reduction using equations}
\theta_1^{k_1}\cdots\theta_n^{k_n}=\sum_{l_1+\cdots+l_n=N-1} c_{il_1\cdots l_n}^{k_1\cdots k_n}y_i\theta_1^{l_1}\cdots\theta_n^{l_n} \hspace{1em} \text{ in } R.$$ From (\[equation–describing a minimal reduction using equations\]), we can further plug in $y_i=r_{i1}\theta_1+\cdots+r_{in}\theta_n$ and write each $c_{il_1\cdots l_n}^{k_1\cdots k_n}$ as $\sum s_j\theta_j$ where $s_j$ are (solutions) in $A$. We thus get many equations over $R$. But then these equations could be written as equations over $A$ using the variables introduced when we describe the multiplication structure on $R$ and extra equations involving $(a_{ij})$, the relation matrix representing $R$ as an $A$-module. We can do exactly the same thing for $z_1,\dots,z_d$ in $S$. Finally, (ii) follows from the fact that we can actually keep track of a finite complex of finitely generated modules as well as its homology [@HochsterNonnegativityfollowingGabber Page 12, (8)] (this was originated from [@PeskineSzpiroDimensionProjective Theorem 6.2], see also [@DuttaAtheoremonSmoothness]). Note that we should be careful here because we also need to keep track that the complex is the Koszul complex of $y_1,\dots,y_d$ on $R$ (resp. the Koszul complex of $z_1,\dots,z_d$ on $S$), but we can add auxiliary equations to describe this property.
By the above discussion, we see that we can use a large family of equations to keep track of (1) and (2), i.e., a counter-example of Theorem \[theorem–main technical theorem\]. Hence by Artin approximation [@ArtinApproximation], these equations have a solution in $B=(K[x_1,\dots,x_n]_{(x_1,\dots,x_n)})^h$, i.e., we have a counter-example $(R,{\mathfrak{m}})\to (S,{\mathfrak{n}})$ where $R$ and $S$ are finite over $B$ with $R/{\mathfrak{m}}=S/{\mathfrak{n}}=K$. Furthermore, since $B$ is a direct limit of local rings essentially of finite type over $K$. We know that we have a counter-example $(R,{\mathfrak{m}})\to (S,{\mathfrak{n}})$ of Theorem \[theorem–main technical theorem\] where both $R$ and $S$ are local rings essentially of finite type over $K$ with $R/{\mathfrak{m}}=S/{\mathfrak{n}}=K$.
Reduction to characteristic $p>0$
---------------------------------
Since $R$, $S$ are local rings essentially finite type over $K=\overline{K}$ with $R/{\mathfrak{m}}=S/{\mathfrak{n}}=K$,[^5] by Nullstellensatz we can do a change of variables if necessary to assume: $$(R,{\mathfrak{m}})\cong\left(\frac{K[y_1,\dots,y_m]}{I}\right)_{(y_1,\dots,y_m)} \text{ and } (S,{\mathfrak{n}})\cong\left(\frac{K[z_1,\dots,z_n]}{J}\right)_{(z_1,\dots,z_n)}.$$ Because $(R,{\mathfrak{m}})\to (S,{\mathfrak{n}})$ is a finite flat extension, we can find $f\in K[y_1,\dots,y_m]$ and $g\in K[z_1,\dots,z_n]$ such that $$\left(\frac{K[y_1,\dots,y_m]}{I}\right)_f\to\left(\frac{K[z_1,\dots,z_n]}{J}\right)_g$$ is a well-defined finite flat extension (note that the above two rings are both finite type over $K$). Now we can adjoin an extra variable to $y_1,\dots,y_m$ and $z_1,\dots,z_n$ respectively, and by Nullstellensatz we can perform a linear change of variables to assume that we have a finite flat extension: $$R_K=\frac{K[y_0,y_1,\dots,y_m]}{I}\to \frac{K[z_0, z_1,\dots,z_n]}{J}=S_K,$$ such that ${\mathfrak{m}}=(y_0, y_1,\dots,y_m)$, ${\mathfrak{n}}=(z_0, z_1,\dots,z_n)$, and $R=(R_K)_{\mathfrak{m}}\to S=(S_K)_{\mathfrak{n}}$ is our counter-example to Theorem \[theorem–main technical theorem\]. Furthermore we can assume that $\underline{y}=y_1,\dots,y_d$ is a minimal reduction of ${\mathfrak{m}}$ and $\underline{z}=z_1,\dots,z_d$ is a minimal reduction of ${\mathfrak{n}}$. We can assume that all the Koszul homology modules $H_i(\underline{y}, R_K)$ and $H_i(\underline{z}, S_K)$ are supported only at ${\mathfrak{m}}$ and ${\mathfrak{n}}$ respectively: they are supported at finitely many maximal ideals, so we can localize $R_K$ and $S_K$ at one extra element respectively (and then adjoin extra variables and perform a linear change of variables to assume $R_K$ and $S_K$ still have the shape as above) to assume they are only supported at ${\mathfrak{m}}$ and ${\mathfrak{n}}$ respectively.
At this point, we pick a finitely generated $\mathbb{Z}$-algebra $W$ of $K$ such that the coefficients of a set of generators of $I$, $J$ are contained in $W$. By generic freeness, we can shrink $W$ if necessary to assume we have a well-defined map between free $W$-algebras: $$R_W=\frac{W[y_0, y_1,\dots,y_m]}{I_W}\to \frac{W[z_0, z_1,\dots,z_n]}{J_W}=S_W.$$
By shrinking $W$, we may assume that $R_W\to S_W$ is still a finite flat extension and $\underline{y}$, $\underline{z}$ are still minimal reductions of ${\mathfrak{m}}$, ${\mathfrak{n}}$ respectively (here we still use ${\mathfrak{m}}$ and ${\mathfrak{n}}$ to denote the ideals ($y_0, y_1,\dots,y_m)$ and $(z_0, z_1,\dots,z_n)$ in $R_W$ and $S_W$ respectively). By generic freeness, we can shrink $W$ further to assume that all the kernels and cokernels of the Koszul complexes $K_\bullet(\underline{y}, R_W)$ and $K_\bullet(\underline{z}, S_W)$ are free $W$-modules, and the homologies are finite free $W$-modules. Therefore we have $$e((R_K)_{\mathfrak{m}})=\chi(\underline{y}, (R_K)_{\mathfrak{m}})=\chi(\underline{y}, R_K)=\sum(-1)^il_{R_K}(H_i(\underline{y}, R_K))=\sum (-1)^i\rank_WH_i(\underline{y}, R_W),$$ $$e((S_K)_{\mathfrak{n}})=\chi(\underline{z}, (S_K)_{\mathfrak{n}})=\chi(\underline{z}, S_K)=\sum(-1)^il_{S_K}(H_i(\underline{z}, S_K))=\sum (-1)^i\rank_WH_i(\underline{z}, S_W).$$
We pick a maximal ideal $Q$ of $W$ and note that $\kappa=W/Q$ is a field of characteristic $p>0$. Tensoring $R_W\to S_W$ with $\kappa$, we obtain a flat extension $R_\kappa\to S_\kappa$ (we can shrink $W$ to assume $S_W/R_W$ is free over $W$ and thus the map is an injection). We claim that $(R_\kappa)_{\mathfrak{m}}\to(S_{\kappa})_{\mathfrak{n}}$ is a counter-example to Theorem \[theorem–main technical theorem\] in characteristic $p>0$, which would contradict our Theorem \[theorem–main theorem in characteristic p>0\]. It is clear that $(R_\kappa)_{\mathfrak{m}}\to(S_{\kappa})_{\mathfrak{n}}$ is a flat local extension. Moreover, we know that $\underline{y}$ and $\underline{z}$ are minimal reductions of ${\mathfrak{m}}$ and ${\mathfrak{n}}$ in $(R_\kappa)_{\mathfrak{m}}$ and $(S_\kappa)_{\mathfrak{n}}$ respectively, because this is the case even in $R_W$ and $S_W$. Therefore, since the kernels, cokernels and homologies of the Koszul complexes $K_\bullet(\underline{y}, R_W)$ and $K_\bullet(\underline{z}, S_W)$ are all free $W$-modules, we have $$H_i(\underline{y}, R_W)\otimes_W\kappa\cong H_i(\underline{y}, R_\kappa) \text{ and } H_i(\underline{z}, S_W)\otimes_W\kappa=H_i(\underline{z}, S_\kappa)$$ for every $i$. Because we can also assume that the homologies of $K_\bullet(\underline{y}, R_W)$ and $K_\bullet(\underline{z}, S_W)$ are annihilated by a power of ${\mathfrak{m}}$ and ${\mathfrak{n}}$ respectively by shrinking $W$ further, $K_\bullet(\underline{y}, R_\kappa)$ and $K_\bullet(\underline{z}, S_\kappa)$ are supported only at ${\mathfrak{m}}$ and ${\mathfrak{n}}$ respectively. Therefore we have $$e((R_\kappa)_{\mathfrak{m}})=\chi(\underline{y}, (R_\kappa)_{\mathfrak{m}})=\chi(\underline{y}, R_\kappa)=\sum(-1)^il_{R_\kappa}(H_i(\underline{y}, R_\kappa))=\sum (-1)^i\rank_WH_i(\underline{y}, R_W),$$ $$e((S_\kappa)_{\mathfrak{n}})=\chi(\underline{z}, (S_\kappa)_{\mathfrak{n}})=\chi(\underline{z}, S_\kappa)=\sum(-1)^il_{S_\kappa}(H_i(\underline{z}, S_\kappa))=\sum (-1)^i\rank_WH_i(\underline{z}, S_W).$$ Thus $e((R_\kappa)_{\mathfrak{m}})=e((R_K)_{\mathfrak{m}})$ and $e((S_\kappa)_{\mathfrak{n}})=e((S_K)_{\mathfrak{n}})$.
Theorem \[theorem–main technical theorem\] is finally proved!
[^1]: The author is partially supported by NSF Grant DMS \#1600198, and NSF CAREER Grant DMS \#1252860/1501102.
[^2]: In equal characteristic, we can simply pick a coefficient field $L$ of $S$ and set $S'=S\widehat{\otimes}_L\overline{L}$.
[^3]: The definition given here is taken from Theorem 6.4 in [@KuranoNumericalequivalenceonChowgroupsoflocalrings], in general, one defines numerically Roberts ring in arbitrary characteristic by letting $\sum_{j=0}^{d-1}\operatorname{ch}_j(G_\bullet)[R]_j=0$ for every $G_\bullet$ bounded complex of finite length homology [@KuranoNumericalequivalenceonChowgroupsoflocalrings Definition 6.1].
[^4]: Hanes essentially proved this result under the additional (strong) hypothesis that $R$ admits a small maximal Cohen-Macaulay module and $R/{\mathfrak{m}}$ is perfect (and the result was only stated in dimension three).
[^5]: In fact we don’t have to assume $K$ is algebraically closed and $R/{\mathfrak{m}}=S/{\mathfrak{n}}=K$ in the process of reduction to characteristic $p>0$. However, assuming these will simply the argument.
|
---
author:
- 'A. Frejsel, M. Hansen and H. Liu'
title: Consistency tests for Planck and WMAP in the low multipole domain
---
Introduction
============
The Cosmic Microwave Background (CMB) is a key source of information on the early Universe. The newly released Planck data [@Planck1] as well as data from WMAP [@WMAP9] play important roles in this study. Whereas Planck greatly improves the current CMB measurement precision, it will still be very interesting and relevant to compare CMB products obtained with Planck and WMAP respectively, like the WMAP ILC maps and the Planck full sky maps.
The recent Planck CMB data release included the derived CMB products NILC, SMICA and SEVEM maps. The three maps are made via different techniques: the NILC map is a needlet variant of the Internal Linear Combination technique, SMICA is made via spectral parameter fitting in the harmonic domain, and SEVEM is constructed through template fitting using the lowest and highest frequency bands (see [@Planck12] for details). It is highly important to understand the differences and similarities of these three maps, when making cosmological conclusions.
Recently, the 9 year CMB data from WMAP has also been released, including the 9yr Internal Linear Combination map (ILC9) [@WMAP9]. The ILC9 is constructed through the combination of data at different frequencies with different weighting coefficients, in order to obtain a full sky map of the CMB with as little contamination (foregrounds, systematics and noise) as possible.
The aim of this paper is twofold. We wish to test the internal consistency and differences between the three released Planck maps (SMICA, SEVEM and NILC) and the differences between the WMAP ILC9 and 7 maps (7 year WMAP ILC data [@WMAP7]). Furthermore, we wish to test the external consistency between the WMAP ILC9 map and the Planck NILC map.
The basic assumption of this paper is that the difference between two maps (the ’difference map’) is composed of non-CMB contributions (such as noise, systematic errors, foregrounds etc.) that ought to be uncorrelated with the true, primordial CMB signal. If the maps do contain remnants of these contaminants they will be correlated with the difference map, the significance of which can be tested by comparing to simulations.
Testing the map of errors
=========================
Since we are faced with several different sky maps, all in principle depicting the CMB signal, we must take under consideration that they contain some element of contamination. Thus, the three Planck maps as well as the WMAP ILC maps are likely not perfect representations of the true primordial CMB, but contain an intrinsic CMB component, $c$, and a small non-cosmological component, $n$, which is due to noise, foregrounds, systematics etc. If we subtract two maps we will be left with a difference map, $d$, which is only composed of a difference in contaminants, $\Delta n$. In a spherical harmonics decomposition we can write this as $$\begin{aligned}
&&a^{map}_{lm}=c_{lm}+n_{lm} \nonumber \\
&&d_{lm}=a^{map1}_{lm}-a^{map2}_{lm}=\Delta n_{lm},
\label{eq:alm+noise}\end{aligned}$$ where $a^{map}_{lm}$ refers to a specific map (WMAP or Planck). If the map is clean of contaminants (i.e. $n_{lm}^{map1}=0$) we would not expect a significant correlation between $a^{map1}_{lm}$ and $d_{lm}$, because a pure CMB signal of course should not be correlated significantly with noise, systematics or foreground residuals. One can consider a cross correlation coefficient in multipole space $K$, where we compare a difference map, $\Delta n_{lm}$, and one of the maps it was created from ($a_{lm}$). When we substitute via Eq. \[eq:alm+noise\], we get the following $$\begin{aligned}
K(l)&=&\frac{\sum\limits_{m}(c_{lm}+n_{lm}^{a})\Delta n_{lm}^{*} + (c_{lm}+n_{lm}^{a})^{*}\Delta n_{lm}} {2(\sum\limits_{m}|c_{lm}+n_{lm}^{a}|^{2} \sum\limits_{m}|\Delta n_{lm}|^{2})^{1/2}} \nonumber \\
&=& \xi_{1} \left( \frac{\sum\limits_{m}|c_{lm}|^{2}}{\sum\limits_{m}|c_{lm}+n_{lm}^{a}|^{2}} \right)^{1/2} + \xi_{2} \left( \frac{\sum\limits_{m}|n_{lm}^{a}|^{2}}{\sum\limits_{m}|c_{lm}+n_{lm}^{a}|^{2}} \right)^{1/2}
\label{eq:cross_large}\end{aligned}$$ where a \* indicates a complex conjugate and $n_{lm}^{a}$ is the noise inherent in map $a$. $\xi_{1}$ is the cross correlation between the primordial CMB ($c_{lm}$) and the difference map ($\Delta n_{lm}$), and is expected to be at the order of $0$. $\xi_{2}$ is the cross correlation between the noise of map $a$ ($n_{lm}^{a}$) and the difference map, which is at the order of $1$ (we expect the noise to correlate strongly with the difference map). Term 1 in Eq. \[eq:cross\_large\] is thus on the order of $0$, while the first part of the second term ($\xi_{2}$) is of order $1$. Thus the interesting term in the equation, the last part of the second term, is essentially a noise to signal ratio and is the term one would effectively test via cross correlations. The case is the same if one performs the test in pixel space.
We estimate the significance of the correlations by comparing the correlation between the sky maps and their respective difference map with the correlation between randomly simulated CMB maps and the difference map. We therefore compare the cross correlations with 10000 random Gaussian simulations based on the $\Lambda$CDM theoretical power spectrum.
We calculate the cross correlation using the following definition of the cross correlation coefficient in pixel space: $$K_p = \frac{\sum\limits_{i} (x_i - \bar{X})(y_i - \bar{Y})}{\sqrt{(\sum\limits_{i} (x_i-\bar{X})^2)(\sum\limits_{i} (y_i-\bar{Y})^2)}},
\label{eq:crossp}$$ where $K_p$ is the total cross correlation coefficient, $x_i$ and $y_i$ are the values of pixel $i$ for the two maps respectively and $\bar{X}$ and $\bar{Y}$ are the mean pixel values for the two maps. We know that some residuals of the galactic plane are definitely still present in the maps. We therefore mask out the galaxy using the WMAP KQ85 9yr mask [@WMAP9], in order to see the effect of contaminants in the rest of the map. Thus, the sum in $i$ is only over unmasked pixels. Note that the sign of the correlation coefficient is not significant, since the choice of which map to subtract from the other is interchangeable. As such, a significant anti-correlation would change to correlation if the two maps were subtracted from each other in the opposite order.
Also, bear in mind that the power of the difference map is not very important for this investigation. What we are truly testing here is the morphology of the maps. We are comparing the same spot in the two maps (pixel $i$), and the estimator $K_p$ is normalized. Thus we are effectively looking at normalized pixel values between $-1$ and $1$, and the estimator $K_p$ shows the similarity of the morphology of the two maps.
![Left: Planck NILC map. Middle: Planck SMICA map. Right: Planck SEVEM map.[]{data-label="fig:planckmaps"}](map_nilc_lmax100.eps "fig:"){width="33.00000%"} ![Left: Planck NILC map. Middle: Planck SMICA map. Right: Planck SEVEM map.[]{data-label="fig:planckmaps"}](map_planck_smica_lmax100.eps "fig:"){width="33.00000%"} ![Left: Planck NILC map. Middle: Planck SMICA map. Right: Planck SEVEM map.[]{data-label="fig:planckmaps"}](map_planck_sevem_lmax100.eps "fig:"){width="33.00000%"}
Consistency tests
=================
Consistency tests of Planck NILC, SMICA and SEVEM maps
------------------------------------------------------
As mentioned in the introduction, we now perform consistency tests of the three Planck CMB maps: SMICA, NILC and SEVEM (shown in figure \[fig:planckmaps\]).
First, we create the difference map by subtracting the various Planck maps from each other in pixel space. We take three representative combinations of difference maps: NILC minus SMICA (denoted NILC-SMICA), NILC minus SEVEM (NILC-SEVEM) and SMICA minus SEVEM (SMICA-SEVEM). The three difference maps are created with $l_{max} = 100$, and a resolution corresponding to $N_{side} = 128$. In figure \[fig:planckdiffmaps\] both the unmasked difference maps and the same maps with the KQ85 9yr mask applied are shown for illustrative purposes. Similar maps can be found in [@Planck12], albeit with a different color scheme and other temperature bounds.
![The difference maps, with the masked map to the right (WMAP KQ85 9yr mask). Top: NILC-SMICA difference map. Middle: NILC-SEVEM difference map. Bottom: SMICA-SEVEM difference map.[]{data-label="fig:planckdiffmaps"}](PLANCK_nilc-smica_lmax100_3deg_smooth_ESTEC.eps "fig:"){width="47.00000%"} ![The difference maps, with the masked map to the right (WMAP KQ85 9yr mask). Top: NILC-SMICA difference map. Middle: NILC-SEVEM difference map. Bottom: SMICA-SEVEM difference map.[]{data-label="fig:planckdiffmaps"}](PLANCK_nilc-smica_lmax100_3deg_smooth_ESTEC_MASKED.eps "fig:"){width="47.00000%"}
![The difference maps, with the masked map to the right (WMAP KQ85 9yr mask). Top: NILC-SMICA difference map. Middle: NILC-SEVEM difference map. Bottom: SMICA-SEVEM difference map.[]{data-label="fig:planckdiffmaps"}](PLANCK_nilc-sevem_lmax100_3deg_smooth_ESTEC.eps "fig:"){width="47.00000%"} ![The difference maps, with the masked map to the right (WMAP KQ85 9yr mask). Top: NILC-SMICA difference map. Middle: NILC-SEVEM difference map. Bottom: SMICA-SEVEM difference map.[]{data-label="fig:planckdiffmaps"}](PLANCK_nilc-sevem_lmax100_3deg_smooth_ESTEC_MASKED.eps "fig:"){width="47.00000%"}
![The difference maps, with the masked map to the right (WMAP KQ85 9yr mask). Top: NILC-SMICA difference map. Middle: NILC-SEVEM difference map. Bottom: SMICA-SEVEM difference map.[]{data-label="fig:planckdiffmaps"}](PLANCK_smica-sevem_lmax100_3deg_smooth_ESTEC.eps "fig:"){width="47.00000%"} ![The difference maps, with the masked map to the right (WMAP KQ85 9yr mask). Top: NILC-SMICA difference map. Middle: NILC-SEVEM difference map. Bottom: SMICA-SEVEM difference map.[]{data-label="fig:planckdiffmaps"}](PLANCK_smica-sevem_lmax100_3deg_smooth_ESTEC_MASKED.eps "fig:"){width="47.00000%"}
Following to Eq. (\[eq:crossp\]) we now calculate $K_p$, where we cross correlate each input map with the difference map. We compare this to the result for 10000 random simulations, in order to asses the significance. The results are presented in figure \[fig:hist\_pix\_crosscorr\]. We see that the Planck maps only correlate weakly with their respective difference maps, and are well consistent with simulations. We attribute this to the high similarity between the Planck maps (see figure \[fig:planckdiffmaps\]) outside the Galactic mask, making the numerator in Eq. (\[eq:crossp\]) small. In conclusion, the three Planck maps are very consistent with each other. The numerical values are shown in table \[tab:planck1\].
![Cross correlations coefficients compared to 10000 simulated maps. Top left is for the NILC-SMICA, top right is for NILC-SEVEM and bottom is for SMICA-SEVEM. Red line: NILC, green line: SMICA, light blue line: SEVEM. All with the respective difference map.[]{data-label="fig:hist_pix_crosscorr"}](PLANCK_NILC-SMICA_with_10000_sims_ESTEC.eps "fig:"){width="35.00000%"} ![Cross correlations coefficients compared to 10000 simulated maps. Top left is for the NILC-SMICA, top right is for NILC-SEVEM and bottom is for SMICA-SEVEM. Red line: NILC, green line: SMICA, light blue line: SEVEM. All with the respective difference map.[]{data-label="fig:hist_pix_crosscorr"}](PLANCK_NILC-SEVEM_with_10000_sims_ESTEC.eps "fig:"){width="35.00000%"} ![Cross correlations coefficients compared to 10000 simulated maps. Top left is for the NILC-SMICA, top right is for NILC-SEVEM and bottom is for SMICA-SEVEM. Red line: NILC, green line: SMICA, light blue line: SEVEM. All with the respective difference map.[]{data-label="fig:hist_pix_crosscorr"}](PLANCK_SMICA-SEVEM_with_10000_sims.eps "fig:"){width="35.00000%"}
-- ----------- ------------ ---------- ------------ ---------- ------------
$K_p$ Percentage $K_p$ Percentage $K_p$ Percentage
0.02294 23.4% 0.01485 36.0%
-0.007667 41.0% 0.01518 35.4%
-0.03536 20.6% -0.04156 16.7%
-- ----------- ------------ ---------- ------------ ---------- ------------
: Numerical values of the cross correlations for figure \[fig:hist\_pix\_crosscorr\]. For each difference map, the table shows the value of $K_p$ for a map cross correlated with the respective difference map, and the percentage of the 10000 simulations that have a higher (or for $K_p<0$: lower) value of $K_p$.
\[tab:planck1\]
![Top left: The difference map, computed as ILC9-ILC7. Top right: the masked difference map (WMAP KQ85 9yr mask). Bottom: Dipole ($l=1$) of the difference map ILC9-7.[]{data-label="fig:diffmap"}](WMAP_ilc9-7_lmax100_3deg_smooth_ESTEC.eps "fig:"){width="47.00000%"} ![Top left: The difference map, computed as ILC9-ILC7. Top right: the masked difference map (WMAP KQ85 9yr mask). Bottom: Dipole ($l=1$) of the difference map ILC9-7.[]{data-label="fig:diffmap"}](WMAP_ilc9-7_lmax100_3deg_smooth_ESTEC_MASKED.eps "fig:"){width="47.00000%"}
![Top left: The difference map, computed as ILC9-ILC7. Top right: the masked difference map (WMAP KQ85 9yr mask). Bottom: Dipole ($l=1$) of the difference map ILC9-7.[]{data-label="fig:diffmap"}](map_ilc_9minus7_dipole.eps){width="47.00000%"}
Consistency tests of WMAP ILC9 and ILC7 maps
--------------------------------------------
The weights in the construction of the ILC9 have been improved from the ILC7 through refinement of the pixel noise, calibrations and beam profiles (see [@WMAP9] for details). The ILC9 map is thus expected to be superior to the ILC7 through 2 additional years of data taking as well as optimization of the method of construction.
Similar to the test for the individual Planck maps, we therefore now turn our attention to the ILC9 in comparison with the ILC7. The ILC9-7 difference map is shown in figure \[fig:diffmap\]. The galactic plane is clearly visible in the ILC9-7 difference map, as are selected point sources. Therefore we mask it with the KQ85 9yr mask, as we did in the case for the Planck maps (figure \[fig:diffmap\], top right). It is immediately clear that the map is dominated by a dipole (see bottom figure), which is closely aligned with the well known kinematic dipole. The same feature is present in the difference map between the 7 year and 5 year ILC map, as discovered in [@WMAP7-1].
We compute the cross correlation between the ILC9 and ILC7 year maps with the ILC9-7 difference map. The procedure is the same as in the previous section. The results are presented in figure \[fig:ilc9-7\_corr\], and the numerical values for the cross correlation are presented in table \[tab:wmap\]. We see that ILC9 cross correlates much stronger with the difference map than ILC7, and that the ILC7 is in agreement with the distribution of the 10000 random simulations, while the ILC9 is not.
Consistency tests of Planck NILC and WMAP ILC9 maps
---------------------------------------------------
Finally, we turn to an investigation of the difference map between the Planck NILC map and the WMAP ILC9 map. We select the Planck NILC map for the comparison, since the NILC method is similar in nature to the ILC method (see for instance [@Planck12]). Since the WMAP ILC9 map has been smoothed at the level of $1^\circ$, we start by smoothing the NILC at the same level. Then the difference map is calculated similar to the procedure in the previous sections, and the result is shown in figure \[fig:diffmap\_planck\_wmap\]. We have enhanced the min and max temperature compared to previous difference maps in the paper, in order to clearly show local features. In the difference map we clearly see the galactic plane, and some feature in the lower left quadrant of the map. This is similar to the dipole clearly seen in figure \[fig:diffmap\], but since we are now subtracting the ILC9 map, the sign of the dipole is changed. Is is evident that the ILC9 contains an enhanced dipole, both in comparison with the WMAP ILC7 and with the Planck NILC map.
![Left: the difference map, computed as NILC-ILC. Right: the masked difference map (WMAP KQ85 9yr mask). []{data-label="fig:diffmap_planck_wmap"}](diffmap_nilcilc9_1deg_nx201.eps "fig:"){width="50.00000%"} ![Left: the difference map, computed as NILC-ILC. Right: the masked difference map (WMAP KQ85 9yr mask). []{data-label="fig:diffmap_planck_wmap"}](diffmap_nilcilc9_1deg_nx201_MASK.eps "fig:"){width="50.00000%"}
In figure \[fig:hist\_pix\_nilc\_ilc9\] we show the result of the cross correlations for the difference map and the NILC (smoothed to $1^\circ$) and ILC9 map respectively, in comparison to 10000 simulated maps. Numerical values are presented in table \[tab:Planck-WMAP\]. The resolution, mask and maximum $l$-value is similar to the previous tests. We see a strong (anti-)correlation for the ILC9 cross correlated with NILC-ILC9, in comparison with the simulations. The cross correlation case for the NILC is in reasonable accordance with the simulations. In comparison with the case for ILC9-7 in the previous section this indicates that the ILC9 map includes a proportion of non-cosmological features.
Summary and conclusions
=======================
In this paper we have investigated the consistency of the Planck NILC, SMICA and SEVEM maps as well as the consistency of the WMAP ILC9 and ILC7 maps, and a cross test between NILC and ILC9. The basic assumption was that the difference between two maps–the difference map–consists only of a non-cosmological signal, which a pure CMB map should not correlate strongly with. We have tested this through comparing pairs of maps to their respective difference map, and the significance of the correlation was determined through comparison with 10000 random simulations of the CMB.
For the consistency test of the derived Planck products, we cross correlated the three Planck maps with the respective difference maps NILC-SMICA, NILC-SEVEM and SMICA-SEVEM at $N_{side}=128$ (all masked with the KQ85 9 year mask). We found that the Planck maps are very well consistent with each other, as well as consistent with random simulations.
Additionally, we find that the ILC9-ILC7 difference map cross correlates significantly with the ILC9 map, whereas the ILC7 cross correlation is consistent with the random simulations (both at $N_{side}=128$). This shows that the ILC9 map matches the morphology of the non-cosmological contributions in the difference map, and the ILC7 map does not significantly.
Finally, we did an external consistency test, comparing the Planck NILC map with the ILC9 map. The difference map between NILC and ILC9 showed a dipole, also visible for the ILC9-7 case, indicating that this is probably an artifact of the ILC9 map. Also, the NILC-ILC9 difference map cross correlates strongly with the ILC9 and much less so with the NILC map.
In conclusion, the Planck NILC, SEVEM and SMICA maps are in very good agreement with each other, and none of them show a significant correlation with their respective difference maps outside the mask. On the other hand, the ILC9 map correlates significantly with the difference map, both in comparison to ILC7, in comparison to Planck NILC and in comparison to random simulations. Thus ILC9 appears to be more contaminated than the ILC7. This should be taken into consideration when using WMAP maps for cosmological analyses.
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful for discussions on this work with P. Naselsky, K. Górski, C. Burigana, J. Tauber, C. Lawrence, M. Bersanelli, P. Natoli and A. Bandy.
This work is based on observations obtained with Planck ([<http://www.esa.int/Planck>]{}), an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada. The development of Planck has been supported by: ESA; CNES and CNRS / INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN and JA (Spain); Tekes, AoF and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); and PRACE (EU). A description of the Planck Collaboration and a list of its members, including the technical or scientific activities in which they have been involved, can be found at the Planck web page[^1].
We acknowledge the use of the NASA Legacy Archive for Microwave Background Data Analysis (LAMBDA). Our data analysis made use of the GLESP package [@Glesp][^2], and of HEALPix [@Healpix_primer]. This work is supported in part by Danmarks Grundforskningsfond which allowed the establishment of the Danish Discovery Center, FNU grants 272-06-0417, 272-07-0528 and 21-04-0355, the National Natural Science Foundation of China (Grant No. 11033003), the National Natural Science Foundation for Young Scientists of China (Grant No. 11203024) and the Youth Innovation Promotion Association, CAS.
[90]{}
Planck Collaboration, 2013
C. L. Bennett, et al., 2012
Planck Collaboration, 2013
Jarosik, N., Bennett, C. L., Dunkley, J., et al. 2011, , 192, 14
Hinshaw, G., Weiland, J. L., Hill, R. S., et al. 2009, , 180, 225
B. Gold, et al., 2011, , 192, 15
A. G. Doroshkevich et al., 2005, 14, 275-290
K. M. Gorski et al. 2005, , 622, 759–771
[^1]: <http://www.sciops.esa.int/index.php?project=planck&page=Planck_Collaboration>
[^2]: <http://www.glesp.nbi.dk/>
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abstract: 'We consider quantum $N=2$ string embedded into the $N=4$ topological framework from the perspective of the old covariant quantisation. Making use of the causality and cyclic symmetry of tree amplitudes we argue that no Lorentz covariant boson emission vertex can be constructed within the $N=4$ topological formalism.'
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c i ł ø u ¶ Ł Ø §
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[**Can one restore Lorentz invariance** ]{}\
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[**in quantum N=2 string?** ]{}\
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[Stefano Bellucci]{} [^1] and [Anton Galajinsky]{} [^2]
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[*INFN–Laboratori Nazionali di Frascati, C.P. 13, 00044 Frascati, Italy*]{} 0.4cm
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PACS: 04.60.Ds; 11.30.Pb\
Keywords: quantum $N{=}2$ string, $N=4$ topological string, Lorentz symmetry
[**1. Introduction**]{}\
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Critical $N=2$ string provides a conventional framework for describing self–dual gauge theory or self–dual gravity in two spatial and two temporal dimensions. Apart from a number of salient features characterising the model like the absence of massive excitations in the spectrum of physical states, continuous family of sectors interpolating between $R$ and $NS$ and connected by spectral flow, the vanishing of scattering amplitudes with more than three external legs to all orders in perturbation theory, there is a fundamental drawback intrinsic to the model. It lacks manifest Lorentz covariance. Technically, in order to construct the $U(1)$ current entering the $N=2$ superconformal algebra (SCA) out of the fermionic fields at hand one has to introduce a complex structure in the target which breaks the full Lorentz group $SO(2,2)$ down to the $U(1,1)$ subgroup.
A way out of the problem has been observed by Siegel [@ws] and elaborated in much more detail by Berkovits and Vafa [@berkvafa] (see also the related works [@ov3; @bvw]). The idea is that one can embed the $N=2$ string in a larger $N=4$ topological framework by adding to the theory two more fermionic currents and two more bosonic ones which extend the $N=2$ SCA to a small $N=4$ SCA. Classically the new constraints prove to be functionally dependent on those forming the $N=2$ SCA [@ws; @bg]. The crucial observation, however, is that extending the $N=2$ algebra to a small $N=4$ algebra one raises also the group of external global automorphisms ($U(1,1)$ in the case on the $N=2$ string) to the full Lorentz group. In other words, within the $N=4$ topological framework one reveals a tempting possibility to restore the Lorentz invariance missing in the $N=2$ formalism. Curiously enough, this resembles the Green–Schwarz superstring, for which the extracting of a functionally independent set of fermionic first class constraints is known to break the manifest Lorentz covariance.
A classical action for the $N=4$ topological extension of the $N=2$ string has been constructed recently in Refs. [@bg; @bg1]. In order to provide the higher global symmetry, on the world–sheet of the string some extra fields are to be introduced, these complementing the $d=2$, $N=2$ conformal supergravity multiplet to $d=2$, $N=4$ one. The global limit of the four local supersymmetry transformations corresponds to a twisted version of the $N=4$ supersymmetry algebra [@bdg]. Interestingly enough, being classically equivalent on a flat background, the theories lead to different geometries when put in a curved space. In contrast to the Kähler geometry characterising the $N=2$ case, for the $N=4$ model a manifold has to admit a covariantly constant holomorphic two–form in order to support the $N=4$ twisted supersymmetry [@bdg]. This restricts the holonomy group to be a subgroup of $SU(1,1)$ and leads to a Ricci–flat manifold already at the classical level.
Turning to the quantum description for the $N=4$ model, the reducibility of the constraints causes a serious complication of the BRST procedure [^3] and a rigorous BRST quantisation of the $N=4$ topological string is unknown. To bypass the problem, in Ref. [@berkvafa] (see also [@ov3; @bvw]) a set of reasonable prescriptions to calculate scattering amplitudes has been proposed and shown to reproduce the results known for the $N=2$ string including an elegant proof of the vanishing theorems. Notice, however, that the prescription essentially relies upon a specific topological twist which does not treat all the currents on equal footing and breaks the Lorentz group down to the $U(1,1)$ subgroup.
The purpose of this paper is to reconsider the issue of the Lorentz invariance in the quantum $N=4$ topological string from the perspective of the old covariant quantisation. Avoiding problematic BRST analysis this still maintains one within a conventional framework. In the next section we briefly discuss the extension of the $N=2$ SCA to a small $N=4$ SCA and give an explicit form of the Lorentz transformations arising within the $N=4$ topological framework. Sect. 3 contains the discussion of vertex operators. In particular, we show that, although the vertex operators describing asymptotic physical states do respect the full Lorentz group, the causality and cyclic symmetry of tree amplitudes prevent one from constructing a Lorentz invariant boson emission vertex. Thus, the conclusion we draw is that, although giving an efficient key to the restoration of the Lorentz invariance at the classical level, the $N=4$ formalism fails to do so at the quantum level.
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[**2. N=2 SCA, small N=4 SCA and Lorentz symmetry**]{}\
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Let us consider a $c=6$ CFT corresponding to the critical closed $N=2$ string. The matter sector involves two complex bosons $x^a(z,\bar z)$, $a=0,1$, and four complex fermions $\p^a (z)$, $\vf^a (\bar z)$, the latter belonging to the right and left movers, respectively. The fields are assigned with the standard propagators[^4] && x\^a(z,|z) [|x]{}\^[|a]{}(z’,|z’) = \^[|a]{}(z,|z) x\^a(z’,|z’) = -\^[|a a]{} ((z-z’)+(|z -|z’)),\
&& \^a(z) [|]{}\^[|a]{}(z’) = \^[|a]{}(z) \^a(z’) = -,\
&& \^a(|z) [|]{}\^[|a]{}(|z’) = \^[|a]{}(|z) \^a(|z’) = -, where $\eta^{\bar a a}=\mbox{diag}~(-,+)$ stands for the Minkowski metric in the target.
Given the matter fields, one can readily construct the $N=2$ superconformal currents (in what follows we discuss only the right movers and use the abbriviation $\vf \bar\chi=\vf^a \eta_{a\bar a}
{\bar\chi}^{\bar a}$) && T(z)=-x |x + (|+|),\
&& G(z)=x |, |G (z)=|x ,\
&& J(z)=|. The corresponding OPE’s are well known && T(z) T(z’)\~ + +,\
&& T(z) G(z’)\~ +,\
&& T(z) |G (z’)\~ +,\
&& T(z) J(z’)\~ +,\
&& G(z) |G (z’)\~ - - +,\
&& G(z) J(z’)\~, |G(z) J(z’)\~-,\
&& J(z) J(z’)\~, which also imply that the fermionic currents $G$ and $\bar G$ carry confromal spin $3/2$ while the bosonic $U(1)$ current $J$ has conformal spin $1$. Notice that with respect to the latter the generators $G(z)$ and $\bar G(z)$ are charged with the charges $-1$ and $+1$, respectively.
A striking point about the $N=2$ algebra is that it admits a continuous automorphism [@schwimmer] with a local parameter $\a(z)$ \[sflow\] && T’=T-iJ+[(i)]{}\^2, J’=J-2i,\
&& G’=e\^[i]{}G,|G’=e\^[-i]{} |G, which relates R and NS sectors (spectral flow) and allows one to stick with a preferred representation. For the rest of the paper we choose to work in the NS representation. Since, by the very construction, each current in the $N=2$ SCA holds invariant under the action of the $U(1,1)$ group, the latter provides a global automorphism of the $N=2$ algebra which also coincides with the global symmetry group intrinsic to the $N=2$ string.
With a closer inspection one can further discover that some extra currents can be constructed out of the matter fields at hand [@ws; @berkvafa] \[top1\] && H(z)=, |H(z)=,\
&& J\^[(1)]{}(z)=, J\^[(2)]{}=, where we denoted $\underline{\vf\p}=\vf^a \e_{ab} \p^b$, $\underline{{\bar\vf}{\bar\p}}={\bar\vf}^{\bar a} \e_{\bar a \bar b}
{\bar\p}^{\bar b}$ and $\e_{ab}={(\e_{\bar a \bar b})}^{*}$, $\e_{01}=-1$, is the Levi-Civita totally antisymmetric tensor. These extend the $N=2$ SCA to a small $N=4$ SCA && T(z) H(z’)\~ +,\
&& T(z) |H (z’)\~ +,\
&& T(z) J\^[(1,2)]{}(z’)\~ +,\
&& G(z) |H(z’)\~-- ,\
&& |G(z) H(z’)\~-- ,\
&& G(z) J\^[(1)]{}(z’)\~-, |G(z) J\^[(2)]{}(z’)=-,\
&& J(z) H(z’)\~, J(z) |H(z’)\~-,\
&& J(z) J\^[(1)]{}(z’)\~, J(z) J\^[(2)]{}(z’)\~-,\
&& H(z) |H (z’)\~- - - -,\
&& H(z) J\^[(2)]{}(z’)\~-, |H(z) J\^[(1)]{}(z’)\~-,\
&& J\^[(1)]{}(z) J\^[(2)]{}(z’)\~+. In checking the OPE’s the identities \[idd\] \_[ab]{}\^[|b b]{}\_[|b |a]{}=\_[a|a]{}, \^[|a a]{} \^[|b b]{} \_[|a |b]{}=\^[ab]{}, prove to be helpful.
Thus, altogether there are four fermionic currents of conformal spin $3/2$. The bosonic triplet $J,J^{(1)},J^{(2)}$ (spin 1) forms an $su(1,1)$ Kac–Moody subalgebra. It is noteworthy that viewed as constraints at the classical level the extra currents prove to be functionally dependent on those forming the $N=2$ algebra [@ws; @bg]. A remarkable fact, however, is that extending the algebra that way, one raises the global automorphism group to the full Lorentz group $SO(2,2)$ and restores the Lorentz invariance in the classical $N=2$ string [@bg; @bg1].
Let us dwell on the issue for the case of the quantum $N=4$ algebra. First of all, it is straightforward to verify that the continuous automorphism (\[sflow\]) remains to hold in the extended version, provided the new currents transform in accord with && H’=e\^[-i]{}H,|H’=e\^[i]{} |H, J\^[(1)’]{}=e\^[-2i]{} J\^[(1)]{}, J\^[(2)’]{}=e\^[2i]{} J\^[(2)]{}. Hence, one can continue to work in a preferred (NS) representation. Turning to global automorphisms, as transformations from the conventional $SU(1,1)$ group leave each current of the $N=4$ algebra invariant, these provide an apparent automorphism. A less obvious point is that within the extended framework the $U(1)$ automorphism one had in the $N=2$ case is accompanied by two extra transformations, altogether forming another ${SU(1,1)}_{outer}$ group (we stick with the terminology of Ref. [@bvw]). For the elementary field combinations $\psi{\bar\vf}$, $\underline{\psi \vf}$ $\underline{{\bar\psi}{\bar\vf}}$ these read (independently of the statistics of the fields involved) $$
[lll]{}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
${SU(1,1)}_{outer}$ $\d_{\o}$ $\d_{\b}$ $\d_{\l}$
--------------------------------- ------------------------------------------------------- --------------------------------------- -------------------------------------------------------- -- -- -- -- -- -- -- -- --
$\p {\bar\vf}$ $\o(\underline{\p\vf}-\underline{{\bar\p}{\bar\vf}})$ 0 $i\l(\underline{\p\vf}+\underline{{\bar\p}{\bar\vf}})$
$\underline{\p\vf}$ $\o(\p{\bar\vf}-{\bar\p}\vf)$ $-2i\b $-i\l(\p\bar\vf -{\bar\p}\vf)$
(\underline{\p\vf})$
$\underline{{\bar\p}{\bar\vf}}$ $-\o(\p{\bar\vf}-{\bar\p}\vf)$ 2i$\b(\underline{{\bar\p}{\bar\vf}})$ $-i\l(\p\bar\vf -{\bar\p}\vf)$
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$$ where $\o$, $\b$ and $\l$ are real infinitesimal constant parameters. We have put the $U(1)$ automorphism mentioned above in the second column. Being applied to the $N=4$ superconformal currents the ${SU(1,1)}_{outer}$ transformations ammount to (we give them in an infinitesimal form) \[automor1\] && T’=T, G’=G-ø|H +øH, |G’=|G+ø|H -øH,\
&& H’=H-ø|G +øG, |H’=|H+ø|G -øG, J’=J+øJ\^[(2)]{} -øJ\^[(1)]{},\
&& J\^[(1)’]{}=J\^[(1)]{}-2øJ, J\^[(2)’]{}=J\^[(2)]{}+2øJ, && T’=T, G’=G, |G’=|G, J’=J, H’=H-2iH,\
&& |H’=|H +2i|H, J\^[(1)’]{}=J\^[(1)]{}-2iJ\^[(1)]{}, J\^[(2)’]{}=J\^[(2)]{}+2iJ\^[(2)]{}, and \[automor3\] && T’=T, G’=G+ił|H +iłH, |G’=|G-ił|H -iłH,\
&& H’=H+ił|G -iłG, |H’=|H+ił|G -iłG, J’=J-iłJ\^[(2)]{} -iłJ\^[(1)]{},\
&& J\^[(1)’]{}=J\^[(1)]{}+2iłJ, J\^[(2)’]{}=J\^[(2)]{}+2iłJ. It is straightforward, although a bit tedious, to verify that Eqs. (\[automor1\])–(\[automor3\]) do provide an automorphism of the $N=4$ SCA.
Since in two spatial and two temporal dimensions the Lorentz group factorizes as $SO(2,2)=SU(1,1)\times SU(1,1)'$ one reveals a tempting possibility to restore the full Lorentz symmetry for the $N=2$ string treated in the enlarged $N=4$ topological formalism. At the classical level this has been achieved [@bg; @bg1] by means of enlarging the world sheet $d=2$, $N=2$ conformal supergravity multiplet to the $d=2$, $N=4$ conformal supergravity multiplet. The extra fields transform nontrivially under ${SU(1,1)}_{outer}$ and render the full action $SO(2,2)$ invariant. At the same time, as classically the constraints $H=0$, $\bar H=0$, $J^{(1)}=0$, $J^{(2)}=0$ prove to be functionally dependent on those forming the $N=2$ SCA [@ws; @bg] one still has the classical equivalence between the two formalisms. In the next section we turn to discuss how far one can get in restoring the Lorentz invariance at the quantum level.
[**3. Vertex operators**]{}
Our main concern in this section will be the structure of tree amplitudes. In particular, we shall study the constraints imposed on physical vertices by causality and cyclic symmetry of the amplitudes. To this end we decompose in modes (as we mentioned above one can choose the NS representation due to the spectral flow; in the relations below $n$ is an integer and $r$ is a half integer) \[n2modes\] && L\_n= dz z\^[n+1]{} :T(z):, G\_r= dz z\^[r+1/2]{} :G(z):,\
&& [|G]{}\_r= dz z\^[r+1/2]{} :|G(z):, J\_n= dz z\^n :J(z):,\
&& H\_r= dz z\^[r+1/2]{} :H(z):, \_r= dz z\^[r+1/2]{} :|H(z):,\
&& J\^[(1)]{}\_n= dz z\^n :J\^[(1)]{}(z):, J\^[(2)]{}\_n= dz z\^n :J\^[(2)]{}(z):, check the unitarity of the representation \^[+]{}=L\_[-n]{}, \^[+]{}=J\_[-n]{}, \^[+]{}=[|G]{}\_[-r]{}, \^[+]{}=[|H]{}\_[-r]{}, \^[+]{}=-J\^[(2)]{}\_[-n]{}, and work out the full algebra \[n4alg\] && \[ L\_n, L\_m \]=(n-m) L\_[n+m]{} + (n+1)n(n-1) \_[n+m,0]{},\
&& \[ L\_n, G\_r \]=( n -r) G\_[n+r]{}, =( n-r) [|G]{}\_[n+r]{},\
&& \[ L\_n, J\_m \]=-m J\_[n+m]{}, =G\_[n+r]{},\
&& { G\_r,[|G]{}\_q } =L\_[r+q]{}-(r-q) J\_[r+q]{}+ (r+)(r-) \_[r+q,0]{},\
&& \[ [|G]{}\_r, J\_n \]=-[|G]{}\_[n+r]{}, =2n\_[n+m,0]{},\
&& \[L\_n, H\_r\]=( n -r) H\_[n+r]{}, \[L\_n, [|H]{}\_r\]=( n -r) [|H]{}\_[n+r]{},\
&& \[ L\_n, J\^[(1)]{}\_m \]=-m J\^[(1)]{}\_[n+m]{}, =-m J\^[(2)]{}\_[n+m]{},\
&& { G\_r, [|H]{}\_q }=(q -r) J\^[(2)]{}\_[r+q]{}, { [|G]{}\_r, H\_q }=(q -r) J\^[(1)]{}\_[r+q]{},\
&& \[G\_r, J\^[(1)]{}\_n\]=-2H\_[r+n]{}, =-2[|H]{}\_[r+n]{},\
&& \[J\_n, H\_r\]=H\_[r+n]{}, =-[|H]{}\_[r+n]{},\
&& \[J\_n, J\^[(1)]{}\_m\]=2J\^[(1)]{}\_[n+m]{}, =-2J\^[(2)]{}\_[n+m]{},\
&& {H\_r,[|H]{}\_q } =-L\_[r+q]{}+(q -r) J\_[r+q]{} -(r+)(r-) \_[r+q,0]{},\
&& \[H\_r, J\^[(2)]{}\_n\]=-2G\_[r+n]{}, =-2[|G]{}\_[r+n]{},\
&& \[J\^[(1)]{}\_n,J\^[(2)]{}\_m\]=4J\_[n+m]{}+4n\_[n+m,0]{}, with other (anti) commutators vanishing. With respect to the $N=2$ algebra physical states are defined to obey the standard relations \[pstates\] && L\_n |=G\_r |= [|G]{}\_r |=J\_n |=0 n,r>0\
&& L\_0 |=J\_0 |=0, where the absence of the normal ordering constants in the last two relations follows from a rigorous BRST analysis[^5] [@bilal]. A simple inspection of the full $N=4$ algebra shows then that the modes $H_r$, ${\bar H}_r$, $J^{(1)}_n$, $J^{(2)}_n$ for $r,n>0$ automatically annihilate the physical states (\[pstates\]) if so do the zero modes of $J^{(1)}$ and $J^{(2)}$. Thus the only extra conditions coming with the new generators are \[pstate1\] J\^[(1)]{}\_0 |=0, J\^[(2)]{}\_0 |=0. Notice that this seems to be more restrictive then the situation at the classical level, where the constraints $H=0$, $\bar H=0$, $J^{(1)}=0$, $J^{(2)}=0$ prove to be functionally dependent and do not imply any new information as compared to that encoded into the $N=2$ superconformal currents. The analysis of the relations (\[pstates\]) is well known [@mukhi] which has also been confirmed by the calculation of the one–loop partition function [@mukhi; @ov]. The spectrum of physical states consists of a single ground state, this being a massless scalar. As the operators $J^{(1)}_0$ and $J^{(2)}_0$ give zero when acting on that state, the $N=2$ and $N=4$ spectra coincide.
We now turn to discuss vertex operators. As the critical intercept equals zero ($L_0=0$ on physical states), a vertex operator valid for describing an asymptotic physical state must carry conformal spin zero. On account of Eqs. (\[pstates\]),(\[pstate1\]) the natural guess is V\_0(k,0)=:e\^[ik [|x]{} +i[|k]{} x]{}: , k|k=0. Worth noting is that the vertex holds invariant with respect to both the conventional $SU(1,1)$ and ${SU(1,1)}_{outer}$ (see the first line in the table above), thus exhibiting [*the full*]{} Lorentz invariance. To construct a vertex operator $V(k,\bar k, z,\bar z)$ which would be capable of describing the emission of the bosonic state it is instructive to analyse the causality of the corresponding amplitudes. This implies, in particular, that spurious physical states decouple from physical processes \[cause\] |V(k,1)| =0, where for simplicity we restricted ourselves to the three point function which proves to be sufficient for our purposes. As is well known, the decoupling of spurious physical states of the form $\sum_{n>0} \langle \chi_n |~L_n$ with \[spur\] \_[n>0]{} \_n | L\_n L\_0=0 \_n | (n+L\_0)=0, requires $V$ to carry conformal spin 1 \[physvertex1\] T(z) V(z’)\~+. Actually, from Eqs. (\[physvertex1\]) and (\[n2modes\]) one finds (L\_n-n-L\_0)V(k,1)=V(k,1)(L\_n-L\_0), which when combined with Eq. (\[spur\]) renders the amplitude (\[cause\]) vanishing. Given the matter fields, the most general ansatz for the boson emission vertex is \[verans\] && V(k,z,|z)=:{|k x +k +ł(|k )(k |) + + +(|k )()\
&& +ø() (k|)}e\^[ik [|x]{} +i[|k]{} x]{}: , k|k=0, with $\a,\b,\l,\m,\g,\r,\o$ some complex constants to be determined below (here $\a,\b,\l,\o$ not to be confused with the infinitesimal parameters involved in the global ${SU(1,1)}_{outer}$ automorphism we discussed in the previous section). The term like $(\underline{k\p})
(\underline{\bar k {\bar\p}})$ one could try to include into the ansatz above proves to be redundant due to the identity \[idd1\] \_[cp]{}\_[|s |k]{}=-\_[c|s]{} \_[p|k]{} + \_[c|k]{} \_[p |s]{}. At this stage we refrain from requiring the vertex to respect the full Lorentz symmetry, while by the very construction the ansatz does respect the conventional $SU(1,1)$.
Consider further the spurious physical states generated by the fermionic currents ${\sum}_{r>0} \langle \chi_r |~G_r$ with \[spur1\] \_[r>0]{} \_r | G\_r L\_0=0 \_r | (r+L\_0)=0. Taking into account the explicit form of $G(z)$ and the fact that $V$ has conformal spin 1, one infers that the corresponding OPE can involve poles no higher than of the second order G(z) V(z’)\~+, with $U_1(z)$ and $U_2(z)$ to be determined below. The amplitude (\[cause\]) then acquires the form \[spur3\] \_[r>0]{} \_r |U\_1(1)(r+)+U\_2(1)| . Assuming further that $U_1$ carries conformal spin $h$ \[spinU\_1\] T(z) U\_1(z’)\~+, and taking into account Eq. (\[spur1\]) one immediately reveals that Eq. (\[spur3\]) vanishes provided h=, U\_2=U\_1. Thus the OPE of the fermionic current $G(z)$ with the boson emission vertex must be of the form \[opegv\] G(z) V(z’)\~+, with $U_1$ having conformal spin $1/2$.
The current $\bar G(z)$ can be treated in the same way yielding the OPE \[opegv1\] |G(z) V(z’)\~+ , with ${\tilde U}_1$ being a conformal field of spin $1/2$.
So far we have considered spurious (bra) states generated by $G_r$, ${\bar G}_r$ and $L_n$ independently. It is instructive then to recall the anticommutator in the algebra $\{G_r,{\bar G}_r \}=L_{2r}$, $r>0$, which relates the states and provides further information on the structure of $V$. Actually, starting with the spurious state of the form $\sum_{r>0} \langle \chi_{2r} |~L_{2r}$ with $\langle \chi_{2r} |(2r+L_0)=0$, one readily finds the identity \[iden\] && \_[r>0]{} \_[2r]{} |(2r+1)V(1)+V(1) | = \_[r>0]{} \_[2r]{} |G\_r{ (r+) [U]{}\_1 (1)+\_1 (1)}|\
&& +\_[r>0]{} \_[2r]{} |[|G]{}\_r { (r+) U\_1 (1)+U\_1 (1) }| . Now we have to guess the OPE of $G(z)$ with ${\tilde U}_1(z')$ and $\bar G (z)$ with $U_1 (z')$. Since these have to match the left hand side of the identity (\[iden\]) the poles of the second order or higher are not allowed (this is also prompted by the conformal spin which carry the fields $U_1 (z)$ and ${\tilde U}_1(z)$) and one finally has to set \[opegv2\] G(z) [U]{}\_1(z’)\~, |G(z) U\_1(z’)\~. Substitutions of this in Eq. (\[iden\]) yields \[opegv3\] V=W+W, which together with Eqs. (\[opegv\]),(\[opegv1\]), (\[opegv2\]) gives a closed set of equations to fix $V$. Finally, given the explicit form of the ansatz (\[verans\]) it is easy to verify the OPE’s J(z) V(z’) \~0, J\^[(1,2)]{} V(z’) \~0, which hold independently of the value of the constant coefficients entering the ansatz. This seems reasonable since, for instance, the $U(1)$ charge is then conserved in the amplitude. The relations above imply also the decoupling of the spurious physical (bra) states generated by $J_n,J^{(1,2)}_n$ with $n>0$. In a similar way, making use of the algebra $[G_r, J^{(1)}_0]=-2H_r$, $[{\bar G}_r,J^{(2)}_0]=-2{\bar H}_r$, $[L_0, J^{(1,2)}_0]=0$ one can show the decoupling of those generated by $H_r$, ${\bar H}_r$ with $r>0$. Thus the decoupling of spurious physical states generated by the currents extending the $N=2$ SCA to the $N=4$ SCA proves to be automatic and do not impose further restrictions on the form of the emission vertex.
In the $N=2$ framework the $U(1,1)$ invariance of the formalism selects only the first three terms in the ansatz (\[verans\]). Then a simple inspection of Eqs. (\[opegv\]),(\[opegv1\]), (\[opegv2\]),(\[opegv3\]) shows that the only essential restriction coming along with the causality conditions reads ł=i(-). This leaves two unspecified constants in the $U(1,1)$ invariant ansatz. Calculating further the three point tree amplitude with physical [*in*]{} and [*out*]{} states A\^[tree]{}\_[right]{} (1,2,3)= V\_0 (k\_1,)V(k\_2,1)V\_0(k\_3,0) =-i (\_2 k\_3 + k\_2 [|k]{}\_3), one finds this to possess cyclic symmetry only when \[ab\] =-. Up to an irrelevant number coefficient this reproduces the vertex found by Ooguri and Vafa [@ov] within the alternative superfield approach \[ovvertex\] V(k,z,|z)=:{ik -i |k x -2(|k )(k|) }e\^[ik [|x]{} +i[|k]{} x]{}: , k|k=0. Given the vertex, tree amplitudes involving more than three legs prove to vanish [@ov] which persists also to higher orders in perturbation expansion [@berkvafa].
Turning to the $N=4$ formalism, one is to consider the full ansatz (\[verans\]). The causality arguments now enforce the conditions ł=i(-), =i, ø=-i, where one has to use the identity (|k [x]{}) - (|k )= -(k|k) , which is a consequence of Eqs. (\[idd\]) and (\[idd1\]). The new terms entering the ansatz bring the extra contribution to the three point function -i(\_2 k\_3 -k\_2 [|k]{}\_3 + +), and the last two terms reveal cyclic symmetry without any restrictions on the coefficients $\g,\m$. Then a simple inspection of the table displaying the action of the ${SU(1,1)}_{outer}$ group on the elementary field combinations shows that the correlator above [*is not*]{} ${SU(1,1)}_{outer}$ invariant at any value of $\a,\m,\g$. A way out could be to allow the constants to transform nontrivially under ${SU(1,1)}_{outer}$. This type of reasoning has been advocated in Ref. [@olaf] where two of the constants were identified with the gravitational and Maxwell string couplings. In our opinion, however, this does not fit well neither the field theory framework nor the string theory framework.
Thus, we conclude that no Lorentz invariant boson emission vertex can be constructed within the $N=4$ topological formalism. The maximal subgroup one can retain proves to be $U(1,1)$ which brings one back to the Ooguri–Vafa vertex (\[ovvertex\]). Ultimately, our conclusion based on the old covariant quantisation proves to be in line with the analysis of Refs. [@berkvafa; @ov3; @bvw] which relied upon the topological prescription for calculating the scattering amplitudes.
[**4. Concluding remarks**]{}
To summarize, in this paper we reconsidered the issue of restoring the Lorentz invariance in the quantum $N=2$ string. We used the passage to the equivalent $N=4$ topological formalism which raises the global automorphism group of the underlying superconformal algebra to the full Lorentz group. Being efficient in restoring the invariance at the classical level, the $N=4$ formalism, however, is not powerful enough to do so in the quantum theory. In particular, the causality and cyclic symmetry of tree amplitudes prevent one from the construction of a Lorentz invariant boson emission vertex. Although asymptotic states still can be described by the Lorentz invariant vertices.
An interesting continuation of this work could be a rigorous BRST analysis, although in the BRST framework we do not see means which could impact the conclusion drawn in this paper.
[**Acknowledgements**]{}\
A.G. thanks E.A. Ivanov for helpful comments. This work was supported by INTAS grant No 00-0254 and by the Iniziativa Specifica MI12 of the Commissione IV of INFN.
[nn]{} W. Siegel, Phys. Rev. Lett. [**69**]{} (1992) 1493. N. Berkovits and C. Vafa, Nucl. Phys. B [**433**]{} (1995) 123. H. Ooguri and C. Vafa, Nucl. Phys. B [**451**]{} (1995) 121. N. Berkovits, C. Vafa and E. Witten, JHEP [**9903**]{} (1999) 018. S. Bellucci and A. Galajinsky, Nucl. Phys. B [**606**]{} (2001) 119. S. Bellucci and A. Galajinsky, hep–th/0107161, Phys. Rev. D (to appear). S. Bellucci, A. Deriglazov and A. Galajinsky, [*The geometry of N=4 twisted string*]{}, hep-th/0110287. A. Schwimmer and N. Seiberg, Phys. Lett. B [**184**]{} (1987) 191. A. Bilal, Phys. Lett. B [**180**]{} (1986) 255. S. Mathur and S. Mukhi, Nucl. Phys. B [**302**]{} (1988) 130. H. Ooguri and C. Vafa, Nucl. Phys. B [**361**]{} (1991) 469. J. Bischoff and O. Lechtenfeld, Phys. Lett. B [**390** ]{} (1997) 153.
[^1]: E-mail: [email protected]
[^2]: On leave from Department of Mathematical Physics, Tomsk Polytechnical University, Tomsk, Russia\
E-mail: [email protected]
[^3]: In contrast to the Green–Schwarz superstring, the constraints intrinsic to the $N=4$ model are scalars. Hence they do not seem to require infinitely many ghost fields.
[^4]: On the cylinder we conjugate as ${(x^a)}^{*}={\bar x}^{\bar a}$, ${(\p^a)}^{*}={\bar \p}^{\bar a}$. The target metric is Hermitian ${\eta^{\bar a b}}^{*}=\eta^{\bar b a}$.
[^5]: In the $N=4$ framework the vanishing of the normal ordering constant for the operator $J_0$ is also required by the last commutator in Eq. (\[n4alg\]).
|
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abstract: 'Structural identity is a concept of symmetry in which network nodes are identified according to the network structure and their relationship to other nodes. Structural identity has been studied in theory and practice over the past decades, but only recently has it been addressed with representational learning techniques. This work presents *struc2vec*, a novel and flexible framework for learning latent representations for the structural identity of nodes. *struc2vec* uses a hierarchy to measure node similarity at different scales, and constructs a multilayer graph to encode structural similarities and generate structural context for nodes. Numerical experiments indicate that state-of-the-art techniques for learning node representations fail in capturing stronger notions of structural identity, while *struc2vec* exhibits much superior performance in this task, as it overcomes limitations of prior approaches. As a consequence, numerical experiments indicate that *struc2vec* improves performance on classification tasks that depend more on structural identity.'
author:
- 'Leonardo F. R. Ribeiro'
- 'Pedro H. P. Saverese'
- 'Daniel R. Figueiredo'
bibliography:
- 'sigproc.bib'
title: '*struc2vec*: Learning Node Representations from Structural Identity'
---
<ccs2012> <concept> <concept\_id>10010147.10010257.10010258.10010260</concept\_id> <concept\_desc>Computing methodologies Unsupervised learning</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010257.10010293.10010319</concept\_id> <concept\_desc>Computing methodologies Learning latent representations</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
|
---
abstract: 'A new calculation of the $\bar{p}/p$ ratio in cosmic rays is compared to the recent PAMELA data. The good match up to 100 GeV allows to set constraints on exotic contributions from thermal WIMP dark matter candidates. We derive stringent limits on possible enhancements of the WIMP [|[p]{}]{} flux: a $m_{\rm WIMP}$=100 GeV (1 TeV) signal cannot be increased by more than a factor 6 (40) without overrunning PAMELA data. Annihilation through the $W^+W^-$ channel is also inspected and cross-checked with $e^+/(e^-+e^+)$ data. This scenario is strongly disfavored as it fails to simultaneously reproduce positron and antiproton measurements.'
author:
- 'F. Donato'
- 'D. Maurin'
- 'P. Brun'
- 'T. Delahaye'
- 'P. Salati'
title: 'Constraints on WIMP Dark Matter from the High Energy PAMELA $\bar{p}/p$ data'
---
The cosmic ray (CR) antiproton and positron fluxes are considered as prime targets for indirect detection of galactic dark matter (DM). A deviation from the predicted astrophysical background has been searched for mostly at low energy (e.g., [@1998PhRvD..58l3503B]). However, in some scenario, heavy Weakly Interacting Massive Particle (WIMP) candidates|either from annihilation of the lightest supersymmetric species, or from the lightest Kaluza-Klein particles in universal extra dimensions|should be able to provide sizeable fluxes beyond a few GeV [@2007PhRvD..75h3006B]. The PAMELA collaboration has recently published the cosmic ray antiproton to proton ratio in the hitherto unexplored $\sim 1-100$ GeV energy range [@PAMELA_PRL]. We find that the background flux, yielded by standard astrophysical processes, can explain the data up to high energies. Adding a contribution from the annihilation of a generic WIMP dark matter halo, we derive stringent upper limits on possible boost factors of the exotic [|[p]{}]{} flux. In addition, in some scenarios, the WIMPs mostly annihilate into $W^+W^-$ pairs, hence giving rise to a copious amount of hard positrons. Recent calculations, in the light of the new PAMELA measurement of the positronic fraction in CRs, showed that the secondary production alone could marginally explain the data [@Delahaye:2008ua]. Should a heavy WIMP be required to better match the positron flux, we also check the viability of such models through the combined constraints from the $e^+$ and [|[p]{}]{}fluxes as also done in [@Cirelli:2008pk].
{width="\columnwidth"} {width="\columnwidth"}
The secondary [|[p]{}]{} flux provided in this paper is an improved calculation of that presented in [@2001ApJ...563..172D], to which we refer for a thorough discussion of the ingredients and the technical details. Antiprotons are yielded by the spallation of cosmic ray proton and helium nuclei over the interstellar medium, the contribution of heavier nuclei being negligible [@2005PhRvD..71h3013D]. Even if only p and He with kinetic energy larger than $6\,m_p$ can produce [|[p]{}]{}, a good description of the p and He interstellar (IS) fluxes is mandatory to correctly provide the [|[p]{}]{}$/p$ ratio in the 0.1 GeV$-$100 GeV range. Following [@2007APh....28..154S], we model the proton and helium IS fluxes as $$\Phi = A \beta^{P_1} R^{-P_2} \;\;{\rm m}^{-2}~{\rm s}^{-1}~{\rm sr}^{-1}~({\rm GeV/n})^{-1},
\label{eq:flux_fit}$$ where $R$ is the rigidity of the particle. The parameterization for the fluxes below 20 GeV/n are taken from the reanalysis of the 1997 to 2002 BESS data [@2007APh....28..154S]: $\{A,\,P_1,\,P_2\}=\{19400,\,0.7,\,2.76\}$ for H and $\{7100,\,0.5,\,2.78\}$ for He. For the high energy range, the combined fit of AMS-01 [@2000PhLB..490...27A; @2000PhLB..472..215A; @2000PhLB..494..193A], BESS98 [@2000ApJ...545.1135S] and BESS-TeV [@2004PhLB..594...35H] demodulated data respectively give $\{A,\,P_1,\,P_2\}=\{24132,\,0.,\,2.84\}$ for H and $\{8866,\,0.,\,2.85\}$ for He. The two fits connect smoothly at 20 GeV/n. Compared to [@2001ApJ...563..172D], we also improve the calculation of the tertiary mechanism [@1999ApJ...526..215B]. The Anderson prescription [@1967PhRvL..19..198A] is used, as described in [@2005PhRvD..71h3013D; @2008PhRvD..78d3506D]. As a result, the low energy tail is more replenished, leading to a larger flux. The framework used to calculate cosmic ray fluxes is the diffusion model with convection and reacceleration. The transport parameters are fixed from the boron-to-carbon (B/C) analysis [@2001ApJ...555..585M] and correspond to i) the diffusion halo of the Galaxy $L$; ii) the normalization of the diffusion coefficient $K_0$ and its slope $\delta$ ($K(E)=K_0 \beta R^\delta$); iii) the velocity of the constant wind directed perpendicular to the galactic disk $\vec{V_c} = \pm V_c \vec{e_z}$; and iv) the reacceleration strength mediated via the Alfvénic speed $V_a$. Strong degeneracies are observed among the allowed parameter sets [@2001ApJ...555..585M], but it has a limited impact on the secondary [|[p]{}]{} flux [@2001ApJ...563..172D]. At variance, the corresponding DM-induced [|[p]{}]{} flux suffers large propagation uncertainties [@2004PhRvD..69f3501D]; this is also the case for the secondary and primary positron fluxes [@2008PhRvD..77f3527D; @Delahaye:2008ua]. Throughout the paper, the fluxes will be shown for the B/C best fit propagation parameters, i.e. $L = 4.$ kpc, $K_0 = 0.0112$ kpc$^2$Myr$^{-1}$, $\delta=0.7$, $V_c = 12.$ km s$^{-1}$ and $V_a = 52.9$ km s$^{-1}$ [@2001ApJ...555..585M].
The secondary IS [|[p]{}]{} flux is displayed in the left panel of Fig. \[fig:pbar\_sec\] along with the data demodulated according to the force-field prescription. We either use the DTUNUC [@2001ApJ...563..172D] [|[p]{}]{} production cross sections (solid line) or those discussed in [@2005PhRvD..71h3013D; @2007PhRvD..75h3006B] (dashed line). The differences between the two curves illustrate the uncertainty related to the production cross sections, as emphasized in [@2001ApJ...563..172D], where a careful and conservative analysis within the DTUNUC simulation settled a nuclear uncertainty of $\sim 25\%$ over the energy range $0.1-100$ GeV. The conclusion is similar here, although the two sets of cross sections differ mostly at low energy. In the right panel, along with the demodulated [|[p]{}]{}$/p$ data, we show the curves bounding the propagation uncertainty on the [|[p]{}]{} calculation based either on the DTUNUC [@2001ApJ...563..172D] [|[p]{}]{} production cross sections (solid lines) or those borrowed from [@2007PhRvD..75h3006B] (dashed lines). The uncertainty arising from propagation is comparable to the nuclear one [@2001ApJ...563..172D]. From a bare eye inspection, it is evident that the secondary contribution alone explains PAMELA data on the whole energetic range. It is not necessary to invoke an additional component to the standard astrophysical one.
Motivated by the accuracy of our predictions and their well understood theoretical uncertainties, as well as by the good statistical significance of PAMELA data, we derive limits on a possible exotic component. We focus on the high energy part of the $\bar{p}/p$ ratio, where solar modulation does not play any role [@1999PhRvL..83..674B]. We assume an additional component of antiprotons produced by annihilation of WIMPs filling the dark halo of the Milky Way. Their distribution is taken as a cored-isothermal sphere with local density $\rho_{\odot} = 0.3$ GeV cm$^{-3}$. The velocity-averaged annihilation cross section is taken as ${\langle \sigma_{\rm ann} v \rangle}= 3 \times 10^{-26}$ cm$^{3}$s$^{-1}$, with an annihilation channel into $b$-$\bar{b}$. According to [@2004PhRvD..69f3501D], the propagated primary [|[p]{}]{} flux is only very mildly dependent on the annihilation channel and the DM distribution function. Therefore, our assumptions can be considered valid for a generic WIMP dark matter candidate except for a rough rescaling factor. Propagation is treated in the same way as for the secondary component [@2004PhRvD..69f3501D; @2008PhRvD..78d3506D]. As a reference case, we employ the best fit transport parameters listed above and recall that the uncertainty on the primary [|[p]{}]{} flux due to propagation spans roughly one order of magnitude above and one below the best fit scenario.
![Upper limits on the enhancement factor to the primary [|[p]{}]{} flux as a function of the WIMP mass, derived from a comparison with PAMELA high energy data. Each curve is labelled according to the corresponding PAMELA energy bin.[]{data-label="fig:boost"}](BF_2sigma.eps){width="1.\columnwidth"}
We add the calculated primary [|[p]{}]{} flux for different WIMP masses to the secondary component and compare the total flux to PAMELA high energy data, namely $T_{\bar{p}}> $ 10 GeV. To be conservative, the background calculated from Bringmann & Salati’s [|[p]{}]{} production cross sections is considered (dashed curves in Fig. \[fig:pbar\_sec\]). We derive the factor by which the DM flux could be enhanced without exceeding experimental data (2$\sigma$ error bars) in any energy bin. The maximum allowed enhancement factor is plotted in Fig. \[fig:boost\] as a function of the WIMP mass: it cannot exceed 6–20–40 for $m_{\rm WIMP}$=100–500–1000 GeV, respectively. These limits can be reinforced as well as relaxed by quite simple modifications of the key ingredients in the flux calculation, just as described above. The boost factor may be ascribed, in principle, to clumpiness in the DM distribution |this contribution being energy-dependent|as well as to an increase of the annihilation cross section as proposed by [@1989PhLB..225..372B] and more recently by [@Hisano:2003ec] using the Sommerfeld effect.
{width="\columnwidth"} {width="0.96\columnwidth"}
Our conclusions have important consequences on the explanations of the positron data based on the annihilation of DM species within the Milky Way halo. The positron fraction suffers from large uncertainties related for instance to the poorly determined electron spectral index above 10 GeV [@Delahaye:2008ua]. Although soft electrons are associated to large values of the positron fraction and to a marginal agreement of the pure secondary positron flux with the measurements, we cannot dismiss the possibility of a hard cosmic ray electron distribution. A spectral index of 3.44 [@2004ApJ...612..262C] leads actually in the left panel of Fig. \[fig:1\_TEV\_WIMP\] to the long-dashed curve featuring a low background case. With a typical annihilation cross section ${\langle \sigma_{\rm ann} v \rangle}$ of $3 \times 10^{-26}$ cm$^{3}$s$^{-1}$, WIMPs do not produce enough positrons to reproduce the increasing trend observed in $e^+/(e^+ + e^-)$ data [@PAMELA_frac],so that a significant enhancement of the annihilation rate is necessary as shown in [@Bergstrom:2008gr]. However the boost factor associated to DM clumps cannot exceed at most a factor of $\sim$ 10 in the standard $\Lambda$-CDM scenario of structure formation . Astrophysics does not provide then a natural explanation for the large boost factors required to fit the positron excess. That is why the Sommerfeld effect [@Hisano:2003ec] has been advocated as a plausible mechanism to significantly increase the WIMP annihilation cross section in the non-relativistic regime prevailing today in galactic haloes. Heavy DM species is a prerequisite. We then consider a generic 1 TeV particle annihilating into $W^{+} W^{-}$ pairs and boost ${\langle \sigma_{\rm ann} v \rangle}$ by a factor of 400 in order to get the solid line in the left panel of Fig. \[fig:1\_TEV\_WIMP\]. Although an annihilation cross section of $1.2 \times 10^{-23}$ cm$^{3}$s$^{-1}$ is possible should non-perturbative effects be involved, the consequences on antiprotons are drastic. The red solid curve in the right panel of Fig. \[fig:1\_TEV\_WIMP\] features an unacceptable distortion of the [|[p]{}]{} spectrum. The DM positron signal cannot be enhanced without playing havoc with the [|[p]{}]{} measurements.
Nonetheless, notice that ways out are possible whose careful investigation is beyond the scope of this Letter. The value of 400 assumed for the positron signal of Fig. \[fig:1\_TEV\_WIMP\] could arise from the combined effects of DM clumpiness and ${\langle \sigma_{\rm ann} v \rangle}$ enhancement. If a generous factor of 10 is assumed for the former|a marginally acceptable value |the latter does not exceed 40. Unlike positrons which are produced locally, the antiprotons detected at the Earth originate from a large region of the Milky Way halo over which substructures may not be as important as in our vicinity. The [|[p]{}]{} flux may not be much enhanced by the presence of DM clumps so that a value of 40 would apply in that case to the antiproton boost. The corresponding blue long-dashed line in the right panel of Fig. \[fig:1\_TEV\_WIMP\] features a fairly acceptable [|[p]{}]{} spectrum. Viable scenarios such as a large black hole population pervading the Galaxy [@Bertone:2005xz; @Brun:2007tn] also lead to large boost factors although it seems difficult a priori to prevent the antiproton flux from being enhanced too much. Notice finally that the cosmic ray propagation model could be different from the one selected here. Once again, positron and antiproton fluxes have different behaviors toward a change in the propagation parameters. For example, the primary [|[p]{}]{} flux could be easily decreased by an order of magnitude without violating B/C data, allowing a Sommerfeld boost of the cross section of 400.
A new calculation for the secondary cosmic antiproton flux and the relevant uncertainties have been presented. The ratio $\bar{p}/p$ has been derived after fitting recent proton data. Our predictions can explain the experimental data, and in particular the recent PAMELA data, which span more than two decades in energy. No exotic contribution|as from annihilating dark matter in the galactic halo|has to be invoked to reproduce experimental results. Analyzing the high energy part of the PAMELA $\bar{p}/p$ we derive strong upper limits on possible enhancements of the exotic [|[p]{}]{} flux as a function of the WIMP mass. Relying on standard assumptions the exotic antiproton flux induced by a $m_{\rm WIMP}=100$ GeV (1 TeV) DM halo cannot be increased by more than a factor 6 (40) without overrunning PAMELA data. Would the Sommerfeld effect ($W^+W^-$ channel) be invoked to explain PAMELA leptonic data, the corresponding enhancement of the [|[p]{}]{} production would lead to an unacceptable distortion of the $\bar{p}/p$ spectrum.\
[*Acknowledgements.*]{} F.D. thanks N. Fornengo and M. Boezio for useful discussions. We acknowledge the support from the French Programme National de Cosmologie and the Explora’doc PhD student program.
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---
abstract: |
Let $n$ be the size of a parametrized problem and $k$ the parameter. We present polynomial-time kernelizations for <span style="font-variant:small-caps;">Cluster Editing/Deletion</span>, <span style="font-variant:small-caps;">Path Contractions</span> and <span style="font-variant:small-caps;">Feedback Vertex Set</span> that run with $O(\poly(k) \log n)$ bits and compute a kernel of size polynomial in $k$. By first executing the new kernelizations and subsequently the best known polynomial-time kernelizations for the problem under consideration, we obtain the best known kernels in polynomial time with $O(\poly(k) \log n)$ bits.
Our kernelization for <span style="font-variant:small-caps;">Feedback Vertex Set</span> computes in a first step an approximated solution, which can be used to build a simple algorithm for undirected $s$-$t$-connectivity (USTCON) that runs in polynomial time and with $O(\poly(k) \log n)$ bits.
author:
- Frank Kammer
- Andrej Sajenko
bibliography:
- 'main.bib'
title: 'FPT-space Graph Kernelizations'
---
Introduction
============
A fixed-parameter tractable problem is a problem whose instances of size $n$ are given with an extra parameter $k$ and can be solved in FPT time, i.e., in time $f(k) \cdot n^{O(1)}$ where $f$ is some computable function independent of $n$. One major field in parameterized complexity is the kernelization, i.e., an algorithm that applies several rules that simplify an instance of size $n$ to an instance, the size of which depends only on $k$. Traditionally researchers in the field of fixed-parameter tractability only focus on time. However, it is natural to ask the question: without increasing the running time to much, can the space of such algorithms be also bound by FPT space, i.e., a space bound by $O(f(k))$ computer or $O(f(k) \log n)$ bits where $f$ is some computable function?
We call the memory where the input of an algorithm is stored [*input*]{} memory and the memory that an algorithm uses to compute its result [*working*]{} memory. Our model of computation is the [*read-only word-RAM*]{} where we assume a word-size of $w = \Omega(\log N)$ bits (with $N$ being the size of the input) and that arithmetic operations ($+, -, \cdot, /$, modulo) and bit-shift operations on bit sequences of $w$ bits take constant time. We have constant-time random access to the input memory. Another model is the [*streaming model*]{} where the input is presented as a linear stream of unordered elements and no other access to the input memory is permitted. It is easy to see that the streaming model is more restricted than the read-only word-RAM, allowing us to take over all upper bound results showed in the streaming model while possibly allowing us to break the lower bounds. Fafianie and Kratsch [@FafK14] presented upper and lower bounds for the question above in the streaming model. For an overview of our results, see Table \[tab:streaming-kernels\].
problem $O(1)$-pass streaming word RAM
------------------------------------------------------------------- -------------------------------- --------------------------------
<span style="font-variant:small-caps;">Vertex Cover</span> $O(\poly(k) \log n)^{\dagger}$ $O(\poly(k) \log n)^{\dagger}$
<span style="font-variant:small-caps;">Cluster Editing</span> $\Omega(n)^{\dagger}$ [ **$O(\poly(k) \log n)$**]{}
<span style="font-variant:small-caps;">Path Contraction</span> ? [ **$O(\poly(k) \log n)$**]{}
<span style="font-variant:small-caps;">Feedback Vertex Set</span> $\Omega(n)^{\dagger}$ [ **$O(\poly(k) \log n)$**]{}
: The results shown by Fafianie and Kratsch are marked with $\dagger$.
\[tab:streaming-kernels\]
Algorithms that treat space as a valuable resource are called space-efficient algorithms. There are several space-efficient algorithms published [@AsaIKKOOSTU14; @ChakSS20; @ElmHK15; @Hag19; @KamMS19; @KamS18c; @KamS20]. We can roughly say that space-efficient algorithms are usually slower in practice than their non space-efficient versions, because they need asymptotically more running time or their hidden constant in their running time is too large. However, if the difference in space between a space-efficient and non space-efficient algorithm is large enough, then less cache-faults occur or less memory has to be copied. The time saved here can compensate for a slightly worse running time. For an example see the implementation and tests [@KamS18g] of a so-called [*space-efficient subgraph stack*]{} [@HagKL19]. The usually huge difference between $k$ and $n$ makes fixed-parameter algorithms with FPT-space interesting.
In the following we denote by $n$ the number of vertices of the given graph and by $k$ the given parameter. A usual kernelization uses $O(\poly(n) \log n)$ bits of working memory and produces a kernel of $O(f(k) \log n)$ bits where $f(k)$ is the kernel size. Our approach is to either modify standard kernelization rules to bound their working memory to $O(\poly(k) \log n)$ bits or find new ones working with the same bound. These rules do not necessarily produce the smallest known kernel, but the kernel size is bounded by $\poly(k)$ and the current best polynomial-time kernelization on that kernel then also performs within our targeted space bound of $O(\poly(k) \log n)$ bits. Moreover, our kernelization for <span style="font-variant:small-caps;">Cluster Editing/Deletion</span> and <span style="font-variant:small-caps;">Path Contraction</span> is almost as fast as the known best kernelization (we have only an extra factor of $\log k$), i.e., our kernelization is space-efficient.
In the next section we give definitions required for this paper and show some auxiliary algorithms used afterwards. These algorithms together with our kernelization result for the so-called <span style="font-variant:small-caps;">Feedback Vertex Set</span> problem easily allow us to solve undirected $s$-$t$-connectivity in an $n$-vertex graph that has an unknown feedback vertex set of size $k$ in polynomial time using $O(\poly(k) \log n)$ bits. Directed and undirected s-t connectivity (STCON and USTCON, resp.) is one of the most essential problems with, e.g., $SL$ is the complexity class of problems that are log-space reducible to USTCON. In a celebrated result, Reingold [@Rei08] showed a log-space algorithm for USTCON and showed so that $SL$ is equal to the complexity class $L$, i.e., the class of decision problems that can be solved using logarithmic space. His algorithm is based on so-called [ *zig-zag products*]{}, which are relatively complicated and difficult to analyze. An alternative algorithm ist presented by Rozenman and Vadhan [@RozV05] by using a so-called [*de-randomized squaring algorithm*]{}, whose construction is also relatively unnatural. Our solution is simple, but can be used to obtain a log-space algorithm only in the case where the given graph has a unknown feedback vertex set of size $O(1)$.
We also want to remark that connectivity is also often solved by a DFS. To the authors knowledge, a polynomial-time sublinear-space DFS exists only on planar graphs and on graphs of treewidth $O(n^{1-\epsilon} )$ for some $\epsilon>0$ [@IzuO20]. Both DFS algorithms require $\Omega(n^{1/2})$ bits.
In Section \[sec:kernels\], we describe our kernelizations, where we begin in Section \[sec:ce\] with the kernelizations for <span style="font-variant:small-caps;">Cluster Editing/Deletion</span>. The standard kernelizations iterate over all vertex triples and use large matrices to store information for each vertex-pair whenever the triple induces a path in the graph (i.e., $\Theta(n^2 \log n)$ bits). Instead, we make sure that an edge is “considered” in at most $k+1$ induced paths.
Furthermore in Section \[sec:pc\], we focus on a kernel for <span style="font-variant:small-caps;">Path Contraction</span>. To find such a kernel, one usually searches for bridges and contracts the endpoints of the bridge. Instead of this reduction rule, we only shrink induced paths as long as their length is greater than $k+3$ and show that by doing so we get a polynomial-sized kernel.
Our main result is our kernelizations for <span style="font-variant:small-caps;">Feedback Vertex Set</span> shown in Section \[sec:fvs\]. It is our most complicated algorithm. As part of the algorithm, we construct an approximated feedback vertex that we can also use for our $s$-$t$-connectivity result in Section \[sect:USTCON\]. Many kernelizations for <span style="font-variant:small-caps;">Feedback Vertex Set</span> search repeatedly for a [*flower*]{}, i.e., a set of $k+1$ cycles that are disjoint except for one common vertex $v$. Since it is not clear how we can determine a flower with $\poly(k \log n)$ bits, we first show how to compute a feedback vertex set $S$ of size polynomial in $k$ and with $O(f(k) \log n)$ bits improving an algorithm from Becker and Geiger [@BeckerG94]. Their algorithms either return a $(2\log n)$-approximation or need to search for cycles where an implementation with few space seems to be complicated. By removing $S$ the graph decomposes in several trees $\mathcal T$. Afterwards, we remove trees that are not relevant and shrink the remaining trees to a size polynomial in $k$. To shrink the trees, we search for a special kind of flowers.
Inplace Algorithms and FPT-space poly-time USTCON {#sect:USTCON}
=================================================
In the following, an instance of a fixed-parameter tractable problem is described by a tuple $(G, k)$ where $G$ is a graph and $k$ is a parameter. We call an instance a [*yes*]{}-instance exactly if there is a solution of size max $k$, otherwise we call it a [*no*]{}-instance. We define the [*size*]{} of a graph as the sum of its vertices and edges.
[(kernel, full kernel)]{} A kernel for a given instance $(G, k)$ of a fixed-parameter tractable problem is an instance $(G', k')$ with the properties that the size of $G'$ and $k'$ is bound by $f(k)$ where $f$ is a computable function (independent of the size of $G$) and $(G', k')$ contains a solution for the problem exactly if $(G, k)$ contains a solution.
In case that the kernel contains the vertices of every solution of size $k$, it is called [*[full]{}*]{}.
To compute a kernel a [*kernelization algorithm*]{} (or short only [*kernelization*]{}) is used. For an easier description, we allow our kernelization to output “no solution”. From that answer, one can easily construct a trivial no-instance. Usually, a kernelization algorithm modifies $G$ step-wise by local modifications described in so-called [*reduction rules*]{}. A reduction rule is [*safe*]{} if it maintains solvability of the given instance. Our kernelizations compute a kernel $(G', k')$ of an instance $(G, k)$ by deleting some vertices and/or edges of $G$ and so obtains $G'$. Furthermore, our kernelizations always allow us to decide which subset $S$ of the deleted vertices or edges have to be part of a solution for $(G, k)$ in addition to the vertices or edges in the solution for $(G', k')$. On the read-only word-RAM and with our space bound we can neither modify the input graph $G$ nor create a mutable copy of it. Instead we compute the information needed on demand, and use it to construct a graph $G'$ representing the kernel. To store $G'$ we use a mutable graph structure where a runtime penalty factor logarithmic in the size of the graph has to be accounted in. Such a graph structure can be realized by using a balanced heap for the vertices $u$ with modifications. The heap then stores a tuple $(u, p_u)$ where $p_u$ is a pointer to another balanced heap that stores $u$’s neighborhood.
We now want to show how our approximation of a feedback vertex set can be used on an $n$-vertex graph with an unknown feedback vertex set of size $k=O(1)$ to solve USTCON in polynomial time and with $O(\log n)$ bits. As a subroutine, we use several depth first searches (DFS). A DFS visits all vertices of a connected component of a graph once as follows: starting with some vertex $r$ it marks $r$ (all other vertices are unmarked) and puts the tuple $(r, 0)$ representing the first edge of $r$ on a stack, which we call [*DFS stack*]{}. Then it repeats until the stack is empty: take a tuple $(u, i)$ from the stack. If $\deg(u) < i$, take the $i$th neighbor $v$ from $u$’s neighborhood and put $(u, i + 1)$ on the stack. Moreover, if $v$ is not marked, mark it, output it as well as add $(v, 0)$ on the stack. We then call edge $\{u, v\}$ a [*tree edge*]{}. All other edges of the graph are called [*back edges*]{}.
A DFS has many applications, one is to compute a so-called DFS-tree. A DFS tree is a spanning tree consisting of all tree edges. We next show that we can run a DFS in a tree with few bits.
\[lem:traverseTree\] Given an $n$-vertex tree $T$ and a node $r$ of $T$ as root there is a linear-time $O(\log n)$-bits algorithm that traverses all vertices of $T$ in depth-first-search manner.
Start the algorithm at $r$ and output it. In an outer loop, iterate over the children $u$ of $r$ and set $p:=r$ (define $p$ as predecessor of $u$). In an inner loop repeat as long as $u \ne r$: Output $u$. Then determine the index $i$ of vertex $p$ in $u$’s neighborhood and take $u'$ as the vertex at index $(i + 1) \mod \op{deg}(u)$ from $u$’s neighborhood—here we assume that the indices run from $0$ to $\deg(u) - 1$. Set $p := u$, $u := u'$ and repeat the inner loop, i.e., visit $u'$.
Observe that, after the algorithm visits all descendants of a vertex $u$ and returns from $p$ to $u$, we want to return to the parent $v$ of $u$. By construction, the index of edge $\{u,p\}$ is one less than the index of edge $\{u,v\}$ at $u$ since we used the edge from $u$ with index one more than edge $\{u,v\}$ to visit a first child of $u$. This allows the algorithm to move to the parent after it visited all its descendants. The inner loop traverses a whole maximum subtree rooted at $r$’s child and the outer loop makes sure that such a traversal is started for every child of $r$. Every edge is traversed twice (once in each direction) leading to a total running time of $O(n)$. The algorithm runs with $O(\log n)$ bits since it has to store only a constant number of vertices and indices.
We next want to describe our algorithm for solving USTCON with a given feedback vertex set $F$. Let $F^*, F', F'' \subseteq F$ be initially empty sets. The vertices of $F$ connect trees of $G[V \setminus F]$. Our approach is to use $F^*$ as a set of already reached vertices of $F$, the set $F'$ as the set from which we explore trees to reach new vertices of $F$, which we store in $F''$. A sketch of the sets is shown in Figure \[fig:ustcon\].
Initially use Lemma \[lem:traverseTree\] to determine the set $F' \subseteq F \setminus F^*$ of vertices that are adjacent to a vertex in the tree of $G[V \setminus F]$ that contains $s$. If $s \in F$, take $F' := \{s\}$.
Then for each $T$ of $G[V \setminus F]$ that is adjacent to a vertex of $F'$, run Lemma \[lem:traverseTree\] to determine the set $F'' \subseteq F \setminus F^*$ of vertices that are adjacent to a vertex of such a tree $T$. If $t$ is part of $F''$ or part of some tree $T$ above, then there exists a path between $s$ and $t$ in $G$. Otherwise and if $F'' \ne \emptyset$ repeat with $F^* := F^* \cup F', F' := F''$ and $F'' = \emptyset$.
Given an $n$-vertex $m$-edge graph $G$ together with a feedback vertex set $F$ of size $k$ in $G$, we can decide USTCON in $O(nm \log k)$ time and with $O(k \log n)$ bits.
It remains to analyze the time and space of the algorithm above. By storing the vertices of $F^*, F'$ and $F''$ in balanced heaps, our algorithm uses $O(k \log n)$ bits, but accessing a vertex in a heap costs $O(\log k)$ time per access. In the worst case we have to explore each tree $T$ of $G[V \setminus F]$ for each edge connecting $T$ with a vertex of $F$. Since we can not store $T$ part of $G[V \setminus F]$, we use $G$, but skip over edges pointing to a vertex of $F$. Thus, the runtime to explore all trees once is $O(n \log k)$ and the total running time is $O(nm \log k)$.
By Theorem \[thm:approxfvs\], we can compute an approximated feedback vertex set of an $n$-vertex $m$-edge graph with unknown feedback vertex set of size $k$ in $O(mn^2 \log k)$ time and $O(k^2\log
n)$ bits.
Given an $n$-vertex $m$-edge graph $G$ with an unknown feedback vertex set of size $k$ in $G$, we can decide USTCON in $O(mn^2 \log k)$ time and with $O(k^2 \log n)$ bits.
To the end of the section, we show that our DFS traversal from the Lemma \[lem:traverseTree\] can be used to check if a given graph $G$ is a tree or to return a non-tree edge.
\[lem:isTree\] Given an $n$-vertex graph $G$ and a vertex $r$, there is an $O(\log n)$-bits algorithm that answers in $O(n^2)$ time if the connected component with $r$ in $G$ is a tree or it returns a back-edge that is a non-tree edge in the DFS tree rooted at $r$.
We want to use our algorithm of Lemma \[lem:traverseTree\]. To do so, let us store the first edge $e$ used by the algorithm. As stated in the lemma, the algorithm only works on trees. However, it is not hard to see that we can run the algorithm as long as it does not discover/use a back edge $\{u, v\}$ to move from $u$ to $v$ where $v$ is not the predecessor of $u$ on the DFS stack, i.e., $\{v, u\}$ is not a tree edge.
Using the invariant that the algorithm has discovered no non-tree back edge, we can check this by running a second DFS from $r$ to $u$. If the check passes, we can go to $v$ and the invariant holds again. If the algorithm returns to $r$ by using $e$ again, return that $G$ is a tree.
The algorithm described uses $O(\log n)$ bits and explores the vertices in the same manner as the algorithm described in Lemma \[lem:traverseTree\]. However, before taking an edge it has to run the algorithm Lemma \[lem:traverseTree\] to check for a back edge, which can be done in $O(n)$ time. Because the algorithm stops immediately after it finds a back edge, the algorithm considers only $O(n)$ edges and so the total running time is $O(n^2)$.
Kernelizations {#sec:kernels}
==============
We start to demonstrate our ideas with an easy kernelization for <span style="font-variant:small-caps;">Cluster Editing</span>.
Cluster Editing {#sec:ce}
---------------
We now focus on <span style="font-variant:small-caps;">Cluster Editing</span>, which is formally defined as follows: given a tuple $(G,k)$ where $G=(V,E)$ is an $n$-vertex $m$-edge graph and $k$ is an integer (i.e., the parameter), is it possible to add or delete at most $k$ edges so that the resulting graph becomes a union of disjoint cliques? Several kernelizations for the problem are known. Gramm et al. [@GraGHN03] presented a kernel with at most $2k^2 + k$ vertices and at most $2k^3 + k^2$ edges that can be computed in $O(n^3)$ time by using several $(n \times n)$ matrices with entries of $\Theta(\log n)$ bits each. Bäcker et al. [@BocBBT09] improved the constants of the kernel size, but also used $(n \times n)$ matrices to check for unaffordable edge modifications. Both kernelizations above have a reduction rule that removes connected components being cliques. The space consumption of that rule is not analyzed in both papers. Furthermore, Fafianie and Kratsch [@FafK14] showed that $t$-pass streaming kernel requires at least $(n-2)/2t$ bits.
We next show that $k$-pass streaming over all graphs $G'$ induced by all vertex-tripes allows us to compute a full kernel by using only $\poly(k)\log n$ bits. We start to describe the details of the first stream and explain the correctness subsequently. Whenever $G'$ is a [*forbidden subgraph*]{}, i.e., consists of exactly two edges between vertices $\{u, v, w\}$, say edges $\{u,v\},\{v,w\}$ and no edge $\{u, w\}$, then allocate initial-zero counters $C_{u,v},C'_{u,v},C_{u,w},C'_{u,w},C_{v,w},C'_{v,w}$—in case that a counter already exists, no allocation of that counter is done—and subsequently increment $C_{u,v},C_{u,w},C'_{v,w}$ by 1 unless one of the counters is already $k+1$. In this case, non of the three counters is changed or allocated. Intuitively, a counter $C_{u,v}$ ($C'_{u,v}$) stores the number of vertex triples $\{u,v,w_1\},\ldots,\{u,v,w_{C_{u,v}}\}$ where either $\{u,v\}$ has to be deleted (added) from $G$ or another edge in each triangle has to be changed. Thus, it suffices to store $C_{u,v}= k+1$ ($C'_{u,v}= k+1$) to know that $\{u,v\}$ will be deleted (added). At the end of the first stream over all vertex triples, if we have more than $(k+1)^2$ vertex pairs with a non-zero counter, we have a no-instance since adding or deleting one edge $\{u,v\}$ can resolve the forbidden subgraph for at most $k+1$ vertex triples—recall that this is the maximum number of forbidden subgraphs we counted above. If in the last stream over all vertex triples, no counter reached $k+1$, then we skipped over no forbidden subgraphs. In other words, our counters indicate exactly those vertices that are not part of a connected component being a clique and we can return the graph induced by all vertices with a non-zero counter as a kernel.
If, in the last stream over all vertex triples, a counter of a vertex pair $\{u,v\}$ reached $k+1$, we modify $G$ by adding and deleting an edge between $u$ and $v$ depending on if $C_{u,v}=k+1$ and $C'_{u,v}=k+1$, respectively. If both counters are $k+1$, we again have a no-instance. After the change, we stream again over all vertex triples. We then find a kernel or have another change. If we are not done after the $k+1$th stream, i.e., after having $k$ required changes, we again have a no-instance.
To store the counters, we use the corresponding edges as keys in a balanced heap. Since we have $O(k^2)$ counters, we can store the counters by using $O(k^2\log n)$ bits and access them in $O(\log k)$ time. To turn the kernel into a full kernel add the $k + 1$ forbidden subgraphs of the $O(k)$ streams.
Given an $n$-vertex $m$-edge instance $(G, k)$ of <span style="font-variant:small-caps;">Cluster Editing</span>, there is an $O(nm\log k)$-time $O(k^2\log n)$-bits kernelization that either outputs a full kernel consisting of $(k+1)^2$ vertices or returns that $(G, k)$ is a no-instance.
Note that $k+1$ streams over all vertex triples allow us to compute a full kernel of size $(k+1)^2$ for <span style="font-variant:small-caps;">Cluster Editing</span>.
We next describe and analyze an implementation of our algorithm above on the read-only word RAM. First of all, we do not need to iterate over all vertex triples $\{u,v,w\}$. Instead, we can iterate over all edges and, for each edge $\{u,v\}$, over all vertices $w$. Thus, using a mutable graph, we can compute a stream in $O(nm\log k)$ time.
Instead of using a complete new stream whenever we add or delete an edge $\{u,v\}$ to the mutable graph, it suffices to update the counters for all triples $\{u,v,w\}$ over all vertices $w$. This can be done in $O(n\log k)$ time. Since we add or delete $O(k)$ edges, the 2nd to $(k+1)$st streams run in $O(nk\log k)=O(nm\log
k)$ time. Clearly, our algorithm uses $O(k^2\log n)$ bits—note that we can assume without loss of generality that $k<m$ or we have a yes-instance.
We finally want to remark that <span style="font-variant:small-caps;">Cluster Deletion</span> where we are allowed to delete edges only, but not to add edges, can be solved in a similar way. A full kernel can be computed almost as described above. The only change is that we return a no-instance for $(G,k)$ whenever a counter $C'_{u,v}=k+1$ for an edge $\{u,v\}$. Counters $C_{u,v}$ are still allowed to increment up to $k+1$.
Path Contraction {#sec:pc}
----------------
Let $G$ be an $n$-node $m$-edge graph and $C$ is a subset of edges of $G$. We write $G/C$ for the graph obtained from $G$ by contracting each edge in $C$. [*Contracting an edge*]{} is done by merging its endpoints and removing any loops or parallel edges afterwards. In the [*path contraction*]{} problem, a graph $G = (V, E)$ is given and the task is to find a set $C \subseteq E$ such that $G/C$ is a path. In the parameterized version, called <span style="font-variant:small-caps;">Path Contraction</span>, a graph $G = (V, E)$ with a parameter $k \in {\mathbb{N}}$ is given, and the task is to find a set $C \in E$ with $|C| \le k$ such that $G/C$ is a path.
The currently best kernelization for <span style="font-variant:small-caps;">Path Contraction</span> runs in linear time and produces a kernel consisting of $3k + 4$ vertices. It is due to Seng and Sun [@ShengS19], and builds on Heggernes et al.’s algorithm [@HegHLLP14]. One reduction rule used in both algorithms is to determine the bridges the removal of which would disconnect the graph into components of size at least $k + 2$. The reduction rule then merges the endpoints of those bridges.
Finding bridges is closely related to the connectivity problem. Instead of computing bridges with our kernelization algorithm, we introduce a new reduction rule below, which is implemented by a breath-first search (BFS). We shortly sketch a usual BFS and the construction of a [*BFS tree*]{}. The BFS visits the vertices of an input graph round-wise. As a preparation of the first round it puts some vertex $v$ into a queue $Q$ and starts a round. In a round it dequeues every vertex $u$ of $Q$, and marks $u$ as visited. Moreover, it puts every unvisited vertex $w \in N(u)$ into a queue $Q'$. We then say that $w$ was first discovered from $u$ and add the edge $\{u, w\}$ to an initial empty BFS tree. If $Q'$ is empty at the end of the round, the BFS finishes. Otherwise, it proceeds with the next round with $Q := Q'$ and $Q' := \emptyset$.
To present our reduction rules, whose correctness is explained subsequently, we define a path $P = v_0, \ldots, v_{\ell - 1}$ ($\ell \in {\mathbb{N}}$) to be [*clean*]{} exactly if its inner vertices $v_1, \ldots, v_{\ell - 2}$ are all of degree $2$. Moreover, we call $v_0$ a [*start*]{} vertex of $P$ and $v_{\ell - 1}$ the [*end*]{} vertex of $P$.
Rule 1
: If $G$ contains more than $n + k$ edges or a tree $T$ as a subgraph such that $T$ has more than $k+2$ leaves, output “no-instance”.
Rule 2
: If there is a clean path $P$ that consists of more than $k + 2$ vertices, contract all but $k + 2$ arbitrary edges of $P$.
The first part of Rule 1 corresponds to the following well-known reduction rule for path contraction: an $n$-node graph with more than $n + k$ edges is a no-instance since $k+1$ edges can not be part of an $n$-vertex path. Concerning the second part, observe: if we have a tree with $k + 2$ leaves, then a path can connect two leaves, but all other leaves must be contracted with its parents. Therefore, we can bound the number of leaves in a BFS tree (and thus the size of each queue $Q$ of the BFS) by $k+2$ or we have a no-instance.
We now consider Rule 2. Observe that either all edges of $P$ must be chosen to be contracted or none at all. Moreover, no algorithm can contract all edges of a clean path with $\ell > k+1$ vertices. Thus, contracting an arbitrary edge of $P$ is a safe reduction rule and we get a full kernel.
Assume in the following that an exhaustive application of our reduction rules does not return “no-instance”. For the time being, let us ignore vertices of degree 2 in our BFS tree by removing them and connecting their two neighbors by an edge. By Rule 1, we have a tree with at most $k+2$ leaves and thus at most $k+1$ edges. Undoing the removal of the degree-2 vertices, each edge can be replaced by a clean path with at most $k + 2$ vertices, i.e., on top of each edge we add at most $k-1$ vertices. Thus, after applying Rule 1 and 2, we get a kernel of size at most $k+2+(k+1)(k-1)=k^2+k+1$.
We next sketch our ideas to run the kernelization in $O(n \log k)$ time by using $O(k^2 \log n)$ bits by basically running a BFS once—see also Fig \[fig:pc\]. However, we change the information on the queue. Instead of storing just vertices $v$ on the BFS queue we store quadruples. Each quadruple $(v, p, i, v^*)$ consists of the vertex $v$ and its predecessor $p$ if $v$ is not the root. Moreover, let $P=(v_1,\ldots,v_i)$ be the clean path induced by the successive ancestors of $v$ in the BFS tree that have degree $2$. Then $i$ is the number of vertices of $P$, i.e., $v=v_i$ and $v^*=v_{k+1}$ is the $(k+1)$th vertex on $P$. This way we can easily check both reduction rules above.
Given an $n$-vertex instance $(G, k)$ of <span style="font-variant:small-caps;">Path Contraction</span>, there is a $O(n \log k)$-time $O(k^2 \log n)$-bits kernelization that either outputs a full kernel consisting $2(k^2 - k + 1)$ vertices and a kernel of $3k + 4$ vertices or returns that $(G, k)$ is a no-instance.
The BFS visits the vertices as usual and updates its quadruples as follows. For each quadruple $(v, p, i, v^*)$, it iterates over $v$’s neighborhood and puts for every unvisited neighbor $v'$ the quadruple $(v', v, 1, \op{null})$ if $v$ is of degree other than two, and otherwise the quadruple $(v', v, i + 1, v^{**})$ where $v^{**}$ is $v$ if $i = k + 1$, otherwise $v^{**} = v^{*}$.
By Rule 1 we can bound the size of the BFS queue by $k+2$ and the size of the kernel by $2k+3$. To ensure Rule 1, we can easily count the number of leaves in the kernel.
We now describe how a kernel $(G', k')$ can be constructed in adherence to Rule 2. Instead of contracting arbitrary edges we contract edges at the end of a clean path. The contraction is realized by not copying the inner vertices and edges at the end of a clean path while the BFS traverses the paths and connecting the $(k+1)$st vertex with the last vertex of the path in the kernel. To avoid adding vertices into the queue that are already visited we maintain the vertices of the previous and the current queue inside two balanced heaps, respectively.
For simplicity, we first ignore a problem that two vertices in the BFS queue may be adjacent (e.g., the BFS starts to explore a clean path from both its borders simultaneously).
For each quadruple $(v, p, i, v^*)$, we add the vertex $v$ into $G'$ if $i \le k + 1$ and if additionally $p \ne \op{null}$, we also add the edge $\{v, p\}$ into $G'$. If the degree of $v$ in $G$ is not two, then $v$ is the [*end of a clean path*]{} and we add the edge $\{v^*, v\}$ if $i > k + 1$ (the bold edges in Fig \[fig:pc\]). Since edges between visited vertices also need to be part of the kernel (in Fig. \[fig:pc\] they are shown dashed), we additionally add for every $u \in N(v)$ with $u\ \textrm{is already visited}$, the edge $\{v, u\}$ into $G'$. To check quickly if $u$ is visited we use our two balanced heaps maintaining the vertices of the queues.
We now consider the case where two vertices $v$ and $v'$ on the BFS queue are connected to each other in $G$. An example with three connected vertices in the queue is showed at the right part of Figure \[fig:pc\]. If such a vertex $v$ is processed, the clean paths end at $v$ and $v'$ and we run the changes as described above. In addition we connect $v$ and $v'$. Moreover, because a clean path can be explored from both its endpoints, our algorithm then can add up to $2(k+2)$ vertices of a clean path into the kernel. In fact, we only guarantee the exhaustive application of the following reduction rule instead of Rule 2.
Rule 2’
: If there is a clean path $P$ that consists of more than $2(k + 2)$ vertices, contract all but $2(k + 2)$ arbitrary edges of $P$.
Thus, we can only conclude to have a no-instance if the kernel size $n'$ exceeds $2(k^2 - k + 1)$ vertices or $n' - 1 + k + 1$ edges. Our kernel is still small enough to apply Seng and Sun’s linear-time kernelization within our space bounds. We obtain a kernel of $3k + 4$ vertices.
It remains to show the space and time bounds of our kernelization. The queue size is bounded by $O(k)$ vertices and thus cannot exceed $O(k \log n)$ bits. The kernel is bounded by $O(k^2)$ vertices and $O(k^2)$ edges and thus uses $O(k^2 \log n)$ bits. In total, we use $O(k^2 \log n)$ bits. Concerting the time bound, a standard BFS runs in $O(n + m)$ time. By Rule 1 $m = O(n)$ or we stop. The algorithm has to check for each vertex if it is in a balanced heap of size at most $k + 2$, which takes $O(\log k)$ time per vertex. In total we have running time of $O(n \log k)$.
Feedback Vertex Set {#sec:fvs}
-------------------
Given a graph $G = (V, E)$ a [*feedback vertex set*]{} $F \subseteq V$ is the set of vertices whose removal from $G$ turns $G$ into an acyclic graph (also so-called forest). In a parameterized version, called <span style="font-variant:small-caps;">Feedback Vertex Set</span>, a tuple $(G,k)$ is given where $k$ is a parameter and we search for a feedback vertex set $F$ of size at most $k$.
Iwata showed a kernelization for <span style="font-variant:small-caps;">Feedback Vertex Set</span> that produces a kernel consisting of at most $2k^2 + k$ vertices and $4k^2$ edges and runs in $O(k^4 m)$ time [@Iwata17]. Among other reduction rules he makes use of some reduction rules described below. For the last rule we need to define a $v$-[*flower of order $d$*]{} as a set of $d$ cycles pairwise intersecting exactly on $v$.
Basic Rule 1
: Remove a vertex $v$ with $\op{deg}(v) \le 1$.
Basic Rule 2
: Remove a vertex $v$ that has only two incident edges $\{v, u\}$ and $\{v, w\}$ (possibly $u = w$), and add the edge $\{u, w\}$.
Flower Rule
: Remove a vertex $v$ if a $v$-flower of order $k + 1$ exists and reduce $k$ by 1.
We start to describe our general idea to find a kernel for <span style="font-variant:small-caps;">Feedback Vertex Set</span> with $\poly(k) \log n$ bits. In a first step, we want to find the approximated feedback vertex set $X$ consisting of $O(k^2)$ vertices. By definition, $G[V \setminus X]$ is a forest, i.e., it consists of connected components that are all trees. In a second step, we analyze the relation between these trees and the vertices of $X$. This allows us to remove several trees and to shrink the remaining ones. Moreover, whenever we get the knowledge that a vertex $v$ must be part of every solution, we put $v$ into a set $F$. Subsequently, we can simplify the instance because every vertex in $F$ with its incident edges can be removed from the given graph. In other words, we consider only the graph $G_F = G[V_F = V \setminus F]$ and analyze the relation between components of the graph $G_X = G[V_F \setminus X]$ and the vertices of $X$. We next present our idea with more details. Our implementation is given after the high-level description. Initially set $F = \emptyset$ and $X = \emptyset$.
[**Part A (approx. FVS $X$ of max size $k^2 + 3k - 2$):**]{} The idea of our approximation algorithm can be described with the following steps sketched in Fig. \[fig:aprox\]: 1) If $G$ has $m > n - 1 + kn$ edges, output that $(G, k)$ is a no-instance. 2) Take $k$ vertices with the largest degree into $X$. 3) Obtain a graph $G_Y$ from $G_X$ by exhaustively applying Basic Rule 1. 4) Add all vertices with degree greater than $2$ of $G_Y$ into $X$. 5) Obtain a graph $G_Z$ from $G_X$ by exhaustively applying Basic Rule 1. Since all vertices of $G_z$ have degree two afterwards, every connected component is a cycle. 6) For each vertex $v$ of $G_Z$ explore its cycle. Select the smallest vertex $v$ of the cycle into the set $X$—intuitively speaking, we cut the component at $v$. If more than $k$ such cycles exist, output that $(G, k)$ is a no-instance. Otherwise, proceed with the next vertex of $G_Z$.
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We next show that the set $X$ obtained by the algorithm is an approximated feedback vertex set for $G$. For the analysis we assume to know the optimal feedback vertex set $F$ of $\ell \le k$ vertices in $(G, k)$. Each of the $\ell$ vertices can be connected with each of the $n$ vertices in the graph. Moreover, the removal of $F$ from $G$ removes at most $kn$ edges and, since $F$ is a feedback vertex set, it turns $G$ into a forest with at most $n - 1$ edges. Thus, if $G$ has more than $n - 1 + kn$ edges, we deal with a no-instance.
After Step 2 the set $X$ consists of $k$ vertices with the largest degree. Assume that the vertices $x_0, x_1, \ldots, x_{k - 1} \in X$ and the vertices $v_0, v_1, \ldots, v_{\ell-1} \in F$ ($\ell \le k$) are ordered by their degree, starting with the largest. We know that $\deg(x_i) \ge \deg(v_i)$ holds for each $i = 0, \ldots, \ell - 1$. We must take into account that some of the vertices of $X$ are connected to each other, meaning that the removal of $x_i$ may reduce the degree of each vertex $x_j$ ($j > i$) by one. Since the removal of $F$ turns $G$ into a forest, i.e., $m - \sum_{i = 0}^{\ell - 1} \deg(v_i) \le n - 1$, $G_X$ has at most $m - \sum_{i = 0}^{k - 1} \deg(x_i) \le m - \sum_{i = 0}^{\ell - 1}(\deg(v_i) + i) = m - \sum_{i=0}^{\ell - 1}\deg(v_i) + \sum_{i = 0}^{\ell - 1}i \le n - 1 + k^2/2 - k/2$ edges.
Step 3 removes every subgraph of $G_X$ that is a tree whose root is a cut vertex in $G$ (the only connection to the rest of $G$). We so obtain a graph $G_Y$. Let $\ell$ be the number of vertices that are removed by Basic Rule 1 to produce $G_Y$. Then $G_Y$ has $n' = n - k - \ell$ vertices that are all of degree greater than $2$ and at most $m' = n - 1 + k^2/2 - k/2 - \ell$ edges. It is not hard to see that $G_Y$ has at most $2(m' - n') = k^2 + k - 2$ vertices of degree greater than $2$. In Step 4, we thus add at most $k^2 + k - 2$ vertices into $X$. What remains in $G_Z$ after Step 5 are separate cycles, each requires us to take one arbitrary vertex into $X$, meaning that at most $k$ such cycles may exist or we have a no-instance. Hence after applying Step 6 on a yes-instance, we get a feedback vertex set $X$ of size $k + (k^2 + k - 2) + k = k^2 + 3k - 2$ vertices. The next lemma shows an implementation of the algorithm above.
\[thm:approxfvs\] Given an $n$-vertex $m$-edge instance $(G, k)$ of <span style="font-variant:small-caps;">Feedback Vertex Set</span>, there is an $O(mn^2 \log k)$-time $O(k^2 \log n)$-bits algorithm that either returns a feedback vertex set $X$ consisting of at most $k^2 + 3k - 2$ vertices or returns that $(G, k)$ is a no-instance.
We show how to realize the non-trivial steps of the algorithm above. Let $V$ be the vertices of the given graph. In Step 2, we store the $k$ vertices of the largest degree in a balanced heap to answer queries for $G_X$ in $O(\log k)$ time by checking for modifications stored in the balanced heap.
In contrast to our description of Step 3, we do not explicitly compute $G_Y$ since we cannot store such a graph. Instead we compute the information required on demand. To run Step 4, we iterate over all vertices $v_1, \ldots, v_n$ of $G$. To determine the degree of $v_i$, we iterate over all neighbors $u$ of $v_i$. For each neighbor $u$, we check with Lemma \[lem:isTree\] if the connected component of $G[V\setminus \{v_i\}]$ containing $u$ is a tree. If so, we count the edge. The degree of $v_i$ in $G_Y$ is equal to the number of edges that we counted. We also do not run Step 5 to remove subtrees from $G_X$. Instead, we need to ignore these subtrees in Step 6. In an outer loop, iterate over the vertices $v_1, \ldots, v_n $ of $G$, but skip over the vertices in $X$. We now want to move over a cycle in $G_Z$ and select the smallest vertex into $X$. Let $v=v_i$ be the currently visited vertex on the cycle, let $z := v$ be the currently smallest vertex found in the cycle yet and $p := v$ be the previously visited vertex on the cycle. In an inner loop repeat until broken: Take the two edges $\{v, w_1\}$ and $\{v, w_2\}$ for which Lemma \[lem:isTree\] finds no back edge in the component of $G[V \setminus X]$. Go to a vertex $w\in \{w_1,w_2\}\setminus \{p\}$ by updating $p:=v$, $v:=w$. If $v = v_i$ break the inner loop, add the vertex with the smallest [<span style="font-variant:small-caps;">ID</span>]{} to $X$, and continue the outer loop. If it is the case that no vertex $w$ is found, then the cycle is already broken in a previous step of the outer loop so that we also continue the outer loop. To avoid adding too many vertices in $X$ during Step 6, count the added vertices and, if necessary, stop and answer “no-instance”. We now focus on the running time and space analysis. Step 1 runs in $O(1)$ time. Step 2 runs in $O(n \log k)$ time: checking for membership, adding, removing and retrieving the largest vertex of $X$ runs in $O(\log k)$ time. This has to be done for up to $n$ elements. runs in $O(mn^2 \log k)$ time: for each vertex the membership in $X$ has to be checked in $O(\log k)$ time and, for each edge, Lemma \[lem:isTree\] has to be called with the adaptation that edges to vertices of $X$ have to be ignored. This sums up to $O(mn^2 \log k)$ time. Observe that have the same running time, which is also the total runtime.
Concerning space, note that we use a constant amount of variables including the algorithm of Lemma \[lem:isTree\]. Thus, $O(\log n)$ bits suffice. The number of vertices in $X$ is bounded by $k^2 + 3k - 2$. In total our algorithm uses $O(k^2 \log n)$ bits.
[**Part B (construct the kernel):**]{} Let us say that a vertex $x \in X$ and a tree are [*in touch once*]{} if $x$ is adjacent to one vertex of the tree. Moreover, we define [*in touch ($\ell\in {\mathbb{N}}$ times)*]{} if $x$ is adjacent to $\ell$ vertices of the tree. For an easier description, we start to define a partition $\mathcal{T}_0, \mathcal{T}_1$ and $\mathcal{T}_2$ of the trees in $G_{X}$ as follows: $\mathcal{T}_0$ is set of trees in $G_{X}$ that are in touch at most once to at most one $x \in X$ and are not in touch to all other $x' \in X$. $\mathcal{T}_1$ is the set of trees in $G_{X} \setminus \mathcal{T}_0$ such that each vertex $x \in X$ is in touch at most once to each tree. $\mathcal{T}_2$ is the set of the remaining trees in $G_{X}$. In other words, each tree in $\mathcal{T}_2$ is in touch at least twice to a vertex $x \in X$.
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To bound the number of trees in our kernel, we use the Reduction Rules 1 - 3 to remove superfluous trees and the Reduction Rules 4 - 6 to shrink the trees themselves.
Rule 1
: Remove every tree of $\mathcal{T}_0$.
Rule 2
: For every pair $x_1, x_2 \in X$, choose up to $k + 2$ trees of $\mathcal{T}_1$ that are in touch with $x_1$ and with $x_2$ (if there are less, then choose all). Remove all remaining trees of $\mathcal{T}_1$.
Rule 3
: If $x \in X$ is in touch with at least $k + 1$ different trees of $\mathcal{T}_2$, then move $x$ from $X$ into $F$, reduce $k$ by $1$ and restart.
Rule 4
: If ${2k}$ or more children of a vertex $v$ are the roots of maximal subtrees in $G_X$ such that each subtree is in touch with the same vertex $x \in X$, then take $v$ into $X$ and the pair $\{v,x\}$ into an initial empty set $Y$. If a vertex $w$ occurs in more than $k+1$ pairs of $Y$, then move $w$ from $X$ into $F$, reduce $k$ by 1, and restart. Otherwise and if $|Y|\ge k^2-1$, we have a no-instance.
Rule 5
: If Rule 4 does not apply and a tree $T$ of $G_X$ is in touch at least ${2k}(k + 1)$ times with the same vertex $x \in X$, then move $x$ from $X$ into $F$, reduce $k$ by 1 and restart.
Rule 6
: Apply Basic Rule 1 and 2 on vertices that are not connected to a vertex of $X$.
Rules 1 – 6 are safe.
Rule 1 is clearly safe since no vertex of a tree being in touch at most once with at most one vertex of $X$ can be part of a cycle. We next argue that Rule 2 is safe. Assume that vertices $x_1, x_2 \in X$ are in touch with a tree of $\mathcal{T}_1$. If another tree exists that is in touch with the same vertices, we have a cycle. If $\ell \ge k + 2$ such trees exist, then we have $\ell$ disjoint paths from $x_1$ to $x_2$ and every feedback vertex set must contain $x_1$ or $x_2$. If we now remove a tree from $\mathcal{T}_1$, then all pairs of vertices that are in touch with the tree have already “enough” common trees. Let $X' \subseteq X$ be the vertices that are in touch with the removed tree. Then all except one vertex of $X'$ are removed in every solution and thus no cycle is passing through the removed tree. Therefore, Rule 2 is safe. We next focus on Rule 3. By definition $x \in X$ forms a cycle with each tree being in touch with $x$ at least twice. Thus, if the rule applies, we have a $(k+1)$-flower and are safe.
For Rule 4, note that at least ${2k}$ internally vertex-disjoint paths exist between vertex $x$ and $v$. Thus, any feedback vertex set must contain $x$ or $v$. Consider two pairs $\{v,x\}, \{ v',x'\}\in Y$ and assume that the latter pair was added after the first pair into $Y$. Then all ${2k}$ paths between $v'$ and $x'$ are in the same subtree of $G[V\setminus (X \setminus \{
v',x'\})]$. This means that at most one of the ${2k}$ paths between $v$ and $x$ has common vertices with the ${2k}$ paths between $v'$ and $x'$. If we ignore that path, then all remaining paths between $v$ and $x$ as well as between $v'$ and $x'$ are pairwise internally vertex disjoint. Assume that we have pairs $\{v_1,w\},\ldots,\{v_{k+1},w\}$ and $w$ is moved into $F$. If we remove one path between all pairs except the last pair, then the paths are internally vertex disjoint to the paths between the last pair. Furthermore, we have to remove one other path between all pairs except the last two pairs to make the paths internally disjoint to the paths between the second last pair, etc. Altogether, we find ${2k}-(k-1)=k+1$ paths between all pairs that are internally vertex disjoint. Therefore, we have a flower of degree $k+1$ and it is correct to add $w$ into $F$. Otherwise and if $|Y| \ge k^2 - 1$, we have $(k^2-1)/(k-1)=k+1$ pairs such that the vertices in all pairs are pairwise disjoint. Again, we have $k+1$ paths between all pairs such that these paths are internally vertex disjoint. Thus, one vertex of each of the $k+1$ pairs must be in a solution and we have a no-instance.
We next consider Rule 5. Since Rule 4 can not be applied, at most ${2k}-1$ children of every vertex $v$ can be the root of a maximal subtree in $T$ that are in touch with a vertex of $X$. Let $U$ be a set of the vertices of $T$ that are adjacent to $x$. Ignoring the parts of $T$ that are not on a path between $u,u'\in U$, we obtain a tree with maximum degree $\Delta={2k}-1$. One can easily see that, given a tree $T = (V_T, E_T)$ with maximum degree $\Delta$ and $U \subseteq
V_T$, we can find $\lfloor |U|/(\Delta + 1)\rfloor$ pairs of vertices in $U$ such that the paths in $T$ between each pair are vertex disjoint[^1]. This means that we have an $x$-flower of order $|U|/({2k})=k+1$ and Rule 5 is safe. Clearly, also Rule 6 is safe.
We next want to show that we get a kernel of size $O(k^9)$ if no rule applies any more. Clearly, $|\mathcal{T}_0| = 0$ by Rule 1. After Rule 2, $|\mathcal{T}_1| \le |X|^2(k + 2)$ holds because we have less than $|X|^2$ pairs, each have chosen at most $k + 2$ trees. Moreover, Rule 3 guarantees us that $|\mathcal{T}_2| \le |X|k$ since every $x$ is connected to at most $k$ trees of $\mathcal{T}_2$. In total, we have $|X|^2(k + 2) + |X|k$ trees left in $G_X$.
By Rule 5, each tree $T$ is in touch less than $2k(k + 1)$ times with each $x\in X$. Thus, less than $2k(k + 1)|X|$ vertices of $T$ are connected to vertices of $X$. By Rule 6, $T$ shrinks to at most $2k(k + 1)|X|$ leaves and at most $4k(k + 1)|X|$ vertices in total. Thus, we have $(|X|^2(k + 2) + |X|k) (4k(k + 1)|X|)$ vertices in all trees, $|X|=k^2 + 3k - 2$, and get a kernel with at most vertices and $(|X|^2(k + 2) + |X|k)(2k(k + 1)|X|) = O(k^9)$ edges.
After computing such a kernel it remains to apply the currently best kernelization algorithm for <span style="font-variant:small-caps;">Feedback Vertex Set</span> on $(G', k)$ to reduce the kernel size to $2k^2 + k$ vertices.
\[th:fbs\] Given an $n$-vertex instance $(G, k)$ of <span style="font-variant:small-caps;">Feedback Vertex Set</span>, there is an $O(mn^2 \log k)$-time $O(k^{9} \log n)$-bits kernelization that either outputs a kernel consisting of $2k^2 + k$ vertices or returns that $(G, k)$ is a no-instance.
[^1]: For a proof see, e.g., Erlebach et al. [@ErlKLSS19 Lemma 2.4].
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---
abstract: 'Any irreducible Dynkin diagram $\Delta$ is obtained from an irreducible Dynkin diagram $\Delta_h$ of type ${\mathrm{ADE}}$ by folding via graph automorphisms. For any simple complex Lie group $G$ with Dynkin diagram $\Delta$ and compact Riemann surface ${\Sigma}$, we give a Lie-theoretic construction of families of quasi-projective Calabi-Yau threefolds together with an action of graph automorphisms over the Hitchin base associated to the pair $({\Sigma}, G)$ . These give rise to Calabi-Yau orbifolds over the same base. Their intermediate Jacobian fibration, constructed in terms of equivariant cohomology, is isomorphic to the Hitchin system of the same type away from singular fibers.'
address: 'FB Mathematik, Universität Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany'
author:
- Florian Beck
bibliography:
- 'bibtex1.bib'
title: 'Calabi-Yau orbifolds over Hitchin bases'
---
Introduction
============
In recent years, it has been realized that the $G$-Hitchin system ([@Hit1], [@Hit2]) $${\bm{h}}:\mathcal{M}_{\mathrm{H}}({\Sigma}, G)\to {\mathbf{B}}({\Sigma}, G),$$ for a compact Riemann surface ${\Sigma}$ of genus $\geq 2$ and a simple complex Lie group $G$ of type ${\mathrm{ADE}}$, are closely related to certain (quasi-projective) Calabi-Yau threefolds ([@DDD], [@DDP], [@Tbranes], [@Tbranes2]). For example, if $G$ is a simple adjoint complex Lie group of type ${\mathrm{ADE}}$, then there exists a family ${\bm{\pi}}:{\mathcal{X}}\to {\mathbf{B}}({\Sigma},G)$ of quasi-projective Gorenstein Calabi-Yau threefolds over the Hitchin base ${\mathbf{B}}({\Sigma}, G)$ such that the intermediate Jacobian $$J^2(X_b)=H^3(X_b,{\mathbb{C}})/\left(F^2H^3(X_b,{\mathbb{C}})+H^3(X_b,{\mathbb{Z}})\right),\quad X_b:={\bm{\pi}}^{-1}(b),$$ is isomorphic to the Hitchin fiber ${\bm{h}}^{-1}(b)$ for generic $b\in {\mathbf{B}}({\Sigma},G)$ ([@DDP]). In this paper, we extend this to a global result for all simple adjoint and simply-connected complex Lie groups of any Dynkin type by constructing families of Calabi-Yau orbifolds/global orbifold stacks[^1] $$\label{intro:fam}
[{\mathcal{X}}/{\mathbf{C}}]\to {\mathbf{B}}({\Sigma}, \Delta)={\mathbf{B}}({\Sigma},G), \quad \Delta=\Delta(G).$$ Here $\Delta(G)$ is the irreducible Dynkin diagram associated to $G$ which is obtained by a unique irreducible Dynkin diagram $\Delta_h$ of type ${\mathrm{ADE}}$ by folding via graph automorphisms ${\mathbf{C}}\subset \operatorname{Aut}(\Delta_h)$, i.e. $\Delta=\Delta_h^{{\mathbf{C}}}$. Moreover, ${\bm{\pi}}:{\mathcal{X}}\to {\mathbf{B}}({\Sigma}, \Delta)$ is a family of quasi-projective Gorenstein Calabi-Yau threefolds with ${\mathbf{C}}$-trivial canonical class. More precisely, we prove in Section \[s:mhs\]:
\[thm:thm1\] Let $[{\mathcal{X}}/{\mathbf{C}}]\to {\mathbf{B}}={\mathbf{B}}({\Sigma},\Delta)$ be a family of global Calabi-Yau orbifolds as in (\[intro:fam\]). Its intermediate Jacobian fibration $$\mathcal{J}^2([{\mathcal{X}}^\circ/{\mathbf{C}}]) \to {\mathbf{B}}^\circ,$$ defined in integral equivariant cohomology, is isomorphic to the $G_{ad}(\Delta)$-Hitchin system over a Zarisiki-open and dense ${\mathbf{B}}^\circ \subset {\mathbf{B}}$ where $G_{ad}(\Delta)$ is the simple adjoint complex Lie group with Dynkin diagram $\Delta$.
The basic idea for constructing ${\mathcal{X}}\to {\mathbf{B}}$ (without the ${\mathbf{C}}$-action) goes back to [@Sz1] and [@DDP]. However, we offer two main novelties. The first one is a consequent use of Lie-theoretic methods, namely Slodowy slices, which makes the link to Hitchin systems from the start. For example, we construct nowhere-vanishing sections $s_b\in H^0(X_b,K_{X_b})$ of the canonical class $K_{X_b}$ of $X_b$, $b\in {\mathbf{B}}$, in terms of the Kostant-Kirillov form. The second one is the construction of a ${\mathbf{C}}$-action on the family ${\mathcal{X}}\to {\mathbf{B}}$ and we prove that the sections $s_b$ are ${\mathbf{C}}$-invariant. Therefore the global quotient family $[{\mathcal{X}}/{\mathbf{C}}]\to {\mathbf{B}}$ is indeed a family of Calabi-Yau orbifolds.\
Another important step in the proof of Theorem \[thm:thm1\] is an isomorphism $$\label{eq:isoMHS}
H^3_{{\mathbf{C}}}(X_b,{\mathbb{Z}}) \cong H^3(X_b,{\mathbb{Z}})^{{\mathbf{C}}}, \quad b\in {\mathbf{B}}^\circ,$$ between Deligne’s ${\mathbb{Z}}$-MHS on the integral equivariant cohomology $H^3_{{\mathbf{C}}}(X_b,{\mathbb{Z}})$ and the ${\mathbb{Z}}$-MHS on the ${\mathbf{C}}$-invariant part $H^3(X_b,{\mathbb{Z}})^{{\mathbf{C}}}$ of the integral cohomology. Even though (\[eq:isoMHS\]) always holds true over ${\mathbb{Q}}$, it is false in general over ${\mathbb{Z}}$ due to torsion. For example, it fails for the minimal resolution of the $\Delta_h$-singularity (Example \[ex:eqcoh\]). Combined with the global isomorphism $$\begin{gathered}
\label{eq:J2C}
\mathcal{J}^2_{{\mathbf{C}}}({\mathcal{X}}^\circ) \cong \mathcal{M}_{\mathrm{H}}({\Sigma}, G_{ad}(\Delta)),
$$ over ${\mathbf{B}^\circ}$ (Corollary 5 in [@Beck]), where $\mathcal{J}_{{\mathbf{C}}}^2({\mathcal{X}}^\circ)\to {\mathbf{B}^\circ}$ is the intermediate Jacobian fibration defined by ${\mathbf{C}}$-invariants in cohomology, we conclude Theorem \[thm:thm1\]. In fact, the isomorphism (\[eq:J2C\]) gave a hint that we have to use with Calabi-Yau orbifolds if ${\mathbf{C}}\neq 1$ instead of ordinary Calabi-Yau threefolds if ${\mathbf{C}}\neq 1$. By working with equivariant compactly supported cohomology instead of equivariant cohomology, we further obtain the Langlands dual version of Theorem \[thm:thm1\] and hence cover all simple simply-connected complex Lie groups as well.\
Besides the ${\mathbf{C}}$-action and construction of ${\mathbf{C}}$-invariant holomorphic volume forms, the Lie-theoretic construction gives a way to study the smooth locus of the family ${\mathcal{X}}\to {\mathbf{B}}$. As an application, we see in Corollary \[cor:comp\] that if $\Delta_h$ is of type ${\mathrm{ADE}}$ and admits non-trivial Dynkin graph automorphisms ${\mathbf{C}}$ (see Remark \[r:dynkin\] in Appendix \[SectFolding\]), then the smooth locus of the family ${\mathcal{X}}\to {\mathbf{B}}({\Sigma}, \Delta_h)$ is much larger than the smooth locus of $\Delta_h$-cameral curves. It follows that the family ${\mathcal{X}}\to {\mathbf{B}}({\Sigma}, \Delta_h)$ does not only geometrically encode the smooth loci of $$\begin{tikzcd}
\mathcal{M}_{\mathrm{H}}({\Sigma}, G_{ad}(\Delta_h)) \ar[r] & {\mathbf{B}}({\Sigma}, \Delta) & \ar[l] \mathcal{M}_{\mathrm{H}}({\Sigma}, G_{sc}(\Delta_h))
\end{tikzcd}$$ but correspondingly for $G_{ad}(\Delta)$ and $G_{sc}(\Delta)$, under the natural inclusion ${\mathbf{B}}({\Sigma}, \Delta)\subset {\mathbf{B}}({\Sigma}, \Delta_h)$ for $\Delta=\Delta_h^{\mathbf{C}}$.
Relation to other works
-----------------------
Our work is intellectually indebted to [@Sz1], [@DDP], [@DDD] and [@Slo]. We relate our approach to the first three works, for example we give explicit equations in the spirit of [@DDD] (also see Theorem \[thm:defX0\]). For that reason, we give two Slodowy slices $S\subset \mathfrak{so}(5,{\mathbb{C}}) $ and $S_h \subset \mathfrak{sl}(4,{\mathbb{C}}) $ and show by a direct computation that they both realize the semi-universal deformation of the $\mathrm{B}_2$-singularity in Appendix \[a:ex\]. Moreover, we show that families of [@Sz1] provide simultaneous resolutions over a cone in the singular locus of ${\bm{\pi}}:{\mathcal{X}}\to {\mathbf{B}}$ (Proposition \[p:simresol\]). Graph automorphisms also appear in [@Sz1] but we use them differently. In particular, orbifold stacks and their cohomology appear in neither of these works. Finally, it is further crucial for our work [@Beck] that the ‘volume forms’ $s_b\in H^0(X_b,K_{X_b})$ induce a period map on ${\mathbf{B}^\circ}$. This period map is a so-called abstract Seiberg-Witten differential which in turn determines the structure of an algebraic integrable system on $\mathcal{J}_{\mathbf{C}}^2({\mathcal{X}}^\circ/{\mathbf{B}^\circ})\to {\mathbf{B}^\circ}$ (cf. Section 4.4 in [@Beck]).
Outline
-------
Since the key ingredient for the local building blocks of the Calabi-Yau orbifolds (\[intro:fam\]) are Slodowy slices, we recall and prove important preparatory results on them in Section \[s:local\]. Section \[s:cy\] begins with a recap on Hitchin bases and cameral curves and continues with the construction of Calabi-Yau orbifolds and related families of quasi-projective Calabi-Yau threefolds. In Section \[s:mhs\] we study mixed Hodge structures on equivariant cohomology and prove Theorem \[thm:thm1\]. Appendix \[SectFolding\] contains our conventions on folding of Dynkin diagrams and Appendix \[a:ex\] gives two examples of Slodowy slices that realize the semi-universal deformations of the same surface singularity.
Acknowledgements
----------------
It is a pleasure to thank Katrin Wendland and Emanuel Scheidegger as well as Ron Donagi and Tony Pantev for discussions and questions on this paper’s content. This work was supported by the DFG Graduiertenkolleg 1821 “Cohomological Methods in Geometry” and through the DFG Emmy-Noether grant on “Building blocks of physical theories from the geometry of quantization and BPS states”, number AL 1407/2-1.
Local preparations {#s:local}
==================
$\Delta$-singularities and Slodowy slices
-----------------------------------------
Let $\Delta$ be an irreducible Dynkin diagram. Then there is a unique pair[^2] $(\Delta_h,AS(\Delta))$ consisting of an ${\mathrm{ADE}}$-Dynkin diagram and the associated symmetry group $AS(\Delta)\subset \operatorname{Aut}(\Delta_h)$ such that $\Delta$ is obtained by folding, i.e. $\Delta=\Delta_h^{AS}$ (see Appendix \[SectFolding\] for more details). A $\Delta$-singularity $(Y,H)$ ([@Slo]) consists of a (germ of a) surface singularity $Y=(Y,0)$ of type $\Delta_h$ and a subgroup $H\subset \operatorname{Aut}(Y)$ with the following properties:
1. $H\cong AS(\Delta)$;
2. the action of $H$ on $Y-\{0\}$ is free;
3. the induced action on the dual resolution graph of the minimal resolution $\hat{Y}\to Y$ coincides with the $AS(\Delta)$-action on $\Delta_h$.
For $\Delta=\Delta_h$, these are just ${\mathrm{ADE}}$-surface singularities. Every $\Delta$-singularity $(Y,H)$ is quasi-homogeneous (cf. Table \[ADEsings\] in Appendix \[SectFolding\]), in particular, $Y$ carries a ${\mathbb{C}}^*$-action that commutes with the $H$-action. A ${\mathbb{C}}^*$-deformation of $(Y,H)$ is therefore a ${\mathbb{C}}^*\times H$-deformation $\mathcal{Y}\to B$ such that $H$ acts trivially on the base. Each $(Y,H)$ has a semi-universal ${\mathbb{C}}^*$-deformation, see Section $2$ in [@Slo].\
The relation to the simple complex Lie algebra ${\mathfrak{g}}={\mathfrak{g}}(\Delta)$ of type $\Delta$ is as follows: Let $x\in {\mathfrak{g}}$ be a subregular nilpotent element and $(x,y,h)$ an $\mathfrak{sl}_2$-triple in ${\mathfrak{g}}$ with semisimple $h$. Then the Slodowy slice through $x$ (associated to the triple $(x,y,h)$) is given by $$S=S(\Delta):=x+\ker \mathrm{ad}(y).$$ It carries a non-trivial action by the group ${\mathbb{C}}^*\times AS(\Delta)$ (cf. Section 6 in [@Slo]). Here the $AS(\Delta)$-action is defined by the action (via adjunction) of the group[^3] $$\label{eq:Cc}
{\mathbf{C}}= C(x,h)/C(x,h)^\circ \cong AS(\Delta)$$ which is the group of connected components of $C(x,h)=\{g\in G_{ad}~|~g\cdot x=x, g\cdot h= h\}$. For a fixed Cartan subalgebra ${\mathfrak{t}}\subset {\mathfrak{g}}$, denote by $\chi:{\mathfrak{g}}\to {\mathfrak{t}}/W$ the adjoint quotient and $$\sigma:=\chi_{|S}: S\to {\mathfrak{t}}/W$$ its restriction to the Slodowy slice $S$. If we let ${\mathbb{C}}^*$ act on ${\mathfrak{t}}/W$ with *twice* the usual weight and ${\mathbf{C}}$ on ${\mathfrak{t}}/W$ trivially, then $\sigma$ is ${\mathbb{C}}^*\times {\mathbf{C}}$-equivariant. Slodowy has shown that $\sigma: S \to {\mathfrak{t}}/W$ with the ${\mathbb{C}}^*\times {\mathbf{C}}$-action is a ${\mathbb{C}}^*$-semi-universal deformation of the $\Delta$-singularity $(\sigma^{-1}(\bar{0}),x)$ ([@Slo], Section 8.7).\
Finally, we recall Grothendieck’s simultaneous resolution of $\chi:{\mathfrak{g}}\to {\mathfrak{t}}/W$, $$\label{eq:Gsimresol}
\begin{tikzcd}
\tilde{{\mathfrak{g}}}\cong G\times^B \mathfrak{b} \ar[r, "\psi"] \ar[d, "\theta"'] & {\mathfrak{g}}\ar[d, "\chi"] \\
{\mathfrak{t}}\ar[r , "q"] & {\mathfrak{t}}/W.
\end{tikzcd}$$ If $S=x+\ker \mathrm{ad}(y)$ is a Slodowy slice, then (\[eq:Gsimresol\]) restricts to the simultaneous resolution $\psi:\tilde{S}\to S$ of $\sigma: S\to {\mathfrak{t}}/W$ (see Chapter 5 in [@Slo]). If $(x,y,h)$ is the $\mathfrak{sl}_2$-triple defining $S$, then the resulting square is a square of ${\mathbb{C}}^*$-spaces if $h\in {\mathfrak{t}}$ (see Remark 1.53 in [@Beck-thesis] for details). Since any two Cartan subalgebras are conjugate to each other, we will assume $h\in {\mathfrak{t}}$ from now on.
Relative symplectic form
------------------------
It is well-known (e.g. Chapter 1, [@ChrissGinzburg]) that the the Kostant-Kirillov form $\nu=\omega_{KK}$ restricts to a symplectic form on each adjoint orbit $O\subset {\mathfrak{g}}$. Consequently, the relative differential form $\hat{\nu}\in \Omega^2_{\chi}({\mathfrak{g}})$ induced by $\nu$ restricts to a relative symplectic form on the locus ${\mathfrak{g}}^{reg}$ of regular elements in ${\mathfrak{g}}$. This holds true for the restriction $\hat{\nu}_{|S^{reg}}\in \Omega^2_{\sigma^{reg}}$ to $S^{reg}:=S\cap {\mathfrak{g}}^{reg}$ as well (cf. [@GG], Appendix 7). Since $S-S^{reg}$ is of codimension at least $2$, $\hat{\nu}_{|S^{reg}}$ extends to a global section of $K_{\sigma}$ which we also denote by $\hat{\nu}\in \Gamma(S,K_{\sigma})$.
\[p:C\*equivariant\] Let $S \subset {\mathfrak{g}}$ be a Slodowy slice and $\hat{\nu}\in \Gamma(S,K_{\sigma})$ be the previously constructed section. Then $\hat{\nu}$ is nowhere vanishing, in particular $K_\sigma$ is trivializable. Moreover, it is ${\mathbb{C}}^*$-equivariant, $$\lambda^*\hat{\nu}=\lambda^2 \hat{\nu},\quad \lambda\in {\mathbb{C}}^*,$$ as well as ${\mathbf{C}}$-invariant.
The proof proceeds in two steps. In the first step, we use Yamada’s realization ([@Yamada]) of $\theta: \tilde{{\mathfrak{g}}}\to {\mathfrak{t}}$ as a relative symplectic reduction to relate the corresponding relative symplectic form $\hat{\omega} \in \Omega^2_{\theta}(\tilde{{\mathfrak{g}}})$ to $\hat{\nu}\in \Omega^2_\chi({\mathfrak{g}})$ over the locus ${\mathfrak{g}}^{reg}\subset {\mathfrak{g}}$ of regular elements. In the second step, we restrict to $S^{reg}$ and extend over the non-regular locus.\
For the first step, fix a maximal torus $T$ and a Borel subgroup $B\supset T$ in the adjoint group $G=G_{ad}$ and denote by $N\subset B$ the nilradical of $B$. Then the symplectic manifold $(M:=T^*(G/N),\omega_c)$, with the canonical symplectic structure $\omega_c$, carries a natural Hamiltonian $G$- and $T$-action induced by left and right multiplication on $G/N$ respectively. The corresponding moment maps descend to $\hat{\mu}_G:M/T \to {\mathfrak{g}}^*$ and $\hat{\mu}_T:M/T \to {\mathfrak{t}}^*$. Yamada ([@Yamada]) identifies the square (\[eq:Gsimresol\]) with the diagram $$\begin{tikzcd}
M/T \ar[r, "\hat{\mu}_G"] \ar[d, "\hat{\mu}_T"'] & {\mathfrak{g}}\ar[d, "\chi"] \\
{\mathfrak{t}}\ar[r] & {\mathfrak{t}}/W
\end{tikzcd}$$ after ${\mathfrak{g}}^*\cong {\mathfrak{g}}$ and ${\mathfrak{t}}^*\cong {\mathfrak{t}}$ via the Killing form (which we will do from now on). Under this identification, the symplectic form $\omega_c$ descends and induces the relative symplectic form $\hat{\omega}\in \Omega^2_{\theta}(\tilde{{\mathfrak{g}}})$. Moreover, for each $t\in {\mathfrak{t}}$ the induced $G$-action on $(\theta^{-1}(t),\hat{\omega}_t)$ is Hamiltonian and the restriction $\psi_t:\theta^{-1}(t) \to {\mathfrak{g}}$ of $\psi: \tilde{{\mathfrak{g}}}\to {\mathfrak{g}}$ is the corresponding moment map (Theorem 2.5 in [@Yamada]). Now let $\psi^{reg}:\tilde{{\mathfrak{g}}}^{reg}\to {\mathfrak{g}}^{reg}$ be the restriction of $\psi: \tilde{{\mathfrak{g}}} \to {\mathfrak{g}}$ to the regular locus. Note that $$\psi^{reg}_t: \tilde{{\mathfrak{g}}}^{reg}\cap \theta^{-1}(t) \to {\mathfrak{g}}^{reg}\cap \chi^{-1}(\bar{t}),\quad q(t)=\bar{t},$$ is an isomorphism for each $t\in {\mathfrak{t}}$. In fact, it is not difficult to see that $\psi^{reg}_t$ is a symplectomorphism if the left-hand side is endowed with the symplectic form $\hat{\omega}_{t}$ and the right-hand side with the Kostant-Kirillov form $\nu$. Since $\psi^{reg}$ is smooth, $
(\psi^{reg})^*\Omega_\chi^2\cong \Omega^2_\theta
$ naturally. Under this isomorphism, we have $$\psi^*\hat{\nu}=\hat{\omega}\in \Gamma(\tilde{{\mathfrak{g}}}^{reg}, \Omega^2_\theta)$$ because $\psi^{reg}_t$ is a symplectomorphism for each $t\in {\mathfrak{t}}$.\
For the second step, we note that the restriction of $\hat{\omega}$ to $\tilde{S}\subset \tilde{{\mathfrak{g}}}$ is again a relative symplectic form (Theorem 4.5 in [@Yamada]), again denoted by $\hat{\omega}\in \Omega^2_{\tilde{\sigma}}(\tilde{S})$. Again we have a natural isomorphism $$\Phi^{reg}:\psi^*K_{\sigma^{reg}}=\psi^*\Omega^2_{\sigma^{reg}}\to \Omega^2_{\tilde{\sigma}^{reg}}$$ over $\tilde{S}^{reg}$ with $\Phi^{reg}(\psi^*\hat{\nu})=\hat{\omega}$. We claim that $\Phi^{reg}$ extends to an isomorphism $\Phi: \psi^*K_{\sigma} \to K_{\tilde{\sigma}}$ satisfying $$\label{eq:nuomega}
\Phi(\psi^*\hat{\nu})=\hat{\omega}.$$ Since $K_\sigma$ and $K_{\tilde{\sigma}}$ are reflexive, it is sufficient to prove $\mathrm{codim}_{\tilde{S}}\tilde{T}\geq 2$ for $\tilde{T}:=\tilde{S}-\tilde{S}^{reg}$. The components of $\tilde{T}$ of highest dimension lie over the hypersurfaces ${\mathfrak{t}}_{\alpha}-\cap_{\beta\neq \alpha} {\mathfrak{t}}_\beta\cap {\mathfrak{t}}_\alpha$. If $t$ lies in such a hypersurface, then $\tilde{\sigma}^{-1}(\bar{t})\cap \tilde{T}$ consists of the exceptional divisor of $\psi_t:\tilde{S}_t\to S_{\bar{t}}$ for $\bar{t}=q(t)$. Hence these components of $\tilde{T}$ have dimension $(r-1)+1=r$ which is of codimension $2$ in $\tilde{S}$. Equation (\[eq:nuomega\]) implies that $\hat{\nu}$ is nowhere vanishing because $\hat{\omega}$ is and $\psi$ is surjective.\
Finally, the ${\mathbb{C}}^*$-equivariance follows from that of $\hat{\omega}$ (Corollary 4.6 in [@Yamada]) whereas the ${\mathbf{C}}$-invariance is a consequence of the $\operatorname{Aut}({\mathfrak{g}})$-invariance of the Kostant-Kirillov form $\nu=\omega_{KK}$.
The morphism $\tilde{\sigma}: \tilde{S}\to {\mathfrak{t}}$ is topologically trivial (cf. [@Slo2]). Hence parallel transport defines canonical isomorphisms $P_t: H^2(\tilde{S}_t,{\mathbb{C}}) \to H^2(\tilde{S}_0, {\mathbb{C}})$, $ t\in {\mathfrak{t}}$.
The period map $$\label{eq:H2per}
P_{\tilde{S}}:{\mathfrak{t}}\to H^2(\tilde{S}_0,{\mathbb{C}})^{\mathbf{C}},\quad t\mapsto P_t([\hat{\omega}_t]),$$ is a $W$-equivariant isomorphism.
The case of ${\mathrm{ADE}}$-Dynkin diagrams, i.e. ${\mathbf{C}}=1$, goes back to [@Yamada]. For ${\mathbf{C}}\neq 1$, it has been checked in [@Beck-thesis], Corollary 1.98.
Derivative of the adjoint quotient {#SloSectRemOnDerivatives}
----------------------------------
Let $b:U\to {\mathfrak{t}}/W$ be a morphism, where $U\subset {\mathbb{C}}$ is a Zariski-open subset. Then we define $\tilde{U}_b$ and $Y_b$ by the fiber products $$\label{LocalCY3Cameral}
\begin{tikzcd}
\tilde{U}_b \arrow[r] \arrow[d, "p_b"'] & {\mathfrak{t}}\arrow[d, "q"] \\
U \arrow[r, "b"] & {\mathfrak{t}}/W \\
Y_b \arrow[r] \arrow[u, "\pi_b"] & S. \arrow[u, "\sigma"']
\end{tikzcd}$$ In Section \[s:cy\] it will become clear that $\tilde{U}_b$ is a local model for a cameral curve and $Y_b$ is a local model for the quasi-projective Calabi-Yau threefolds that we construct in loc. cit. Both $\tilde{U}_b$ and $Y_b$ are non-singular if $b$ is transversal to $q$ and $\sigma$. This condition in particular implies that the rank of $dq$ may not be less than $r-1$ for $r=\mathrm{rk}({\mathfrak{g}})$.
\[ExSL2\] We consider the simplest example, ${\mathfrak{g}}=\mathfrak{sl}_2({\mathbb{C}})$. Of course, a Slodowy slice $S$ has to be all of ${\mathfrak{g}}$, ${\mathfrak{t}}={\mathbb{C}}\cong {\mathfrak{t}}/W$ and $$\begin{gathered}
q:{\mathbb{C}}\to {\mathbb{C}}, \quad z\mapsto z^2,\\
\sigma: S\to {\mathbb{C}}, \quad A\mapsto \det(A). \end{gathered}$$ A morphism $b:U\to {\mathfrak{t}}/W$ is transversal to $q$ iff it has only zeros of multiplicity $1$. If this is the case $\tilde{U}_b=\{ (x,y)\in U\times {\mathbb{C}}~|~y^2-b(x)=0\}$ is non-singular and a branched double covering of $U$. It branches precisely over the zeros of $b$.\
Identifying $\mathfrak{sl}_2({\mathbb{C}})\cong {\mathbb{C}}^3$, we write $\sigma(u,v,w)=-u^2-vw$. Again using that $b$ has simple zeros only, we see that $Y_b=\{ ((u,v,w),x)\in {\mathbb{C}}^3\times U~|~-u^2-vw-b(x)=0\}$ is non-singular. Note however, that if $b(x)=0$, then $\pi^{-1}_b(x)$ is an $\mathrm{A}_1$-singularity.
\[RKdq\] Let $x=h$ be semisimple and $t\in {\mathfrak{t}}$ with $\chi(h)=q(t)$. Then $\mathrm{im}(dq_t)\subset \mathrm{im}(d\chi_h)$ and $$\mathrm{rk}(dq_t)=\dim \bigcap_{\alpha \in R, \alpha(t)=0} \ker \alpha=\dim C(Z_{\mathfrak{g}}(h))$$ where $R$ are the roots corresponding to ${\mathfrak{t}}\subset {\mathfrak{g}}$.
Let ${\mathfrak{t}}(h)\subset {\mathfrak{g}}$ be a Cartan subalgebra that contains $h$. Since ${\mathfrak{t}}(h)$ is conjugate to ${\mathfrak{t}}$ we may assume that $h\in {\mathfrak{t}}$ and in fact even $t=h$ by the Ad-invariance of $\chi$ and $q$. The first claim is obvious because $\chi_{|{\mathfrak{t}}}=q$.\
For the second claim we may assume $t=h$ as before. Then the first equality is proven in [@Steinberg]. For the second equality we claim $$\begin{aligned}
\label{CenterOfCentralizer}
\bigcap_{\alpha \in R, \alpha(t)=0} \ker \alpha=C(Z_{\mathfrak{g}}(h))\end{aligned}$$ which can be seen as follows: Let ${\mathfrak{g}}={\mathfrak{t}}\oplus \bigoplus_{\alpha\in R}{\mathfrak{g}}_\alpha$ be the root space decomposition with respect to ${\mathfrak{t}}$. Then we get $$Z_{\mathfrak{g}}(h)={\mathfrak{t}}\oplus \bigoplus_{\alpha \in R, \alpha(h)=0} {\mathfrak{g}}_\alpha.$$ Since $[{\mathfrak{t}},{\mathfrak{g}}_{\alpha}]\neq \{0\}$ and $[h,{\mathfrak{g}}_\alpha]=0$ iff $\alpha(h)=0$, the equality (\[CenterOfCentralizer\]) follows.
If $x=h$ is semisimple and $t\in {\mathfrak{t}}$ with $q(t)=\chi(h)$, then $\mathrm{im}(dq_t)=\mathrm{im}(d\chi_h)$.
If $x=h+v$ is the Jordan decomposition of $x\in {\mathfrak{g}}$, for $h$ semisimple and $v$ nilpotent, then (see [@Rich]) $$\begin{aligned}
\label{CaseOfInterestDerivatives}
\mathrm{rk}(d\chi_x)=\dim C(Z_{\mathfrak{g}}(h))+ \mathrm{rk}(d\chi_{1,v}).\end{aligned}$$ Here $\chi_1:{\mathfrak{g}}_1:=[Z_{{\mathfrak{g}}}(h), Z_{{\mathfrak{g}}}(h)] \to {\mathfrak{t}}_1/W_1$ is the induced adjoint quotient of the semisimple Lie algebra ${\mathfrak{g}}_1$ and $C(Z_{{\mathfrak{g}}}(h))$ is the cetner of the centralizer of $h$ in ${\mathfrak{g}}$. By the previous lemma, it remains to show that $d\chi_0=0$ for any semisimple adjoint quotient $\chi:{\mathfrak{g}}\to {\mathfrak{t}}/W$. This follows from the fact that the degrees $d_i$ of any basis $\hat{\chi}_i$ of $G_{ad}$-invariant polynomials are greater or equal $2$ because ${\mathfrak{g}}$ is semisimple ([@BourbakiLie4-6]).
\[p:transversal\] Let $b:U\to {\mathfrak{t}}/W$ be a morphism from an open $U\subset {\mathbb{C}}$ which is transversal to $q:{\mathfrak{t}}\to {\mathfrak{t}}/W$. Then it is also transversal to $\chi:{\mathfrak{g}}\to {\mathfrak{t}}/W$ and $\sigma:S\to {\mathfrak{t}}/W$.
Let $x=h+v\in {\mathfrak{g}}$ and $t\in {\mathfrak{t}}$ such that $\chi(x)=q(t)$. The previous corollary together with (\[CaseOfInterestDerivatives\]) implies that $$r \geq \mathrm{rk}(d\chi_x)\geq \mathrm{rk}(dq_t)\geq r-1.$$ If $x=h$ is semisimple, then $\chi$ is also transversal to $b$ at $x$ by the previous lemma. So we are left with the case $x=h+v$ and $\mathrm{rk}(d\chi_x)=r-1=\mathrm{rk}(dq_t)$. We claim that $v=0$ and $x=h$ must be semisimple which concludes the proof. Without loss of generality we assume again that $h\in {\mathfrak{t}}$. Since $\dim C(Z_{\mathfrak{g}}(h))=\mathrm{rk}(dq_t)=r-1$ by Lemma \[RKdq\], it follows that $Z_{\mathfrak{g}}(h)={\mathfrak{t}}\oplus {\mathfrak{g}}_\alpha \oplus {\mathfrak{g}}_{-\alpha}$ for a root $\alpha$ with respect to ${\mathfrak{t}}$. Therefore the derived algebra is $$[Z_{\mathfrak{g}}(h),Z_{\mathfrak{g}}(h)]=\langle h_\alpha, {\mathfrak{g}}_{\pm \alpha}\rangle \cong \mathfrak{sl}_2({\mathbb{C}}),$$ where $h_\alpha$ generates the commutator $[{\mathfrak{g}}_\alpha,{\mathfrak{g}}_{-\alpha}]\subset {\mathfrak{t}}$. As a consequence, $v$ can be considered as a nilpotent element in $\mathfrak{sl}_2({\mathbb{C}})$ because $v\in {\mathfrak{g}}_\alpha\oplus {\mathfrak{g}}_{-\alpha}\subset Z_{\mathfrak{g}}(h)$. By formula (\[CaseOfInterestDerivatives\]) and $\mathrm{rk}(dq_t)=r-1$ we must have $\mathrm{rk}(d\chi_{1,v})=0$ for the adjoint quotient $\chi_1=\det:\mathfrak{sl}_2({\mathbb{C}})\to {\mathfrak{t}}_1/W_1$. But $d_A \det=(-2a,-c,-b)$ for $A=aH+bX+cY\in \mathfrak{sl}_2({\mathbb{C}})$ in the standard basis $H,X,Y$ of $\mathfrak{sl}_2({\mathbb{C}})$. Hence we must have $v=0$, i.e. $x=h$ is semisimple (and subregular).
Calabi-Yau orbifolds over Hitchin bases {#s:cy}
=======================================
Hitchin bases and cameral curves {#ss:hitchinbase}
--------------------------------
This subsection mainly fixes notation and we refer to [@DG], [@DP] for more details. As before, we let $G$ be a simple complex Lie group with Dynkin diagram $\Delta=\Delta(G)$ and ${\mathfrak{t}}\subset {\mathfrak{g}}={\mathfrak{g}}(\Delta)$ a Cartan subalgebra in ${\mathfrak{g}}=\mathrm{Lie}(G)$. If ${\Sigma}$ is a compact connected Riemann surface of genus $\geq 2$, we denote by $\mathcal{M}_{\mathrm{H}}({\Sigma},G)$ the moduli space of semistable $G$-Higgs bundles of degree $1\in \pi_1(G)$ and by $${\bm{h}}:\mathcal{M}_{\mathrm{H}}({\Sigma},G)\to {\mathbf{B}}({\Sigma}, G)$$ the Hitchin map. It maps to the Hitchin base $${\mathbf{B}}\colon={\mathbf{B}}({\Sigma}, G)=H^0({\Sigma}, {\bm{U}})$$ for the bundle $
u\colon{\bm{U}}=K_{\Sigma}\times_{{\mathbb{C}}^*} {\mathfrak{t}}/W\to {\Sigma}$ of cones where ${\mathbb{C}}^*$ acts by the standard action on ${\mathfrak{t}}/W$. Choosing generators $\chi_1,\dots ,\chi_r\in {\mathbb{C}}[{\mathfrak{t}}]^W$ of Weyl group invariants gives ${\bm{U}}$ the structure of a vector bundle via the isomorphism $${\bm{U}}\cong \bigoplus_{j=1}^r K_{\Sigma}^{d_j}$$ for the degrees $d_j=\deg(\chi_j)$. In particular, the Hitchin base ${\mathbf{B}}$ inherits a vector space structure. By its construction, ${\mathbf{B}}$ only depends on the Lie algebra ${\mathfrak{g}}$ of $G$ so that the notation $${\mathbf{B}}({\Sigma},\Delta):={\mathbf{B}}({\Sigma}, G)$$ is justified. The total space of the vector bundle $
\tilde{u}\colon{\bm{\tilde{U}}}=K_{\Sigma}\otimes {\mathfrak{t}}\to {\Sigma}$ maps to ${\bm{U}}$ via the natural quotient map $\bm{q}\colon{\bm{\tilde{U}}}\to {\bm{U}}$. The fibers of the universal cameral curve $$\bm{p}:{\bm{\tilde{\Sigma}}}:=ev^*{\bm{\tilde{U}}}\to {\mathbf{B}},$$ for the evaluation map $ev:{\Sigma}\times {\mathbf{B}}\to {\bm{U}}$, are the cameral curves ${\tilde{\Sigma}}_{b}:=\bm{p}^{-1}(b)$. These are smooth over the Zariski-open and dense $${\mathbf{B}^\circ}:=\{ b\in {\mathbf{B}}~|~b \mbox{ intersects }\mathrm{disc}(\bm{q})\mbox{ transversally} \}$$ and the generic Hitchin fibers ${\bm{h}}^{-1}(b)$, $b\in {\mathbf{B}^\circ}$, are generalized Prym varieties defined in terms of ${\tilde{\Sigma}}_b$.\
For later reference, we further need the cone $$\label{eq:B/W}
\tilde{{\mathbf{B}}}/W\hookrightarrow {\mathbf{B}}, \quad \tilde{{\mathbf{B}}}:=H^0({\Sigma}, {\bm{\tilde{U}}}),$$ where the Weyl group $W$ acts pointwise on sections. Away from $0\in {\mathbf{B}}$, this is the locus of completely reducible but reduced cameral curves, i.e. $${\tilde{\Sigma}}_b=\coprod_{w\in W} {\tilde{\Sigma}}_{b,w}, \quad {\tilde{\Sigma}}_{b,w}\cong {\Sigma}.$$ The irreducible components ${\tilde{\Sigma}}_{b,w}$ intersect over the points of the divisor of the section $$\prod_{\alpha \in R} \alpha(b)\in H^0({\Sigma}, K_{\Sigma}),$$ where $R=R(\Delta)$ is the corresponding root system (which makes sense because $\prod_{\alpha\in R}\alpha \in {\mathbb{C}}[{\mathfrak{t}}]^W$).
Surfaces
--------
For the following constructions, we fix a spin bundle $L\in \mathrm{Pic}^{g-1}({\Sigma})$, $L^2=K_{{\Sigma}}$ and denote by $\alpha=\alpha_L\in H^1({\Sigma}, {\mathcal{O}}^*)$ its cohomology class. Moreover, let $\Delta$ be an irreducible Dynkin diagram with associated symmetry group ${\mathbf{C}}=AS(\Delta)$ and $S=S(\Delta)\subset {\mathfrak{g}}(\Delta)$ a Slodowy slice. Twisting the ${\mathbb{C}}^*$-spaces $S$ and ${\mathfrak{t}}/W$ by $L$, we obtain[^4] $$\label{eq:Scb}
{\bm{S}}:=L\times_{{\mathbb{C}}^*} S, \quad L\times_{{\mathbb{C}}^*} {\mathfrak{t}}/W \cong {\bm{U}}.$$ By its ${\mathbb{C}}^*$-equivariance, the morphism $\sigma:S\to {\mathfrak{t}}/W$ and its simultaneous resolution $\tilde{\sigma}:\tilde{S}\to {\mathfrak{t}}$ glue to give the following commutative diagram $$\label{eq:tScb}
\begin{tikzcd}
{\tilde{\bm{S}}}\ar[r, "\bm{\psi}"] \ar[d, "{\tilde{\bm{\sigma}}}"'] & {\bm{S}}\ar[d, "{\bm{\sigma}}"] \\
{\bm{\tilde{U}}}\ar[r, "\bm{q}"] & {\bm{U}}\end{tikzcd}$$ of ${\mathbf{C}}$-spaces where ${\mathbf{C}}$ acts trivially on ${\bm{\tilde{U}}}$ and ${\bm{U}}$. However, it takes more care to glue the sections $\hat{\omega}\in \Gamma(\tilde{S},\Omega^2_{\tilde{\sigma}})$ and $\hat{\nu}\in \Gamma(S, K_{\sigma})$.
\[l:bhatnu\] With the notation of (\[eq:Scb\]) and (\[eq:tScb\]), the following holds:
1. The section $\hat{\omega}$ glues to a section $\hat{{\bm{\omega}}}\in \Gamma({\tilde{\bm{S}}},\Omega^2_{{\bm{\sigma}}}\otimes (\tilde{u}\circ {\tilde{\bm{\sigma}}})^*K_{\Sigma})$ which is ${\mathbf{C}}$-invariant. It induces a fiberwise period map $$\bm{\eta}:{\bm{\tilde{U}}}\to \tilde{u}^*{\bm{\tilde{U}}}$$ which coincides with the tautological section $\bm{\tau}\in \Gamma({\bm{\tilde{U}}}, \tilde{u}^*{\bm{\tilde{U}}})$.
2. Likewise, the section $\hat{\nu}$ glues to a ${\mathbf{C}}$-invariant section $\hat{\bm{\nu}}\in \Gamma({\bm{S}}, K_{{\bm{\sigma}}}\otimes (u\circ {\bm{\sigma}})^*K_{\Sigma})$ such that $$\bm{\psi}^*\hat{\bm{\nu}}=\hat{\bm{\omega}}$$ under the natural isomorphism $\bm{\psi}^*K_{{\bm{\sigma}}}\cong \Omega^2_{{\tilde{\bm{\sigma}}}}$.
In particular, the sections $\hat{\bm{\sigma}}$ and $\hat{\bm{\nu}}$ give respective isomorphisms $$\Omega^2_{{\bm{\sigma}}}\cong (\tilde{u}\circ {\tilde{\bm{\sigma}}})^*K_{{\Sigma}}^{-1} \quad \mbox{and} \quad K_{{\bm{\sigma}}} \cong (u\circ {\bm{\sigma}})^*K_{{\Sigma}}^{-1}.$$
To construct the section $\hat{\bm{\omega}}$, we need the gluing data for the sheaf $\Omega^2_{{\tilde{\bm{\sigma}}}}$. Let $\alpha=\alpha_L$ be the cocycle corresponding to $L$ and denote by $(\alpha_{ij})$ a Čech representative for a fixed open covering $(D_{ij})$ of ${\Sigma}$. In particular, $(\beta_{ij})=(\alpha_{ij}^2)$ is a cocycle for $K_{\Sigma}$. Trivializing ${\tilde{\bm{S}}}$ over each $D_{i}$ gives rise to the commutative diagram $$\begin{tikzcd}
& {\tilde{\bm{S}}}_{ij} \arrow[r, "\psi_i"] & D_{ij}\times \tilde{S} \ar[r, "\mathrm{id}\times \tilde{\sigma}"] & D_{ij}\times {\mathfrak{t}}\\
{\bm{\tilde{U}}}\ar[ru, leftarrow, "\tilde{{\bm{\sigma}}}_{ij}"] \ar[rd, leftarrow, "\tilde{{\bm{\sigma}}}_{ij}"']
\\
& {\tilde{\bm{S}}}_{ij} \arrow[r, "\psi_j"] \arrow[uu, equal] & D_{ij}\times \tilde{S} \arrow[uu, "g_{ij}"'] \ar[r, "\mathrm{id}\times \tilde{\sigma}"]& D_{ij}\times {\mathfrak{t}}\ar[uu, "h_{ij}"']
\end{tikzcd}$$ over $D_{ij}=D_i\cap D_j$ with $$\begin{aligned}
&g_{ij}(x,s)=(x, \alpha_{ij}(x)\cdot s)=(x, \mu(\alpha_{ij}(x),s)),\\
&h_{ij}(x,t)=(x, \alpha_{ij}(x)\cdot t)=(x, \beta_{ij}(x)t).
\end{aligned}$$ for $(x,s)\in D_{ij}\times \tilde{S}$ and the action map $\mu:{\mathbb{C}}^*\times \tilde{S}\to \tilde{S}$. On each $D_i\times \tilde{S}$ we have the sheaves $\mathcal{E}_i:=\Omega^2_{\mathrm{id}\times \tilde{\sigma}}\cong \mathrm{pr}_{2,i}^*\Omega^2_{\tilde{\sigma}}$ together with the sections $\mathrm{pr}_{2,i}^*\hat{\omega}$. Clearly, $\mathcal{E}_i$ and $\mathcal{E}_j$ are canonically isomorphic over $D_{ij}$. Now $\Omega^2_{{\tilde{\bm{\sigma}}}}$ is glued from[^5] $\psi_i^*\mathcal{E}_i$ on $D_{ij}$ via the isomorphisms $$\begin{tikzcd}
\varphi_{ij}:=\psi_i^*dg_{ji}^t: \psi_j^*\mathcal{E}_j=\psi_i^*g_{ji}^*\mathcal{E}_j \arrow[r] & \psi_i^*\mathcal{E}_i
\end{tikzcd}$$ over $D_{ij}$. Here we denote by $dg_{ji}^t:g_{ij}^*\mathcal{E}_j\to \mathcal{E}_i=\mathcal{E}_j$ the natural morphism (over $D_{ij}$). Observe that we have $\varphi_{ij}\circ \varphi_{jk}=\varphi_{ik}$ and $(\varphi_{ij})$ is the gluing (or descent) datum for $\Omega^2_{{\tilde{\bm{\sigma}}}}$. Indeed, we can write this composition as $$\begin{tikzcd}
\psi_k^*\mathcal{E}_k\ar[rr, "\varphi_{kj}"] \ar[d, "\cong"] && \psi_j^*\mathcal{E}_j \ar[rr, "\varphi_{ji}"] \ar[d, "\cong"] && \psi_i^*\mathcal{E}_i \ar[d, "\cong"] \\
\psi_i^*(g_{ji}^*) g_{kj}^*\mathcal{E}_j \ar[rr, "\psi_i^* g_{ji}^* dg_{kj}^t"] && \psi^*_i g_{ji}^*\mathcal{E}_j \ar[rr, "\psi_i^*dg_{ji}^t"] && \psi_i^*\mathcal{E}_j.
\end{tikzcd}$$ The lower line is $\varphi_{ik}$ by the chain rule, showing the cocycle condition for $(\varphi_{ij})$.
Define the local sections $$\hat{\omega}_i:=\psi_i^*\mathrm{pr}_{2,i}^*\hat{\omega}\in \Gamma(\mathcal{S}_i,\psi_i^*\mathcal{E}_i).$$ On the overlaps ${\tilde{\mathcal{S}}}_{ij}$, they transform as follows $$\label{TrafoRuleSections}
\varphi_{ij}(\hat{\omega}_j)=((\mathrm{pr}_{1,i}\circ \psi_i)^*\beta_{ji}) \, \hat{\omega}_i.$$
Before we prove this claim, let us see how it yields the desired section. Observe that $(\mathrm{pr}_{1,i}\circ\psi_i)^* \beta_{ji}$ is a cocycle for $(\tilde{u}\circ {\tilde{\bm{\sigma}}})^*K_{\Sigma}^{-1}$. Hence in order to obtain a well-defined global section on $\tilde{\mathcal{S}}$, we have to tensor with $(\tilde{u}\circ {\tilde{\bm{\sigma}}})^*K_{\Sigma}$. More precisely, let $\zeta_i\in \Gamma(D_i,K_\Sigma)$ be the local frames of $K_{\Sigma}$ over $D_i$ so that $\zeta_i=\beta_{ij}~\zeta_j$ on $D_{ij}$. Letting $\hat{\zeta}_i:=\psi_i^*\mathrm{pr}_{1,i}^*\zeta_i$, we see that the local sections $$\hat{\omega}_i\otimes \hat{\zeta}_i\in\Gamma({\tilde{\bm{S}}}_i,\Omega^2_{{\tilde{\bm{\sigma}}}}\otimes (\tilde{u}\circ {\tilde{\bm{\sigma}}})^*K_\Sigma)$$ glue to give the global section $\bm{\hat{\omega}}\in \Gamma({\tilde{\bm{S}}},\Omega^2_{{\tilde{\bm{\sigma}}}}\otimes (\tilde{u}\circ {\tilde{\bm{\sigma}}})^*K_\Sigma)$ as claimed.\
\
We still have to give a proof of (\[TrafoRuleSections\]). To simplify notation, we drop the subscript $ij$ if not necessary and only write $g:D\times \tilde{S}\to D\times \tilde{S}$ etc. Then the second component of $dg:TD\oplus T\tilde{S}\to TD\oplus T\tilde{S}$ at $(x,s)\in D\times \tilde{S}$ is given by $$\label{SecondComponent}
d\mu_{\alpha(x)\cdot s}(d\alpha_x(v),w)=d\mu_{\alpha(x)\cdot s}(d\alpha_x(v),0)+d\mu_{\alpha(x)\cdot s}(0, w).$$ Note that $d\mu_{\alpha(x)\cdot s}(0,w)=d(\mu_{\alpha(x)})_s(w)$ where $\mu_{\alpha(x)}=\mu(\alpha(x),-)$. Now let $p\in {\tilde{\mathcal{S}}}_{ij}$ and $\psi_i(p)=(x,s)\in D_{ij}\times \tilde{S}$. Then we clearly have $$\ker d_{(x,s)}(\mathrm{id}\times \tilde{\sigma})=0\oplus \ker d_s\tilde{\sigma}\subset T_xD_{ij}\oplus T_s\tilde{S}.$$ In particular, the first summand in (\[SecondComponent\]) plays no role for our discussion. For $w_k\in \ker d_s\tilde{\sigma}$ ($k=1,2$) one computes $$\begin{aligned}
&\quad~\varphi_{ij}(\hat{\omega}_j)_p\Big((0,w_1),(0,w_2)\Big)\\
&=\mathrm{pr}_{2,j}^*\hat{\omega}_{g_{ji}(x,s)}\circ dg_{ji, (x,s)}\Big((0,w_1),(0,w_2)\Big) & (\psi_j\circ \psi_i^{-1}=g_{ji}) \\
&=\beta_{ji}(x) \, \hat{\omega}_s(w_1,w_2) & ({\mathbb{C}}^*\text{-equivariance and }(\ref{SecondComponent}))\\
&=(\mathrm{pr}_{1,i}\circ \psi_i)^*\beta_{ji}(p) \, (\hat{\omega}_i)_p\Big((0,w_1),(0,w_2)\Big).
\end{aligned}$$ Here we have used the ${\mathbb{C}}^*$-equivariance of the relative form $\hat{\omega}$ showing (\[TrafoRuleSections\]).\
To construct the period map $\bm{\eta}$ out of $\hat{{\bm{\omega}}}$, observe that it induces the morphism $$(x,t)\mapsto P_{\tilde{S}}(t)\otimes ((x,t),\zeta_i(x))$$ in each trivialization ${\bm{\tilde{U}}}_{|D_i} \cong D_i\times {\mathfrak{t}}$ where $P_{\tilde{S}}$ is as in (\[eq:H2per\]). These morphisms glue to give the morphism $$\bm{\eta}: {\bm{\tilde{U}}}\to {\mathfrak{t}}\otimes\tilde{u}^*K_\Sigma$$ which coincides with the tautological section $\bm{\tau}\in H^0({\bm{\tilde{U}}},\tilde{u}^*{\bm{\tilde{U}}})$ by construction.\
The existence of the global section $\bm{\hat{\nu}}\in H^0({\bm{S}},K_{{\bm{\sigma}}}\otimes (u\circ {\bm{\sigma}})^*K_\Sigma)$ works similarly: There exists a section $\bm{\hat{\nu}}^{reg}\in \Gamma({\bm{S}}^{reg}, K_{{\bm{\sigma}}})$, which is constructed as $\hat{\bm{\omega}}$ by replacing $\hat{\omega}$ with $\hat{\nu}$. Here ${\bm{S}}^{reg}\subset {\bm{S}}$ is the locus which is glued from $S^{reg}\subset S$. Using a codimension argument as in the proof of Proposition \[p:C\*equivariant\], we see that it uniquely extends to a section $\bm{\hat{\nu}}\in \Gamma({\bm{S}}, K_{{\bm{\sigma}}})$. It satisfies $\bm{\psi}^*\bm{\hat{\nu}}=\bm{\hat{\omega}}$ under the isomorphism $\bm{\psi}^*K_{{\bm{\sigma}}}\cong K_{{\tilde{\bm{\sigma}}}}$ by construction, since this holds for the corresponding local sections.
Calabi-Yau threefolds
---------------------
The family ${\bm{\sigma}}: {\bm{S}}\to {\bm{U}}$ of surfaces over ${\bm{U}}$ pulls back to the product ${\Sigma}\times {\mathbf{B}}$, ${\mathbf{B}}={\mathbf{B}}({\Sigma}, \Delta)$ via the evaluation map ${ev}:{\Sigma}\times {\mathbf{B}}({\Sigma},\Delta)\to {\bm{\tilde{U}}}$. Projecting to the second factor yields the family $$\begin{tikzcd}
{\mathcal{X}}_L(\Delta) \ar[rr, bend left, "{\bm{\pi}}_L"] \ar[r, "{\bm{\pi}}_{1,L}"] & {\Sigma}\times {\mathbf{B}}\ar[r, "{\bm{\pi}}_{2,L}"] & {\mathbf{B}}\end{tikzcd}$$ of threefolds with a ${\mathbf{C}}$-action over the Hitchin base ${\mathbf{B}}$. Analogously, pulling back ${\tilde{\bm{S}}}$ along the natural map ${\bm{\tilde{\Sigma}}}\to {\bm{\tilde{U}}}$ gives the family $$\begin{tikzcd}
\tilde{{\mathcal{X}}}_L(\Delta) \ar[rr, bend left, "\tilde{{\bm{\pi}}}_L"] \ar[r, "\tilde{{\bm{\pi}}}_{1,L}"] & {\bm{\tilde{\Sigma}}}\ar[r, "\tilde{{\bm{\pi}}}_{2,L}"] & {\mathbf{B}}\end{tikzcd}$$ of threefolds with ${\mathbf{C}}$-action over ${\mathbf{B}}$.
\[thm:cy3s\] Let $\Delta$ be an irreducible Dynkin diagram and $L$ be a spin bundle of ${\Sigma}$. Then $$\begin{tikzcd}
{\mathcal{X}}_L(\Delta)\ar[r] & {\mathbf{B}}({\Sigma}, \Delta) & \ar[l] \tilde{{\mathcal{X}}}_L(\Delta)
\end{tikzcd}$$ are algebraic families of quasi-projective Gorenstein threefolds with ${\mathbf{C}}$-trivial canonical class. They are smooth over the locus ${\mathbf{B}^\circ}({\Sigma},\Delta)\subset {\mathbf{B}}({\Sigma}, \Delta)$ of smooth cameral curves.
The projections ${\bm{U}}\to {\Sigma}$ and ${\bm{S}}\to {\Sigma}$ are quasi-projective morphisms because they are (affine) bundles over the projective curve ${\Sigma}$. Therefore the morphism ${\bm{\sigma}}: {\bm{S}}\to {\bm{U}}$ which factorizes as $$\begin{tikzcd}
{\bm{S}}\ar[d, "{\bm{\sigma}}"] \ar[dr] & \\
{\bm{U}}\ar[r ] & {\Sigma}\end{tikzcd}$$ is quasi-projective as well. Hence ${\bm{\pi}}:{\mathcal{X}}\to {\mathbf{B}}$ is quasi-projective. Using the fact that ${\bm{\sigma}}: {\bm{S}}\to {\bm{U}}$ is a Gorenstein morphism, a similar argument shows that each $X_b={\bm{\pi}}^{-1}(b)$, $b\in {\mathbf{B}}$, is Gorenstein.\
To see the statement about the canonical class, let $j:{\mathcal{X}}\to {\bm{S}}$ be the natural morphism obtained from base change. Then the section $\hat{\bm{\nu}}$ of Lemma \[l:bhatnu\] pulls back to (dropping $L$ from the notation) $$\bm{s}:=j^*\hat{\bm{\nu}}\in H^0({\mathcal{X}}, K_{{\bm{\pi}}_1} \otimes (\mathrm{pr}_1\circ {\bm{\pi}}_1)^*K_\Sigma).$$ Base change and the adjunction formula yields the isomorphism $$(K_{{\bm{\pi}}_1}\otimes (\mathrm{pr}_1\circ {\bm{\pi}}_1)^*K_{\Sigma})_{|X_b}\cong K_{\pi_b}\otimes \pi_b^*K_{\Sigma}\cong K_{X_b}.$$ Hence each restriction $s_b:=\bm{s}_{|X_b}$ is a nowhere vanishing section of the locally free sheaf $X_b$. Since the ${\mathbf{C}}$-action on $X_b$ is pulled back from the ${\mathbf{C}}$-action on ${\bm{S}}$, $s_b$ is ${\mathbf{C}}$-invariant by Lemma \[l:bhatnu\]. Finally, the statement about smoothness follows from Proposition \[p:transversal\].\
The proof for $\tilde{{\mathcal{X}}}_L(\Delta)$ works in complete analogy by replacing $\bm{s}$ with $$\tilde{\bm{s}}:=\tilde{j}^*\hat{{\bm{\omega}}}\in H^0(\tilde{{\mathcal{X}}}, K_{\tilde{{\bm{\pi}}}_1}\otimes (\mathrm{pr}_1\circ \bm{p}_1)^*K_{\Sigma})$$ for the natural morphism $\tilde{j}:\tilde{{\mathcal{X}}}\to {\tilde{\bm{S}}}$.
We close this subsection with the deformation-theoretic meaning of the Hitchin base ${\mathbf{B}}={\mathbf{B}}({\Sigma}, \Delta)$ for the family ${\mathcal{X}}\to {\mathbf{B}}$ (see [@DDD] for the $\mathrm{A}_1$-case). To this end, observe that the central fiber $$X_0=L\times_{{\mathbb{C}}^*} S_{\bar{0}}$$ has a curve of $\Delta$-singularities along ${\Sigma}\hookrightarrow X_0$. It is easily seen from the construction that $X_b$, $b\neq 0$, only has isolated singularities. In particular, the deformation ${\mathcal{X}}\to {\mathbf{B}}$ of $X_0$ is not a locally trivial one. More precisely, we have:
\[thm:defX0\] Let $X=X_0$ be the central fiber of a family ${\mathcal{X}}\to {\mathbf{B}}$ of quasi-projective Gorenstein Calabi-Yau with ${\mathbf{C}}$-trivial canonical class. Then $$\mathrm{Ext}^1(\Omega_X^1,{\mathcal{O}}_X)/H^1(X,T_X) \cong {\mathbf{B}},$$ i.e. the space of ${\mathbf{C}}$-deformations of $X$ modulo locally trivial ${\mathbf{C}}$-deformations is isomorphic to the corresponding Hitchin base.
We first consider the case ${\mathbf{C}}=1$ and set $r:=\mathrm{rk}({\mathfrak{g}})$. Since $\pi:X\to {\Sigma}$ is affine, the Leray spectral sequence implies that $H^2(X,T_X)=0$. Therefore the local-to-global spectral sequence implies $$\mathrm{Ext}^1(\Omega_X^1, {\mathcal{O}}_X)/H^1(X,T_X)\cong H^0(X, {\mathscr{E}\hspace{-3.3pt}\mathcalligra{xt}_{\,\mathcal{O}_X}}^1(\Omega_X^1,{\mathcal{O}}_X)).$$ If $k:{\Sigma}\hookrightarrow X$ is the closed imbedding, then we claim $$\label{eq:ext1}
\mathscr{E}:={\mathscr{E}\hspace{-3.3pt}\mathcalligra{xt}_{\,\mathcal{O}_X}}^1(\Omega_X^1,{\mathcal{O}}_X)\cong k_*{\bm{U}}.$$ As a first step, we realize $X$ as a regular imbedding. Let $\pi_{{\bm{S}}}:{\bm{S}}\to {\Sigma}$ be the natural projection and $\pi_{{\bm{S}}}^*{{\bm{U}}} \to {\bm{S}}$ the induced vector bundle. Then there exists a unique section $\tau:{\bm{S}}\to \pi^*_{{\bm{S}}} {\bm{U}}$ such that $pr_{{\bm{U}}}\circ \tau={\bm{\sigma}}$ for the natural projection $pr_{{\bm{U}}}:\pi^*_{{\bm{S}}}{\bm{U}}\to {\bm{U}}$. In a trivialization of ${\bm{S}}_{|D}\cong D\times S$ for an affine $D\subset {\Sigma}$, the section is given by $$D\times S \to (D\times S) \times {\mathfrak{t}}/W, \quad (x,s)\mapsto ((x,s),\sigma(s)).$$ Therefore the vanishing locus $Z(\tau)\subset {\bm{S}}$ coincides with $i:X\hookrightarrow {\bm{S}}$ and is a regular imbedding. The latter implies that the normal sheaf satisifes $$\label{eq:normalsheaf}
\mathcal{N}_{X/{\bm{S}}}\cong \pi^*_{{\bm{S}}}{\bm{U}}_{|X},$$ cf. [@FultonLang], Chapter IV. In order to compute $\mathscr{E}$, let $\mathcal{J}$ be the ideal sheaf defining $X\subset {\bm{S}}$. Since the latter is a regular imbedding, we have the locally free resolution $$\begin{tikzcd}
0 \ar[r] & \mathcal{J}/\mathcal{J}^2 \ar[r] & i^*\Omega^1_{{\mathcal{S}}} \ar[r] & 0
\end{tikzcd}$$ of $\Omega_X^1$. Dualizing gives the sequence $$\begin{tikzcd}
0 \ar[r] & {\mathscr{H}\hspace{-5pt}\mathcalligra{om}}_{{\mathcal{O}}_X}(i^*\Omega^1_{{\bm{S}}},{\mathcal{O}}_X) \ar[r, "\varphi"] & \mathcal{N}_{X/{\bm{S}}} \ar[r] & 0
\end{tikzcd}$$ so that $\mathrm{coker}(\varphi)={\mathscr{E}\hspace{-3.3pt}\mathcalligra{xt}_{\,\mathcal{O}_X}}^1(\Omega_X^1,{\mathcal{O}}_X)$ which is supported on ${\Sigma}\subset X$. In order to prove (\[eq:ext1\]) it therefore suffices to show that $\mathscr{E}$ has constant rank $r$ along ${\Sigma}$ in light of (\[eq:normalsheaf\]).\
Let $U:=D\times S_{\bar{0}}\subset X$ and $D\subset {\Sigma}$ be an affine open in $X$ and ${\Sigma}$ respectively. Hence $$U=\mathrm{Spec}(B), \quad B={\mathbb{C}}[t,x,y,z]/\langle f(x,y,z) \rangle,\quad D=\mathrm{Spec}(C), \quad C={\mathbb{C}}[t,x,y,z],$$ where $f$ defines the $\Delta$-singularity $S_{\bar{0}}$. The sheaf $\mathscr{E}$ corresponds to the $B$-module $$\mathrm{Ext}^1_B(\Omega^1_B, B) \cong {\mathbb{C}}[t,x,y,z]/\langle f, \partial f \rangle.$$ As a $C$-module, it is isomorphic to $C[t]^r$. This concludes the proof for ${\mathbf{C}}=1$. If ${\mathbf{C}}\neq 1$, the same proof goes through if we work with ${\mathbf{C}}$-invariants and recall that ${\mathbf{B}}_h^{\mathbf{C}}={\mathbf{B}}$.
Relation between and
---------------------
We next compare the two families $$\begin{gathered}
{\mathcal{X}}={\mathcal{X}}_L(\Delta) \to {\mathbf{B}}={\mathbf{B}}({\Sigma}, \Delta)\\
{\mathcal{X}}_h={\mathcal{X}}_L(\Delta_h)\to {\mathbf{B}}_h={\mathbf{B}}({\Sigma}, \Delta_h)\end{gathered}$$ in case $\Delta\neq \Delta_h$. As a first step, we construct a non-trivial $AS(\Delta)$-action on ${\mathcal{X}}_h$.\
Let $S_h:=S(\Delta_h)=x+\ker(\mathrm{ad}(y))\subset {\mathfrak{g}}_h:={\mathfrak{g}}(\Delta_h)$ be a Slodowy slice. Then the group $$\label{eq:CA}
CA(x,h):=\{ \phi \in \operatorname{Aut}({\mathfrak{g}}_h) ~|~\phi(x)=x, \phi(h)=h \}$$ acts on $S_h$. There is a subgroup ${\mathbf{CA}}\subset CA(x,h)$ which is isomorphic to $AS(\Delta)$ (see [@Slo], 7.5). Of course, the definition of ${\mathbf{CA}}$ and ${\mathbf{C}}$ makes sense for $S$ and $S_h$ respectively. Then ${\mathbf{CA}}\cong {\mathbf{C}}$ in the former and ${\mathbf{C}}=1$ in the latter case.
Even though ${\mathbf{C}}\cong AS(\Delta) \cong {\mathbf{CA}}$, we often write ${\mathbf{C}}$ and ${\mathbf{CA}}$ to emphasize how the $AS(\Delta)$-action is realized. To illustrate the relation between $S$ and $S_h$ together with the corresponding $AS(\Delta)$-actions, we give a worked out example in Appendix \[a:ex\].
The ${\mathbf{CA}}$-action on $S_h$ induces a ${\mathbf{CA}}$-action on ${\mathfrak{t}}_h/W_h$ such that $\sigma_h:S_h\to {\mathfrak{t}}_h/W_h$ is equivariant. Since the ${\mathbf{CA}}$-action commutes with the ${\mathbb{C}}^*$-action on $S_h$, we obtain a ${\mathbf{CA}}$-action on ${\mathcal{X}}_h$. Lemma \[l:bhatnu\] and Theorem \[thm:cy3s\] analogously hold for the ${\mathbf{CA}}$-action (cf. [@Beck-thesis] for details). However, ${\mathbf{CA}}$ acts non-trivially on the base so that a general member $X_{h,b}$ does not have ${\mathbf{CA}}$-trivial canonical bundle.
\[cor:comp\] With the previous notation, $$\label{eq:incl}
{\mathbf{B}}\cong {\mathbf{B}}_h^{{\mathbf{CA}}}\hookrightarrow {\mathbf{B}}_h$$ but ${\mathbf{B}}^\circ \cap {\mathbf{B}}_h^\circ =\emptyset$. The family ${\mathcal{X}}\to {\mathbf{B}}$ is the restriction of the family ${\mathcal{X}}_h\to {\mathbf{B}}_h$ under the inclusion (\[eq:incl\]) and the $AS(\Delta)$-action on ${\mathcal{X}}$ is induced by the $AS(\Delta)$-action on ${\mathcal{X}}_h$. In particular, ${\mathcal{X}}_h$ is smooth over a Zariski open and dense subset containing ${\mathbf{B}^\circ}\sqcup {\mathbf{B}^\circ}_h\subset {\mathbf{B}}_h$.
It is not difficult to see that ${\mathfrak{t}}/W\cong ({\mathfrak{t}}_h/W_h)^{{\mathbf{CA}}}$ which gives the inclusion $i:{\mathfrak{t}}/W \hookrightarrow {\mathfrak{t}}_h/W_h$ and hence (\[eq:incl\]). To see that ${\mathbf{B}}^\circ \cap {\mathbf{B}}_h^\circ=\emptyset$, let $\alpha\in \Delta$ be a long root. Under folding, it corresponds to an $AS(\Delta)$-orbit $O(\beta)$ of length $\geq 2$ for some $\beta\in \Delta_h$. If ${\mathfrak{t}}_\alpha\subset {\mathfrak{t}}$ is the fixed point locus of $s_\alpha\in W$ (and analogously for ${\mathfrak{t}}_h$), then $$\label{eq:talpha}
{\mathfrak{t}}_\alpha=\bigcap_{\beta'\in O(\beta)} {\mathfrak{t}}_{h,\beta'}\subset {\mathfrak{t}}_h$$ under the inclusion ${\mathfrak{t}}\subset {\mathfrak{t}}_h$. If $b\in {\mathbf{B}^\circ}$, then $b$ necessarily maps to the smooth locus $\mbox{disc}(\bm{q})^{sm}$ of the discriminant of $\bm{q}:{\bm{\tilde{U}}}\to {\bm{U}}$. Since ${\tilde{\Sigma}}_b$ necessarily intersects $K_{\Sigma}\otimes {\mathfrak{t}}_{\alpha}\subset {\bm{\tilde{U}}}$ and because of (\[eq:talpha\]), $b$ cannot map to $\mbox{disc}(\bm{q}_h)^{sm}$ under the inclusion ${\bm{U}}\subset {\bm{U}}_h$. Hence ${\mathbf{B}^\circ}\cap {\mathbf{B}^\circ}_h=\emptyset$.\
Slodowy has proven (Chapter 8.8 in [@Slo]) that $i^*S_h\cong S$ over ${\mathfrak{t}}/W$ as ${\mathbb{C}}^*$-deformations of the $\Delta$-singularity over $\bar{0}\in {\mathfrak{t}}/W$. By the ${\mathbb{C}}^*$-equivariance, this isomorphism induces the isomorphism ${\mathcal{X}}\cong {\mathcal{X}}_h$ over ${\mathbf{B}}\subset {\mathbf{B}}_h$.
Equations
---------
To connect to similar constructions in the physical mathematics literature ([@DDD], [@Tbranes], [@Tbranes2]), we give explicit equations of the families ${\mathcal{X}}\to {\mathbf{B}}$ in appropriate bundles over ${\Sigma}$.\
If we identify $S=\cong {\mathbb{C}}^{r+2}$ appropriately, then the weights of the ${\mathbb{C}}^*$-action are given by $w_1=2d_1,\dots, w_r=2d_r, w_{r+1},w_{r+2},w_{r+3}$ for the degrees $d_j=\mathrm{deg}(\chi_j)$ as in Section \[ss:hitchinbase\] and weight $w_{r+1},w_{r+2},w_{r+3}$ as in Table \[ADEsings\] in Appendix \[SectFolding\]. It follows that $${\bm{S}}=L\times_{{\mathbb{C}}^*} S \cong \bigoplus_{j=1}^{r} K^{d_j} \oplus \bigoplus_{j=1}^3 L^{w_{r+k}}.$$ Let $f\in {\mathbb{C}}[x,y,z]$ define the $\Delta$-singularity and let $g_1,\dots, g_r\in {\mathbb{C}}[x,y,z]^{{\mathbf{C}}}$ be representatives of generators of the ${\mathbf{C}}$-invariant Jacobi ring ${\mathbb{C}}[x,y,z]^{{\mathbf{C}}}/(f,\partial f)$. Then the family ${\mathcal{X}}\to {\mathbf{B}}$ reads as $$\label{eq:explicit}
{\mathcal{X}}=\{ (x,y,z,\underline{b})~|~f(x,y,z)+\sum_{i=1}^r b_i g_i(x,y,z)=0 \in \mathrm{tot}(K^d) \}$$ in $\mathrm{tot}(\bigoplus_{i=1}^3 L^{w_{r+k}})\times {\mathbf{B}}$ with the obvious projection and where $d$ is again as in Table \[ADEsings\].
\[ex:B2\] Let us give at least one simple example (also compare with Appendix \[a:ex\]). In this case $f(x,y,z)=x^4-yz$ and ${\mathbf{C}}={\mathbb{Z}}/2{\mathbb{Z}}$ acts as $(x,y,z)\mapsto (-x,z,y)$. Therefore (\[eq:explicit\]) reads as $${\mathcal{X}}=\{ (x,y,z, (b_1,b_2))~|~x^4-yz+b_1 x^2+ b_2=0 \in \mathrm{tot}(K^4) \}$$ in $\mathrm{tot}(K\oplus K^2\oplus K^2)\times {\mathbf{B}}$ with ${\mathbf{B}}=H^0({\Sigma}, K^2)\oplus H^0({\Sigma}, K^4)$.
Simultaneous resolutions
------------------------
The families ${\bm{\pi}}:{\mathcal{X}}\to {\mathbf{B}}$ and $\tilde{{\bm{\pi}}}:\tilde{{\mathcal{X}}}\to {\mathbf{B}}$ share many features of the Slodowy slice $S$. It seems unlikely though that our methods provide an explicit description of a simultaneous resolution for these families. However, we succeed over the locus $\tilde{{\mathbf{B}}}/W\subset {\mathbf{B}}$ (see (\[eq:B/W\])) which is in the singular locus of ${\bm{\pi}}$.\
To see this, let $\tilde{ev}:{\Sigma}\times \tilde{{\mathbf{B}}}\to \tilde{{\mathbf{B}}}$ be the evalution map which gives rise to the family $$\begin{tikzcd}
\tilde{ev}^*\tilde{{\bm{S}}}=\hat{{\mathcal{X}}}_L \ar[rr, bend left, "\hat{{\bm{\pi}}}_L"] \ar[r, "\hat{{\bm{\pi}}}_{1,L}"] & {\Sigma}\times \tilde{{\mathbf{B}}} \ar[r, "\hat{{\bm{\pi}}}_{2,L}"] & \tilde{{\mathbf{B}}}
\end{tikzcd}$$ of threefolds with a non-trivial ${\mathbf{C}}$-action. With the methods of the proof of Theorem \[thm:cy3s\], one proves that $\hat{{\bm{\pi}}}_L:\hat{{\mathcal{X}}}_L\to \tilde{{\mathbf{B}}}$ is a *smooth* family of quasi-projective threefolds with ${\mathbf{C}}$-trivial canonical class.
\[p:simresol\] The family $\hat{{\bm{\pi}}}:\hat{{\mathcal{X}}}\to \tilde{{\mathbf{B}}}$ is a simultaneous ${\mathbf{C}}$-resolution of $p^*{\mathcal{X}}$ where $p:\tilde{{\mathbf{B}}}\to \tilde{{\mathbf{B}}}/W\subset {\mathbf{B}}$ is the natural projection. It descends to a simultaneous *small* ${\mathbf{C}}$-resolution over $\tilde{{\mathbf{B}}}/W-\{0\} \subset {\mathbf{B}}$.
Using the fiber product property, we obtain a unique ${\mathbf{C}}$-equivariant morphism $\hat{{\mathcal{X}}}\to p^*{\mathcal{X}}$ over ${\Sigma}\times \tilde{{\mathbf{B}}}$ which makes the obvious diagrams commute.\
For the second claim, observe that the ${\mathbf{C}}$-family $
\hat{{\mathcal{X}}}/W\to \tilde{{\mathbf{B}}}/W
$ is still smooth over ${\mathbf{B}}^*:=\tilde{{\mathbf{B}}}/W-\{0\}\subset {\mathbf{B}}$. Then the morphisms $\hat{{\mathcal{X}}}\to p^*{\mathcal{X}}$ descends to the ${\mathbf{C}}$-morphism $$\hat{{\mathcal{X}}}/W\to p^*{\mathcal{X}}/W\cong {\mathcal{X}}$$ over ${\mathbf{B}}^*$. It remains to show that $(\hat{X}/W)_{b}\to X_b$ is a small resolution for $b\in {\mathbf{B}}^*$. Let $\tilde{b}$ with $p(\tilde{b})=b$. Then $X_b$ is only singular at the singularities of the fibers $p_b^{-1}(x)$ for $$x\in \mathrm{Div}(\prod_{\alpha\in R} \alpha(b) )$$ and the projection $p_b:X_b\to {\Sigma}$. By assumption $\prod_{\alpha\in R} \alpha (b)\neq 0$, i.e. the singularities are isolated, so that $(\hat{X}/W)_b\to X_b$ is a small resolution.
The last statement is false if we include $0\in {\mathbf{B}}$: Then $X_0\cong L\times_{{\mathbb{C}}^*} Y$ for the $\Delta$-singularity $Y$ and $\hat{X}_0\cong L\times_{{\mathbb{C}}^*} \hat{Y}$ for its minimal resolution $\hat{Y}\to Y$. Hence it is not a small resolution.
The singularities of $X_b$, $b\in {\mathbf{B}}^*$, are precisely the Gorenstein threefold singularities studied in [@KatzMorr]. The simplest (local) example is $$\begin{tikzcd}
&\hat{X}\ar[d] =\{ (x,y,z,t,[u:v])\in {\mathbb{C}}^4\times \mathbb{P}^1~|~x^2+y^2+z^2-t^2=0~,~xv=u(z+t)\}
\\
&X =\{ (x,y,z,s)\in {\mathbb{C}}^4~|~x^2+y^2+z^2-s=0 \}
\end{tikzcd}$$ for the small resolution is $(x,y,z,t,[u:v])\mapsto (x,y,z,t^2)$.
Calabi-Yau orbifolds and Hitchin systems {#s:mhs}
========================================
Let ${\mathcal{X}}\to {\mathbf{B}}$ be a family of quasi-projective Calabi-Yau threefolds with ${\mathbf{C}}$-trivial canonical class as in the previous section. For each $b\in {\mathbf{B}}$, we denote by $[X_b/{\mathbf{C}}]$ the corresponding quotient stack and refer to it as a Calabi-Yau orbifold. By construction, they fit into the family $
{\bm{\pi}}_{{\mathbf{C}}}^\circ:[{\mathcal{X}}/{\mathbf{C}}]\to {\mathbf{B}}$ of Calabi-Yau orbifolds. Before we study the *integral* (equivariant) cohomology groups $$H^3([X_b/{\mathbf{C}}],{\mathbb{Z}})=H^3_{{\mathbf{C}}}(X_b,{\mathbb{Z}}),\quad b\in {\mathbf{B}^\circ},$$ we need a general result.
MHS on equivariant cohomology {#ss:mhseq}
-----------------------------
Let $G$ be a finite group acting on a locally compact topological space $X$. We replace the orbifold stack $[X/G]$ by the simplicial space $$[X/G]_\bullet=((G^{p+1}\times X)/G)_{p\geq 0}$$ where $G$ acts on $G^{p+1}\times X$ via $$g\cdot (g_0,\dots, g_p, x)=(g_0 g^{-1},\dots, g_p g^{-1}, g\cdot x).$$ The simplicial structure maps of $[X/G]_\bullet$ are induced by the standard simplicial structure on $G^{\bullet+1}$. The importance of $[X/G]_\bullet$ is that it is a simplicial model for $X_G:=X \times_G EG$ which defines equivariant cohomology $$H_G^k(X,R)=H^k(X_G,R),\quad R={\mathbb{Z}},{\mathbb{Q}}.$$ More precisely, we have $$\label{eq:eqcohorb}
H^k([X/G]_\bullet, R) \cong H^k([X/G],R)= H_G^k(X,R), \quad R={\mathbb{Z}},{\mathbb{Q}}.$$ An important tool to compute the left-hand side is the spectral sequence $$\label{EqCohSS}
E_1^{pq}=H^q([X/G]_p,R) \Rightarrow E_\infty^k=H^k([X/G]_\bullet, R)=H_G^k(X,R),\quad R={\mathbb{Z}},{\mathbb{Q}}.$$ By [@DeligneIII], this spectral sequence is a spectral sequence of MHS if $X$ is a complex algebraic variety (considered in the analytic topology) on which $G$ acts algebraically. In particular, the equivariant cohomology groups $H^k_G(X,R)$ carry natural MHS.\
The Leray (or Serre) spectral sequence for the fibration $X_G\to BG$ implies that $H_G^k(X,{\mathbb{Q}})\cong H^k(X,{\mathbb{Q}})^G$ as abelian groups. Of course, both sides carry MHS and the next lemma shows that they agree. This is well-known to experts[^6] but for lack of a reference, we give a proof here for completeness.
\[lem:EqCohInv\] Let $G$ be a finite group acting on a complex algebraic variety $X$. Then the spectral sequence (\[EqCohSS\]) degenerates on the $E_2$-page and yields an isomorphism $$H_G^k(X,{\mathbb{Q}})\cong H^k(X,{\mathbb{Q}})^G$$ of ${\mathbb{Q}}$-MHS for each $k\geq 0$.
We express $E_1^{pq}=H^q((G^{p+1}\times X)/G,{\mathbb{Z}})$ of (\[EqCohSS\]) as follows: $$\begin{aligned}
E_1^{pq}=& H^q(G^{p+1}\times X,{\mathbb{Z}})^G & (\mbox{freeness of action}) \\
=& \operatorname{Hom}_{{\mathbb{Z}}[G]} ({\mathbb{Z}}, H^q(G^{p+1}\times X,{\mathbb{Z}})) & \\
=& \operatorname{Hom}_{{\mathbb{Z}}[G]} ({\mathbb{Z}}, H^0(G^{p+1},{\mathbb{Z}})\otimes_{{\mathbb{Z}}} H^{q}(X,{\mathbb{Z}}) ) & (\mbox{K\"unneth formula}) \\ =& \operatorname{Hom}_{{\mathbb{Z}}[G]} ({\mathbb{Z}}, \operatorname{Hom}_{{\mathbb{Z}}} (H_0(G^{p+1},{\mathbb{Z}}), H^q(X,{\mathbb{Z}})))) & (\mbox{freeness of }H_0(G^{p+1},{\mathbb{Z}})) \\
=& \operatorname{Hom}_{{\mathbb{Z}}}({\mathbb{Z}}\otimes_{{\mathbb{Z}}[G]} H_0(G^{p+1},{\mathbb{Z}}), H^q(X,{\mathbb{Z}})) & (\mbox{adjunction}) \\
=& \operatorname{Hom}_{{\mathbb{Z}}[G]}(H_0(G^{p+1},{\mathbb{Z}}),H^q(X,{\mathbb{Z}})).\end{aligned}$$ In the last line we have used the adjunction between the trivial module functor and $-\otimes_{{\mathbb{Z}}} {\mathbb{Z}}[G]$ ([@Weibel]). Since the complex $H_0(G^{\bullet+1}, {\mathbb{Z}})$ coincides with the bar resolution $B_\bullet \to {\mathbb{Z}}$, we see that $$E_1^{\bullet q}=\operatorname{Hom}_{{\mathbb{Z}}[G]}(B_\bullet, H^q(X,{\mathbb{Q}}))$$ as complexes for each $q\geq 0$. In particular, $E_1^{\bullet q}$ computes group cohomology $H^k(G, H^q(X,{\mathbb{Z}}))$ for each $q\geq 0$ so that $E_2^{pq}\cong H^p(G,H^q(X,{\mathbb{Q}}))$. But $G$ is finite and we work over ${\mathbb{Q}}$ so that $H^p(G,H^q(X,{\mathbb{Q}}))=0$ for all $p\geq 1$. Hence the spectral sequence (\[EqCohSS\]) of MHS degenerates on the $E_2$-page to give isomorphisms $$H^k(X,{\mathbb{Q}})^G\cong H^k([X/G]_\bullet,{\mathbb{Q}})=H_G^k(X,{\mathbb{Q}})$$ of MHS.
CY orbifolds and Hitchin systems
--------------------------------
We begin with a general result on the Leray spectral sequence for equivariant maps.
\[lem:leraygroup\] Let $G$ be a discrete group acting on topological spaces $X,Y,Z$. Further let $f:X\to Y$, $h:Y\to Z$ be $G$-equivariant morphisms. Then the Leray spectral sequence $$\label{LeraySS}
R^pg_* R^qf_*A_X \Rightarrow R^{p+q}(g\circ f)_*A_X$$ for any constant sheaf $A_X$ of an abelian group $A$ on $X$ lifts via the forgetful functor $\mathrm{For}:Sh_G(Z)\to Sh(Z)$ to $G$-equivariant abelian sheaves $Sh_G(Z)$ on $Z$.
First of all, the exact forgetful functor $\mathrm{For}:Sh_G(W)\to Sh(W)$ for $W=X,Y,Z$ commutes with the equivariant direct image functor $f^{\mathrm{eq}}_*$, e.g. $\mathrm{For}\circ f^{\mathrm{eq}}_*\simeq f_*\circ \mathrm{For}$. Since $Sh_G(W)$ has enough injectives, we can form derived direct image functors, e.g. $$Rf^{\mathrm{eq}}_*:D^b_G(X)\simeq D^b(Sh_G(X))\to D^b_G(Y)\simeq D^b(Sh_G(Y)).$$ Again they commute with the exact forgetful functors which implies the claim.
\[p:isoeq\] Let $X=X_b$, $b\in {\mathbf{B}^\circ}$, be a smooth quasi-projective Calabi-Yau threefold with ${\mathbf{C}}$-action as in Section \[s:cy\]. Then there is a natural isomorphism $$H^3([X/{\mathbf{C}}],{\mathbb{Z}})=H^3_{{\mathbf{C}}}(X,{\mathbb{Z}})\cong H^3(X,{\mathbb{Z}})^{{\mathbf{C}}}$$ of ${\mathbb{Z}}$-MHS.
We emphasize that we work over the *integers* giving a stronger result as Lemma \[lem:EqCohInv\]. In general, such a result is false due to torsion, cf. Example \[ex:eqcoh\] below.
As in the proof of Lemma \[lem:EqCohInv\], we employ the spectral sequence (\[EqCohSS\]). In that proof we have seen that its $E_2$-page is of the form $$E_2^{pq}=H^p({\mathbf{C}}, H^q(X,{\mathbb{Z}})).$$ We show that it degenerates on the $E_3$-page in this situation by proving $$\label{E3deg}
H^p({\mathbf{C}},H^q(X,{\mathbb{Z}}))=0,\quad \forall p\geq 1, q\notin \{3,4 \}.$$ Of course this is automatic if we worked over ${\mathbb{Q}}$. To show it over the integers, we observe that $$\begin{gathered}
H^0(X,{\mathbb{Z}})\cong H^0({\Sigma}, \pi_*{\mathbb{Z}}),\quad H^1(X,{\mathbb{Z}})\cong H^1({\Sigma},\pi_*{\mathbb{Z}}), \quad H^2(X,{\mathbb{Z}})\cong H^2({\Sigma},\pi_*{\mathbb{Z}}), \\
H^3(X,{\mathbb{Z}})\cong H^1({\Sigma}, R^2\pi_*{\mathbb{Z}}), \quad H^4(X,{\mathbb{Z}})\cong H^2({\Sigma}, R^2\pi_*{\mathbb{Z}}), \quad H^q(X,{\mathbb{Z}})=0 \quad\forall q\geq 5\end{gathered}$$ for the projection $\pi: X\to {\Sigma}$. This is seen by using the Leray spectral sequence (cf. [@DDP], [@Beck]). Since ${\mathbf{C}}$ acts trivially on $\pi_*{\mathbb{Z}}$, it follows that ${\mathbf{C}}$ acts trivially on $H^q(X,{\mathbb{Z}})$ for $q\notin \{ 3,4\}$ by Lemma \[lem:leraygroup\]. This yields (\[E3deg\]) so that the spectral sequence (\[EqCohSS\]) gives the isomorphism $$E_3^{0,3}=H^3(X,{\mathbb{Z}})^{\mathbf{C}}\cong E_\infty^3=H^3_{{\mathbf{C}}}(X,{\mathbb{Z}})$$ of ${\mathbb{Z}}$-MHS.
\[ex:eqcoh\] The analogous statement is *false* for the minimal resolution $\hat{Y}\to Y$ of the $\Delta$-singularity $Y$ (assuming $\Delta\neq \Delta_h$): A group cohomology computation shows $$H^2_{{\mathbf{C}}}(\hat{Y},{\mathbb{Z}})\cong H^2(\hat{Y},{\mathbb{Z}})^{{\mathbf{C}}}\oplus {\mathbb{Z}}/2{\mathbb{Z}}.$$
Since $H^3(X_b,{\mathbb{Z}})$, $b\in {\mathbf{B}^\circ}$, is up to a Tate twist a polarizable ${\mathbb{Z}}$-HS (see [@Beck], Lemma 5) of weight $1$, it follows that the orbifold intermediate Jacobian $$\label{eq:eqj2}
J^2([X_b/{\mathbf{C}}])=H^3([X_b/{\mathbf{C}}],{\mathbb{C}})/\left(F^2H^3([X_b/{\mathbf{C}}],{\mathbb{C}})+H^3([X_b/{\mathbf{C}}],{\mathbb{Z}})\right)$$ of the Calabi-Yau orbifold $[X_b/{\mathbf{C}}]$ is an abelian variety.\
To globalize the previous discussion, we consider the augmented simplicial morphism associated to the family ${\bm{\pi}}_{{\mathbf{C}}}:[{\mathcal{X}}/{\mathbf{C}}]\to {\mathbf{B}}$ of Calabi-Yau orbifolds. By abuse of notation, it is again denoted by $$\begin{tikzcd}
{\bm{\pi}}_{{\mathbf{C}}}:[{\mathcal{X}}/{\mathbf{C}}]_\bullet \ar[r, "{\bm{\pi}}_{{\mathbf{C}},\bullet}"] & {\mathbf{B}}_\bullet \ar[r, "a"] & {\mathbf{B}}.
\end{tikzcd}$$ where the last arrow is the augmentation. In accordance with (\[eq:eqcohorb\]), we set $$R^k{\bm{\pi}}_{{\mathbf{C}},*} {\mathbb{Z}}= \bm{s} R^k({\bm{\pi}}_{{\mathbf{C}},\bullet})_*{\mathbb{Z}},$$ compare [@DeligneIII], Section 5.2, where $\bm{s}$ stands for the total complex.
Let ${\bm{\pi}}_{{\mathbf{C}}}^\circ:[{\mathcal{X}}^\circ/{\mathbf{C}}] \to {\mathbf{B}^\circ}$ be the family of orbifolds associated with the smooth family ${\bm{\pi}}:{\mathcal{X}}^\circ\to {\mathbf{B}^\circ}$. Then $R^3{\bm{\pi}}_{{\mathbf{C}},*}^\circ {\mathbb{Z}}$ carries the structure of a polarizable ${\mathbb{Z}}$-VMHS such that $$(R^3{\bm{\pi}}_{{\mathbf{C}},*}^\circ {\mathbb{Z}})_b\cong H^3_{{\mathbf{C}}}(X_b,{\mathbb{Z}}), \quad b\in {\mathbf{B}^\circ},$$ naturally as $Z$-MHS.
This is a consequence of the more general treatment in [@BeckVMHS] on VMHS for simplicial smooth and quasi-projective morphisms (which are topologically locally trivial on each level). Alternatively, the V(M)HS-structure can be constructed directly in this present case by using an approximation of the classifying space $B{\mathbf{C}}$ of ${\mathbf{C}}$ by finite-dimensional smooth projective varieties.
Therefore the orbifold intermediate Jacobians (\[eq:eqj2\]) fit into the family $$\mathcal{J}^2([{\mathcal{X}}^\circ/{\mathbf{C}}])\to {\mathbf{B}^\circ}$$ of abelian varieties. Using the methods of [@Beck], it is seen to be an algebraic integrable system.
Let $\Delta$ be an irreducible Dynkin diagram, ${\mathbf{C}}=AS(\Delta)$ and $G=G(\Delta)$ the corresponding simple adjoint complex Lie group. Further let ${\mathcal{X}}\to {\mathbf{B}}({\Sigma}, \Delta)$ be one of the families of Calabi-Yau threefolds of Section \[s:cy\] with ${\mathbf{C}}$-action. Then $$\mathcal{J}^2([{\mathcal{X}^\circ}/{\mathbf{C}}]) \cong {\mathbf{Higgs}}_1^\circ({\Sigma}, G)$$ $$\mathcal{J}_2([{\mathcal{X}^\circ}/{\mathbf{C}}]) \cong {\mathbf{Higgs}}_1^\circ({\Sigma}, ^L G).$$ as algebraic integrable systems over ${\mathbf{B}^\circ}({\Sigma}, \Delta)$.
The spectral sequence (\[EqCohSS\]) globalizes to a spectral sequence of polarizable ${\mathbb{Z}}$-VMHS. Hence Proposition \[p:isoeq\] implies $$R^3{\bm{\pi}}_{{\mathbf{C}},*}^\circ {\mathbb{Z}}\cong (R^3{\bm{\pi}}_*{\mathbb{Z}})^{\mathbf{C}}$$ as polarizable ${\mathbb{Z}}$-VHS of weight $1$ (up to a Tate twist). In particular, $$\mathcal{J}^2([{\mathcal{X}^\circ}/{\mathbf{C}}]) \cong \mathcal{J}^2_{{\mathbf{C}}}({\mathcal{X}^\circ})$$ over ${\mathbf{B}^\circ}$ where the right-hand side is the intermediate Jacobian fibration defined by ${\mathbf{C}}$-invariants in cohomology. Now the claim follows from [@Beck], Theorem 6.\
Replacing ${\bm{\pi}}_{{\mathbf{C}},*}$ with ${\bm{\pi}}_{{\mathbf{C}},!}$ and using Theorem 8 of [@Beck] gives the second isomorphism.
Folding {#SectFolding}
=======
Let $\Delta$ be an irreducible Dynkin diagram. We follow [@Slo] and define the associated symmetry group of $\Delta$ via$$\label{ASDelta}
AS(\Delta):=
\begin{cases}
1, & \Delta\mbox{ is of type }{\mathrm{ADE}}, \\
{\mathbb{Z}}/2{\mathbb{Z}}, & \Delta\mbox{ is of type }\mathrm{B}_{k}, \mathrm{C}_{k}, \mathrm{F}_4, \\
S_3, & \Delta\mbox{ is of type }\mathrm{G}_2,
\end{cases}$$ for $k\geq 2$. There is a unique irreducible ${\mathrm{ADE}}$-Dynkin diagram $\Delta_h$ such that $AS=AS(\Delta)\subset \operatorname{Aut}(\Delta_h)$ and $\Delta=\Delta_h^{AS}$. Here $\Delta_h^{AS}$ stands for the Dynkin diagram which is obtained by taking $AS(\Delta)$-invariants $R_h^{AS}$ in the root space $R_h=R(\Delta_h)$ associated with $\Delta_h$ (cf. [@Beck-thesis], Chapter 1.2, or [@Springer])
\[r:dynkin\] To obtain a reasonable notion of folding on the level of root spaces, it is important that we only work with Dynkin graph automorphisms $\operatorname{Aut}_D(\Delta_h)\subset \operatorname{Aut}(\Delta_h)$. These are graph automorphisms $a\in \operatorname{Aut}(\Delta_h)$ such that $a(v)$ and $v$ are not direct neighbors for each vertex $v\in \Delta_h$. If $\Delta_h\neq \mathrm{A}_{2n}$, then $\operatorname{Aut}_D(\Delta_h)=\operatorname{Aut}(\Delta_h)$ and $\operatorname{Aut}_D(\Delta_h)=1$ if $\Delta_h=\mathrm{A}_{2n}$.
Restricted to Dynkin diagrams of type $\mathrm{B}_k, \mathrm{C}_k, \mathrm{F}_4, \mathrm{G}_2$ (*${\mathrm{BCFG}}$-Dynkin diagrams for short*), we obtain a bijection $$\label{FoldingBijection}
\begin{aligned}
\{ \Delta ~\text{ of type }{\mathrm{BCFG}}\} &\to \{ (\Delta_h,{\mathbf{C}})~|~\Delta_h~{\mathrm{ADE}},~1\neq {\mathbf{C}}\subset \operatorname{Aut}_D(\Delta_h)\}\\
\Delta&\mapsto (\Delta_h,AS(\Delta)) \\
\Delta=\Delta^{{\mathbf{C}}}_h&\mapsfrom (\Delta_h, {\mathbf{C}}).
\end{aligned}$$ We say that $\Delta=\Delta_{h}^{\mathbf{C}}$ is obtained from $(\Delta_h,{\mathbf{C}})$ (or simply $\Delta_h$) by *folding*. For convenience we summarize the corresponding types in the following table $$\label{FoldingDynkin}
\begin{array}{c|c|c}
\Delta & \Delta_h & AS(\Delta) \\ \hline
\mathrm{B_{k+1}} & \mathrm{A}_{2k+1} & {\mathbb{Z}}/2{\mathbb{Z}}\\
\mathrm{C}_{k} & \mathrm{D}_{k+1} & {\mathbb{Z}}/2{\mathbb{Z}}\\
\mathrm{F}_4 & \mathrm{E}_6 & {\mathbb{Z}}/2{\mathbb{Z}}\\
\mathrm{G}_2 & \mathrm{D}_4 & S_3 \\
\end{array}$$ Finally, we collect the weights of the ${\mathrm{ADE}}$-singularities that are naturally induced by their Lie-theoretic realization.
Dynkin type of $\Gamma$ and equation $(w_{r+1},w_{r+2},w_{r+3}; d)$
----------------- -------------------------------------- --------------------------------
$\mathrm{A}_k$: $x^{k+1}-yz=0$ $(2,k+1,k+1; 2(k+1))$
$\mathrm{D}_k$: $x(x^{k-2}-y^2)-z^2=0$ $(2,k-2,k-1;2k-2)$
$\mathrm{E}_6$: $x^4+y^3+z^2=0$ $(6,8,12;24)$
$\mathrm{E}_7$: $x^3y+y^3+z^2=0$ $(8,12,18;36)$
$\mathrm{E}_8$: $x^5+y^3+z^2=0$ $(12,20,30;60)$
\[ADEsings\]
Here $r$ is the rank of the corresponding simple complex Lie algebra. Note that by definition, this table contains the natural weights of all $\Delta$-singularities.
Slodowy slices for $\mathrm{B}_2$-singularities {#a:ex}
===============================================
Let $\Delta=\mathrm{B}_2$ and[^7] $(\Delta_0,{\mathbf{C}})=(\mathrm{A}_3,{\mathbb{Z}}/2{\mathbb{Z}})$ be the associated pair. Further we let $${\mathfrak{g}}_0={\mathfrak{g}}_(\Delta_0)=\mathfrak{sl}(4,{\mathbb{C}}), \quad {\mathfrak{g}}={\mathfrak{g}}(\Delta)=\mathfrak{so}(5,{\mathbb{C}})$$ be the corresponding simple complex Lie algebras. We give two Slodowy slices $S_0\subset {\mathfrak{g}}_0$ and $S\subset {\mathfrak{g}}$ together with the ${\mathbf{CA}}$- and ${\mathbf{C}}$-action respectively (see (\[eq:Cc\] and (\[eq:CA\]) and directly compute that they both realize a semi-universal deformation of the $\mathrm{B}_2$-singularity.
Extrinsic case (${\mathfrak{g}}_0$)
-----------------------------------
We consider the following $\mathfrak{sl}_2$-triple in ${\mathfrak{g}}_0$: $$x_0=
\begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 \\
\end{pmatrix},\quad
y_0=
\begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
\end{pmatrix},\quad
h_0=
\begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 \\
\end{pmatrix}.$$ The Slodowy slice with respect to $(x_0,y_0,h_0)$ is given by $$\label{A3S0}
S_0=x_0+\ker \mathrm{ad}(y_0)=\left\{
\begin{pmatrix}
-3a & b & 0 & 0 \\
0 & a & 1 & 0 \\
0 & c & a & 1 \\
d & e & c & a
\end{pmatrix}~:~a,b,c,d,e\in {\mathbb{C}}\right\}$$ Let $\phi\in \mathrm{Aut}({\mathfrak{g}}_0)$ be given by $\phi(A)=-A^t$ and $$g_0=
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 \\
0 & 0 & 1 & 0 \\
0 & -1 & 0 & 0
\end{pmatrix}$$ (actually the class in $G_0=PGL(4,{\mathbb{C}})$). Then it is immediate to check that $$\tau:=\mathrm{Ad}(g_0)\circ \phi\in CA(x,h)$$ and $\tau^2=1$. It generates the subgroup $\mathbf{CA}\subset CA(x,h)$ which is isomorphic to ${\mathbb{Z}}/2{\mathbb{Z}}$. Its action on $S_0$ is given by $$\label{CAaction}
\tau\cdot \begin{pmatrix}
-3a & b & 0 & 0 \\
0 & a & 1 & 0 \\
0 & c & a & 1 \\
d & e & c & a
\end{pmatrix}
=
\begin{pmatrix}
3a & d & 0 & 0 \\
0 & -a & 1 & 0 \\
0 & c & -a & 1 \\
b & -e & c & -a
\end{pmatrix}.$$ To determine its action on the base, we compute $\sigma_0:S_0\to {\mathfrak{t}}_0/W_0$. We take as homogeneous generators the coefficients $\chi_2,\chi_3,\chi_4$ of degree $2,3,4$ of the characteristic polynomial. If $B\in S_0$ is as in (\[A3S0\]), then $$\sigma_0(B)=(\chi_2(B),\chi_3(B),\chi_4(B))
= (-6a^2-2c,8a^3-4ac-e,-3a^4+6a^2c-bd-3ae).$$ It can be readily checked that $\sigma^{-1}(0,0,0)$ is an $\mathrm{A}_3$-singularity given by $$\label{A3inS0}
c=-3a^2,\quad e=20a^3,\quad -81a^4-bd=0.$$ in the notation of (\[A3S0\]). Moreover, we have $$\sigma_0(\tau\cdot B)=(\chi_2(B),-\chi_3(B),\chi_4(B)).$$ Hence if we restrict to the subspace $$\begin{tikzcd}
({\mathfrak{t}}_0/W_0)^{\mathbf{CA}}=\{(s,0,t)\in {\mathbb{C}}^3\cong {\mathfrak{t}}_0/W_0~|~s,t\in {\mathbb{C}}\}\ar[r, hook, "i"] & {\mathfrak{t}}_0/W_0,
\end{tikzcd}$$ then $\mathbf{CA}$ preserves the fibers of $\sigma_0$. Since $i^*S_0\subset S_0$ is defined by $e=4ac-8a^3$, we obtain $$\sigma_0(B)=(-6a^2-2c,0,-27a^4+18a^2c-bd),\quad B\in i^*S_0.$$ Finally, we see that $\mathbf{CA}$ acts on the $\mathrm{A}_3$-singularity $\sigma_0^{-1}(0,0,0)$ as described in Example \[ex:B2\] by setting $x=-3a,y=b,z=d$ in the notation of (\[A3inS0\]).
Intrinsic case (${\mathfrak{g}}$)
---------------------------------
For this example, we use the conventions and methods, e.g. the classification of nilpotent orbits in terms of weighted Dynkin diagrams, of [@CollingwoodMcGovern], Chapter 5. First of all, let $${\mathfrak{g}}=\mathfrak{so}(5,{\mathbb{C}})\cong
\left\{
\begin{pmatrix}
0 & u_1 & u_2 & v_1 & v_2 \\
-v_1 & a_1 & a_2 & 0 & b \\
-v_2 & a_3 & a_4 & -b & 0 \\
-u_1 & 0 & c & -a_1 & -a_3 \\
-u_2 & -c & 0 & -a_2 & -a_4
\end{pmatrix}~:~
u_i,v_j,a_k,b,c\in {\mathbb{C}}\right\}$$ This form of $\mathfrak{so}(5,{\mathbb{C}})$ has the advantage that diagonal matrices give a Cartan ${\mathfrak{t}}\subset {\mathfrak{g}}$. Then we consider the following $\mathfrak{sl}_2$-triplet $(x,y,h)$ where $x\in {\mathfrak{g}}$ is a subregular nilpotent element: $$x=\begin{pmatrix}
0 & 0 & 0 & 1 & 0 \\
-1 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0
\end{pmatrix},\quad
y=\begin{pmatrix}
0 & -2 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
2 & 0 & -2 & 0 & 0 \\
0 & 2 & 0 & 0 & 0
\end{pmatrix},\quad
h=\begin{pmatrix}
0 & 0 & 0 & 0 & 0 \\
0 & 2 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -2 & 0 \\
0 & 0 & 0 & 0 & 0 \\
\end{pmatrix}.$$ The corresponding Slodowy slice $S=x+\ker \mathrm{ad}~y\subset {\mathfrak{g}}$ is given by $$\label{SB2}
S=
\left\{C=
\begin{pmatrix}
0 & - a & -b & 1 & 0 \\
-1 & 0 & 1 & 0 & 0 \\
0 & c & -b & 0 & 0 \\
a & 0 & d & 0 & -c \\
b & -d & 0 & -1 & b
\end{pmatrix}~:~a,b,c,d\in {\mathbb{C}}\right\}$$ Again using the coefficients $\chi_2,\chi_4$ of the characteristic polynomial of degree $2$ and $4$ respectively, the adjoint quotient restricted to $S$ is given by $$\sigma(C)=(\chi_2(C),\chi_4(C))=(-b^2-2a-2c,2ab^2+2b^2c+2ac+c^2+2cd).$$ In the notation of (\[SB2\]), we see that $\sigma^{-1}(0,0,0)$ is given by $$\label{B2eq}
a=-c-\frac{b^2}{2},\quad b^4+c(b^2-2d+c)=0$$ which is immediately seen to be an $\mathrm{A}_3$-singularity.\
Using the fact that $Z_G(h)$ is the subgroup in $G\cong PSO(5,{\mathbb{C}})$ generated by the diagonal maximal torus $T$ and the unipotent subgroup $U_{\alpha_2}$ corresponding to the short root $\alpha_2\in \Delta$, we compute $$C(x,h)=Z_G(x)\cap Z_G(h)=
\left\{
D^k=
\begin{pmatrix}
1 & 0 & 0 & 0 & 2 \\
0 & -1 & 0 & 0 & 0 \\
2 & 0 & -1 & 0 & 2 \\
0 & 0 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & -1
\end{pmatrix}^k
~:~k=0,1
\right\}$$ (actually the class in $G$). Observe that in this case $\mathbf{C}= C(x,h)$, cf. (\[eq:Cc\]). Finally, the action of ${\mathbf{C}}$ on $S$ is given by $$D\cdot C=
\begin{pmatrix}
0 & a+2d & b & 1 & 0 \\
-1 & 0 & 1 & 0 & 0 \\
0 & 2a+c+2d & b & 0 & 0 \\
-a-2d & 0 & d & 0 & -2a-c-2d \\
-b & -d & 0 & -1 & -b
\end{pmatrix}.$$ It is readily seen from (\[B2eq\]) that $(\sigma^{-1}(0),\mathbf{C})$ is indeed a $\mathrm{B}_2$-singularity.
[^1]: Since the Hitchin base ${\mathbf{B}}({\Sigma}, G)$ only depends on the Dynkin diagram $\Delta(G)$, it is sufficient to write ${\mathbf{B}}({\Sigma}, \Delta)$, see \[ss:hitchinbase\]).
[^2]: The subindex of $\Delta_h$ stands for homogeneous.
[^3]: We often use ${\mathbf{C}}$ to emphasize how the $AS(\Delta)$-action is realized.
[^4]: Recall that we usually let ${\mathbb{C}}^*$ act on ${\mathfrak{t}}/W$ by twice the natural weights.
[^5]: Note that $\psi_i^*\mathcal{E}_i\cong \Omega^2_{\tilde{{\bm{\sigma}}}|{\tilde{\mathcal{S}}}_i}$.
[^6]: We kindly acknowledge the help of Donu Arapura via MathOverflow.
[^7]: Here we use the notation $\Delta_0$ instead of $\Delta_h$ in the main text for notational reasons.
|
---
author:
-
- '[^1]'
-
-
-
bibliography:
- 'refs.bib'
title: ' Update on $\varepsilon_K$ with lattice QCD inputs '
---
Introduction {#sec:intro}
============
This paper is an update of our previous papers [@Bailey:2015tba; @Bailey:2015frw; @Lee:2016xkb; @Bailey:2016dzk]. In the standard model, the indirect CP violation parameter $\epsK$ of neutral kaons can be expressed as follows, $$\begin{aligned}
\label{eq:epsK_def}
\epsK
& \equiv \frac{\mathcal{A}(K_L \to \pi\pi(I=0))}
{\mathcal{A}(K_S \to \pi\pi(I=0))}
\nonumber \\
& = e^{i\theta} \sqrt{2}\sin{\theta}
\Big( C_{\eps} \hat{B}_{K} X_\text{SD}
+ \frac{ \xi_{0} }{ \sqrt{2} } + \xi_\text{LD} \Big)
+ \mathcal{O}(\omega\eps^\prime)
+ \mathcal{O}(\xi_0 \Gamma_2/\Gamma_1) \,, \end{aligned}$$ where $$\begin{aligned}
C_{\eps}
&= \frac{ G_{F}^{2} F_K^{2} m_{K^{0}} M_{W}^{2} }
{ 6\sqrt{2} \; \pi^{2} \; \Delta M_{K} } \,,
\qquad
\xi_0 = \frac{\Im A_0}{\Re A_0} \,,
\qquad
\xi_\text{LD} = \frac{m^\prime_\text{LD}}{\sqrt{2} \; \Delta M_K} \,,
\label{eq:xiLD}
\\
m^\prime_\text{LD}
&= -\Im \left[ \mathcal{P} \; \sum_{C}
\frac{\mate{\wbar{K}^0}{H_\text{w}}{C} \mate{C}{H_\text{w}}{K^0}}
{m_{K^0}-E_{C}}
\right]
\label{eq:mLD}
\\
X_\text{SD} &= \Im\lambda_t \Big[ \Re\lambda_c \eta_{cc} S_0(x_c)
-\Re\lambda_t \eta_{tt} S_0(x_t) - (\Re\lambda_c -
\Re\lambda_t) \eta_{ct} S_0(x_c,x_t) \Big] \,.
\label{eq:X_SD}\end{aligned}$$ Here, $X_\text{SD}$ represents the short distance contribution from the box diagrams. The formulas for $\lambda_i$, $S_0$, $x_i$, and $\eta_{ij}$ with $i,j =
\{c,t\}$ are given in Ref. [@Bailey:2015tba]. The $\xi_0$ and $\xi_\text{LD}$ represent the long distance effects from the absorptive part and the dispersive part, respectively.
Since Lattice 2016, there has been a major update on the Wolfenstein parameters: $\lambda$, $\bar{\rho}$, and $\bar{\eta}$ from the angle-only fit (AOF) of the UTfit collaboration as well as those from the global unitarity triangle (UT) fit of the CKMfitter and UTfit collaborations. Since Lattice 2016, the HFLAV group has also updated results of $\abs{\Vcb}$ and $\abs{\Vub}$. Hence, it is time to update the current status of $\epsK$ in lattice QCD.
Input parameters {#sec:input-para}
================
Wolfenstein parameters, $\hat{B}_K$, $\lvert V_{cb} \rvert$, $\xi_0$, and $\xi_\text{LD}$ are separately discussed in the following subsections. Other input parameters in Eq. are the same as in Ref. [@Bailey:2015tba] except for the charm quark mass $m_c(m_c)
= 1.2733(76)\;\mathrm{GeV}$, which is taken from the HPQCD collaboraion result [@Chakraborty:2014aca]. They are given in Tables \[tab:eta\] and \[tab:other\].
Wolfenstein parameters {#ssec:wolf-para}
----------------------
The CKMfitter and UTfit collaboration provide the Wolfenstein parameters ($\lambda$, $\bar{\rho}$, $\bar{\eta}$, $A$) determined by the global unitarity triangle (UT) fit. The 2017 results are summarized in Table \[tab:wp\]. Here, $\epsK$, $\BK$, and $|V_{cb}|$ are used as inputs to the global UT fit. Hence, the Wolfenstein parameters extracted by the global UT fit contain unwanted correlation with $\epsK$. In order to avoid this correlation, we take another input set from the angle-only fit (AOF) suggested in Ref. [@Bevan2013:npps241.89]. The AOF does not use $\epsK$, $\BK$, and $|V_{cb}|$ as input to determine the UT apex ($\bar{\rho}$, $\bar{\eta}$). We can determine $\lambda$ independently from $|V_{us}|$ which comes from the $K_{\ell 3}$ and $K_{\ell 2}$ decays using lattice QCD inputs. The 2017 results of the AOF are summarized in Table \[tab:wp\].
Input parameter $\BK$ {#ssec:bk}
---------------------
Recently, FLAG has reported results for $\BK$ calculated in lattice QCD with $N_f=2+1$ flavors [@Aoki:2016frl]. $$\begin{aligned}
\BK &= 0.7625(97) \qquad \text{from FLAG-2017.}\end{aligned}$$ This is the global average over the results of BMW-2011 [@Durr2011:PhysLettB.705.477], Laiho-2011 [@Laiho:2011np], RBC-UK-2016 [@Blum:2014tka], and SWME-2016 [@Jang:2015sla].
Input parameter $|\Vcb|$ {#ssec:V_cb}
------------------------
In Table \[tab:Vcb\], we summarize updated results for both exclusive $|\Vcb|$ and inclusive $|\Vcb|$. Recently HFLAV reported them [@Amhis:2016xyh]. At present, we find that there exists a $4.1\sigma$ tension between exclusive and inclusive $|\Vcb|$ when we use the combined averages given in Table \[tab:Vcb\]. We use these combined results when we evaluate $\epsK$.
In Fig. \[fig:Vcb\], we present results for $|\Vcb|$ and $|\Vub|$. The big change is that, as of Lattice 2016, the result for exclusive $|\Vcb|$ from $\bar{B} \to D \ell \bar{\nu}$ was about one sigma away from that from $\bar{B} \to D^{*} \ell \bar{\nu}$ (refer to Ref. [@Lee:2016xkb; @Bailey:2016dzk] for more details), but as of Lattice 2017, they are on top of each other, as shown in Fig. \[fig:Vcb\]. In 2017, due to the addition of more data from ALEPH, CLEO, OPAL, and DELPHI to the HFLAV analysis for exclusive $|\Vcb|$, the results from $\bar{B} \to D^{*} \ell \bar{\nu}$ do not change visibly, but those from $\bar{B} \to D \ell \bar{\nu}$ shift downward by about $1.0\sigma$. For more details, refer to Ref. [@Amhis:2016xyh].
Regarding $|\Vub|/|\Vcb|$, in 2016, we used the results of lattice QCD in Ref. [@Detmold:2015aaa], but in 2017, we use the results of HFLAV in Ref. [@Amhis:2016xyh]. Due to the addition of more data to the HFLAV analysis, the results shift downward by $\frac{3}{4}\sigma$ in 2017.
Regarding $|\Vub|$, in 2016, we used the results of Ref. [@DeTar:2015orc] obtained using a combined fit of the results in Refs. [@Lattice:2015tia; @Flynn:2015mha] for $B \to \pi \ell \nu$ decay over the data subset from BABAR and BELLE, but in 2017, we use the results of HFLAV in Ref. [@Amhis:2016xyh] obtained using the exclusive decay results of $B \to \pi \ell \nu$ from BABAR and BELLE. Both results are used as input to determine $|\Vcb|$ and $|\Vub|$ from the combined fit with all the exclusive decay results for $B$ mesons and $\Lambda_b$ baryons. The shift in $|\Vub|$ is downward by about $0.1\sigma$ in 2017.
Input parameters $\xi_0$ and $\xi_\text{LD}$ {#ssec:xi0}
--------------------------------------------
The absorptive part of long distance effects in $\epsK$ is parametrized into $\xi_0$. It is related to $\eps'/\eps$ and $\xi_2$ as follows, $$\begin{aligned}
\xi_0 &= \frac{\Im A_0}{\Re A_0}, \qquad
\xi_2 = \frac{\Im A_2}{\Re A_2}, \qquad
\Re \left(\frac{\eps'}{\eps} \right) =
\frac{\omega}{\sqrt{2} |\eps_K|} (\xi_2 - \xi_0) \,.
\label{eq:e'/e:xi0}\end{aligned}$$
There are two independent methods to determine $\xi_0$ in lattice QCD: one is the indirect method and the other is the direct method. In the indirect method, we determine $\xi_0$ using Eq. with lattice QCD input $\xi_2$ and with experimental results for $\eps'/\eps$, $\epsK$, and $\omega$. In the direct method, we can determine $\xi_0$ directly using lattice QCD results for $\Im A_0$ combined with experimental results for $\Re A_0$. Here, we prefer the indirect method to the direct method thanks to two reasons. The first reason is that the lattice QCD calculation of $\Im A_0$ is much noisier than that of $\Im A_2$ due to many disconnected diagrams. The second reason is that the S-wave phase shift $\delta_0$ of the $\pi-\pi$ scattering in Ref. [@Bai:2015nea] is lower by $3.0\sigma$ than the conventional determination of $\delta_0$ in Ref. [@Colangelo:2001df; @GarciaMartin:2011cn; @DescotesGenon:2001tn], which indicates that there might be some issues unresolved at present. For more details on the second reason, refer to Ref. [@Lee:2016xkb; @Bailey:2016dzk]. In Table \[tab:xi0\], we present results of $\xi_0$ determined using both indirect and direct methods. Here, we use the value of $\xi_0$ with the indirect method.
The $\xi_\text{LD}$ parameter represents the long distance effect from the dispersive part. There has been an on-going attempt to calculate it in lattice QCD [@Christ:2014qwa]. However, this attempt [@Christ:2015phf] is an exploratory study rather than a high precision measurement at present. Hence, in this paper we use the rough estimate of $\xi_\text{LD}$ in Ref. [@Christ:2014qwa]. It is given in Table \[tab:xi0\].
Results for $\epsK$ with lattice QCD inputs
===========================================
In Fig. \[fig:epsK\], we present results for $\epsK$ calculated directly from the standard model with the lattice QCD inputs described in the previous section. In Fig. \[fig:epsK\], the blue curve represents the theoretical evaluation of $\epsK$ with FLAG $\BK$, AOF Wolfenstein parameters, and exclusive $|\Vcb|$ which corresponds to ex-combined in Table \[tab:Vcb\]. The red curve in Fig. \[fig:epsK\] represents the experimental value of $\epsK$. In Fig. \[fig:epsK\], the blue curve represents the same as in Fig. \[fig:epsK\] except for using the inclusive $|\Vcb|$ which corresponds to in-combined in Table \[tab:Vcb\].
The updated results for $\epsK$ are, in units of $1.0\times
10^{-3}$, $$\begin{aligned}
|\epsK| &= 1.58 \pm 0.16 & & \text{for exclusive $|\Vcb|$ (lattice QCD)}
\\
|\epsK| &= 2.05 \pm 0.18 & & \text{for inclusive $|\Vcb|$ (QCD sum rule)}
\\
|\epsK| &= 2.228 \pm 0.011 & & \text{(experimental value)} \end{aligned}$$ This indicates that the theoretical evaluation of $\epsK$ with lattice QCD inputs (with exclusive $|\Vcb|$) has $4.0\sigma$ tension with the experimental result, while there is no tension in the inclusive $|\Vcb|$ channel (heavy quark expansion based on the OPE and QCD sum rules).
In Table \[tab:epsK-budget\], we present the error budget for $\epsK^\text{SM}$. Here, we find that the uncertainty from $|V_{cb}|$ is dominant in the error budget, while the errors from $\bar{\eta}$ and $\eta_{ct}$ are sub-dominant. Hence, if we are to see a gap $\Delta \epsK$ greater than $5.0\sigma$, it is essential to reduce the error in $|V_{cb}|$ significantly. To reduce this error as much as possible, a project to calculate $\bar{B} \to D^{(\ast)} \ell \bar{\nu}$ form factors using the OK action is underway. Note that the OK action is improved up to the $\lambda^3$ order[^2] in the HQET power counting, while the original Fermilab action is improved up to the $\lambda^1$ order. For more details on this issue, refer to Refs. [@SWPark:2017; @JHLeem:2017; @Bailey:2016wza; @Jeong:2016eot].
In Fig. \[fig:epsK-his\], we plot the $\Delta \epsK =
\epsK^\text{Exp} - \epsK^\text{SM}$ in units of $\sigma$ (= the total error in the estimate of $\Delta\epsK$) as the time evolves starting from 2012. We started to monitor $\Delta \epsK$ in 2012 when several lattice QCD results for $\BK$ obtained using different methods became consistent with each other within one sigma. In 2012, $\Delta \epsK$ was $2.5\sigma$ but now it is $4.0\sigma$. To understand the time evolution over the past 5 years, we have performed an additional analysis on the error and average.
In Fig. \[fig:epsK-err-his\], we plot the chronological evolution of $\Delta\epsK$ and its error $\sigma_{\Delta\epsK}$. Here, we find that the value for $\Delta\epsK$ has increased with some fluctuations by 25% during the period of 2012—2017, and its error $\sigma_{\Delta\epsK}$ has decreased monotonically by 24% in the same period. These two effects interfere constructively so as to produce the $4.0\sigma$ tension in $\Delta\epsK$ in 2017. The monotonic decrease in the error $\sigma_{\Delta\epsK}$ reflects the fact that lattice QCD calculations are becoming more precise, and the experimental results also are becoming more accurate. In Fig. \[fig:epsK-err-his\], we show the time evolution of $\epsK^\text{SM}$ and $\Delta\epsK$ in units of $\epsK^\text{Exp}$. Here, we find that the evaluation of $\epsK^\text{SM}$ with lattice QCD inputs explains only 71% of the experimental value, and the gap of about 29% in $\Delta\epsK$ cannot be described in the standard model with lattice QCD inputs.
Acknowledgement
===============
We would like to express our sincere gratitude to Jon Bailey for help with the manuscript. We would like to express sincere gratitude to Carleton DeTar, Aida El-Khadra, and Andreas Kronfeld for helpful discussion. We also would like to express sincere gratitude to Guido Martinelli for providing to us most updated results of UTfit. The research of W. Lee is supported by the Creative Research Initiatives Program (No. 2017013332) of the NRF grant funded by the Korean government (MEST). W. Lee would like to acknowledge the support from the KISTI supercomputing center through the strategic support program for the supercomputing application research \[No. KSC-2014-G2-002\]. Computations were carried out in part on the DAVID GPU clusters at Seoul National University.
[^1]: Speaker,
[^2]: $\lambda \approx \dfrac{\Lambda}{2 m_Q} \approx
\dfrac{1}{8} $ for the charm quark.
|
---
abstract: 'We have analyzed H$\alpha$ intensity images obtained at a 1 minute cadence with the Global Oscillation Network Group (GONG) system to investigate the properties of oscillations in the 0–8 mHz frequency band at the location and time of strong M- and X-class flares. For each of three sub-regions within two flaring active regions, we extracted time series from multiple distinct positions, including the flare core and quieter surrounding areas. The time series were analyzed with a moving power map analysis to examine power as a function of frequency and time. We find that, in the flare core of all three sub-regions, the low-frequency power ($\sim$1–2 mHz) is substantially enhanced immediately prior to and after the flare, and that power at all frequencies up to 8 mHz is depleted at flare maximum. This depletion is both frequency and time dependent, which probably reflects the changing depths visible during the flare in the bandpass of the filter. These variations are not observed outside the flare cores. The depletion may indicate that acoustic energy is being converted into thermal energy at flare maximum, while the low-frequency enhancement may arise from an instability in the chromosphere and provide an early warning of the flare onset. Dark lanes of reduced wave power are also visible in the power maps, which may arise from the interaction of the acoustic waves and the magnetic field.'
author:
- Teresa Monsue
- Frank Hill
- 'Keivan G. Stassun'
title: 'Temporal Evolution of Chromospheric Oscillations in Flaring Regions – A Pilot Study'
---
Introduction {#sec:intro}
============
The relationship between solar acoustic oscillations, active regions, and energetic flares remains an open question in solar physics. A fundamental question is whether acoustic oscillations are enhanced, suppressed, or perhaps both, in active regions and by flares, and how these power enhancements and/or suppressions behave as functions of time before, during, and after energetic flaring events. Solar flares release great amounts of energy and so in principle are capable of exciting acoustic oscillations in the magnetically active sunspot regions, perhaps by exciting velocity oscillations in regions where a higher-class solar flare has taken place [@Kumar2006]. The first clear observations of helioseismic waves produced by a flare were by Kosovichev and Zharkova [-@Kosovichev1998] using data from the Michelson Doppler Imager on board the Solar and Heliospheric Observatory space mission. It is expected that this production of waves will change the characteristics, such as the power, of the p-modes [@Ambastha2002; @Ambastha2003; @Ambastha2003B; @Kumar2006; @Kumar2006I; @Kumar2010; @Kumar2011].
In addition, Braun *et al.* [-@Braun1987] and others have observed that sunspots absorb acoustic power in the photosphere. These authors found that outgoing waves were reduced in amplitude by 50% compared to incoming waves [@Braun1987].
Observations of acoustic modes in the chromosphere using the H$\alpha$ line were first carried out by Elliott [-@Elliott1969] and later by Harvey [-@Harvey1993]. Observations of H$\alpha$ intensity oscillations in solar flares were performed by @Jain1999. Chromospheric oscillations in the 3.4 mHz and 5.6 mHz p-mode frequencies were found, confirming Elliott’s study of their existence and also agreeing with @Kneer1983. They surveyed 18 locations around two flares with Fourier power spectra and clearly found prominent 5 and 3 minute modes in H$\alpha$ [@Jain1998]. Furthermore, the observation of short-time variations of the Sun’s chromosphere observed in H$\alpha$, contributes to our understanding of the atmospheric dynamics and could reveal progenitors for chromospheric heating mechanisms [@Kneer1985].
We are interested in studying the acoustic frequency spectrum to investigate energy transfer in the chromosphere [@Kneer1983]. Our method incorporates a short time-series analysis to study the time variations in the Sun’s chromosphere observed in H$\alpha$. Studying the chromosphere in the Fourier spectrum, and observing the temporal evolution gives us an advantage in that it provides a way to trace the temporal behavior of specific structures involved in the flare, which can be rather difficult to follow over the course of several minutes or hours otherwise [@Kneer1985].
A technique for studying the p-mode excitation due to solar flares in a three-dimensional Fourier analysis was devised by Jackiewiez and Balasubramaniam [-@Jackiewicz2013] in their *“frequency-filtered amplitude movies (FFAMs)"*. This method is simple and powerful in that it preserves the initial 3-dimensional information in the inputted time series by incorporating a moving power map method. Here we employ this novel approach to the Global Oscillation Network Group H$\alpha$ data set to create power-map movies (PMMs). From these PMMs we investigate the H-alpha oscillatory modes across the frequency band (0 < $\nu$ < 8.33 mHz) and characterize the temporal and spatial evolution of these oscillations in flaring regions.
In Section \[sec:data\] we describe the X-ray flare data and the GONG H$\alpha$ time-series data that we employ in our analysis. We specifically study three active regions, directly over sunspots, in which M- or X-class flares occurred on 2012 June 13 and 2012 July 12. Section \[sec:methods\] describes our data reduction and analysis methods. In Section \[sec:results\] we then present the results of how the frequency distribution evolves temporally and spatially by constructing a PMM of each region. We find that, in the core regions of all three flares, the low-frequency power ($\sim$1–2 mHz) is substantially enhanced immediately prior to and after the flare, and that power at all frequencies up to 8 mHz is depleted at flare maximum. This depletion is both frequency- and time- dependent These variations are not observed outside the flaring region. In addition, dark lanes of reduced wave power are also visible in the power maps. Finally, in Section \[sec:discussion\] we conclude with a discussion of these results. The observations may indicate that acoustic energy is being converted into thermal energy at flare maximum, while the low-frequency enhancement may arise from an instability in the chromosphere and provide an early warning of the flare onset. We also suggest that the dark lanes observed may arise from the interaction of the acoustic waves and the magnetic field. Suggestions for next steps to advance this work are also provided.
Observational Data\[sec:data\]
==============================
The data set used here comprises GONG H$\alpha$ intensity images centered at a wavelength of 6562.8. The GONG H$\alpha$ network became operational in 2010; it consists of a set of six detectors placed around the Earth for nearly continuous solar full-disk observations. The images have a cadence of 1 minute and a format of 2048 $\times$ 2048 pixels.
Our initial study analyzes two individual data sets taken from two different GONG stations; El Tiede and Cerro Tololo. Each data set covers approximately two hours of time (121 images) around two solar flare events that occurred on 2012 June 13 at 13:19UT and 2012 July 12 at 16:53UT (Table \[table1\]). Each data set was centered in time on flare maximum, and started one hour before the event. The June 13 flare had an NOAA classification of M1 in active region AR11504, and the July 12 flare was an X1.4 in active region AR11520. The two detector data sets were not missing any time frame images.
We also use X-ray flux from the NOAA Space Weather Prediction Center (SWPC), obtained every minute with the Solar X-ray Imager (SXI) instrument on board the NOAA Geostationary Operational Environmental Satellite 15 (GOES-15) spacecraft. We compare the GONG time-series data with the X-ray flux in the 1-8 wavelength band.
We carry out an analysis on active regions directly over sunspots during a flare. There are a total of three regions of interest: AR1 and AR2 in the June 13 flare (Figure \[figure1\]), and AR3 in the July 12 flare (Figure \[figure2\]). The regions were chosen to be located close to the central meridian to reduce projection effects [@Gizon2005]. We restrict the GONG data to the same day of universal time (UT), and to one site for image stability.
\[T-observation\] \[table1\]
![Analysis was done on solar flare regions directly over sunspots where the magnetic field was concentrated. a) The event is the M1 flare that occurred on 2012 June 13 at 13:19UT. b) Our region of interest is AR1, an ideal candidate with an active flaring region directly over a sunspot in AR11504, and the smaller active region AR2, a companion sunspot within that region. c) The main regions of interest have a rectangular area of approximately between 55$''$ $\times$ 55$''$ for AR1, and 65$''$ $\times$ 65$''$ for AR2.[]{data-label="figure1"}](figure1.pdf){width="95.00000%"}
![Analysis was done on solar flare regions directly over sunspots where the magnetic field was highly concentrated. a) The event is the X1.4 flare that occurred on 2012 July 12 at 16:53UT. b) Our region of interest, AR3, was a nearly spherical flaring region directly over a sunspot in active region AR11520. The sunspot morphology of AR11520 is depicted in the GONG continuum image below. c) The main region of interest, AR3, has a rectangular area of approximately 48$''$ $\times$ 48$''$.[]{data-label="figure2"}](figure2.pdf){width="95.00000%"}
DATA ANALYSIS AND REDUCTION METHODS\[sec:methods\]
==================================================
We employ a technique similar to the *“FFAMs"* [@Jackiewicz2013] to construct a PMM. In summary, a PMM consists of a time series of acoustic power maps that show wave power as a function of frequency and space, but with an initial starting time that is systematically offset. For a given starting time, the power maps are created by applying a fast Fourier transform (FFT) in the temporal direction for each pixel in the region of interest. The starting time is then shifted by one minute and the procedure is repeated.
In detail, we start with a time series of GONG H$\alpha$ intensity images, $I(x,y,t)$ in a data cube with a cadence of 1 minute. We define a region of interest ($I_{ROI}$) covering the area $x_1 \le x \le x_2$ and $y_1 \le y \le y_2$ within the large data cube. We extract the time series at each ROI pixel over a temporal length $N$ starting at time *t~i~*. Missing images are replaced by a value of zero. The input data for the PMM are constructed by incrementing $t_i$. Explicitly, we have $$\label{equation1}
I_{\emph{ROI}}(x',y',t',t_i) = I(x_1 \le x \le x_2, \ y_1 \le y \le y_2, \ t_i \le t \le t_i + N)$$ For our analysis here, $t_i$ is incremented by one minute, $0 \le t_i \le 60$, and $N = 60$, producing a set of 61 time series, each an hour long, that start at one hour prior to flare maximum, and end at one hour after flare maximum.
To create a single frame, $P$, in the PMM, we apply an FFT in the temporal direction to the time series for each spatial pixel in $I_{ROI}$, and then take its modulus, producing $$\label{equation2}
P(x',y',\nu, t_i) = | FFT (I_{\emph{ROI}}(x',y',t',t_i)) | ^2$$ where $\nu$ is the temporal frequency. The power $P$ is then averaged in frequency bins $\nu_i$, producing one frame in the PMM, Equation (\[equation3\]). $$\label{equation3}
PMM(x', y', \nu_j, t_i) = {{\Delta \nu} \over {\nu_2 - \nu_1}}
\sum_{\nu_1 < \nu_j < \nu_2} P(x', y', \nu, t_i)$$ where $\Delta \nu$ is the frequency resolution of the power spectrum (277.8 $\mu$Hz) and $\nu_1$ and $\nu_2$ are the lower and upper limits respectively of frequency bin $j$. We define a set of seven bins $\nu_j$ of width 1 mHz starting at 1 mHz, with the highest band covering 7–8.33 mHz.
Within the areas AR1–AR3, we extract a number of smaller regions of 3 $\times$ 3 pixels, or approximately 3.2$''$ in length, and then average the PMM values within these small areas. Averaging the pixels over a smaller region improves the signal-to-noise ratio and allows us to isolate different physical conditions. We kept all subregions the same size to maintain consistency within the experiment.
To compare the PMM data with the intensity, we carry out a 60 minute running average of the intensity that is incremented by one minute to provide the same sampling as the PMM. Figure \[figure3\] shows a power-map frame for region AR3 for each $\nu_j$ along with the averaged intensity. The sampling window in Figure \[figure3\] is from 32 to 92 minutes in the total time series.
![The flare intensity and frequency-binned PMM frames of the X1.4 flare on 2012 July 12 in region AR3. The PMM frames are in the time window of 32–92 minutes. Dark lanes are seen in each frame and are sharper in the 1–2 mHz and 2–3 mHz ranges. The power is measured in arbitrary instrumental units.[]{data-label="figure3"}](figure3.pdf){width="\textwidth"}
RESULTS\[sec:results\]
======================
In this section, we present the results of our time-series power spectrum analysis on each of the three flaring active regions defined in Section \[sec:data\]. Within each of the three main analysis regions—AR1, AR2, and AR3—we define several subregions in and around the centers of the active regions in order to relate the temporal behavior of the oscillation power to different spatial positions. As we discuss below, the overall results for all three analysis regions are qualitatively similar, so we describe the results for one of the three regions in detail and then more briefly summarize the similar results for the other two regions. We find that, in the core regions of all three flares, the low-frequency power ($\sim$1–2 mHz) is substantially enhanced immediately prior to and after the flare, and that power at all frequencies up to 8 mHz is depleted at flare maximum. This depletion is both frequency- and time-dependent. We also observe dark lane features in the power maps. We defer a discussion and interpretation of these general findings to Section \[sec:discussion\].
AR3 {#S-AR3}
---
There are three main groups of subregions for area AR3. Figure \[figure4\] shows seven rectangular subregions, each approximately 3.2$''$ $\times$ 3.2$''$, distributed around the solar flare and the dark lanes. The subregions are selected to probe the temporal behavior at various levels of overall power and at various positions relative to the center of the active region. We present the results by grouping the subregions accordingly. Obviously these are not the only subregions that could be selected; we emphasize that this selection of subregions is made arbitrarily, and on the basis of visual impression, but in an attempt to sample the active region at various representative locations relative to the peak of activity. The regions with the lowest power (log power 0.5 to 2.0), are AR3\_1 and AR3\_6 (Figure \[figure7\] (a) and (b)). The regions with mid-level power levels (log power 2.0 to 3.0) are AR3\_2 and AR3\_5 (Figure \[figure6\] (a) and (b)). The regions located in the middle of the flare (AR3\_3, AR3\_4 and AR3\_7) exhibited the greatest amount of power (Figure \[figure5\](a)–(c)), with logarithmic values from 3.0 to 4.5.
![Region AR3 has a rectangular area of approximately 48.15$''$ $\times$ 48.15$''$. There are seven subregions sampled in AR3 for the X1 solar flare on 2012 July 12. The subregions are approximately 3.2$''$ $\times$ 3.2$''$ in size. In the above figure representing AR3, the PMM frame depicted is in the time window of 34–94 minutes.[]{data-label="figure4"}](figure4.pdf){width="\textwidth"}
### Inner Flaring Regions – *Locations 3, 4, and 7 in AR3*
These three subregions are placed along the dark lanes and in a bright region in the flare. Figure \[figure5\](a)–(c) shows the power as a function of time and frequency in these subregions. We observe an increase in power across the entire frequency band once the flare begins. However, as the intensity increases in each subregion, we observe a suppression of power that begins first at the higher frequencies, and then moves toward lower frequencies as time progresses. Maximum power suppression at all frequencies occurs at the time of the maximum local intensity. The power then increases back toward pre-flare levels, with the lower frequencies recovering first. The result is the appearance of a “V”-shaped feature in the color plots of Figure \[figure5\]. For each subregion, the lowest frequency band (1–2 mHz) shows the greatest power.
The overall temporal extent of the “V”-shaped decrease in power is shortest for subregion AR3\_4, which might be a consequence of the narrow spatial width of the dark lane in that region. The suppression in power in AR3\_3 (Figure \[figure5\](a)) is wider in time, perhaps due to the larger width of the dark lane in that area. AR3\_7 (Figure \[figure5\](c)), free of dark lanes, shows the longest time period of power suppression. Figure \[figure5\](d) shows the total average intensity variations for each of the three regions along with the *GOES* X-ray flux.
![Time-frequency power plots of the three subregions AR3\_3 (a), AR3\_4 (b) and AR3\_7 (c) along with the corresponding average intensity (d). The *GOES* X-ray flux is scaled as a reference for the overall solar flare event. The red dashed line indicates the time of flare maximum in the time series at 39 minutes.[]{data-label="figure5"}](figure5.pdf)
### Outer Flaring Regions – *Locations 2 and 5 in AR3*
Regions AR3\_2 and AR3\_5 both lie on the outer edge of the dark lanes, as shown in Figure \[figure4\]. The overall power for these two regions are in the middle range of log power, 2.0–3.0. The corresponding time-frequency plots are shown in Figure \[figure6\].
These plots have a substantially different qualitative nature compared to those in Figure \[figure5\]. Here, there is no sign of the “V” shape in these time-frequency images. For region AR3\_2 (Figure \[figure6\](a)) there is an overall increase in power for the 4–5 mHz frequency band, with logarithmic values in the range of 2.6 to 3.0 - also observed in region AR3\_1 (Figure \[figure7\](a)). A decrease in power for frequencies above 5 mHz (Figure \[figure6\](a)) is present at the start of the time series and persists to around 35 minutes. There appears to be a trend of increasing power late in the time series in the frequency band of 1–2 mHz. For region AR3\_5 (Figure \[figure6\](b)) there are three areas of power suppression within the 1–2 mHz band. A period of fluctuating suppression is also observed in the 5–6 mHz band.
Figure \[figure6\] shows that these two subregions have substantially lower levels of average intensity variations than those in Figure \[figure5\]. Thus suggests that rapid changes in the average intensity are at least partially responsible for the power variations in the oscillations.
![Analysis of the two outer flare regions, AR3\_2 (a) and AR3\_5 (b), along with the corresponding pixel intensity and *GOES* X-ray flux (c). The *GOES* X-ray flux is scaled as a reference for the overall solar flare event. The red dashed line indicates the solar flare event in the time series at 39 minutes.[]{data-label="figure6"}](figure6.pdf)
### Quiescent Regions – *Locations 1 and 6 in AR3*
Regions AR3\_1 and AR3\_6 both lie on the outer edge of the solar flare and in very dark outer regions. The overall logarithmic power for these two regions is in the lowest range of 0.6–2.0, Figure \[figure7\].
These areas also do not show the “V”-shape suppression feature. For region AR3\_1 (Figure \[figure7\](a)) there is an overall increase in the power in the frequency band of 1–6 mHz. Some power suppression is apparent in the time period of about 8–42 minutes. The overall power has a maximum in the 5–6 mHz band.
Region AR3\_6 (Figure \[figure7\](b)) shows an overall suppression of power in all frequency bands that starts at the time of maximum X-ray intensity. This region also shows a substantially higher power level at frequencies of 1–3 mHz than at 6–8 mHz.
As in Figure \[figure6\](c), the curves of average intensity curves in Figure \[figure7\](c) do not show much relative variation. This is consistent with the hypothesis that the “V” shape is related to the presence of strongly varying intensity.
![Analysis of the two quiescent flare regions: AR3\_1 (a) and AR3\_6 (b), along with the corresponding pixel intensity and *GOES* X-ray flux (c). The *GOES* X-ray flux is scaled as a reference for the overall solar flare event. The red dashed line indicates the solar flare event in the time series at 39 minutes.[]{data-label="figure7"}](figure7.pdf)
AR1 {#S-AR1}
---
Region AR1 has an area of approximately 55$''$ $\times$ 55$''$ (Figure \[figure8\]). As with AR3, we select seven subregions intended to probe the temporal evolution of the flaring event at various spatial locations relative to the center of the active region. Each of the seven subregions samples a rectangular area with 3.2$''$ per side. AR1 displays similar behavior to that of AR3 and so only the most significant of results (Figure \[figure9\]) will be discussed. We present the results for locations AR1\_5, AR1\_6 and AR1\_7, the regions where the M1 flare on 2012 June 13 was the most intense.
![AR1 has an area of approximately 55$''$ $\times$ 55$''$. There are seven subregions in AR1 for the M1 solar flare occurring on 2012 June 13. In this figure, the PMM frame depicted is in the time window of 32 to 92 minutes.[]{data-label="figure8"}](figure8.pdf)
The subregions where the M1 flare in AR1 was the strongest were AR1\_5, AR1\_6, and AR1\_7. The relationship between the peak of the intensity and the suppression of power is clearly observed in Figure \[figure9\]. The X-ray flux of the flare is scaled and plotted along with the intensity to show the time of maximum overall flaring activity (Figures \[figure9\](d)). The suppression of power is observed as a “V” shape for locations AR1\_5 and AR1\_6 (Figure \[figure9\](a) and (b)) with results similar to AR3\_3, AR3\_4 and AR3\_7 (Figure \[figure5\]). Again we see a relatively large variation in average intensity, with the point of the “V” corresponding to the intensity maximum in Figure \[figure9\](d).
Figure \[figure9\](c), shows a rather different situation with a clear indication that the power is suppressed not only at the peak of the intensity but also near a minimum. This will be discussed later.
Once again, as seen in almost all of our time-frequency power plots, the power is greatest at low frequencies below 2 mHz.
![Time-frequency power plots for regions AR1\_5, AR1\_6, and AR1\_7, showing the suppression of power in plots (a)–(c), correlating with the peak intensity (d). All three regions exhibit a peak in power around 3.75 at the lowest frequency bands of 1–2 mHz and then gradually decreasing as the frequency bands increase. The *GOES* X-ray flux is scaled as a reference for the overall solar flare event in the time series at 36 minutes.[]{data-label="figure9"}](figure9.pdf)
AR2 {#S-AR2}
---
Region AR2 is approximately 66$''$ $\times$ 66$''$ in size (Figure \[figure10\]). This was the smaller sunspot region involved in the M1 solar flare on 2012 June 13 (Figure \[figure1\]). Once again, we defined several subregions in order to probe the temporal behavior of the activity at various spatial positions relative to the visual center of the activity. Six subregions of 3.2$''$ in size were analyzed. AR2 displayed results similar to those of both AR1 and AR3. Here we present an analysis of two of the subregions close to the center of the active region (Figure \[figure11\], (a) and (b)), and a control quiet subregion for comparison (Figure \[figure11\](c)).
![AR2 has a rectangular area of approximately 66$''$ $\times$ 66$''$. There are six subregions in total in AR2 for the M1 solar flare on 2012 June 13. In this figure, the PMM frame depicted is in the time window of 32–92 minutes.[]{data-label="figure10"}](figure10.pdf)
Region AR2 provides observations of the oscillatory power behavior when the average intensity is slowly varying. In Figure \[figure11\](d) the intensity for AR2\_3 is slowly increasing, while it is decreasing for AR2\_4. The corresponding time-frequency images show power suppression slowly increasing for AR2\_3 and decreasing for AR2\_4, suggesting that the rate of change of the intensity is related to the rate of change of the oscillatory power. Again the lower frequency bands below 3 mHz show higher power levels than the bands above 3 mHz. The relatively low signal-to-noise ratio in these panels can be increased by enlarging the size of the selected subregions.
Region AR2\_5 (Figure \[figure10\]) is a quiet region that provides an observation of the acoustic power characteristics outside flaring regions. This control region shows a constant oscillatory power with no systematic temporal changes within the frequency bands (Figure \[figure11\] (c)). While it does show higher power at lower frequencies, the enhancement is much lower than that seen in the flare regions. Quasi-periodic fluctuations are also seen with a periods of $\sim$7 minutes in the 1–2 mHz band and $\sim$1 minute above 5.0 mHz. However, these may be a result of a low signal-to-noise ratio.
![Acoustic power observations for regions AR2\_3, AR2\_4, and AR2\_5, showing the suppression of power in plots (a) and (b) and a quiet region for reference (c). The *GOES* X-ray flux is scaled as a reference for the overall solar flare event in the time series at 36 minutes.[]{data-label="figure11"}](figure11.pdf)
DISCUSSION AND CONCLUSIONS\[sec:discussion\]
============================================
The results in this paper demonstrate that H$\alpha$ observations of chromospheric oscillations in the p-mode band can provide information about the physical processes occurring in flaring regions. In particular, variations in the oscillatory power as a function of frequency, spatial position, and time can be used to probe energy transport at different heights within a flare.
Figures \[figure5\] and \[figure9\] show a suppression of power that first migrates in time from high to low frequencies in a flare, with a subsequent restoration of the power starting at low frequencies and progressing back to high frequencies. This produces a “V”-shaped feature in the images of the power as a function of frequency and time. The shape can be understood as a consequence of the nature of the observations, the behavior of the H$\alpha$ spectral line during a flare, and the height dependence of the frequency of the maximum oscillation amplitude. Wang *et al.* [-@Wang2000], with their high-cadence H$\alpha$ observations from the Big Bear Solar Observatory, found high-frequency fluctuations that correlate with HXR elementary bursts. These hard X-ray emissions could be signatures of sites of fine structures where individual magnetic reconnection processes are taking place [@Wang2000].
One of the earliest results of studies of solar oscillations showed that the frequency at which the waves reach their maximum amplitude increases with height in the solar atmosphere [@Noyes1962]. In the photosphere, the maximum amplitude occurs at periods around five minutes (frequency near 3.3 mHz), while in the chromosphere the maximum occurs at periods of three minutes (frequency near 5.5 mHz). It is also known that the wings of a spectral line are formed at lower heights in the solar atmosphere than the core of the line, thus observations of oscillations obtained in the wing of a spectral line will be dominated by lower-frequency power than observations in the core of the line.
The observations discussed here are obtained with a filter that is centered on the wavelength of the H$\alpha$ line core in the quiet Sun. During the course of the flare, the motion of the plasma will change the wavelength of the line due to the Doppler effect, so that the filter bandpass will admit a higher proportion of light from the wings of the spectral line rather than from the core. Since the wings of the line are formed at lower heights in the solar atmosphere, and since the peak amplitude of the p-modes occurs at lower frequencies at lower heights, the net effect is to reduce power at high frequencies. This reduction moves to lower frequencies as the flare progresses and the spectral line is increasingly Doppler-shifted. The overall observed intensity also increases as the brighter wings contribute a larger portion of the signal. As the flare energy decreases, the solar plasma motions die out and the spectral line core moves back toward the center of the filter bandpass, restoring the visibility of the high-frequency power and decreasing the overall intensity. A plasma velocity of 5 km s^-1^, easily created in a flare, would move the line core by 0.1Å , which is a substantial fraction of the 0.6Å bandpass of the GONG filter.
The dark lanes in wave power, also observed by Jackiewiez and Balasubramaniam [-@Jackiewicz2013], could be where the magnetic field absorbs or scatters the acoustic waves. In the photosphere, sunspots are known to be areas of suppressed acoustic mode power [@Braun1987]. The appearance of the dark lanes depends on frequency, as seen in Figure \[figure3\], which may provide information on the structure of the magnetic field as well as aspects of energy transport during the flare. In addition, the time-frequency maps of subregions located on dark lanes show diverse behavior (e.g. AR3\_3 and AR3\_2) further suggesting that there is a variation in the underlying magnetic field. Several deductions can be made about the dark appearance of the lanes, which could indicate that energy is being removed from the observed wave frequencies and perhaps converted into the thermal energy of the flare, or scattered into other wave modes with frequencies higher than 8 mHz, or absorbed by the magnetic field in the flare, or dampened by magnetic reconnection. We believe that the correct explanation is that the wave energy is being converted into thermal energy, due to the simultaneous increase in both *GOES* X-ray flux and H$\alpha$ intensity. These possibilities can be investigated by applying the PMM technique to simultaneous magnetograms acquired by GONG. Furthermore, there is a trade-off between signal-to-noise ratio and spatial resolution. Larger subregions increase the signal-to-noise ratio but decrease the spatial resolution. In this paper we chose to have higher spatial resolution in order to investigate oscillations within the narrow dark lanes.
Generally, there is a tendency for an excess of power at low frequencies below 2 mHz compared to higher frequencies. This excess can be as much a factor of 30 for the regions in Figure \[figure5\], but it is also present in the quiet region in Figure \[figure11\] at a much lower level (a factor of about 4). Since these are ground-based intensity observations, it is quite possible that some of this excess is caused by fluctuations in the Earth’s atmospheric transparency. However, the marked increase in the flaring regions suggests that low-frequency power is enhanced during a flare. If this low-frequency power excess is a feature of strong flares, it may arise from an instability in the chromosphere and provide an early warning of the flare onset.
This pilot project demonstrates that the application of PMMs to H$\alpha$ intensity observations opens up a number of new avenues to explore the physical processes in flares. The temporal and spatial variations of acoustic wave power show intriguing features that contain information about the energy transport and magnetic field variations as a function of height within flaring regions.
There are several paths to follow that will further develop the method. The most informative step is the comparison of the results with the GONG magnetograms. The correlation of changes in oscillatory power with the characteristics of the magnetic field should provide additional information on the underlying physical processes. In addition, the PMM method can also be applied to the magnetograms since they are simultaneously observed at a cadence of once per minute. Additional steps will be the analysis of larger subregions, extension of the PMM method to earlier times to search for flare precursors; application of the method to longer time spans to increase the frequency resolution, analysis of additional cases of strong flares, and accumulation of statistically significant measurements. Eventually, we may be able to construct a new picture of the physics of a flare based on its acoustic signatures.
The authors acknowledge the advice given by Dr. Jason Jackiewicz and Dr. Nathan De Lee. This work utilized data from the GONG H$\alpha$ network, operated by The National Solar Observatory (NSO) and The Association of Universities for Research in Astronomy (AURA), and which was originally commissioned by The Air Force Weather Agency (AFWA). T.M. acknowledges support from the Fisk-Vanderbilt Masters-to-PhD Bridge Program, including specifically funding support through NSF PAARE grant AST-1358862 and a Harriett Jenkins Graduate Fellowship from NASA.
, A., [Basu]{}, S., & [Antia]{}, H. M. 2002, in ESA Special Publication, Vol. 508, From Solar Min to Max: Half a Solar Cycle with SOHO, ed. A. [Wilson]{}, 43–46
, A., [Basu]{}, S., & [Antia]{}, H. M. 2003, in ESA Special Publication, Vol. 517, GONG+ 2002. Local and Global Helioseismology: the Present and Future, ed. H. [Sawaya-Lacoste]{}, 219–222
, A., [Basu]{}, S., & [Antia]{}, H. M. 2003, Bulletin of the Astronomical Society of India, 31, 319
, D. C., [Duvall]{}, Jr., T. L., & [Labonte]{}, B. J. 1987, , 319, L27
, I. 1969, , 6, 28
, L., & [Birch]{}, A. C. 2005, Living Reviews in Solar Physics, 2, 6
, J. W., [Duvall]{}, Jr., T. L., [Jefferies]{}, S. M., & [Pomerantz]{}, M. A. 1993, in Astronomical Society of the Pacific Conference Series, Vol. 42, GONG 1992. Seismic Investigation of the Sun and Stars, ed. T. M. [Brown]{}, 111
, J., & [Balasubramaniam]{}, K. S. 2013, , 765, 15
, R., [Tripathy]{}, S., [Bharti]{}, L., & [Kumar]{}, B. 1999, in Astronomical Society of the Pacific Conference Series, Vol. 183, High Resolution Solar Physics: Theory, Observations, and Techniques, ed. T. R. [Rimmele]{}, K. S. [Balasubramaniam]{}, & R. R. [Radick]{}, 531
, R., & [Tripathy]{}, S. C. 1998, , 181, 113
, F., & [von Uexkuell]{}, M. 1983, , 119, 124
—. 1985, , 144, 443
, A. G., & [Zharkova]{}, V. V. 1998, , 393, 317
, B., [Mathur]{}, S., [Garc[í]{}a]{}, R. A., & [Venkatakrishnan]{}, P. 2010, , 711, L12
, B., & [Ravindra]{}, B. 2006, Journal of Astrophysics and Astronomy, 27, 425
, B., [Venkatakrishnan]{}, P., [Mathur]{}, S., [Tiwari]{}, S. K., & [Garc[í]{}a]{}, R. A. 2011, , 743, 29
, B., [Venkatakrishnan]{}, P., & [Venugopalan]{}, K. 2006, in ESA Special Publication, Vol. 624, Proceedings of SOHO 18/GONG 2006/HELAS I, Beyond the spherical Sun
, R. B., [Noyes]{}, R. W., & [Simon]{}, G. W. 1962, , 135, 474
, H., [Qiu]{}, J., [Denker]{}, C., [et al.]{} 2000, , 542, 1080
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abstract: |
We report resistivity measurements from 0.03 K to 10 K in a dilute high mobility 2D electron system. Using an undoped GaAs/AlGaAs heterojunction in a gated field-effect transistor geometry, a wide range of densities, $0.16 \times 10^{10} \mbox{cm}^{-2}$ to $7.5 \times 10^{10}
\mbox{cm}^{-2}$, are explored. For high densities, the results are quantitatively shown to be due to scattering by acoustic phonons and impurities. In an intermediate range of densities, a peak in the resistivity is observed for temperatures below 1 K. This non-monotonic resistivity can be understood by considering the known scattering mechanisms of phonons, bulk and interface ionized impurities. Still lower densities appear insulating to the lowest temperature measured.
author:
- 'M. P. Lilly'
- 'J. L. Reno'
- 'J. A. Simmons'
- 'I. B. Spielman'
- 'J. P. Eisenstein'
- 'L. N. Pfeiffer'
- 'K. W. West'
- 'E. H. Hwang'
- 'S. Das Sarma'
title: Resistivity of dilute 2D electrons in an undoped GaAs heterostructure
---
The resistivity of 2D electrons at zero magnetic field has been used to probe scattering and quantum processes for many years. Interest in the conducting behavior of 2D electrons and holes was further heightened in 1994 when Kravchenko and coworkers discovered an apparent metal-insulator transition of 2D electrons in Si MOSFETs[@metal:si]. While 2D systems are expected to exhibit insulating behavior (weak or strong localization) at sufficiently low temperatures and/or densities, the metallic side of the metal-insulator transition is more puzzling. A true metal conducts at $T=0$, however here we use the term “metallic” as referring to $d\rho/dT \ge 0$ at low temperature, where $\rho$ is the resistivity. Metallic behavior has been seen in a number of different material systems, including electrons in Si MOSFETs[@metal:si], holes in GaAs[@metal:pgaas1; @metal:pgaas2; @metal:pgaas3] and electrons in GaAs[@metal:ngaas1; @metal:ngaas2] provided the carrier density is reasonably low (but not so low that the system is in the insulating phase) and the mobility relatively high. While all of these experimental systems exhibit metallic behavior in certain regimes of density ($n$) and temperature, the quantitative behavior varies widely. As the temperature is lowered, $\rho$ can decrease by a factor of 10 in Si MOSFETs[@metal:si] compared to only a few percent for electrons in GaAs[@metal:ngaas1]. The drop in resistivity is typically observed when the dimensionless parameter $r_s$ (ratio of Coulomb to kinetic energy, $\sim n^{-1/2}$) is much larger than 1, suggesting that electron-electron interactions may be important. The issue here is understanding the physical mechanisms underlying the metallic behavior and elucidating the roles of disorder and interaction.
In this Letter, we present the temperature dependence of the resistivity of a dilute 2D electron gas (2DEG) in GaAs. Electron-electron interactions can be extremely important in this system at low temperatures due to the low amount of disorder present in GaAs heterostructures. In the sample discussed here, the density can be continuously tuned from $0.16 \times 10^{10}$ cm$^{-2}$ ($r_s=13.7$) to $7.5 \times 10^{10}$ cm$^{-2}$ ($r_s=2.0$). With the ability to achieve very high $r_s$ in low disorder samples, these resistivity measurements provide a test of the importance of Coulomb interactions in the metallic regime. In particular, the $r_s$ ($\stackrel{>}{_\sim} 10$) values of our 2DEGs are very comparable to those in the best Si MOSFETs whereas the effective disorder in our sample (as measured by $k_F \ell$, $k_F$ is the Fermi wavevector and $\ell$ the mean free path) is more than two orders of magnitude lower. Earlier experimental studies of the metallic behavior in 2D electrons in GaAs systems were restricted to higher densities and lower mobilities, and the drop in resistivity was much weaker than the 20% drop observed here. Finally, the wide range of $n$ and $T$ reported here allow several scattering regimes to be identified.
We find that our measured temperature dependence of the resistivity can be qualitatively well understood solely within the framework of Fermi liquid theory. At high density, the resistivity is linear with temperature, and the slope agrees both with previous measurements[@phonons:exp1; @phonons:exp2] and established theory for acoustic phonon scattering[@phonons:sds]. For intermediate densities a low temperature peak in the resistance is observed. The non-monotonic resistivity agrees qualitatively with the predicted temperature dependence of ionized impurity scattering (screening at low $T$ and the transition from a degenerate Fermi gas to a classical system at high $T$).
![Plot of mobility as a function of electron density. The top axis indicates $r_s$. Inset is a schematic cross section of the undoped heterojunction with self-aligned ohmic contacts.[]{data-label="fig:mobility"}](fig1.ps)
The sample used in this study is a high mobility 2DEG confined at the interface of a GaAs/Al$_{0.3}$Ga$_{0.7}$As heterojunction. The interface (see Fig. \[fig:mobility\] inset) is separated by undoped AlGaAs from a bulk doped GaAs cap that serves as an integrated top gate. A $2 \times 2$ mm$^2$ square with 16 NiGeAu ohmic contacts is fabricated in a field-effect transistor geometry[@kane]. The ohmic contacts are self-aligned to the top gate, and carriers are drawn into the channel by applying voltage between the gate and contacts. Once present, the density of the 2DEG is linearly proportional to the gate voltage. A calibration of the density is determined using Shubnikov-de Haas oscillations for a number of gate voltages. Although using undoped heterostructures requires complicated processing, an ultra-low density can be achieved without the penalty of a substantial fixed disorder potential arising from remote delta-doping.
The mobility, $\mu = 1/ne\rho$, is determined at $T=30$ mK where we empirically observe that the resistivity no longer changes with temperature. The mobility is extremely high throughout the entire density range, as is shown in Fig. \[fig:mobility\]. At the highest density for this sample the mobility is $8.2 \times
10^6$ cm$^2$/Vs ($k_F \ell = 2500$). In other devices that operate at higher density, we have measured a mobility of $1.4 \times 10^7$ cm$^2$/Vs at a density of $2.5 \times 10^{11}$ cm$^{-2}$, a value to which the data in Fig. \[fig:mobility\] extrapolate. Since the mobility continues to increase at high density, interface roughness scattering is not relevant for data presented here[@ir_scattering].
![The main figure show the resistivity for densities (from the top) 0.16, 0.20, 0.23, 0.29, 0.36, 0.42, 0.55, 0.68, 0.80 and 1.06 $\times 10^{10}$ cm$^2$. The inset emphasizes the phonon component of the temperature dependence as straight parallel lines in inverse mobility (in units of $10^{-6}$ Vs/cm$^{2}$) for densities (from the top) 0.68, 0.80, 1.06, 1.57 and 3.36 $\times 10^{10}$ cm$^{-2}$. For clarity, the highest two densities in the inset are omitted from the main figure. The dotted line indicates $T_F$ and the dashed line in the inset indicates $T_{BG}$. Note the log-scale in the main panel and the linear scale in the inset.[]{data-label="fig:rho"}](fig2.ps)
The temperature dependence of the resistivity for a number of different densities is shown in Fig. \[fig:rho\]. The locus of Fermi temperatures is indicated with a dotted line. Three regimes can be identified. First, at high density ($n>1.0 \times 10^{10}$ cm$^{-2}$) the resistivity is constant at low temperature and then increases as $T$ increases. For intermediate densities ($0.29$ to $1.0 \times 10^{10}
\mbox{cm}^{-2}$) the resistivity is non-monotonic with temperature. As $T$ increases, $\rho$ is initially constant (see Fig. \[fig:r\]), then increases and decreases to form a local maximum below $T = 1$ K. The third regime occurs at the lowest densities, with the resistivity decreasing as the temperature increases in the experimentally accessible temperature range.
The phonon contribution to the resistivity at high density is better shown in the inset of Fig. \[fig:rho\], where $\mu^{-1} = n e \rho$ is plotted for $n=0.68$ to $3.36 \times 10^{10}$ cm$^{-2}$. The inverse mobility for $T \stackrel{>}{_\sim} 2$ K is linear, and each density has roughly the same slope of $\sim 3.2 \times 10^{-8}$ Vs/Kcm$^2$). Acoustic phonon scattering with both piezoelectric and deformation potential coupling leads to a linear dependence of $\mu^{-1}$ above the Bloch-Grüneisen temperature ($kT_{BG} = 2k_F c \hbar$, $c$ is the phonon speed of sound). Previously reported experimental[@phonons:exp1; @phonons:exp2] and theoretical[@phonons:sds] studies of higher density 2DEGs in conventional GaAs heterostructures [*quantitatively*]{} agree with slope found here. This demonstrates that the resistivity of the high density 2DEG is accurately captured using well known scattering mechanisms. Note that the high temperature resistivity of the low density data in the inset of Fig. \[fig:rho\] remains consistent with phonon scattering even as the non-monotonic features begin to appear at low temperatures.
As the density is lowered below $n=1 \times 10^{10}$ cm$^{-2}$, the resistivity becomes non-monotonic. A peak in $\rho$ appears for $T <
T_F$, and the position of the peak shifts to lower temperature for lower densities. The resistivity at the maximum can be up to 23% larger than the low temperature ($T=30$ mK) value of the resistivity. The non-monotonic peak occurs for $T<T_{BG}$, where phonon scattering is strongly suppressed. In this regime of $n$ and $T$ the temperature dependence of ionized impurity scattering must be considered. Screening leads to $\rho(T) \sim T$ for $T \ll T_F$[@sds_ionized], and the crossover from a degenerate Fermi gas to a classical system leads to $\rho(T) \sim 1/T$ for $T>T_F$[@sds_ionized]. Competition between these effects will lead to a low $T$ resistivity peak. Detailed calculations including both ionized impurity and acoustic phonon scattering are presented below (Fig. \[fig:theory\]).
For the lowest density that can be attained in our samples $\rho$ decreases monotonically as $T$ increases. While it appears that the 2D electron system is insulating for these densities, comparison to the non-monotonic resistivities indicates a different interpretation is possible. Clearly, if the data were only taken to 0.2 K, some of the non-monotonic curves at higher density would also appear insulating. Similarly, it is possible that the low $n$ insulating behavior could become non-monotonic if the electron temperature were lowered further. From the results in Fig. \[fig:rho\], it is clear that a separatrix cannot be identified in this system. The functional form of the “insulating” temperature dependence for $n \le
0.23 \times 10^{10} \mbox{cm}^{-2}$ is more consistent with a power law dependence than an activated or Mott variable range hopping behavior.
![Four-wire resistances of the corner contacts for densities shown. The diagram labels the contacts. In defining $R_A$ and $R_B$, the subscripts indicate the current and voltage contacts.[]{data-label="fig:r"}](fig3.ps)
In performing $\rho$ measurements at low $n$ great care must be taken to insure that features do not arise as a result of inhomogeneities in the current flow. The temperature dependence of the resistances is shown in Fig. \[fig:r\]. Two resistances, $R_A$ and $R_B$, are measured for current flowing, on average, in orthogonal directions. A constant current of 0.3 nA (for the lowest density) to 20 nA (for the highest density) is used in a standard lock-in amplifier measurement at a frequency of 7 Hz. At these currents, transport measurements (e.g. quantum Hall minima) indicate that the electrons cool below $\sim$ 50 mK. The resistivity is determined from the resistances using a procedure prescribed by Van der Pauw[@vdp]. At high density ($n=1.57 \times
10^{10}$ cm$^{-2}$, Fig. \[fig:r\]a), the curves $R_A(T)$ and $R_B(T)$ almost lie on top of each other and the temperature dependence is due to phonon scattering. At lower density (Fig. \[fig:r\]b,c), non-monotonic features appear in both $R_A$ and $R_B$, but the resistances are no longer identical. Here we observe $R_A/R_B$ can be as large as 1.6. Earlier designs of undoped samples exhibited a severe ratio; $R_A/R_B$ approached 30. This effect originates in the sample fabrication, with the higher resistance always related to the location of the metallic gate contact. We speculate that the gate contact created a locally strained region above the square 2DEG causing inhomogeneous current flow. After redesigning the sample geometry to make the gate contact far from the square measurement region, $R_A/R_B$ was reduced by more than a factor of 10. While this effect is still visible in Fig. \[fig:r\]b and c, the remaining ratio has a very small ($<$ 10%) impact on the resistivity[@vdp]. Both $R_A$ and $R_B$ are non-monotonic, with resistance peaks located at nearly the same temperature. The qualitative nature of the non-monotonic resistivity is due to the underlying physics of ionized impurity scattering and not inhomogeneous current flow.
![Result from scattering calculations are remarkably similar to the data in Fig. 3. The density (from the top) is 0.42, 0.55 0.68,0.80, 1.06 and $1.57 \times 10^{10}$ cm$^{-2}$. Calculations include acoustic phonons, bulk and interface impurities.[]{data-label="fig:theory"}](fig4.ps)
To understand the low temperature ($T<1$ K) data, where the phonons contribute little to the resistivity as the system enters the Bloch-Grüneisen regime, we have carried out a microscopic transport calculation using the Boltzmann theory, where we include effects of charged impurity scattering on the electronic resistivity. Our calculation includes the following effects: (1) scattering by ionized charged impurities in the bulk GaAs layer and at the GaAs/AlGaAs interface; (2) temperature dependent screening, where the electron-charged impurity scattering is taken to be the temperature dependent statically screened Coulomb interaction with the screening calculated within the finite wavevector random phase approximation; (3) acoustic phonon scattering including both deformation potential and piezoelectric coupling; (4) the quasi-2D nature of the system through subband form-factors calculated in the Fang-Howard-Stern variational scheme. This approach is valid for $k_F \ell > 1$.
We show our calculated results in Fig. \[fig:theory\] for several experimental intermediate and high densities. Reasonable values are chosen for bulk and interface impurities to allow a comparison of the calculated results to the experimental data shown in Fig. \[fig:rho\]. Note that the overall resistivity scale depends on the unknown impurity densities, but the qualitative trends in $\rho(T,n)$ are real and arise from very basic aspects of the underlying scattering mechanisms. The rise in $\rho$ with increasing $T$ at low temperatures is a direct effect of the thermal weakening of screening. The non-monotonicity of $\rho(T)$ at the intermediate temperatures arises from the competition among temperature dependent screening, quantum-classical crossover and phonon scattering. The rise at higher $T$ is a phonon effect.
Quantitative understanding requires a more sophisticated theory which includes higher order electron-electron interactions and disorder induced localization corrections. Recently, electron-electron interaction corrections have been considered by Zala and coworkers[@zala]. They find that for a range of temperature, $\hbar/\tau \ll T \ll T_F$ where $\tau$ is the transport relaxation time, $\rho(T)$ is expected to be linear. From the slope, the Fermi liquid parameter $F^0$ can be determined. In the data presented here there is no significant range over which linear $\rho$ is observed for $T \ll T_F$ (except for the phonon effect at higher $T$), and therefore we do not attempt to determine $F_0$.
In conclusion, we have made resistivity measurements on low disorder gatable 2DEGs over a wide range of densities and temperatures. The resistivity data of high density 2DEGs agree quantitatively with both experimental and theoretical accounts of acoustic phonon scattering in GaAs. For the lowest densities, insulating behavior is observed. At intermediate densities, the resistivity becomes non-monotonic, with a peak appearing for $T<T_F$. At low temperature the competition between screening ($\sim T$) and a crossover from a degenerate Fermi system to a classical 2D gas ($\rho \sim T^{-1}$) leads to the formation of a peak. Theoretical scattering calculations including both ionized impurities and acoustic phonons in GaAs show excellent qualitative agreement with the data. This agreement between experiment and theory suggests that the underlying physics involved is conventional Fermi liquid physics.
We acknowledge outstanding technical assistance from W. Baca and R. Dunn. This work has been supported by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, US Department of Energy. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000.
S. V. Kravchenko, G. V. Kravchenko, J. E. Furneaux, V. M. Pudalov, and M. D’Iorio, Phys. Rev. B [**50**]{}, 8039 (1994).
Y. Hanein, U. Meirav, D. Shahar, C. C. Li, D. C. Tsui, and H. Shtrikman, Phys. Rev. Lett. [**80**]{}, 1288 (1998).
J. Yoon, C. C. Li, D. Shahar, D. C. Tsui and M. Shayegan, Phys. Rev. Lett. [**82**]{}, 1744 (1999); H. Noh, M. P. Lilly, D. C. Tsui, J. A. Simmons, L. N. Pfeiffer and K. W. West, preprint, cond-mat/0206519.
A. P. Mills, Jr., A. P. Ramirez, L. N. Pfeiffer and K. W. West, Phys. Rev. Lett. [**83**]{}, 2805 (1999).
Y. Hanein, D. Shahar, J. Yoon, C. C. Li, D. C. Tsui, and H. Shtrikman, Phys. Rev. B [**58**]{}, 13338 (1998).
C. E. Yasin, M. Y. Simmons, A. R. Hamilton, N. E. Lumpkin, R. G. Clarke, L. N. Pfeiffer, and K. W. West, preprint, cont-mat/0204519.
H. L. Stormer, L. N. Pfeiffer, K. W. Baldwin and K. W. West, Phys. Rev. B [**41**]{}, 1278 (1990).
J. J. Harris, C. T. Foxon, D. Hilton, J. Hewett, C. Roberts and S. Auzoux, Surf. Sci. [**229**]{}, 113 (1990).
T. Kawamura and S. Das Sarma, Phys. Rev. B [**42**]{}, 3725 (1990); Phys. Rev. B [**45**]{}, 3612 (1992).
B. E. Kane, L. N. Pfeiffer, K. W. West, and C. K. Harnett, Appl. Phys. Lett. [**63**]{}, 2132 (1993).
J. Heremans, M. B. Santos, K. Hirakawa, and M. Shayegan, J. Appl. Phys. [**76**]{}, 1980 (1994).
S. Das Sarma and E. H. Hwang, Phys. Rev. Lett. [**83**]{}, 164 (1999).
L. J. van der Pauw, Philips Res. Reports [**13**]{}, 1 (1958).
G. Zala, B. N. Narozhny and I. L. Aleiner, Phys. Rev. B [**64**]{}, 214204 (2001).
|
---
abstract: |
Ultra-compact dwarf galaxies (UCDs) are stellar systems with masses of around $10^7$ to $10^8$ M$_\odot$ and half mass radii of 10-100 pc. They have some properties in common with massive globular clusters, however dynamical mass estimates have shown that UCDs have mass-to-light ratios which are on average about twice as large than those of globular clusters at comparable metallicity, and tend to be larger than what one would expect for old stellar systems composed out of stars with standard mass functions.
One possible explanation for elevated high mass-to-light ratios in UCDs is the existence of a substantial amount of dark matter, which could have ended up in UCDs if they are the remnant nuclei of tidally stripped dwarf galaxies, and dark matter was dragged into these nuclei prior to tidal stripping through e.g. adiabatic gas infall. Tidal stripping of dwarf galaxies has also been suggested as the origin of several massive globular clusters like Omega Cen, in which case one should expect that globular clusters also form with substantial amounts of dark matter in them.
In this paper, we present collisional N-body simulations which study the co-evolution of a system composed out of stars and dark matter. We find that the dark matter gets removed from the central regions of such systems due to dynamical friction and mass segregation of stars. The friction timescale is significantly shorter than a Hubble time for typical globular clusters, while most UCDs have friction times much longer than a Hubble time. Therefore, a significant dark matter fraction remains within the half-mass radius of present-day UCDs, making dark matter a viable explanation for the elevated M/L ratios of UCDs. If at least some globular clusters formed in a way similar to UCDs, we predict a substantial amount of dark matter in their outer parts.
author:
- |
H. Baumgardt$^{1}$ and S. Mieske$^2$\
$^1$ Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany\
$^2$ European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching bei München, Germany
date: 'Accepted ????. Received ?????; in original form ?????'
title: 'High mass-to-light ratios of UCDs - Evidence for dark matter ?'
---
\[firstpage\]
stellar dynamics, methods: N-body simulations, galaxies: star clusters
Introduction {#sec:intro}
============
Ultra-compact dwarf galaxies (UCDs) were discovered in the late 1990s in spectroscopic surveys of the Fornax galaxy cluster [@hetal99; @detal00] and have since then been found in other nearby galaxy clusters as well [@hetal05; @metal05; @metal07; @jetal06; @fetal07; @retal07]. They are bright ($-11 < M_V < -13.5$) and compact ($7 < r_h < 100$ pc) stellar systems which have ages of at least several Gyr and possibly up to 10 Gyr [@metal06; @eetal07]. The masses and sizes of UCDs are larger than those of Galactic globular clusters, but similar to those of nuclei in dwarf elliptical galaxies (Drinkwater et al. 2003, Bekki et al. 2003).
One of the most remarkable properties of UCDs is that their dynamical mass-to-light ratios are on average about twice as large than those of globular clusters of comparable metallicity, and also tend to be larger than what one would expect based on simple stellar evolution models that assume a standard stellar initial mass function, like e.g. @k01 [@hetal05; @dhk08; @metal08]. If not due to a failure of stellar evolution models, this points either to unusual stellar mass functions [@mk08; @dbk08] or possibly to the presence of a significant amount of dark matter in UCDs. We note that most methods used to determine M/L ratios for UCDs rely on the assumptions that mass follows light and isotropic velocity dispersions. How well these assumptions are fulfilled is currently not known. Also due to the large distances of UCDs, only integrated velocity dispersions can be obtained, which in most cases are intermediate between the central and the global velocity dispersions. In order to determine the mass-to-light ratio, the mass modeling has to take the density profiles of the UCDs as well as the effects of seeing and a finite slit size into account, as done for example in @hetal07.
Several formation scenarios have been discussed for UCDs, like e.g.that UCDs are simply massive globular clusters and form in the same way [@hetal99; @eetal07; @fetal08], that they are the nuclei of tidally stripped, originally much more extended galaxies [@bcd01; @bcd03; @tde08; @getal08], or that they are merged globular clusters [@ol00; @fk02]. @getal08 have shown that funneling of dark matter to the central region of a disk galaxy, due to gas-infall, can significantly increase the M/L ratios in the nuclear region, and hence may explain the elevated M/L ratios of UCDs, provided that UCDs formed by tidal stripping. Indeed, it has been suggested that also GCs may have originated as centers of individual primordial dark matter halos (e.g. Carraro & Lia 2000, Lee et al. 2007, Bekki et al. 2007). If dark matter funneling is an efficient mechanism [@getal08], one may therefore expect both UCDs and GCs to be formed with a significant fraction of dark matter. It is important to note that such an increase of dark matter density by some kind of funneling mechanism is necessary to explain a significant amount of dark matter in UCDs or GCs, since their present-day stellar (and hence implied dark matter) densities are up to 2-3 orders of magnitude higher than expected for cuspy dark matter halos of dwarf galaxy mass [@giletal07]. This is shown in Fig. \[fig:density\].
![The mean mass density within the half-mass radius of the joint sample of GCs and UCDs from Fig. \[fig:mtol2\] is plotted vs. their relaxation time [@metal08]. The dotted line indicates the approximate central ($r\lesssim10$pc) dark matter densities expected for cuspy dwarf galaxy CDM halos [@giletal07]. []{data-label="fig:density"}](UCD_trelax_density.eps){width="8.5cm"}
In this paper, we start from the working hypothesis that both GCs and UCDs are formed with the same non-zero dark-to-stellar-mass-fraction. We then investigate how the dynamical co-evolution of dark matter and stars changes the observed dark matter fraction as a function of time. We assess whether the observed rise of M/L ratios from the regime of GCs to that of UCDs can be explained by our working hypothesis and the subsequent dynamical evolution.
The models {#sec:Nbody}
==========
In our simulations, we assume that stars and dark matter particles follow the same density distribution initially, which was given by a @p11 model. Determinations of mass-to-light ratios of globular clusters or UCDs rely mainly on stars located inside the half-mass radius [@hetal07; @metal08] or even closer within [@mcetal05]. Tidal effects are therefore not likely to have a strong influence on determined mass-to-light ratios. Also, due to their high mass and corresponding large dissolution times, tidal interactions probably play only a minor role for UCDs. In our simulations we therefore neglect the influence of an external tidal field.
We assume that the stars initially follow a @k01 mass function with lower and upper mass limits of 0.1 and 100 M$_\odot$. Stellar evolution changes the mass function of stars, however most of this change happens within the first $10^9$ yrs (see e.g. the grid of models by @bm03), i.e. on a timescale short against the lifetime of globular clusters or UCDs. We therefore also neglect stellar evolution and immediately transform stars to the assumed age of GCs and UCDs, T=12 Gyrs. For the transformation, we assume that stars with mass larger than 25 M$_\odot$ form black holes and assume that the black hole mass is 10% of the mass of the initial star. This way, black hole masses in our models are compatible with observed masses for stellar mass black holes (e.g. @c06). Stars with masses between 8 M$_\odot$ and 25 M$_\odot$ are assumed to form neutron stars with a mass of $m=1.3$ M$_\odot$ [@tc99]. Stars between 0.8 M$_\odot$ and 8 M$_\odot$ are assumed to form white dwarfs due to stellar evolution. The masses of the white dwarfs are obtained from @k08, who found, based on observations of white dwarfs in star clusters, the following relation between the initial and final mass of white dwarfs: $m_{wd} = 0.109 m + 0.394 M_\odot$. Stars less massive than 0.8 M$_\odot$ are still on the main sequence and we assume that they have not yet lost any mass. The following table summarises our initial-to-final mass relation: $$m_{rem} =
\left\{ \begin{array}{l}
m, \;\;\;\; m < 0.8 M_\odot \\[+0.2cm]
0.109 \frac{m}{M_\odot} + 0.394 , \;\;\;\; 0.8 M_\odot < m < 8 M_\odot \\[+0.2cm]
1.35, \;\;\;\; 8 M_\odot < m < 25 M_\odot \\[+0.2cm]
0.1 m, \;\;\;\; 25 M_\odot < m \end{array} \right.$$
Neutron stars and probably also black holes receive kicks at the time of their birth due to asymmetric supernova explosions. The size of these kicks is a few hundred km/sec [@ll94], which is large enough that most will be lost from globular clusters or UCDs [@p02]. We therefore assume only a small neutron star and black hole retention fraction of 20% in our simulations. With these assumptions, the mean mass of stars in our models is 0.344 M$_\odot$.
The dark matter is also modeled as point mass particles. In our reference simulation we assume a mass of 0.03 M$_\odot$ for the dark matter particles, but we also make simulations with masses of 0.15 M$_\odot$ and 0.01 M$_\odot$ to study the influence of the adopted particle mass on our results. Theoretically, the dynamical friction of stars should not depend on the mass of the dark matter particles as long as the mass ratio between stars and dark matter is high enough [@bt87]. Also, with the adopted masses, the self-interaction of the dark matter particles is still unimportant over the timescales studied here. We will further investigate the influence of the mass of the dark matter particles in sec. \[sec:variations\].
All runs are performed with the collisional $N$-body code NBODY4 [@a99] on the GRAPE6 computers [@mfkn03] of Bonn University and the results are expressed in $N$-body units [@hh02], in which the constant of gravity and total cluster mass are equal to 1 and the total potential energy is equal to -0.5. Table 1 gives an overview of the runs performed.
Results {#sec:results}
=======
Mass segregation timescale: analytical estimate
-----------------------------------------------
In the following we derive an analytical estimate of the mass segregation time scale in compact stellar systems.
Massive stars will segregate against dark matter particles and lighter stars as a result of dynamical friction and energy equipartition. Since the masses of stars are much higher than the mass of the dark matter particles, the frictional drag on the stars is given by (see @bt87 Eq.7-18): $$\frac{d\vec{v}}{dt} = -\frac{-4 \pi \ln{\Lambda} G^2 \rho(r) m}{v^3}
\left[ {\rm erf}(X) - \frac{2 X}{\sqrt{\pi}} e^{-X^2} \right] \vec{v}
\label{dynf}$$ where $\rho(r)$ is the background density of dark matter and stars, $m$ the mass of an inspiraling star, $\ln{\Lambda}$ the Coulomb logarithm, and $X=\vec{v}/(\sqrt{2}\sigma)$ is the ratio between the velocity of a star and the (1D) stellar velocity dispersion $\sigma$. If we assume $v \approx \sigma$, it follows that $X=1/\sqrt{2}$. Setting $\ln{\Lambda}=12$ for globular clusters (@bt87 Tab. 7-1), the eq. \[dynf\] can be rewritten as: $$\frac{dv}{dt} = - 29.96 \frac{G^2 \rho(r) m}{v^2}\,.$$ The resulting energy change is $\frac{dE}{dt}= \frac{d}{dt}
(\frac{1}{2} m v^2) = m v \frac{dv}{dt}$. For a distribution of stars in virial equilibrium, an energy change $dE$ corresponds to a change in potential energy $m d\Phi =2 dE$. Hence $$\frac{d\Phi}{dt} = - 59.93 \frac{G^2 \rho(r) m}{v}\,.$$ For a Plummer model, the density $\rho$, circular velocity $v_c$ and specific potential $\Phi$ at point $r$ are given by: $$\begin{aligned}
\nonumber \rho(r) & = & \frac{3 M_{Tot} a^2}{4 \pi} \left( a^2 + r^2 \right)^{-5/2} \\
\Phi(r) & = & - \frac{G M_{Tot}}{\left( a^2 + r^2 \right)^{1/2}} \\
\nonumber v_c(r) & = & \frac{\sqrt{G M_{Tot} r^2}}{\left( a^2 + r^2 \right)^{3/4}} \end{aligned}$$ where $M_{Tot}$ is the total cluster mass and $a$ is the scale radius of the Plummer model. With these equations, the above relation can be rewritten as: $$\frac{d r}{dt} = -14.31 \frac{\sqrt{G} a^2 m}{\sqrt{M_{Tot}} r^2 \left( a^2 + r^2 \right)^{1/4}}$$ For an order of magnitude estimate of the inspiral time scale, one can approximate $\left( a^2 + r^2 \right)^{1/4} \approx a^{1/2}$. One can then solve the above relation and obtains as time which a star starting at radius $R_0$ needs to reach the centre: $$\nonumber T_{\rm Fric} = 0.023 \frac{\sqrt{M_{Tot}}}{\sqrt{G} a^{3/2} m} R_0^3$$ For a Plummer model, $a=0.766 R_H$, so the inspiral time for stars near the half-mass radius is given by: $$\nonumber T_{\rm Fric} = 0.035 \frac{\sqrt{M_{Tot}} R_H^{3/2}}{\sqrt{G} m}
\label{tfric}$$ For a globular cluster or UCD we thus obtain: $$\nonumber T_{\rm Fric} = 5.86 \left(\frac{M_{Tot}}{10^6 M_\odot}\right)^{1/2} \left(\frac{R_H}{5 pc}\right)^{3/2}
\left(\frac{m}{M_\odot} \right)^{-1} \rm{Gyr}
\label{tfricgc}$$ The resulting dynamical friction time agrees to within 20% with what Binney & Tremaine found for the inspiral time of an isothermal sphere (their eq. 7-26). Eq. \[tfricgc\] predicts a dynamical friction time scale of 4-5 Gyr for a typical GC (3$\times 10^5$M$_{\odot}$, $r_h=3$pc), and about 400 Gyr for a typical UCD ($10^7$M$_{\odot}$, $r_h=20$pc). That is, after a Hubble time most of the dark matter in globular clusters should have been pushed out of the centre, while in UCDs the inspiral of stars should be far from complete and a significant fraction of DM should still reside in their centres, leading to high mass-to-light ratios.
$N$-body results {#sec:res-nbody}
----------------
Fig. \[fig:lagrad\] shows the evolution of Lagrangian (lag.) radii, i.e. radii which contain a certain fraction of the total mass, of stars and dark matter particles in our first simulation, which had a dark matter particle mass of $m=0.03$ M$_\odot$ and an equal amount of mass in stars and dark matter. The effect of dynamical friction and mass segregation is clearly visible since the lag. radii of dark matter particles increase with time while those of the stars shrink. In $N$-body units, the total cluster mass and constant of gravity are both unity. With a mean stellar mass of $m=6.0 \cdot 10^{-5}$, we predict a dynamical friction timescale of $T_{\rm Fric} = 391$ in $N$-body units according to eq. \[tfric\]. It can be seen that by this time the cluster center is indeed nearly free of dark matter: inside the lag. radius of 10% of the stars, only 1% of the dark matter particles are located. At the end of the simulation, the 10% lag. radius of the dark matter particles is almost equal to the half-mass radius of the stars, i.e. only 20% of the cluster mass is still made up of dark matter inside the (visible) half-mass radius of the cluster. The core of the cluster is even stronger depleted and is nearly free of dark matter by the end of the simulations. It would therefore be difficult to detect the remaining dark matter by its effect on stellar velocities if mainly stars from the cluster center or inside the half-mass radius are used to determine the line-of-sight velocity dispersion.
![Evolution of Lagrangian radii, i.e. radii which contain a certain fraction of the total mass, of stars (red solid lines) and dark matter particles (blue dashed lines) in the first run from Table 1. The left panel depicts the evolution as a function of $N$-body time, the right panel as a function of current relaxation time, where the relaxation time is calculated based on the distribution of stars only. Inside the half-mass radius of the cluster, less than 20% of the total mass is made up out of dark matter after the clusters are ten apparent relaxation times old.[]{data-label="fig:lagrad"}](rad_trc.eps){width="8.5cm"}
The right panel of Fig. \[fig:lagrad\] depicts the evolution of lag. radii as a function of the ratio of cluster age to the actual relaxation time. In order to allow for a better comparison with observations, the relaxation time is calculated from the stellar component according to [@s87]: $$T_{RH} = 0.138 \frac{\sqrt{M_*} R_{H *}^{3/2}}{\sqrt{G} m_* \ln{\gamma N_*}}
\label{eq:trels}$$ where $M_*$ is the total stellar mass of the cluster, $R_{H *}$ the half-mass radius of the stellar distribution, $m_*$ and $N_*$ are the mass and number of stars and $\gamma$ a constant in the Coulomb logarithm which is taken to be $\gamma=0.11$ [@gh94]. Eq. \[eq:trels\] would be the relaxation time inferred by an observer who can only determine the stellar distribution and does not know about the dark matter. It has the same dependence on cluster mass and radius as the friction timescale and can therefore also be used to judge the dynamical state of a cluster.
The right panel of Fig. \[fig:lagrad\] shows that once a cluster is two to three apparent relaxation times old, the centre is free of dark matter and by the time the cluster has become ten relaxation times old, there is little dark matter left inside the half-mass radius. Since most globular clusters have relaxation times of only a few Gyr, their mass-to-light ratios should be within 20% of those of pure stellar populations if mainly stars inside the clusters half-mass radius are used to determine the velocity dispersion. Since this is within the uncertainty of measured mass-to-light ratios and current stellar population models, such a small dark matter cannot be detected kinematically in globular clusters. UCDs on the other hand have relaxation times significantly larger than a Hubble time and should therefore still have large mass-to-light ratios if they formed as a mix of dark matter and stars.
This is confirmed by the upper panel of Fig. \[fig:mcore\], which depicts the dark matter fraction inside the cluster core (assumed to be the region inside the 5% lag. radius of the stars) and inside the half-mass radius of the cluster stars. It can be seen that after about 1.5 to 2 friction times, the cluster core is almost completely free of dark matter. Within the half-mass radius, only about 30% of the initial dark matter amount remains after this time. The lower panel of Fig. \[fig:mcore\] depicts the evolution of the average mass of stars in the core and inside the half-mass radius. At both radii, the average mass of stars is increasing since, while stars segregate against the dark matter particles, heavy-mass stars also segregate against the lighter ones. After about 2 dynamical friction timescales, the mass of stars has reached a near constant value of about 0.65 M$_\odot$, which is significantly higher than the average mass of stars. Clusters which have expelled dark matter out of their centres should therefore also be mass segregated.
![Dark matter fraction (upper panel) and average mass of stars (bottom panel) as a function of time. Core values are shown by solid lines, values inside the half-mass radius by dashed lines. The core radius is assumed to be equal to the 5% lag. radius of the stellar distribution. The dark matter is depleted from the centre within 1 to 2 friction times. At the same time, heavy mass stars segregate against the light mass stars and the average mass of stars increases in the centre. Clusters where the centers are depleted of dark matter should therefore also be mass segregated in their centre.[]{data-label="fig:mcore"}](mseg.eps){width="8.5cm"}
Comparison with observations {#sec:comp}
----------------------------
Fig. \[fig:velfac\] shows the effect which the decreasing dark matter fraction in the center has on the projected velocity dispersion of stars. In order to determine this effect, we first calculate the velocity dispersion profile $\sigma_{Obs}(r)$ of bright stars with masses in the range $0.6 < m < 0.9$ M$_\odot$ as a function of projected radius. We restrict ourselves to this mass range since in a globular cluster or UCD, these would be the stars which dominate the cluster light. After determing the velocity dispersion profile of bright stars, we calculate the expected velocity dispersion profile based on the stellar density distribution according to (Binney & Tremaine 1987, eq. 4-54): $$\sigma^2(r) = - \frac{1}{\rho(r)} \int \rho(r') \left. \frac{d\Phi}{dr}\right|_{r=r'} dr'
\label{eq:jeans}$$ where $\rho(r)$ is the (3D) density distribution of bright stars and $\Phi(r)$ is the potential coming from the stars alone. Eq. \[eq:jeans\] assumes a spherical cluster potential and an isotropic velocity dispersion of stars. After projecting $\sigma(r)$ we can calculate the correction factor $f$ needed so that the predicted velocity dispersion matches the true velocity dispersion of the clusters in the $N$-body simulations, i.e. $f(r)=\sigma_{Obs}(r)/\sigma_{Pred}(r)$. The resulting correction factor is plotted in Fig. \[fig:velfac\] for the first run from Table 1. Initially, dark matter and stars follow the same density distribution, so the velocity dispersion is a factor $f(r)=\sqrt{(M_{DM}+M_*)/M_*}=1.41$ higher than predicted by eq. \[eq:jeans\]. As the cluster evolves, dark matter is removed from the center, so the velocities of stars in the center are determined more and more by the stars alone and $f$ approaches unity. After 3 relaxation times, the central velocity dispersion is only 10% higher than what one would expect based on the stars alone and after 10 relaxation times the difference is less than 1%. Most globular clusters should therefore have central M/L ratios which are close to those predicted by stellar population models. Beyond 10 half-mass radii, $f$ remains close to the initial value even after 10 relaxation times. As long as dark matter is not removed by tidal effects [@ms05], it should therefore be detectable in globular clusters through the observation of stellar velocities in the outer cluster parts.
![Evolution of the ratio $f$ of observed velocity dispersion to predicted one based on the stellar distribution alone for 4 different ratios of cluster age to apparent relaxation time calculated according to eq. \[eq:trels\]. Initially, dark matter and stars follow the same distribution and there is an equal amount of mass in dark matter and stars, so $\sigma_{Obs} = \sqrt{2} \sigma_{Pred}$ independent of radius. As the dark matter is expelled from the centre, stellar velocities in the inner parts are increasingly determined by the stars alone and $f$ drops towards unity in the cluster centre. Globular clusters should therefore have central mass-to-light ratios in agreement with stellar population models.[]{data-label="fig:velfac"}](velfac.ps){width="8.5cm"}
We finally discuss the influence of the dark matter on the global mass-to-light ratios. In order to compare our simulations with observed clusters, we again calculate true and expected velocity dispersions of stars with masses in the range $0.6 < m < 0.9$ M$_\odot$. Since mass-to-light ratios of UCDs are determined from stellar velocities covering a significant fraction of the cluster area (see e.g. discussion in @hetal07) and measured mass-to-light ratios of globular clusters are based mainly on stars in the inner cluster parts [@mcetal05], we determine global velocity dispersions in the simulations for all stars located inside the projected half-light radius. The resulting mass-to-light ratios of our model clusters are then given by $$M/L = f^2 M/L|_*$$ where $M/L|_*$ is the mass-to-light ratio which a pure stellar population would have and $f$ is again $f=\sigma_{Obs}/\sigma_{Pred}$.
Fig. \[fig:mtol2\] depicts the evolution of $M/L$ with cluster age for runs 1 and 2 of Table 1, and compares it with observed M/L ratios of UCDs and GCs from @metal08. Note that the literature M/L estimates are normalised to the same (solar) metallicity, to allow direct intercomparison. Time is again expressed in terms of age divided by the relaxation time as determined from the stars alone.
![Mass-to-light ratios of UCDs (circles) and globular clusters (triangles) as a function of their age divided by their relaxation time. There is a clear trend towards lower M/L values for dynamically more evolved systems. The red solid and blue dashed curves show predicted M/L values for two of our runs calculated from the velocity dispersion of bright stars inside the clusters half-mass radius, assuming stellar mass-to-light ratios of $M/L|_*=2.0$ and $M/L|_*=2.5$ for the two runs. It can be seen that the resulting theoretical curves provide a good fit to the combined globular cluster/UCD sample.[]{data-label="fig:mtol2"}](ml_new.eps){width="8.5cm"}
The observed normalised M/L ratios show a clear trend in the sense that dynamically more evolved systems have on average lower M/L values. The mass-to-light ratios in our simulations also decrease as the dynamic age increases, since, as the dark matter is depleted from the cluster centers, the velocity dispersion is determined more and more by the stars alone, so M/L approaches $M/L|_*$. Depending on whether a stellar $M/L|_*$ of 2.5 or 2.0 is assumed, a run with a primordial dark matter content equal to or twice as high as the stellar mass provides an acceptable fit to the data, making dark matter a viable alternative to explain the elevated mass-to-light ratios of UCDs. We note that if the observed decrease of M/L with dynamical age is due to the dynamical depletion of non-luminous particles from the cluster centers, the dark matter particles have to be of lower mass than the stars, ruling out e.g. a central concentration of black holes in UCDs as the explanation for their high M/L ratios.
The stellar mass-to-light ratios we have to assume in order to fit the data for GCs are marginally lower than predicted by simple stellar population models for 12 Gyr old, solar-metallicity star clusters. For example, the @bc03 models predict an M/L of 2.5 for a 8-9 Gyr old stellar population. The difference to a 12 Gyr old population (M/L=3.5) is still within individual error bars for the literature estimates, and there is also some uncertainty in the underlying stellar mass functions and stellar population models. Nevertheless, we note that the slightly too low M/L ratios may also be interpreted as signs of preferential loss of low-mass stars in the Galactic tidal field [@bm03; @kr08]. If this was the case, then less dark matter would be needed to explain the elevated mass-to-light ratios of UCDs, implying a DM mass of $\sim$50-80% of the stellar mass. However, it is unclear whether the actual dissolution times of the GCs with available M/L measurements are short enough to have experienced significant evaporation [@metal08]. A case-by-case analysis for Galactic GCs will be necessary to assess this, based on measured absolute proper motions and orbital parameters [@aetal06].
![Evolution of Lagrangian radii of stars (left panel) and dark matter particles (right panel) in simulations which assume dark matter particle masses of $m=0.1$ M$_\odot$ (blue lines), $m=0.03$ M$_\odot$ (red lines, the default mass for the simulations presented in Figs. \[fig:lagrad\] to \[fig:mtol2\]) and $m=0.015$ M$_\odot$ (green lines). The agreement between the different curves, especially for light dark matter particles is very good, showing that the adopted mass $m=0.03$ M$_\odot$ of the dark matter particle should not influence our results.[]{data-label="fig:dmpar_var"}](dmpar_var.eps){width="8.5cm"}
Scaling issues {#sec:variations}
--------------
We finally discuss a possible biasing of our results due to the finite mass of the dark matter particles. Fig. \[fig:dmpar\_var\] depicts the evolution of Lagrangian radii in simulations with different dark matter particle masses. All simulations had an equal amount of mass in stars and dark matter and the mass of the dark matter particles was set to be $m=0.1$ M$_\odot$ (blue lines), $m=0.03$ M$_\odot$ (red lines) and $m=0.01$ M$_\odot$ (green lines). Since the mass of heavy stars which drive the inspiral is in all cases much higher than the mass of the dark matter particles, eq. \[tfric\] should still apply for the inspiral timescale. In all three simulations, the mass of individual stars if expressed in $N$-body units was the same, so according to eq. \[tfric\], the inspiral timescale of the stars should be the same in the three simulations.
It can be seen that the inspiral of stars and the ejection of dark matter particles happens in all three clusters in a very similar way. The agreement is especially good between the two simulations with the lightest dark matter particles. We therefore conclude that the adopted mass $m=0.03$ M$_\odot$ for the dark matter particle does not influence the results presented in Figs. \[fig:lagrad\] to \[fig:mtol2\]. Our simulations should therefore give a correct picture of the dynamical ejection of dark matter from the centers of globular clusters and UCDs.
Conclusions {#sec:concl}
===========
We have performed collisional $N$-body simulations of the evolution of compact systems composed out of a mix of stars and dark matter particles. Our simulations show that dark matter is depleted from the centers of these systems due to dynamical friction and energy equipartition between stars and dark matter particles. The inspiral time of stars is short enough that only 20% of the original dark matter would remain within the half-mass radius in typical globular clusters. If mainly stars from the inner cluster parts are used to determine mass-to-light ratios, the resulting increase in the mass-to-light ratio is within the errors with which mass-to-light ratios are typically determined for globular clusters and would therefore be difficult to detect.
If not tidally stripped, dark matter should also reside in the outer parts of globular clusters. For a number of globular clusters, @setal07 have indeed reported a flattening of the velocity disperion in the outer cluster parts. This could however be due to a number of reasons like contamination of the sample by background stars or the tidal interaction of a star cluster with the gravitational field of the Milky Way [@detal98; @cetal05]. Detailed simulations would be necessary to exclude these possibilities and confirm that the observed flattening is due to a dark matter halo.
UCDs on the other hand have inspiral times significantly longer than a Hubble time and therefore still contain most of the dark matter in their centers. Dark matter therefore seems a viable explanation for the elevated M/L ratios of UCDs, provided that UCDs originate from the centers of dark matter halos and have seen their dark matter content being increased by dark matter funneling, through e.g. adiabatic gas infall [@getal08].
A prediction of our simulations, which can in principle be tested by observations, is that globular clusters which have expelled the dark matter from their centers should also be mass segregated. Non-mass segregated clusters with velocity dispersions and mass-to-light ratios in agreement with simple stellar population models, would therefore have formed without significant amounts of dark matter in their centers.
Also, if dark matter existed in a globular cluster at the time of its formation, it should still reside in its outer parts, especially if tidal stripping due to external tidal forces from the host galaxy [@ms05] and relaxation driven internal mass loss was not important for the cluster evolution. In this case, the measured mass-to-light ratio should increase towards the outer cluster parts, which can in principle be detected with dedicated radial velocity or proper motion surveys. The future astrometric satellite [*GAIA*]{} would be an excellent tool for such a search since it will provide accurate proper motions for thousands of stars in the halos of nearby globular clusters.
Acknowledgements {#acknowledgements .unnumbered}
================
Aarseth, S. J., 1999, PASP, 111, 1333
Allen, C., Moreno, E., & Pichardo, B. 2006, ApJ, 652, 1150
Baumgardt, H., Makino, J., 2003, MNRAS, 340, 227
Bekki, K., Couch, W.J., Drinkwater, M.J., 2001, ApJ, 552, L105
Bekki, K., Couch, W.J., Drinkwater, M.J., Shioya, Y., 2003, MNRAS, 344, 399
Binney, J., Tremaine, S., 1987, Galactic Dynamics, Princeton Univ. Press, Princeton, p. 425
Bruzual, G., Charlot, S., 2003, MNRAS, 344, 1000
Carraro, G., Lia, C., 2000, A&A, 357, 977
Casares, J., 2006, in [*Observational evidence for stellar-mass black holes*]{}, IAU Symposium 238 in press, arXiv:astro-ph/0612312
Capuzzo Dolcetta, R., Di Matteo, P., Miocchi, P., 2005, AJ, 129, 1906
Dabringhausen, J., Hilker, M. & Kroupa, P., 2008, MNRAS, 386, 864
Dabringhausen, J., Baumgardt, H. & Kroupa, P., 2008, MNRAS submitted
Drinkwater, M.J., Jones, J.B., Gregg, M.D. & Phillipps, S., 2000, PASA, 17, 227
Drukier, G., et al., 1998, AJ, 115, 708
Evstigneeva, E.A., Gregg, M.D., Drinkwater, M.J. & Hilker, M., 2007, AJ, 133, 1722
Fellhauer, M., Kroupa, P., 2002, MNRAS, 330, 642
Firth, P. et al. 2007, MNRAS, 382, 1342
Forbes, D. et al. 2008, MNRAS submitted, arXiv:0806.1090
Giersz, M., Heggie, D.C., 1994, MNRAS, 268, 257
Gilmore, G. et al. 2007, ApJ, 663, 948
Goerdt, T. et al., 2008, MNRAS, 385, 2136
Hasegan, M. et al., 2005, ApJ, 627, 203
Heggie, D.C., Hut, P., 2002, The Gravitational Million-Body Problem, Cambridge University Press, p. 8
Hilker, M., Infante, L., Viera, G., Kissler-Patig, M. & Richtler, T., 1999, A&AS, 134, 75
Hilker, M., et al., 2007, A&A, 463, 119
Jones, J. B et al. 2006, AJ, 131, 312
Kalirai, J.S. et al., 2008, ApJ, 676, 594
Kroupa, P., 2001, MNRAS 322, 231
Kruijssen, J. M. D. 2008, A&A letters in press, arXiv:0806.0852
Lyne, A.G., Lorimer, D.R., 1994, Nature, 369, 127
Makino, J., Fukushige, T., Koga, M., & Namura, K., 2003, PASJ, 55, 1163
Mashchenko, S., Sills, A. 2005, MNRAS, 619, 258
McLaughlin, D. E & van der Marel, R. P. 2005, ApJS, 161, 304
Mieske, S., et al., 2005, A&A, 430L, 25
Mieske, S., Hilker, M., Infante, L. & Jordán, A. 2006, AJ, 131, 2442
Mieske, S., Hilker, M., Jordán, A., Infante, L. & Kissler-Patig, M. 2007, A&A, 472, 111
Mieske, S., et al., 2008, A&A in press, arXiv:0806.0374
Mieske, S., Kroupa, P., 2008, ApJ, 677, 276
Oh, K.S., Lin, D.N.C., 2000, ApJ, 543, 620
Pfahl, E., Rappaport, S., Podsiadlowski, P., 2002, ApJ, 573, 283
Plummer, H.C., 1911, MNRAS 71, 460
Rejkuba, M., Dubath, P., Minniti, D., & Meylan, G. 2007, A&A, 469, 147
Scarpa, R., Marconi, G., Gilmozzi, R., & Carraro, G. 2007, ESO Messenger 128, 41, arXiv:0707.2459
Spitzer, L., 1987, [*Dynamical Evolution of Globular Clusers*]{}, Princeton University Press
Thomas, P.A., Drinkwater, M.J. & Evstigneeva, E., 2008, MNRAS in press, arXiv:0801.4840
Thorsett, S.E., Chakrabarty, D., 1999, ApJ, 512, 288
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abstract: 'Nearby (D$ \leq$ 100 Mpc) luminous blue starburst galaxies frequently show morphological evidence for recent involvement in mild collisions, or minor mergers where the disk survives. As a consequence UV-bright starbursts are preferentially seen in near face-on galaxies, and the postburst systems may become star-forming late-type galaxies. If this starburst evolutionary channel is important at moderate redshifts, then descendants of the faint blue galaxies could be the very common Sm-Sd field galaxies.'
author:
- 'J.S. Gallagher, C.J. Conselice, and N. Homeier'
- The WFPC2 Investigation Definition Team
title: 'Structure and Evolution of Nearby UV-bright Starburst Galaxies'
---
Objectives
==========
Starburst galaxies provide unique opportunities to study rapid phases of galactic evolution. The Wisconsin starburst project focuses on nearby ultraviolet-bright starbursts selected from the spectrophotometric sample of Gallagher et al. (1989). They resemble intermediate redshift ($z \approx$ 0.5-1) “compact narrow emission line galaxies” (CNELGs, e.g. Guzeman et al. 1998) in terms of their small sizes, blue luminosities, and \[OII\] emission line equivalent widths; e.g., M$_B < -$18, and EW(\[OII\]) $>$ 25 Å with $U-B< -$0.2.
Our strategy is based on the analysis of multi-band imaging and area spectroscopy obtained with the WIYN 3.5-m telescope [^1] and high angular resolution WFPC2 and FOC [*HST*]{} images. Images provide global morphologies, colors, and the small scale structures of star-forming sites. Spectra from the WIYN Densepak multi-fiber array are used to measure global velocity fields in the ionized gas (eg., Homeier & Gallagher 1999; HG99).
Interactions and Starbursts
===========================
Interactions are well-established starburst triggers. Because of the huge range of possible interaction parameters, we expect and observe considerable variance in the responses of galaxies to interactions. Our emphasis is on weak interactions, such as minor mergers, or mild collisions, in which at least one member of the colliding pair survives with its initial structure relatively intact.
The signatures of recent galaxy interactions are well-known, and in the optical include: luminous tails of tidal debris, or ripples in post-interaction systems. These features often have low surface brightnesses and/or small angular sizes; imaging with telescopes like WIYN, that combine excellent surface brightness sensitivity with good angular resolution (0.7 arcsec seeing), is valuable for determining the structures of these types of systems. The signposts of interactions evolve on galaxy orbital time scales of $\sim$10$^8$ yr with infall of tidal debris possibly continuing for $\sim$10$^9$ yr; after this, evidence of a collision will be difficult to detect.
Starbursts are found from their prominent populations of high mass stars, which are the products of upward spikes in star formation rates. This is obvious when the starburst is optically visible, and gives rise to high surface brightnesses, blue optical colors, and strong emission lines. Starbursts also produce supergiant HII complexes, massive and compact “super star clusters”, and the peculiar kpc-scale “clumps” of OB stars, whose existence was emphasized more than 20 years ago by J. Heidmann and collaborators as symptoms of ‘hyperactive star formation’. While the evolutionary time scales of individual star-forming complexes within starbursts are not well known, data from M82 suggest this occurs in less than 10$^8$ yr (Satyapal et al. 1997; Gallagher & Smith 1999). The internal structure of a starburst evolves more quickly than structural perturbations induced by the dynamical influence of a colliding galaxy.
Four Examples of Luminous Blue Starbursts
=========================================
The optical structures of four luminous starbursts within 100 Mpc provide some examples of star formation produced in a variety of collisions:
[**Markarian 8**]{} is a strongly interacting system (Figure 1). It apparently consists of two small disk galaxies. Our WFPC2 images show one of these to be strongly disturbed, while the other remains an organized, coplanar system (Gallagher et al. 2000). Both objects have high inclinations, but the system still has blue colors and strong \[OII\] emission.
[**Haro 1**]{}, a nearly face-on galaxy, can be seen in Figure 2 to be experiencing a weak interaction with a dwarf companion. Even though only mild optical distortions are present in Haro 1, it is in a late phase of an intense starburst (the integrated spectrum shows strong Balmer absorptions).
[**NGC 3310**]{} is one of the nearest examples of a luminous UV-bright starburst and has been extensively studied in the optical and IR and mapped in HI. The starburst probably was caused by a minor merger (Mulder et al. 1995, Conselice et al. 2000). Large scale organization of star formation in NGC 3310 is located in a nearly symmetric two-arm spiral pattern, and around a mildly distorted circum-nuclear ring, as shown in the color map in Figure 2. Evidently a strong starburst was initiated with little collisional disruption to the surviving gas-rich disk galaxy, seen at low inclination.
[**NGC 7673**]{}, an archetypical ‘clumpy irregular’, has a highly disturbed inner structure dominated by four huge star-forming complexes (Figure 2). WFPC2 optical and UV imaging resolve the clumps into clusters of compact star clusters, similar to conditions in M82 (O’Connell et al. 1995). The clumps are located within a faint, relatively symmetric outer disk that also contains a faint stellar arc, or ripple (HG99). This starburst is a remnant of a past interaction with NGC 7678 (Nordgren et al. 1997), or a minor merger.
HG99 measured the H$\alpha$ velocity field of NGC 7673 with WIYN using the Densepak array. They found that the inner part of the galaxy is almost exactly face-on and dynamically cold. The wings of the line are broadened; most likely by gas outflows from the disk, and the integrated H$\alpha$ line profile closely resembles those observed in CNELGs by Guzman et al. (1996). However, in this case we know that we are [*not*]{} seeing a spheroidal system in formation, but rather a nearly face-on disk.
Summary of Observed Properties
==============================
Our data demonstrate the production of intense starbursts in weak interactions in which at least one of the original disks survives. One channel for the production of compact luminous galaxies therefore modifies rather than destroys stellar disks, and some post-starburst systems will be small disk galaxies.
Disk geometries introduce a prejudice for optically- or UV- bright starbursts to be seen at low inclinations where disk obscuration is minimized. M82 is a nearby example; from our perspective it is an IR-starburst, but the brilliant UV reflection nebula seen above the disk in FOCAS images imply that when seen at low inclination, M82 is a UV-bright starburst. This is a statistical trend; some galaxies, such as Markarian 8 or chain galaxies at moderate redshifts, are blue and inclined. A bias to select blue starbursts in low inclination disks will also lead to narrow velocity line widths in integrated optical spectra; these are not necessarily signatures of low mass dwarf galaxies.
Our data do not show a simple relationship between the stage of an interaction and starburst properties; intense starbursts occur in both the ongoing Markarian 8 and very late NGC 7673 interactions. This complicates our ability to distinguish collisionally induced starbursts, since the clear signatures of an ongoing interaction may not be present during the peak starburst phase. This problem is especially severe when dealing with faint blue galaxies at moderate redshifts, where information is often sparse (Ellis 1997).
Discussion
==========
A simple model assumes that luminous blue starbursts take place in gas-rich late-type disk galaxies that remain as disks after the event. Many nearby starbursts are rich in HI and have the fuel to support post-burst star-formation. These types of post-burst objects would be less easily distinguished than extreme cases where star formation is truncated after a burst, giving rise to distinctive ‘E $+$ A’ spectra. The ingredients of our simple recipe for luminous starbursts follows:
1\. Store gas in a dynamically cold, low density disk. Such disks are excellent storage places to keep gas as they have low intrinsic rates of evolution, and populations of such thin, cold disk galaxies exist at the current epoch (Matthews et al. 2000).
2\. Perturb the disk with an interaction; a merger with a galaxy having 10% or less of the total mass, or even a near miss from a comparable or larger neighbor should suffice. What we require is that gas be driven to the central few kpc of the disk where it can support intense star formation. Gas transport can be produced directly by the perturber, or indirectly through production of a bar.
3\. Orient galaxy to be nearly face-on and observe a blue, UV-bright starburst with narrow emission lines. Otherwise, observe an edge-on system to see an M82-type moderately-luminous IR-starburst.
4\. Wait a few Gyr until done. The result may be a late-type system with a moderately thick stellar disk, a product of collisional disk heating during the interaction (Reshetnikov & Combes 1997). The interaction also could produce an off-center bar, yielding a Magellanic irregular. This channel for making Magellanic systems is observationally suggested by the preference for such galaxies to have companions (Odewahn 1994) and theoretical models that produce characteristic off-center bars during collisions (Levine & Sparke 1998).
The Wisconsin exploration of starburst galaxies has been supported as part of the WFPC2 Investigation Team’s research program. We are also pleased to acknowledge support through General Observer Space Telescope grant, and from a Vilas Associateship award to JSG.
[^1]: The WIYN Observatory is a joint facility of the University of Wisconsin-Madison, Indiana University, Yale University, and the National Optical Astronomy Observatories.
|
---
abstract: 'We present spectro-photometry spanning 1–5 $\mu$m of 51 Eridani b, a 2–10 M$_\text{Jup}$ planet discovered by the Gemini Planet Imager Exoplanet Survey. In this study, we present new $K1$ (1.90–2.19 $\mu$m) and $K2$ (2.10–2.40 $\mu$m) spectra taken with the Gemini Planet Imager as well as an updated $L_P$ (3.76 $\mu$m) and new $M_S$ (4.67 $\mu$m) photometry from the NIRC2 Narrow camera. The new data were combined with $J$ (1.13–1.35 $\mu$m) and $H$ (1.50–1.80 $\mu$m) spectra from the discovery epoch with the goal of better characterizing the planet properties. 51 Eri b photometry is redder than field brown dwarfs as well as known young T-dwarfs with similar spectral type (between T4–T8) and we propose that 51 Eri b might be in the process of undergoing the transition from L-type to T-type. We used two complementary atmosphere model grids including either deep iron/silicate clouds or sulfide/salt clouds in the photosphere, spanning a range of cloud properties, including fully cloudy, cloud free and patchy/intermediate opacity clouds. Model fits suggest that 51 Eri b has an effective temperature ranging between 605–737 K, a solar metallicity, a surface gravity of $\log$(g) = 3.5–4.0 dex, and the atmosphere requires a patchy cloud atmosphere to model the SED. From the model atmospheres, we infer a luminosity for the planet of -5.83 to -5.93 ($\log L/L_{\odot}$), leaving 51 Eri b in the unique position as being one of the only directly imaged planet consistent with having formed via cold-start scenario. Comparisons of the planet SED against warm-start models indicates that the planet luminosity is best reproduced by a planet formed via core accretion with a core mass between 15 and 127 M$_{\oplus}$.'
author:
- Abhijith Rajan
- Julien Rameau
- 'Robert J. De Rosa'
- 'Mark S. Marley'
- 'James R. Graham'
- Bruce Macintosh
- Christian Marois
- Caroline Morley
- Jennifer Patience
- Laurent Pueyo
- Didier Saumon
- 'Kimberly Ward-Duong'
- 'S. Mark Ammons'
- Pauline Arriaga
- 'Vanessa P. Bailey'
- Travis Barman
- Joanna Bulger
- 'Adam S. Burrows'
- Jeffrey Chilcote
- Tara Cotten
- Ian Czekala
- Rene Doyon
- 'Gaspard Duch[ê]{}ne'
- 'Thomas M. Esposito'
- 'Michael P. Fitzgerald'
- 'Katherine B. Follette'
- 'Jonathan J. Fortney'
- 'Stephen J. Goodsell'
- 'Alexandra Z. Greenbaum'
- Pascale Hibon
- 'Li-Wei Hung'
- Patrick Ingraham
- 'Mara Johnson-Groh'
- Paul Kalas
- Quinn Konopacky
- 'David Lafreni[è]{}re'
- 'James E. Larkin'
- 'J[é]{}r[ô]{}me Maire'
- Franck Marchis
- Stanimir Metchev
- 'Maxwell A. Millar-Blanchaer'
- 'Katie M. Morzinski'
- 'Eric L. Nielsen'
- Rebecca Oppenheimer
- David Palmer
- 'Rahul I. Patel'
- Marshall Perrin
- Lisa Poyneer
- 'Fredrik T. Rantakyr[ö]{}'
- 'Jean-Baptiste Ruffio'
- Dmitry Savransky
- 'Adam C. Schneider'
- Anand Sivaramakrishnan
- Inseok Song
- 'R[é]{}mi Soummer'
- Sandrine Thomas
- Gautam Vasisht
- 'J. Kent Wallace'
- 'Jason J. Wang'
- Sloane Wiktorowicz
- Schuyler Wolff
title: 'Characterizing 51 Eri from 1–5 $\mu$: a partly-cloudy exoplanet'
---
Introduction
============
Until recently, most of the imaged planetary mass companions detected were typically orbiting their parent star at large orbital separations, $>$30 au. However, new instrumentation with second generation adaptive optics such as the Gemini Planet Imager [GPI, @Macintosh:2014] and Spectro-Polarimetric High-contrast Exoplanet REsearch [SPHERE, @Beuzit:2008] are now routinely obtaining deep contrasts ($>10^5-10^6$) in the inner arcsecond (5–30 au). The recent detection of new companions [@Macintosh:2015; @Konopacky:2016; @Wagner:2016; @Milli:2017] and debris disks [@Currie:2015; @Wahhaj:2016; @Blanchaer:2016; @Bonnefoy:2017] showcase the advances made by these next generation AO systems. Direct imaging, unlike non-direct methods such as radial velocity and transits, measures light from companions directly, which permits measuring the atmospheric spectrum, with the caveat that the final calibration is dependant on complete understanding of the stellar properties. These new AO instruments combine excellent image stability and high throughput with IFU spectrographs, enabling the measurement of a spectrum of the planet in the near infrared (IR) wavelength range. Combining the near-IR spectra with mid-IR photometry from instruments such as Keck/NIRC2, MagAO/Clio or LBT/LMIRCam, provides valuable constraints on the effective temperature and non-equilibrium chemistry when undertaking comprehensive modeling of the exoplanet spectral energy distribution.
In this study we focus on the planetary companions, 51 Eridani b [51 Eri b; @Macintosh:2015]. 51 Eri b is the first planet discovered by the Gemini Planet Imager Exoplanet Survey (GPIES), a survey targeting 600 young and nearby stars using GPI to search for exoplanets. The planet orbits 51 Eri A, a young F0IV star that is part of the $\beta$ Pic moving group [@Zuckerman:2001]. In this study, we adopt an age of of $26\pm3$ Myr for the $\beta$ Pic moving group [@Nielsen:2016]. However, the age of the group is a topic of considerable debate and has been revised several times e.g. $21\pm4$ [@Binks:2014], $23\pm3$ [@Mamajek:2014], $20\pm6$ [@Macintosh:2015], $24\pm3$ [@Bell:2015]. The primary is part of a hierarchical triple with two M-star companions, GJ 3305AB, separated from the primary by $\sim$2000 au [@Feigelson:2006; @Kasper:2007; @Montet:2015]. 51 Eri A is known to have an IR excess, and a debris disk was detected in *Herschel Space Observatory* 70 and 100 $\mu$m bands with very low IR luminosity of L$_\text{IR}$/L$_{\star} = 2 \times 10^{-6}$ and an a lower limit on the inner radius of 82 au [@Marichalar:2014] as well as a detection at 24 $\mu$m with the *Spitzer Space Telescope* [@Rebull:2008]. The debris disk was not detected in @Macintosh:2015, which, given the low fractional luminosity would be extremely challenging. The analysis of the atmosphere of 51 Eri b by @Macintosh:2015 was based on GPI $JH$ spectra (1.1–1.8 $\mu$m) and Keck $L_P$ photometry (3.76 $\mu$m), using two different model atmosphere grids to estimate planet properties. While the models agreed on the temperature and luminosity, they were highly discrepant in terms of best fitting surface gravity with one grid suggesting low surface gravity and youth while the other required a high surface gravity and an old planet. Similarly, one grid best fit the atmosphere when using a linear combination of cloudy and clear models while the other best fit the data with clear atmosphere. These discrepancies indicate that more data is required to fully constrain the planet parameters.
In this paper, we present new observations and revised data analysis that can be used to discriminate between some of the disagreements. In Section \[sec:obs\], we present the first $K1$ (1.90–2.19 $\mu$m) and $K2$ (2.10–2.40 $\mu$m) spectrum of the planet taken with GPI. We also present updated $L_P$ photometry and new observations of the planet in the $M_S$-band (4.67 $\mu$m). In Section \[sec:res\], we present new near-IR photometry of the star and revise the stellar spectral energy distribution (SED) used in the rest of the analysis. In Section \[sec:ana\], we examine the near- and mid-IR photometry of 51 Eri b in relation to that of other field and young brown dwarfs through the brown dwarf color-magnitude diagram. We also compare the near-IR spectrum of 51 Eri b to field brown dwarfs, and planetary-mass companions to estimate the best fitting spectral type of the planet. Finally, in Section \[model\] we model the planet SED using two different grids spanning effective temperatures from 450K to 1000K with deep iron/silicate clouds or sulfide/salt clouds. The 1–5 $\mu$m spectral energy distribution in combination with these two model grids with help refine the planet properties and clarify whether the atmosphere is best fit by clouds, or not.
[cccccccc]{} 2015 Jan 30 & GS/GPI & $J$ & 70 & 23.8 & 1.15 & 0.52 & 3.26\
2014 Dec 18 & GS/GPI & $H$ & 38 & 37.7 & 1.14 & – & –\
2015 Nov 06 & GS/GPI & $K1$ & 55 & 30.5 & 1.17 & 0.38 & 1.56\
2015 Dec 18 & GS/GPI & $K2$ & 103 & 71.7 & 1.22 & 0.69 & 0.94\
2016 Jan 28 & GS/GPI & $K1$ & 97 & 55.5 & 1.15 & 0.86 & 4.40\
2015 Oct 27 & Keck/NIRC2 & $L_P$ & 100 & 74.2 & 1.10 & – & –\
2016 Jan 02 & Keck/NIRC2 & $M_S$ & 139 & 115.7 & 1.18 & – & –\
2016 Jan 21 & Keck/NIRC2 & $M_S$ & 174 & 116.0 & 1.21 & – & –\
2016 Feb 04 & Keck/NIRC2 & $M_S$ & 148 & 101.4 & 1.21 & – & –\
2016 Feb 05 & Keck/NIRC2 & $M_S$ & 142 & 102.1 & 1.21 & – & –\
Observations and Data Reduction {#sec:obs}
===============================
GPI K1 and K2
-------------
51 Eri b was observed with the Integral Field Spectrograph (IFS) of GPI through the $K1$ filter on 2015 November 06 UT and 2016 January 28 and through the $K2$ filter on 2015 December 18 UT (see Table \[tab:phot\]). Standard procedures, namely using an argon-arc lamp, were used to correct the data for instrumental flexure. To maximize the parallactic rotation for Angular Differential Imaging [ADI; @Marois:2006a], the observations were centered on meridian passage. All the GPI datasets underwent the same initial data processing steps using the GPI Data Reduction Pipeline v1.3.0 [DRP; @Perrin:2014]. The processing steps included dark current subtraction, bad pixel identification and interpolation, this is followed by compensating for instrument flexure using the argon arc spectrum [@Wolff:2014]. Following this step, the microspectra are extracted to generate the IFS datacubes [@Maire:2014]. During the process of generating the 3D (x, y, $\lambda$) cubes, the microspectra data are resampled to $\lambda/\delta \lambda$ = 65, and 75 at $K1$ and $K2$, respectively, after which they are interpolated to a common wavelength scale and corrected for geometric distortion [@Konopacky:2014]. The datacubes are then aligned to a common center calculated using the four satellite spots [@Wang:2014]. The satellite spots are copies of the occulted central star, generated by the use of a regular square grid printed on the apodizer in the pupil plane [@Sivaramakrishnan:2006; @Marois:2006b; @Macintosh:2014]. The satellite spots also help convert the photometry from contrast units to flux units. No background subtraction was performed since the following steps of high-pass filtering and PSF subtraction efficiently remove this low frequency component.
Further steps to remove quasi-static speckles and large scale structures were executed outside the DRP. Each datacube was filtered using an unsharp mask with a box width of 11 pixels. The four satellite spots were then extracted from each wavelength slice, and averaged over time to obtain templates of star point spread function (PSF). The Linear Optimized Combination of Images algorithm [LOCI, @Lafreniere:2007] was used to suppress the speckle field in each frame using a combination of aggressive parameters: $dr=5$ px, $N_A$=200 PSF full width at half maximum (FWHM), $g=0.5$, and $N_\delta=0.5-0.75$ FWHM for the three datasets. Where $dr$ is the radial width of the optimization zone, $N_A$ is the number of PSF FWHM that can be included in the zone, $g$ is the ratio of the azimuthal and radial widths of the optimization zone, and $N_\delta$ defines the maximum separation of a potential astrophysical source in FWHM between the target and the reference PSF. The residual image of each wavelength slice was built from a trimmed ($10\%$) temporal average of the sequence.
![Final PSF subtracted images of 51 Eri b. (Top) LOCI-reduced GPIES images at $K1$ (2016 Jan 28, left) and $K2$ band (2015 Dec 18, right). (Bottom) [pyKLIP]{}-reduced NIRC2 images, smoothed with a box of width of 2 pixels, at $L_P$ (2015 Oct 27, left), and a combined image of all four $M_S$ datasets (right). The images are scaled linearly, but are different in each panel in order to saturate the core of the planet PSF. \[fig:gpiesK1K2LpMs\]](Fig1a.pdf "fig:"){width="4.0cm"}![Final PSF subtracted images of 51 Eri b. (Top) LOCI-reduced GPIES images at $K1$ (2016 Jan 28, left) and $K2$ band (2015 Dec 18, right). (Bottom) [pyKLIP]{}-reduced NIRC2 images, smoothed with a box of width of 2 pixels, at $L_P$ (2015 Oct 27, left), and a combined image of all four $M_S$ datasets (right). The images are scaled linearly, but are different in each panel in order to saturate the core of the planet PSF. \[fig:gpiesK1K2LpMs\]](Fig1b.pdf "fig:"){width="4.0cm"}\
![Final PSF subtracted images of 51 Eri b. (Top) LOCI-reduced GPIES images at $K1$ (2016 Jan 28, left) and $K2$ band (2015 Dec 18, right). (Bottom) [pyKLIP]{}-reduced NIRC2 images, smoothed with a box of width of 2 pixels, at $L_P$ (2015 Oct 27, left), and a combined image of all four $M_S$ datasets (right). The images are scaled linearly, but are different in each panel in order to saturate the core of the planet PSF. \[fig:gpiesK1K2LpMs\]](Fig1c.pdf "fig:"){width="4.0cm"}![Final PSF subtracted images of 51 Eri b. (Top) LOCI-reduced GPIES images at $K1$ (2016 Jan 28, left) and $K2$ band (2015 Dec 18, right). (Bottom) [pyKLIP]{}-reduced NIRC2 images, smoothed with a box of width of 2 pixels, at $L_P$ (2015 Oct 27, left), and a combined image of all four $M_S$ datasets (right). The images are scaled linearly, but are different in each panel in order to saturate the core of the planet PSF. \[fig:gpiesK1K2LpMs\]](Fig1d.pdf "fig:"){width="4.0cm"}
Final $K1$ and $K2$ broad-band images were created using a weighted-mean of the residual wavelength frames according to the spectrum of the planet, examples of which can be found in Figure \[fig:gpiesK1K2LpMs\]. These broad-band images were used to extract the astrometry of the planet in each dataset thanks to higher signal-to-noise ratio (SNR) than in individual frames. To do so, a negative template PSF was injected into the raw data at the estimated position and flux of the planet before applying LOCI and reduced using the same matrix coefficients as the original reduction [@Marois:2010]. The process was iterated over these three parameters (x position, y position, flux) with the amoeba-simplex optimization [@NelderMead:1965] until the integration squared pixel noise in a wedge of 2$\times$2 FWHM was minimized. The best fit position was then used to extract the contrast of the planet in each dataset. The same procedure was executed in the non-collapsed wavelength residual images but varying only the flux of the negative template PSF and keeping the position fixed to prevent the algorithm from catching nearby brighter residual speckles in the lower SNR spectral slices. To measure uncertainties, we injected the template PSF with the measured planet contrast into each datacube at the same separation and 20 different position angles. We measured the fake signal with the same extraction procedure. The contrasts measured in the 2015 Nov 06 and 2016 Jan 28 $K1$ datasets agreed within the uncertainties, the latter having significantly better SNR, and were combined with weighted mean to provide the final planet contrasts.
### Spectral covariances {#sec:covariance}
Estimation of a directly imaged planets properties from its measured spectrum is complicated by the fact that spectral covariances are present within the extracted spectra. In the GPI data these are caused by the residual speckle noise in the final PSF-subtracted image, and the oversampling of the individual microspectra during the initial data reduction process. Atmosphere modeling without properly accounting for these covariances can lead to biased results. We present the derivation of the correlation using the parameterization of @Greco:2016 in the Appendix \[app:cov2\].
We use the spectral covariance when carrying out comparison of the planet spectrophotometry against other field and young dwarfs as well as during model fitting. The covariance helps correctly account for the correlation in the spectra while also increasing the importance of the photometry, and thus the use of the covariance tends to move the best fits towards cooler temperatures when compared to using the variance directly.
Keck $L_P$
----------
We observed the 51 Eri system on 2015 Oct 27 in the $L_P$ filter with the NIRC2 camera [@McLean:2003] at the Keck-II observatory (Program ID - U055N2). The observations were taken in ADI mode, starting $\sim$1 hour prior to meridian crossing to maximize the field of view rotation. The target was observed for $\sim$3 hours total, with 100 min of on-source integration. The observations were acquired using the 400mas focal plane mask and the circular undersized “incircle” cold stop. To calibrate the planet brightness unsaturated observations of the star were taken at the end of the observing sequence. The images were dark and flat field corrected. We used the $K_S$-band lamp flats to build the flatfield and masked hot and bad pixels. As these observations were taken after the April 2015 servicing of NIRC2, the geometric distortion was corrected using the solution presented in @Service:2016 (updating the original @Yelda:2010 solution), with an updated plate scale of 9.971$\pm$0.004 mas pixel$^{-1}$ and the offset angle $\beta$ ($0.262\pm0.020$) which is required when calculating the position angle prior to rotating the images to put north up [@Yelda:2010]. Post-processing of the data was carried out using the Python version of the Karhunen-Loève Image Projection algorithm [KLIP, @Soummer:2012; @Amara:2012], `pyKLIP` [@Wang:2015]. As part of this study, we included a NIRC2 module in the `pyKLIP` codebase that is publicly available for users. [^1] The algorithm accepts aligned images and performs PSF subtraction using KLIP where the image can be divided into sections both radially and azimuthally. Aside from the choice of zones, there are two main parameters that were adjusted, the number of modes used in the Karhunen?-Loève (KL) transform and an exclusion criterion for reference PSFs, similar to $N_\delta$ mentioned above, that determines the number of pixels an astrophysical source would move due to the rotation of the reference stack. We carried out a parameter search where the four parameters mentioned were varied to optimize the signal to noise in the planet signal. The planet photometry was estimated using the method described above for the $K1$ and $K2$ filters, using a negative template PSF. The $L_P$ magnitude contrast for the star-planet is 11.58$\pm$0.15 mag which agrees very well with the photometry in the original epoch, 11.62$\pm$0.17 mag. The weighted mean of both measurements is used in the rest of the analysis.
Keck $M_S$
----------
Observations of 51 Eri b were taken in the $M_S$-band filter over four separate half nights on 2016 Jan 02, 21 and 2016 Feb 04, 05 with Keck/NIRC2 Narrow camera. The details of the observations are presented in Table \[tab:phot\]. Each night the target was observed for a period of $\sim$6 hours, as part of two separate NASA and UC Keck observing programs (Program ID - N179N2, U117N2). The data were obtained in ADI mode, with the field of view rotating at the sidereal rate. To reduce the effects of persistence and enable accurate thermal background correction, the star was nodded across the detector in four large dithers centered in each quadrant of the detector. Furthermore to prevent saturation of the detector by the thermal background, the exposures were limited to 0.3s with 200 co-adds, without using an occulting spot. The images were dark and flat field corrected with twilight sky flats, followed by hot and bad pixel correction. As with the $L_P$ data, the solution provided by @Service:2016 was used to correct the NIRC2 Narrow camera geometric distortion. Finally, all the images were rotated to put north up.
An additional step required for the $M_S$-band data that is not as critical for the other datasets is the background subtraction. Since the thermal background at 5$\mu$m is large and highly time variable, rather than median combine, or high pass filter to remove the background we adopted the least-squares sky subtraction algorithm proposed in @Galicher:2011. For each point in the dither pattern, the algorithm uses the images where the star is in one of the other three positions to construct a reference library. We used a ring centered on the star to estimate the thermal background in each image, with an inner annulus of 24 pixels and an outer annulus of 240 pixels. The final calibration step involved aligning the background corrected PSFs. Since the core of the PSF is saturated in the data, we aligned the data using two different methods, a) fitting a 2D Gaussian to the wings of the stellar PSF to estimate the center of the star and then shifting the PSF to a pre-determined pixel value to align all the images and b) using the rotation symmetry of the PSF using the method described in @Morzinski:2015. To compare the two methods, we calculated the residuals between images aligned using the methods and compared the noise in the residuals and found them to be similar and chose to go with the 2D Gaussian which is computationally faster.
The procedure used for the PSF subtraction for the $M_S$ data was similar to the $L_P$ data. The planet is not detected in each of the individual half-night datasets, requiring a combination of all four half-nights to increase the signal to noise ratio to detect the planet flux. To correctly combine the planet flux across the multiple epochs, we adjusted the PA to account for the astrophysical motion of the planet around the star, for which we used the best fitting orbit presented in @DeRosa:2015. In the month between the first and last dataset, the planet rotated $\sim$0.48 degrees or $\sim$0.4 pixel, which is a sufficiently large correction that it must be included in the data reduction. Each night’s data was reduced individually to generate 603 PSF subtracted images. These images were then combined by dividing each image into 13 annuli which were combined using a weighted mean, where the weights are the inverse variance in each annulus. As seen in Figure \[fig:gpiesK1K2LpMs\], we detect the planet signal at $\sim$2–3 sigma. To confirm that we are detecting the planet, we rotated the data to match the PA value of the $L_P$ epoch to find that the flux peak in the $M_S$-band matches the location of the planet in $L_P$. We measured a star to planet contrast of 11.5 mag using the same procedure as described for the $L_P$ data. We injected 25 fake PSFs that were scaled to match the contrast measured for the planet and detected the fakes at the same contrast as the planet. The final magnitude of the planet-star contrast in the $M_S$ is 11.5$\pm$0.5 mag.
Results {#sec:res}
=======
To estimate stellar parameters of 51 Eri A, @Macintosh:2015 made use of Two Micron All-Sky Survey photometry [2MASS; @Cutri:2003; @Skrutskie:2006hla]. However, the $J$ and $H$-band photometry for the star are flagged as ‘E’, indicating that the photometry is of the poorest quality and potentially unreliable (as compared to an ‘A’ flag for the the $K$-band photometry). Further, the study used photometry taken with the *Wide-field Infrared Survey Explorer* (WISE; [@Wright:2010in]) in the $W1$ filter ($\lambda_\text{eff}$=3.35 $\mu$m, $\Delta\lambda$=1.11 $\mu$m) as an approximation for the $L_P$-band magnitude of the primary star. The photometry for 51 Eri A in $W1$, from the AllWISE catalog [@Cutri:2013], has large errors and contributes to more than half the error budget of the final planet photometry. In this study, we thus chose to re-observe the star in the $JHK_S$ filters and fit all the available photometry to estimate the photometry in filters where no calibrated stellar data exists.
Revised Stellar Photometry at $J$,$H$,$K_S$ {#sec:sed}
-------------------------------------------
The 2MASS near-IR colors of 51 Eri A were compared to empirical colors for young F0 stars taken from @Kenyon:1995, where an F0IV star should have a $J-H$ = 0.13 mag and $H-K$ = 0.03 mag. The colors of 51 Eri A estimated using the 2MASS photometry are however discrepant, with $J-H$ = $-0.03\pm0.08$, and $H-K$ = $0.23\pm0.08$ mag. The discrepant near-IR colors combined with poor quality flags suggest that the published photometry is potentially incorrect.
We observed the star 51 Eri A using the 6.5-m MMT on Mt. Hopkins with the ARIES instrument [@McCarthy:1998] on 2016 Feb 28 UT under photometric conditions. We obtained data in the MKO $JHK_S$ broadband filters [@Tokunaga:2002], for a total of 3.4 minutes in each filter. To flux calibrate these observations, we observed a photometric standard star at a similar airmass as 51 Eri A, HR 1552 [@Carter:1990]. The raw images for both targets were processed through a standard near-IR reduction pipeline, performing dark current subtraction, flat field calibration, and bad pixels correction. Aperture photometry was performed on both targets, with the curve of growth used to select an aperture which minimized the error on the measured flux. The measured brightness of 51 Eri A is presented in Table \[tab:props\].
Converting the MKO $K_S$-band measurement into the 2MASS system using empirical relations[^2] yields $K_{S, \text{2MASS}} = 4.551\pm0.032$ mag, which is within 1-$\sigma$ of the published 2MASS photometry. Furthermore, the $J-H$ and $H-K$ colors estimated from the revised photometry are $0.128\pm0.037$ mag and $0.016\pm0.039$ mag which are consistent with the empirical expectations.
![ \[fig:oldvnew\] A comparison of the $JH$ spectra of 51 Eri b using the literature 2MASS values against the new photometry measured in this study. The updated photometry increases the planet flux by $\sim$10% in $J$ and $\sim$15% in the $H$-band. The updated stellar photometry is used in the remainder of this study. However the final stellar spectrum used to correct the planet spectrum does not depend on individual filter photometry, as in @Macintosh:2015 and shown in this plot, but is generated by modeling the full stellar SED prior to converting the planet spectra from contrast to flux units. ](Fig2.pdf){width="\columnwidth"}
The published 51 Eri b spectrum in @Macintosh:2015 was calibrated using the Pickles stellar models [@Pickles:1998] to estimate the spectrum of the primary, where each band was scaled using the published 2MASS photometry. In Figure \[fig:oldvnew\] we present a comparison between the published spectrum and one scaled using the new MKO photometry, using the same stellar models. The revised photometry scales the planet spectrum higher by $\sim$10% in the $J$-band and $\sim$15% in the H-band, which is significant given the high SNR of the $H$-band data.
Fitting the Spectral Energy Distribution of 51 Eri A
-----------------------------------------------------
To mitigate the effects of incorrect photometry, rather than scale the spectrum in pieces using the relevant broadband photometry, we decided to fit the full SED of 51 Eri A using literature photometry and colors, including Geneva $U,B_1,B,B_2,V_1,V,G$ [@Rufener:1988], *Tycho2* $B_T,V_T$ / *Hipparcos* $H_P$ [@Hog:2000; @ESA:1997], MKO $JHK_S$ (this work), and *WISE* $W1,W2$ [@Cutri:2013] measurements. We made use of the Geneva color relations as constraints to the full SED fit since the published Geneva $V$ magnitude, which anchors the colors to estimate the remaining photometry, appears to be offset by $\sim$5% when compared to the *Tycho2* photometry. The *WISE* $W2$ photometry was corrected using the @Cotten:2016 relation for bright stars. We combine the photometry with model stellar atmospheres from the [<span style="font-variant:small-caps;">B</span>T-NextGen]{} grid[^3] [@Allard:2012fp], we estimated the stellar spectrum using a five parameter MCMC grid search. The best fit atmosphere was found with $T_\text{eff} = 7331\pm30$ K, $\log g = 3.95\pm0.04$, \[$M/H$\] = $-0.12\pm0.06$, and a stellar radius, $R = 1.45\pm0.02$ $R_\odot$ (assuming a parallax of $33.98\pm0.34$ mas; [@vanLeeuwen:2007]). No correction for extinction is performed as the extinction in the direction of 51 Eri is negligible ($A_V=0.00$; [@Guarinos:1992um]). These values are consistent with previous literature estimates [e.g. @Koleva:2012]. The final SED of 51 Eri A is shown in Figure \[fig:sed\_A\], which highlights the significantly discrepant 2MASS $JH$-band photometry that was used previously to calibrate the spectrum of 51 Eri b. We extracted MKO $K$, NIRC2 $L_P$ and $M_S$ photometry from the SED fit using the filter response functions presented in @Tokunaga:2002, see Table \[tab:props\].
![ \[fig:sed\_A\] (top panel): Photometry of 51 Eri A from the literature, and from the results presented in this study (filled symbols). One hundred models were randomly selected from the MCMC search, and are plotted (translucent black curves). For each model, the synthetic magnitude was calculated for each filter. The median value for each filter is shown as an open square. The 2MASS photometry points are plotted to illustrate the offset relative to the new MKO measurements, and are not included in the fit. For the plotted Geneva photometry, we computed the Geneva $V$-band photometry using the best fit spectrum and then used the color relations to calculate the photometry in the remaining filters. (bottom panel): The fractional residuals relative to the median model.](Fig3.pdf){width="\columnwidth"}
### Confirming the stellar $L_P$ photometry
51 Eri b emits a substantial amount of flux in the mid-IR and $L_P$ photometry in @Macintosh:2015 was used to constrain the effective temperature of the planet. There exists no $L_P$ flux measurement for the star and thus they used the $W1$ magnitude reported in the AllWISE catalog ($W1 = 4.543\pm0.210$; [@Cutri:2013]), and assumed a color of $W1 - L_P=0$ based on the F0IV spectral type of 51 Eri [@Abt:1995eo]. The $L_P$ photometry we estimated via the SED fits for 51 Eri is $L_P$ = $4.604\pm0.014$ mag, which is consistent with the value reported in @Macintosh:2015 (4.52$\pm$0.21 mag) but with significantly smaller uncertainties.
As a final check for consistency, the 2MASS $K_S$ magnitude of 51 Eri ($K_\text{S, 2MASS} = 4.537\pm0.024$) was used instead as a starting point. The $K_S-L_P$ color for early F-type dwarfs and subgiants was estimated by folding model stellar spectra ($7200 \le T_\text{eff}/\text{K} \le 7400$, $4.0 \le \log g \le 4.5$, $[M/H]=0$) from the <span style="font-variant:small-caps;">BT-Settl</span> model grid through the relative spectral response of the 2MASS $K_S$ [@Cohen:2003gg] and NIRC2 $L_P$ filters. Over this range of temperatures and surface gravities, the color was calculated as $K_S-L_P = -0.001\pm0.001$. In order to realistically assess the uncertainties on this color, the near to thermal-IR spectra of F-type dwarfs and subgiants within the IRTF library [@Rayner:2009ki] were processed in the same fashion, resulting in a $K_S-L_P = 0.014\pm0.055$. A color of $K_S-L_P = -0.001\pm0.055$ was adopted based on the color calculated from the model grid, and the uncertainty calculated from the empirical IRTF spectra. This color, combined with the $K_{S, \text{2MASS}}$ magnitude of 51 Eri, gives an $L_P$ apparent magnitude of $4.538\pm0.060$. Each estimate for the stellar $L_P$ magnitude are within 1-$\sigma$ of each other, and thus we adopt the value derived from the SED fit i.e. $L_P$ = $4.604\pm0.014$ mag.
51 Eri b Spectral Energy Distribution
-------------------------------------
![ \[fig:sed\_b\] Final spectral energy distribution of the directly imaged exoplanet 51 Eri b. The new $K1$ and $K2$ GPI spectra along with the updated $L_P$ and new $M_S$ photometry are shown with red squares. The GPI $J$ and $H$ spectra, updated to account for the revised stellar flux, from the discovery paper [@Macintosh:2015] are plotted with blue circles. The filter extent is shown with the horizontal line over each band. To reduce crowding in the spectra, the errors for one out of every two data points are plotted.](Fig4.pdf){width="\linewidth"}
We present the final spectral energy distribution of the planet 51 Eri b in Figure \[fig:sed\_b\] and use it to analyze the system properties in the following sections. Using the stellar SED estimated earlier, we have updated the $J$ and $H$ spectra that were published in @Macintosh:2015. In Table \[tab:props\], we present the properties of the system, including updated MKO $JHK$ and NIRC2 $L_{P}M_{S}$ photometry for both the star and the planet. A future study will refine the orbital solution presented in @DeRosa:2015.
[lll]{} Distance (pc) &\
Age (Myr) &\
Spectral type & F0IV & T6.5$\pm$1.5\
$\log (L/L_\odot$) & $0.85^{+0.06}_{-0.07}$ & ${-5.83^{+0.15}_{-0.12}}$ to ${-5.93^{+0.19}_{-0.14}}$\
$T_\text{eff}$ & 7331$\pm30$ K & 605–737 K\
$\log g$ & 3.95$\pm$0.04 & 3.5–4.0\
$J_\text{MKO}$ & 4.690$\pm$0.020 & 19.04$\pm$0.40\
$H_\text{MKO}$ & 4.562$\pm$0.031 & 18.99$\pm$0.21\
$K_{S, MKO}$ & 4.546$\pm$0.024 & 18.49$\pm$0.19\
$K_\text{MKO}$ & 4.600$\pm$0.024 & 18.67$\pm$0.19\
$L_P$ & 4.604$\pm$0.014 & 16.20$\pm$0.11\
$M_S$ & 4.602$\pm$0.014 & 16.1$\pm$0.5\
Analysis {#sec:ana}
========
{width="\linewidth"}
Comparison against field brown dwarfs {#ssec:comp}
-------------------------------------
![Comparison of L5 to T9 field (gray circles) and young (yellow stars) brown dwarf $JHK$ spectra to 51 Eri b using the reduced $\chi^2$. The standard brown dwarf for each spectral bin is plotted with a red cross [@Kirkpatrick:2010; @Burgasser:2006a; @Cushing:2011]. The dashed and dotted vertical lines give the best fitting spectral type, and corresponding uncertainty. (Top) Each spectral band of the comparison was allowed to float to find the lowest chi-square while fitting the planet spectrum. (Bottom) The spectrum was allowed to float up/down in flux, but was penalized by the spot ratio uncertainty in each respective band. \[fig:fits\_all\] ](Fig6){width="\columnwidth"}
We plot a series of color magnitude diagrams (CMD) for ultracool objects in Figure \[fig:CMD\], and compare the photometry of field M, L, and T dwarfs and young brown dwarfs and imaged companions to that of 51 Eri b (red star). The colors of 51 Eri b seems to match the phase space of the late-T dwarfs. To classify the spectral type of 51 Eri b, we do a chi-square comparison of the GPI $JHK1K2$ spectrum of 51 Eri b to a library of brown dwarf spectra compiled from the IRTF [@Cushing:2005], SpeX [@Burgasser:2014], and Montreal [e.g. @Gagne:2015a; @Robert:2016] Spectral Libraries. Only a small sub-sample of the brown dwarfs have corresponding mid-IR photometry and thus we choose to restrict our comparison to the near-IR. The spectra within the library were convolved with a Gaussian kernel to match the spectral resolution of GPI.
To compute the chi-square between the spectrum of 51 Eri b and the objects within the library, we use two different equations. The first method permits each individual filter spectrum to vary freely (unrestricted fit). In the unrestricted fit, we compute the $\chi^2$ statistic for the $j^\text{th}$ object within the library as
$$\chi^2_j = \sum_{i=1}^4 \left({\bm S}_i - \alpha_{i,j}{\bm F}_{i,j}\right)^T{\bm C}_i^{-1}\left({\bm S}_i - \alpha_{i,j}{\bm F}_{i,j}\right),
\label{chi-square_unrestricted}$$
where ${\bm S}_i$ is the spectrum of the planet, ${\bm C}_i$ is the covariance matrix calculated in Section \[app:cov2\], and ${\bm F}_{i,j}$ is the spectrum of the $j^\text{th}$ comparison brown dwarf, all for the $i^\text{th}$ filter. For each object, the scale factor $\alpha_{i, j}$ that minimizes $\chi^2$ is found using a downhill simplex minimization algorithm. In this method the scale factor for each object, $\alpha_{i, j}$, is allowed to vary between the four filters ($JHK1K2$). This is equivalent to allowing the near-IR colors to vary freely up and down in order to better fit the object [e.g. @Burningham:2011].
In the second method the individual filter spectra are still allowed to vary, only within the satellite spot brightness ratio uncertainty (restricted fit), thereby restricting the scale factor for each filter. For the restricted fit the scale factor is split into two components. The first, $\alpha_j$, is independent of filter, and accounts for the bulk of the difference in flux between 51 Eri b and the comparison object due to differing distances and radii. The second, $\beta_{i,j}$, is a filter-dependent factor that accounts for uncertainties in the satellite spot ratios given in @Maire:2014. Equation \[chi-square\_unrestricted\] is modified to include an additional cost term restricting the possible values of $\beta_{i,j}$,
$$\begin{gathered}
\chi^2_j = \sum_{i=1}^4 \left[\left({\bm S}_i - \alpha_j\beta_{i,j}{\bm F}_{i,j}\right)^T{\bm C}_i^{-1}\left({\bm S}_i - \alpha_j\beta_{i,j}{\bm F}_{i,j}\right) \right.\\ + \left. N_i \left(\frac{\beta_{i,j}-1}{\sigma_{i}}\right)^2\right]
\label{chi-square}\end{gathered}$$
where $N_i$ is the number of spectral channels in the 51 Eri b spectrum for the $i^\text{th}$ filter, and $\sigma_i$ is the uncertainty on the satellite spot flux ratio given in @Maire:2014 for the same filter. The second term in Equation \[chi-square\] penalizes values of the scale factor, $\beta_{i,j}$, that are very different from the satellite spot uncertainty and thus increases the chi-square for objects significantly different from 51 Eri b.
The spectral type of 51 Eri b was estimated for both fits from the $\chi^2$ of the L5–T9 near-IR spectral standards [@Kirkpatrick:2010; @Burgasser:2006a; @Cushing:2011]. To compute the weighted mean and standard deviation of 51 Eri b, we converted the spectral type to a numerical value for the standard brown dwarfs, i.e. L5 = 75, T5 = 85. Each numerical spectral type when compared to 51 Eri b, is weighted according to the ratio of its $\chi^2$ to the minimum $\chi^2$ for all standards (e.g., [@Burgasser:2010]), and the lowest value was adopted as the spectral type of 51 Eri b. A systematic uncertainty of one half subtype was assumed for the standards. We find that the two estimates are consistent with one another i.e. T$6.3\pm1.3$ and $6.1\pm1.4$ for unrestricted and restricted fits, see Figure \[fig:fits\_all\]. We adopt a spectral type for 51 Eri b of T$6.5\pm1.5$ from the unrestricted fit, rounded to the nearest half subtype.
![ \[fig:fit\_mid\_Ts\] Comparing the spectra, using the restricted fit, of the best fitting T4.5 to T7.5 field brown dwarfs to 51 Eri b. The spectra shown in this figure are a subset of the data plotted in Figure \[fig:fits\_all\]. The brown dwarf spectral fits plotted here use the restricted chi-square equations presented in Equation \[chi-square\]. The T4.5 and T5.0 spectra are from @Looper:2007, the T5.5 is from @Burgasser:2008, the T6.0 is from @Burgasser:2004, the T6.5 and T7.5 are from @Burgasser:2006b, and the T7.0 is from @Dupuy:2012.](Fig7.pdf){width="\columnwidth"}
The best-fit object for both the unrestricted and restricted fits was G 204-39 B (SDSS J175805.46+463311.9; $\chi^2_{\nu} = 1.033$ and $1.209$), a T6.5 brown dwarf common proper motion companion to the nearby M3 star G 204-39 A [@Faherty:2010]. G 204-39 B has marginally low surface gravity based on photometric ($\log g \approx 4.5$; [@Knapp:2004]) and spectroscopic measurements ($\log g = 4.7$–$4.9$; [@Burgasser:2006b]), indicative of it being younger than the field population. While the binary system is not thought to be a member of any known young moving group [@Gagne:2014], the stellar primary can be used to provide a constraint on the age of the system. Combining the X-ray and chromospheric activity indicators for the M dwarf primary, and a comparison of the luminosity of the secondary with evolutionary models, @Faherty:2010 adopt an age of 0.5–1.5Gyr for the system. 51 Eri b is redder than the spectrum of G 204-39 B (Figure \[fig:fit\_mid\_Ts\]), especially in terms of the $H-K$ color, which is a photometric diagnostic of low surface gravity among T-dwarfs [e.g., @Knapp:2004]. This is consistent with the younger age of 51 Eri b, and the most likely cause for this is that it has lower surface gravity than that of G 204-39 B.
Additional good matches to the 51 Eri b spectrum include 2MASS J22282889–4310262 (2M 2228–43, $\chi^2_{\nu} = 1.07$ and $1.26$ for the two fits) and 2MASS J10073369–4555147 (2M 1007–45, $\chi^2_{\nu} = 1.07$ and $1.33$). 2M 2228–43 a well-studied T6 brown dwarf that exhibits spectrophotometric variability in multiple wavelengths indicative of patchy clouds in the photosphere [@Buenzli:2012; @Yang:2016]. 2M 1007–45 is a T5 brown dwarf at a distance of $17\pm2$ pc [@Smart:2013]. It was identified by @Looper:2007 as a low surface gravity object based on its H$_2$O$-J$ vs $K/H$ spectral ratios defined in @Burgasser:2006b; comparisons against solar-metallicity models imply an age of between 200 and 400 Myrs [@Looper:2007].
The best fit object for each spectral type between spectral types T4.5 and T7.5 using the restricted fit are plotted in Figure \[fig:fit\_mid\_Ts\]. While the quality of the fits were generally good, none of the objects were able to provide a good match across all of the bands simultaneously, being too luminous in either the $J$ or $K$-bands. Differences in surface gravity, effective temperature, and/or metallicity could be the cause [e.g., @Knapp:2004]. The poor fit to the color of 51 Eri b is especially apparent in the CMDs plotted in Figure \[fig:CMD\], with 51 Eri b having unusually red near-IR colors relative to similar spectral type objects.
Comparison against young brown dwarfs
-------------------------------------
![ \[fig:fits\_yng\_Ts\] Comparing the spectra of known young T-dwarfs to that of 51 Eri b. Similar to the field sequence, the fits presented here were computed using the restricted chi-square. From top to bottom, the four spectra were sourced from @Luhman:2007, @Naud:2014, @Burgasser:2006b, and @Delorme:2012. ](Fig8.pdf){width="\columnwidth"}
Searches for young companions and moving group objects have resulted in detections of several tens to hundred of million year old L-type brown dwarf and planetary mass companions as well as the identification of L-dwarf sub-classes based on youth [e.g. @Allers:2013; @Filippazzo:2015; @Faherty:2016; @Liu:2016]. In comparison, there exist relatively few known (or suspected) young T-dwarf brown dwarfs. In Figure \[fig:fits\_yng\_Ts\], we plot the known young T-dwarfs and compare them in a similar manner to what was done above for field brown dwarfs. The chi-square for the fits is not much better than what is seen for the field dwarfs which is likely due to the absence of young T-dwarfs of similar spectral type to 51 Eri b.
The brown dwarf SDSS J1110+0116 with a spectral type of T5/T5.5 is the best fitting young comparison object. It has been identified as a *bona fide* member of the AB Doradus moving group and is thus young (110–130 Myr) and low mass (10–12M$_\text{Jup}$) [@Gagne:2015b]. The other young field object that closely matches the near-IR spectrum of 51 Eri b is the T7 peculiar brown dwarf, CFBDSIR J2149-0403 [@Delorme:2012]. CFBDSIR J2149-0403 was originally suggested to be a member of of the AB Doradus moving group, however @Delorme:2017 find that the parallax and kinematics of the free-floating object rule out its membership to any known young moving group. However, despite the lack of proof of youth, medium resolution spectroscopy examining the equivalent width of the KI doublet at 1.25$\mu$m suggests that the object has low surface gravity, and is most likely a young planetary mass object (2–13 M$\text{Jup}$). An alternative solution is that it is a higher mass, 2–40 M$\text{Jup}$, brown dwarf with high metallicity. CFBDSIR J2149-0403 shows stronger methane absorption features in the red end of the $H$-band spectrum as compared to 51 Eri b. However, it is worth pointing out that while both young objects, SDSS J1110+0116 and CFBDSIR J2149-0403, are reasonable matches across the $J$ and $H$ spectra of 51 Eri b, they appear to be under-luminous in the $K$-band. A likely reason for this is that 51 Eri b is much younger than both the comparison companions and thus has the lowest surface gravity amongst the three objects [@Burgasser:2006b].
A very red T6 or an L-T transition planet? {#ssec:LT}
------------------------------------------
![ \[fig:cmd\_wtracks\] The $J$ vs $J-H$ brown dwarf and imaged exoplanet color magnitude diagram reproduced from Figure \[fig:CMD\]. The photometry for 51 Eri b is shown with the red star. Also plotted on the CMD are the evolutionary tracks for 5 and 14 M$_\text{Jup}$ objects [@Marley:2012], with the solid red line and dashed black line respectively. The models assume a simple gravity dependence for the initiation of the transition. A few ages for the 5 M$_\text{Jup}$ track have been over plotted. The L-T transition for the 5 M$_\text{Jup}$ planet starts at approximately 900 K and 20 Myr, but for a lower mass planet such as 51 Eri b will occur at younger ages. ](Fig9.pdf){width="\columnwidth"}
Based on the position of 51 Eri b in Figure \[fig:CMD\], it appears that the trend of planetary mass objects having redder colors compared to the field, seen in young L-type brown dwarfs and planetary mass companions [@Faherty:2016; @Liu:2016], possibly continues for the T-type companions. Note that the $K-L_P$ CMD shows little reddening, which is natural if clouds are causing the effect. The effect of clouds is negligible in the $K$ and $L_P$ bands. Across both the near and mid-IR CMDs, 51 Eri b is one of the reddest T type objects and within its spectral classification it has the reddest colors. This trend in the 51 Eri b colors was originally noted in @Macintosh:2015 where they compared the $L_P$ vs $H-L_P$ color for the planet and noted that it was clearly redder than the field. Rather than simply being redder than the field T-dwarfs due to the presence of clouds, we present a second possible interpretation for the red colors of 51 Eri b, of the planet still undergoing the process of transitioning from L-type to T-type. This hypothesis assumes that the evolutionary track followed is gravity dependant with examples for higher mass objects shown in Figure \[fig:cmd\_wtracks\]. In this scenario, 51 Eri b transitions at fainter magnitudes than that seen for field L-T transition brown dwarfs and it has not yet completed its evolutionary transition to reach the blue colors typical of field, mid-T dwarfs.
In Figure \[fig:cmd\_wtracks\], we re-plot the $J$ vs $J-H$ panel from the series of CMDs shown in Figure \[fig:CMD\]. In addition to the photometry of 51 Eri b and the field and young brown dwarfs we also over-plot two low mass, 5 and 14 M$_\text{Jup}$, evolutionary model tracks (assuming hot-start conditions) from @Saumon:2008 [@Marley:2012]. If the L-T transition is gravity dependent, as multiple lines of evidence now suggest [@Leggett:2008; @Dupuy:2009; @Stephens:2009], then lower mass objects may turn blue at fainter absolute magnitudes than field objects. In Figure \[fig:cmd\_wtracks\], we show a simple model in which the L to T transition begins at 900 K at $\log g = 4$ (solid red line) instead of 1200 K at $\log g = 5.3$ (dashed black line). In the case of a $5 \text{M}_\text{Jup}$ planet the L to T transition begins and ends about 1 magnitude fainter in $J$ band than observed for the field population. Furthermore the congruence of the spectrum of SDSS J1110+0116 with 51 Eri b (Figure \[fig:fits\_yng\_Ts\]) is interesting as SDSS J1110+0116 lies just short of the blue end of the field L to T transition, although it does so at an absolute magnitude just slightly fainter than the field transition magnitude. While these simple models explain the fainter absolute magnitudes of the transition, their colors are too blue and appear to miss the younger brown dwarf and free-floating planets. Similarly the models are too blue to match the T dwarf sequence. Clearly more sophisticated modeling of evolution through the L to T transition, accounting for inhomogeneous cloud cover and a gravity dependent transition mechanism as well as a range of initial conditions is required. Testing this hypothesis is difficult and would require knowledge of the true mass of the companion as well as the formation mechanism. If this hypothesis is true, then the only objects that are brighter on the CMD should be higher mass objects. There should not be any lower mass objects above and to the left of 51 Eri b on the $J$ vs $J-H$ CMD shown in Figure \[fig:cmd\_wtracks\].
[cccccc]{} Iron/Silicate Cloud Grid & 600–1000 & 3.25 & 0.0 & 2 & 0–75\
Sulfide/Salt Cloud Grid & 450–900 & 3.5–5.0 & 0.0, 0.5, 1.0 & 1, 2, 3, 5 & –\
Cloudless Grid & 450–900 & 3.5–5.0 & 0.0, 0.5, 1.0 & no cloud & –\
Modeling the atmosphere of 51 Eri
==================================
For the purpose of modeling the complete SED of 51 Eri b we made use of two updated atmospheric model grids from the same group, focusing on different parameter space (see Table \[tab:model\_params\]). The first grid, described in @Marley:1996 [@Marley:2002; @Marley:2010] focused on the higher effective temperature atmospheres (L-dwarfs) and includes iron and silicates clouds in the atmosphere. The second grid, described in @Morley:2012 [@Morley:2014] and @Skemer:2016, is designed for lower effective temperatures (T and Y dwarfs) and include salt and sulfide clouds in the atmosphere, which are expected to condense in the atmospheres of mid to late-T dwarfs.
The methodology used to fit the models to the data is the same for both model grids. To fit the models to the data, we bin the model spectra to match the spectral resolution of the GPIES spectra across each of the $JHK1K2$ filters. For the photometry we integrated the model flux through the Keck/NIRC2 $L_P$ and $M_S$ filter profiles respectively. The estimation of the best fitting model is done by computing the chi-square value for each model in the grid compared to the data using Equation \[chi-square\]. We made use of the covariance matrices estimated for the four spectral channels described in the appendix and also included the variance for each of the two photometric data points to compute the chi-square statistic. Note that we use the restricted fit equation in the computation of the best fitting model. This equation permits each of individual filters to scale within the 1-$\sigma$ error of the satellite spot ratios. We also did the fitting without the scaling factor and found that the results are similar.
As stated earlier in section \[sec:covariance\], the use of the covariance affects the model fitting where the peak of the posterior distribution occurs at slightly cooler effective temperatures, consistent within the errors. Due to the high spectral correlation in the $J$-band (see Figure \[fig:psi\]), when using the covariance the best fitting models are not models that pass through the data but rather models that have lower flux in the $J$-band than the data. We present the specific modeling details in the following text. \[model\]
Iron and Silicates Cloud Models
-------------------------------
![ \[fig:MM\_modelfits\] Spectral Energy Distribution of 51 Eri b with the best fitting iron and silicates cloudy model. ](Fig10.pdf){width="\columnwidth"}
{width="12cm"}
In sec. \[ssec:LT\], we suggested that 51 Eri b, rather than having completely evolved to T-type, could be transitioning from L-to-T. In this scenario the cloud composition of the planetary atmosphere might still be influenced by the deep iron and silicates condensate grains and patchy cloud atmosphere. Therefore, we compared the planet SED to a grid of models with a fixed low surface gravity and solar metallicity, where the key variable is cloud hole fraction and the unique aspect of this grid is the presence of iron/silicate clouds in an atmosphere with clear indications of methane absorption. The clouds are modelled using the prescription presented in @Ackerman:2001, where cloud thickness is parameterized via an efficiency factor (f$_\text{sed}$). Where small values of f$_\text{sed}$ indicate atmospheres with thick clouds while large values of f$_\text{sed}$ are for atmospheres with large particles that rain out of the atmosphere leaving optically thinner clouds. As mentioned early the primary condensate species in this grid are iron, silicate, and corundum clouds, molecules that are expected to dominate clouds in L-dwarfs [@Saumon:2008; @Stephens:2009]. At the L-T transition clouds are expected to be patchy, thus for each T$_\text{eff}$, the models went from fully cloudy i.e. f$_\text{sed}$ = 2 and 0% holes to an atmosphere with f$_\text{sed}$ = 2 and 75% holes (patchy clouds). The methodology used to calculated the flux emitted from the patchy cloud atmosphere include both cloud and cloud-free regions simultaneously in the atmosphere using a single, global temperature-pressure profile and are not created via a linear combination of two models as is sometimes done in the literature @Marley:2010. The iron and silicates cloud grid models use solar metallicity [@Lodders:2003]. The opacity database used for the absorbers are described in @Freedman:2008, including updated molecular line lists for ammonia and methane [@Yurchenko:2011; @Yurchenko:2014]. The models span effective temperatures from 600K to 1000K for solar metallicity (\[M/H\] = 0.0) and low surface gravity ($\log g$ = 3.25, 3.50) (see Table \[tab:model\_params\]).
![ \[fig:radpos\] The figure shows the effect of applying a Gaussian radius prior when modeling with the iron/silicates grid. The prior shown by the green line is centered on the radius given by evolutionary models i.e. 1.29 R$_\text{Jup}$ [@Marley:2007; @Fortney:2008]. Also plotted are the likelihood (black) and posterior distribution (red). ](Fig12.pdf){width="\columnwidth"}
Presented in Figure \[fig:MM\_modelfits\] is the best fitting model to the SED of 51 Eri b. Stated in the figure are the model parameters along with the radius of the planet required to scale the model spectrum to match the planet SED. This scaling factor is required since the model spectra are typically computed to be the emission at the photosphere or at 10pc from the object. One of the free parameters in most model fitting codes is the term R$^2$/d$^2$ to scale the model flux to match the SED, where R is the radius of the planet and d is the distance to the object. For 51 Eri, the distance is known to better than 2% (see Table \[tab:props\]) and thus we only fit the radius term. Shown in Figure \[fig:MM\_posterior\] is the posterior distribution for the radius where we find that the best fitting radii are significantly smaller than that predicted by evolutionary models, e.g. 1.33–1.14 R$_\text{Jup}$ for a 2–10 M$_\text{Jup}$ hot/cold start planet at the age of 51 Eri [@Marley:2007; @Fortney:2008]. This discrepancy has been noted previously as well for the HR8799 planets [@Marois:2008; @Bowler:2010; @Barman:2011; @Currie:2011; @Marley:2012], $\beta$ Pic b [@Morzinski:2015] and for 51 Eri b itself in the discovery paper [@Macintosh:2015]. In an attempt to circumvent this issue, while modeling the SED we adopted a Bayesian prior probability density function for the radius in the form of a Gaussian centered on the expected radius from evolutionary models (green line in Figure \[fig:radpos\]), with the width chosen to include the radius of Jupiter. Without the prior (i.e. using a uniform prior), the median radius is 0.68 R$_\text{Jup}$ and T$_\text{eff} \sim 740$K, with the prior the median radius value is forced closer to the predictions of evolutionary models (red line in Figure \[fig:radpos\]) at 0.98 R$_\text{Jup}$, and T$_\text{eff}$ $\sim$ 690K, biasing the luminosity of the planet to larger values. When fitting the SED, the term that is conserved is the luminosity rather than the effective temperature or the radius. Adopting the evolutionary radius and marginalizing over the uncertainty in radius raises the luminosity ($\log L/L_\odot$) from -5.83 to -5.65. Since observational constraints on the radius for young planets are unavailable, we chose to use an uninformative prior.
{width="\linewidth"}
Plotted in Figure \[fig:MM\_posterior\] are the normalized posterior distributions for each of the model parameters varied in the model fit, along with the covariances to show how each of the parameters are affected. Since the grid only had a few models with $\log g$ = 3.5, with the majority being 3.25, we marginalized over the the surface gravity. The irregular shape of the effective temperature posterior is caused by the missing models in the grid. The median effective temperature, 737 K, estimated from the grid falls right in between the range of best fitting temperatures from the models in the @Macintosh:2015 paper (700–750K). However, based on the shape of the posterior and the covariances, the peak of the effective temperature distribution extends to cooler temperatures. Since the L to T transition has been suggested to arise from holes or low opacity patches appearing in an initially more uniform cloud deck [@Ackerman:2001; @Burgasser:2002; @Marley:2010], our finding here that partly cloudy models best fit the 51 Eri b spectrum is consistent with this interpretation. In general, however the models struggled to fit the entire planet SED, typically being able to fit either the near or mid IR portions of the SED. The inability to fit mid-IR photometry suggests that chemical equilibrium models are not appropriate. Disequilibrium chemistry predicts less CH$_4$ in the atmosphere and could explain higher flux at 1.6$\mu$m and in the $L_P$ band. It would also introduce CO, accounting for lower flux in the $M_S$ band.
Sulfide and Salt Cloud Models
-----------------------------
In Section \[ssec:comp\], we showed that the best fitting spectral type of 51 Eri b is a mid-to-late T-dwarf. At the effective temperatures of mid to late T-dwarfs, Cr, MnS, Na$_2$S, ZnS, and KCl are expected to condense and form clouds high in the photosphere. The second grid we tested the planet SED against made use of a model grid which includes salt and sulfide clouds to test additional parameters such as the surface gravity and metallicity (which were varied, unlike the iron/silicates grid) and the properties of clouds typically associated with T-dwarfs. The grid was designed specifically for lower temperature objects [$450 \sim 900$ K @Morley:2012; @Morley:2014] and has been successfully to reproduce the SED of GJ 504 b [@Skemer:2016], a cool low mass companion with a similar spectral type (late-T) which is comparable to 51 Eri b [@Kuzuhara:2013]. Note that the use of this cloud grid does not preclude the possibility of the planet transitioning from L-to-T.
Also included as part of this grid are the clear atmosphere models from @Saumon:2008, the ranges for which are presented in Table \[tab:model\_params\]. The range of parameters varied are presented in Table \[tab:model\_params\], including temperatures, surface gravities, metallicities, and sedimentation factor (f$_\text{sed}$) ranging from cloudy (f$_\text{sed}$ = 1) to cloud free. The cloud model used in the sulfide/salt grid is the same as the one described above. In addition to the opacity updates mentioned above, opacity effects due to alkali metals (Li, Na, K) have been included using the results from @Allard:2005. Between effective temperatures of 450–775 K, the grid is complete with models available for every step of the varied parameters. For effective temperatures between 800–900K, the temperature steps switch from increments of 25K to 50K and there are no models with f$_\text{sed}$ values of 1 and 2. This grid does not include opacity effect due to iron and silicates condensates. A future series of paper describing an extended atmosphere model grid will describe the updates, however the present grid extends the models to greater than solar metallicites.
{width="\linewidth"}
In Figure \[fig:CM\_modelfits\], we present the four best fitting model atmospheres for 51 Eri b. Presented in each panel are the atmosphere with the lowest reduced chi-square in one of four cases, namely, solar and cloudless (top-left), solar and cloudy (top-right), non-solar and cloudless (bottom-left), non-solar and cloudy (bottom-right). Both cloudless model atmospheres are warmer and thus fit the near-IR spectrum of the planet while completely missing the $L_P$ photometry. The cloudy atmosphere model fits are cooler and do a much better job of fitting the overall SED of the 51 Eri b and the best fitting atmosphere for both solar and non-solar metallicity have very similar reduced chi-square values.
The normalized posterior distributions for the different parameters varied as part of the model fitting are shown in Figure \[fig:CM\_posterior\]. The best fitting T$_\text{eff}$ (${605}_{-66}^{+61}$ K) is much cooler in comparison to the iron/silicates grid, but the values are within 2-$\sigma$ of each other. We also note out that the median might not be the best estimate for the effective temperature PDF in the iron/silicates grid where the peak extended to cooler temperatures. For the surface gravity and metallicity posterior distributions, we present the median values and error bar assuming a Gaussian distribution, though they may not be Gaussian. The surface gravity PDF suggests that the planet has high surface gravity. However 51 Eri b is clearly a low mass companion indicating that the data does not constrain the gravity. A prior might help constrain the distribution, but there are currently no physically motivated priors available for the surface gravity of young planets. Similarly, the PDF for the metallicity is also unconstrained and higher resolution spectra in the $K$-band might help provide greater constraints on the metallicity of 51 Eri b [@Konopacky:2013].
A difference between the iron/silicate and salt/sulfide atmosphere grids is in the planet radius, where the best fit radii for the cloudy models and the median radius of the PDF for the salt/sulfide models are much closer to evolutionary model predictions. A possible explanation for this discrepancy is that fitting the lower effective temperatures while still matching the bolometric luminosity, requires a larger radius. If the iron/silicates models extended to lower temperatures, assuming the continued presence of these clouds at these colder temperatures, it is likely that the radius discrepancy would not be as apparent. The sedimentation factor was fixed (at f$_\text{sed}$=2) in the iron/silicates grids, but had varying hole fractions (h$_\text{frac}$). In the sulfide/salt grid, f$_\text{sed}$ was varied and the median value for the distribution is f$_\text{sed}$=2.48. If we equate the h$_\text{frac}$ from the iron/silicates model with the f$_\text{sed}$ as the physics controlling the emission of flux from the photosphere then for both model grids the best fitting models tend to be favoring the presence of clouds over cloud free atmospheres. Furthermore, in both cases the best fitting models were not the fully cloudy atmospheres, with the smallest h$_\text{frac}$/f$_\text{sed}$. While the cloud compositions in both models are different, fitting either grid require cloud opacity. This can be achieved in one of two ways: either make the deep iron/silicates clouds be very vertically extended (small f$_\text{sed}$) or introduce a new cloud layer in the form of the sulfide/salt clouds.
![ \[fig:JWST\] The ten best fitting cloudy (red) and cloudless (blue) atmospheres over the wavelength range of the *James Webb Space Telescope*. The median of the models is plotted with a thicker line. The models indicate the divergence between the model fits over the wavelength covered by *JWST*. ](Fig15.pdf){width="\columnwidth"}
The cloudy model atmosphere fits presented in Figure \[fig:CM\_modelfits\] match the $H$ through $K$ spectrum while being slightly under luminous in the $J$ and over luminous $M_S$ bands. Given the large photometric errors in the $M_S$ data, the model photometry lies within 2-$\sigma$ of the data. $JWST$ and other future low background mid-IR instruments will better constrain the 3–24 $\mu$m SED, a further test of current models. In Figure \[fig:JWST\], we show ten of the best fitting models assuming cloudy (sulfide/salt clouds) or cloudless atmospheres extended out to 20$\mu$m. It is clear from these models that observations with the coronagraph on Near InfraRed Camera (NIRCam), spanning the 3–5 $\mu$m wavelength will add significant constraints on the atmosphere of the planet. If the planet can be studied with the Mid Infrared Instrument (MIRI), it could be used to apply constraints on chemical disequilibrium in the atmosphere through observations NH$_3$ in the 10–11$\mu$m range.
Luminosity of the planet
------------------------
The two different grids used in this study have produced similar luminosity predictions for the planet despite the different cloud compositions. From the iron/silicates grid we infer a bolometric luminosity of $\log L/L_\odot$ = $-5.83^{+0.15}_{-0.12}$, and $\log L/L_\odot$ = $-5.93^{+0.19}_{-0.14}$ from the sulfide/salt model atmospheres. We compare these luminosity estimates to predictions of evolutionary models to infer the planet mass and discuss its initial formation conditions.
### Standard cold- and hot-start models
![ \[fig:Lbol\] Luminosity of imaged planetary mass companions as a function of age. For 51 Eri b, the red star is the inferred luminosity from the Iron/Silicates model grid while the green square is the inferred luminosity from the Sulfide/Salt model grid. Also plotted are evolutionary tracks assuming different starting conditions i.e. the “hot-start” (dotted lines) and “cold-start” (solid lines) models of @Marley:2007 [@Fortney:2008]. 51 Eri b is consistent with both hot- and cold-start formation models. A subset of known directly imaged companions are plotted in figure illustrating the difference between 51 Eri b and other imaged planets. Data for the companions, 2M1207 b, HR8799 bcde, HD 95086 b, GJ 504 b were taken from @Patience:2010 [@Rajan:2015; @Zurlo:2016; @DeRosa:2016; @Skemer:2016] respectively.](Fig16.pdf){width="\columnwidth"}
{width="\linewidth"} {width="\linewidth"}
In Figure \[fig:Lbol\] we compare the bolometric luminosity to evolutionary models for planets formed via the two extreme scenarios namely, hot-start and cold-start models [@Burrows:1997; @Marley:2007]. In the hot-start scenario, planets are formed with high initial-entropy and are very luminous at birth. This scenario is usually associated with rapid formation in the circumstellar disk through disk instabilities. Alternatively, in the cold-start scenario, which is often associated with current 1D models of the core-accretion mechanism, planets start with a solid core that accretes gas from the stellar disk. The accreting gas loses energy via a radiatively efficient accretion shock and form with low initial-entropy and thereby lower post-formation luminosity.
The other directly imaged companions plotted in Figure \[fig:Lbol\] can all be considered as having formed via the hot-start scenario. Despite the older age assessment for the companion in this study 26$\pm$3 Myr [@Nielsen:2016] compared to 20$\pm$6 Myr [@Macintosh:2015], the revised luminosity when compared to the system age places 51 Eri b in a location where either cold or hot initial conditions are possible. Based on the hot-start tracks, it would have an inferred mass between 1–2 M$_\text{Jup}$. However, for the cold-start case the planet mass could lie anywhere between 2–12 M$_\text{Jup}$, since the model luminosity is largely independent of mass at the age of 51 Eri b. Dynamical mass estimates for the planet could help clarify the formation mechanism especially if the planet mass $>2$M$_\text{Jup}$.
### Warm-start models
@Spiegel:2012 proposed a complete family of solutions existing between the hot- and cold-start extreme cases. Warm-start models[^4] explore a wide range of initial entropies aimed at covering the possible range of initial parameters that govern the formation of planets. In Figure \[fig:SvM\], we compare the inferred bolometric luminosity and the planet SED to models from @Spiegel:2012. The @Spiegel:2012 models are evolutionary tracks calculated assuming different initial entropies for the planet, between 8 and 13 k$_\text{B}$/baryon, where k$_\text{B}$ is Boltzmann’s constant, with steps of 0.25 k$_\text{B}$/baryon and masses between 1 and 15 M$_\text{Jup}$ with steps of 1 M$_\text{Jup}$. Four different model atmospheres are considered in combination with the evolutionary model: cloud-free and solar metallicity to fully cloudy with 3$\times$ solar metallicity [@Burrows:2011]. The bolometric luminosity of each point in the grid for each of the four atmosphere scenario was computed by integrating the SED over the wavelength range. Because of the sparse sampling of the grid, we linearly interpolate the evolutionary tracks with steps of 0.06 k$_\text{B}$/baryon and 0.2 M$_\text{Jup}$.
In the top row of Figure \[fig:SvM\], we plot the probabilities for each grid point measured by comparing the average of the inferred bolometric luminosities from the SED fit ($\log L/L_\odot$ = $-5.87\pm0.15$) to the predictions of the @Spiegel:2012 models with the four atmosphere conditions. For the bottom row in Figure \[fig:SvM\], the surface is calculated by fitting the planet SED to the @Spiegel:2012 model atmosphere grid, using Equation \[chi-square\]. For both comparisons, luminosity and SED, we chose the age of the evolutionary grid best matching the age of 51 Eri (25 Myr), to minimize the number of interpolations, and only varied the mass of the planet and initial entropy for the models.
@Mordasini:2013 find that the luminosity of a planet that underwent accretion through a super-critical shock (the standard cold-start core accretion hypothesis), is highly dependent on the mass of the core, M$_\text{core}^{2-3}$. Therefore, the continuum of warm-start models can also be explained by similar bulk mass planets with increasing core mass. These models suggest that the entropy of 51 Eri b can be explained via core-accretion, with a core mass ranging between 15 and 127 M$_{\oplus}$, which can reproduce the planet luminosity with various initial entropies.
The four panels generated by fitting the inferred luminosity (upper four panels) appear highly consistent and in agreement with the results from Figure \[fig:Lbol\]. The 1$\sigma$ contour encompasses the entire available entropy space, where for intermediate and high entropies the most likely mass for the planet is between 2 and 3 M$_\text{Jup}$ and for low initial entropy the most likely mass for the planet increases, making distinguishing between cold-, warm- and hot-start difficult.
When we compare the model spectra directly to the planet SED, the surface is qualitatively similar to that made with the luminosity but shifted to higher mass and with the 1$\sigma$ contours and best fit models favoring lower entropy. According to the @Mordasini:2013 models, the fits presented here would be consistent with a planet having core masses ranging from 15–127 M$_{\oplus}$.
Conversely to other directly imaged companions [see figures in @Marleau:2014], 51 Eri b is the only planet compatible with very low initial entropy and the cold-start case. Tighter constraints on the bolometric luminosity and/or higher signal to noise data will help to reduce the width of the two branches and independent mass constraints, from dynamical measurements, will enable to infer the initial entropy and possible formation route. Atmospheric retrievals and/or higher resolution spectra aimed at exploring and characterizing the planets chemical composition might also help understand whether the planet has higher C/O ratios compared to the star, since planetary C/O can be used to understand planet formation [@Oberg:2011; @Konopacky:2013].
Conclusion
==========
In this paper, we have presented the first spectrum of 51 Eridani b in the K-band obtained with the Gemini Planet Imager (K1 and K2 bands) as well as the first photometric measurement of the planet at $M_S$ obtained with the NIRC2 Narrow camera. We also obtained an additional $L_P$ photometric point that agrees very well with the $L_P$ measurement taken in the discovery paper [@Macintosh:2015]. In addition, we revised the stellar photometry by observing the star in the near IR and estimating its photometry in the mid IR through an SED fit. The new data are combined with the published $J$, and $H$ spectra and the $L_P$ photometry to present the spectral energy distribution spanning 1–5 $\mu$m for the planet.
As part of the data analysis, we calculated the covariance for each of the spectral datasets i.e. $J, H, K1$, and $K2$ using the formalism presented in @Greco:2016. The spectral covariance was used in all the chi-squared minimization performed as part of this study, in combination with the photometric variance. Using the covariance ensured that the photometric points were weighted in a suitable manner and resulted in cooler effective temperatures for the best fits.
We compared the planet photometry to field and young brown dwarfs by fitting their near-IR spectra to 51 Eri b to estimate a spectral type of T$6.5\pm1.5$. Due the relative paucity of known young T-dwarfs, our comparison of the planet spectrum to young T-dwarfs only included a handful of objects, and amongst the sample 51 Eri b appears to have the lowest surface gravity based on a comparison of their spectral shape and amplitude.
In a comparison of the near and mid IR photometry for the planet to the field and young brown dwarf population via a range of color magnitude diagrams we note that 51 Eri b is redder than brown dwarfs of similar spectral types. This was also noted in the discovery paper, and it was proposed that this might be due to presence of clouds, similar to young L-type planetary mass companions. In this study, we extended this idea to suggest that a possible reason for the presence of clouds (compared to the field), is that the planet is still transitioning from the L-type to the T-type. This would occur at a lower $J$ magnitude than field brown dwarfs due to its lower mass when including a gravity-dependent transition in the evolution [@Saumon:2008].
We also fit the planet SED with two different model atmosphere grids that varied in the composition of molecules that could condense in the atmosphere. The best fitting models in both cases, were those that contained large amount of condensates in the atmosphere as compared to cloud free atmospheres. Through the iron/silicates grid, we estimate that the planet has a patchy atmosphere with 10–25 % hole fraction in the surface cloud cover, which is consistent with the f$_\text{sed}$ values of 2–3 resulting from the sulfide/salt grid. The median effective temperature from the two grids is ${737}_{-46}^{+39}$ K and ${605}_{-66}^{+61}$ K for iron/silicates and sulfide/salt respectively. This value is slightly cooler, compared to @Macintosh:2015, where the best fit models had temperatures of 700K and 750K respectively. The surface gravity and metallicity both appear to be unconstrained by the data, but empirical fits to young T-dwarfs suggest that the planet has lower surface gravity.
The two atmosphere grids provide similar luminosity estimates which were compared to hot-, warm- and cold-start models. 51 Eri b appears to be one of the only directly imaged planet that is consistent with the cold-start scenario and a comparison of the planet SED to a range of initial entropy models indicates that cloudy atmospheres with low initial entropies provide the best fit to the planet SED.
Following the submission of this study for publication, a paper on 51 Eri b using spectrophotometry taken with the VLT/SPHERE was published by @Samland:2017. Their study includes new $YJH$ spectra as well as $K1K2$ photometry in addition to the $H$ spectrum and $L_P$ photometry from @Macintosh:2015. Their results are consistent in parts with ours, although we note that the SPHERE $J$ band spectrum is fainter than the GPI $J$ spectrum, while their $K1K2$ photometry are brighter than the GPI spectrum (and corresponding integrated GPI photometry). These differences could very well be caused by the application of different algorithms, where @Samland:2017 demonstrate that different algorithms can result in spectra with a range of flux values including ones that agree with the GPI $J$ spectrum. Future studies will need to analyze all the available datasets using a common pipeline for data processing and analysis to understand whether the differences arise from the algorithms or due to other causes.
With future space missions such as the *James Webb Space Telescope*, the 3–24 $\mu$m SED of this planet could be observed at higher SNR, providing tests of current atmospheric models. The best fitting atmosphere models further indicate that the planet might have a cloudy atmosphere with patchy clouds, making 51 Eri b a prime candidate for atmospheric variability studies that might be possible with future instrumentation. Further analysis of this data using methods such as atmosphere retrievals could permit an exploration of other planet parameters that were not considered in this study such as chemical composition of the atmosphere and the thermal structure.
The authors thank Gabriel Marleau for discussion on warm-start models. The Gemini Observatory is operated by the AURA under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciéncia, Tecnologia e Inovaçāo (Brazil), and Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina). Supported by NSF grants AST-1411868 (A.R, J.L.P., B.M.), AST-1518332 (R.J.D.R.), and DGE-1311230 (K.W.D.). F.M and E.N are supported by NASA Grant NNX14AJ80G. Supported by Fonds de Recherche du Québec (J.R, R.D, D.L). K.M.M and T.S.B are supported by the NASA Exoplanets Research Program (XRP) by cooperative agreement NNX16AD44G. G.V and J.K.W acknowledge JPL?s ESI program for GPI related funding. The results reported herein benefitted from collaborations and/or information exchange within NASA’s Nexus for Exoplanet System Science (NExSS) research coordination network sponsored by NASA’s Science Mission Directorate and the NExSS grant NNX15AD95G. Portions of this work were performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. This research has benefited from the SpeX Prism Library (and/or SpeX Prism Library Analysis Toolkit) maintained by Adam Burgasser at <http://www.browndwarfs.org/spexprism>, the IRTF Spectral Library maintained by Michael Cushing, and the Montreal Brown Dwarf and Exoplanet Spectral Library maintained by Jonathan Gagn[é]{}.
Derivation of Spectral Covariance {#app:cov2}
=================================
We follow the method described in @Greco:2016 to measure the inter-pixel correlation within the PSF-subtracted images, and convert these into a covariance matrix. For each image ($J$, $H$, $K1$, and $K2$), the correlation $\psi_{ij}$ between pixel values at wavelengths $\lambda_i$ and $\lambda_j$ within a 1.5 $\lambda/D$ annulus was estimated as $$\psi_{ij} = \frac{\langle I_i I_j\rangle}{\sqrt{\langle I^2_i \rangle \langle I^2_j \rangle}}$$ where $\langle I_i\rangle$ is the average intensity within the annulus at wavelength $\lambda_i$. This was repeated for all wavelength pairs, and at five different separations: 350, 454 (the separation of 51 Eri b), 550, 650, and 750 mas. To avoid biasing the measurement, 51 Eri b was masked in the 454 mas annulus.
![ \[fig:psisep\] Example of the correlation function at the various angular separations included in the fit for the $H$-band spectral cube. The different colors correspond to the angular separations, with the circles being the value of the correlation for all the wavelength pairs and the lines of the same color indicate the best fit to distribution calculated using Equation \[eqn:model\]. ](FigA1.pdf){width="12cm"}
![ \[fig:psi\] Presenting the correlation matrices calculated for each of the four $JHK1K2$ spectra. Going from $J$-band through $K2$ the correlation length can be seen to change as a function of speckle vs background noise. The spectra are highly correlated at $J$ with up to 5 channels showing high correlation values, down to $\sim$3 at $K2$ which is a consequence of the spectral re-sampling.](FigA2.pdf){width="10cm"}
The measurements of the correlation $\psi_{ij}$ at the eight different separations within the final image were used to fit the parametrized correlation model of @Greco:2016, $$\label{eqn:model}
\psi_{ij}\approx A_{\rho}\exp\left[-\frac{1}{2}\left(\frac{\rho}{\sigma_{\rho}}\frac{\lambda_i - \lambda_j}{\lambda_c}\right)^2\right] + A_{\lambda}\exp\left[-\frac{1}{2}\left(\frac{1}{\sigma_{\lambda}}\frac{\lambda_i - \lambda_j}{\lambda_c}\right)^2\right] + A_{\delta}\delta_{ij}$$ where the symbols are as in @Greco:2016. This model is based on the assumption that the correlation consists of three components. The first two terms model the contribution of the speckle noise and the correlation induced by the interpolation within the reduction process. The third models uncorrelated noise, such as read noise, which do not contribute to the off-diagonal terms of the correlation matrix. The amplitude of the first two terms ($A_{\rho}$, $A_{\lambda}$) were allowed to vary with separation, while the two correlation lengths ($\sigma_{\rho}$, $\sigma_{\lambda}$) were fixed. As the sum of the amplitudes must equal unity, $A_{\delta}$ was derived from the other amplitudes. Figure \[fig:psisep\] shows an example of the spectral correlation as a function of the angular separation for the $H$-band spectral cube, $\lambda_c$ is the central wavelength of the spectrum (1.65 $\mu$m for $H$). The colored lines in the plot are the best fits to Equation \[eqn:model\].
Due to the high dimensionality of the problem, we use a parallel-tempered Markov Chain Monte Carlo algorithm [@Foreman-Mackey:2013] to find the global minimum. The best fit parameters at the separation of 51 Eri b within the PSF-subtracted image at each band is given in Table \[tab:cov\_params\]. Using these parameters, the covariance matrix, $C$, was constructed for each band. The diagonal elements contained the square of the uncertainties of the spectrum of the planet, and the off-diagonal elements were calculated using $$\psi_{ij} \equiv \frac{C_{ij}}{\sqrt{C_{ii}C_{jj}}}$$
[cccccccc]{} $J$ & 0.43 & 0.43 & 0.14 & 0.44 & 0.05\
$H$ & 0.70 & 0.27 & 0.03 & 0.45 & 0.01\
$K1$ & 0.51 & 0.41 & 0.07 & 0.68 & 0.004\
$K2$ & 0.30 & 0.62 & 0.08 & 0.43 & 0.004\
The fitted parameters in Table \[tab:cov\_params\] demonstrate that the primary cause of correlation at the shorter wavelengths is speckle noise, with the correlation induced by interpolation becoming more significant in the $K1$ and $K2$ images. In each case the amplitude of the speckle noise term (A$_{\rho}$) is significantly higher than seen for HD 95086 b [@DeRosa:2016]. This can be attributed to the fact that 51 Eri A is approximately two magnitudes brighter at $K1$ (than HD 95086 A), leading to a significantly brighter speckle field. The typical correlation lengths in the PSF-subtracted image for each band are visualized in Figure \[fig:psi\], with the data being highly correlated at $J$ band at wavelengths separated by up to five spectral channels.
Abt, H. A., & Morrell, N. I. 1995, , 99, 135 Ackerman, A. S., & Marley, M. S. 2001, , 556, 872 Allard, N. F., Allard, F., & Kielkopf, J. F. 2005, , 440, 1195 Allard, F., Homeier, D., & Freytag, B. 2012, Philosophical Transactions of the Royal Society of London Series A, 370, 2765 Allers, K. N., & Liu, M. C. 2013, , 772, 79 Amara, A., & Quanz, S. P. 2012, , 427, 948 Barman, T. S., Macintosh, B., Konopacky, Q. M., & Marois, C. 2011, , 733, 65 Bell, C. P. M., Mamajek, E. E., & Naylor, T. 2015, , 454, 593 Beuzit, J.-L., Feldt, M., Dohlen, K., et al. 2008, , 7014, 701418 Binks, A. S., & Jeffries, R. D. 2014, , 438, L11 Bonnefoy, M., Marleau, G.-D., Galicher, R., et al. 2014, , 567, L9 Bonnefoy, M., Milli, J., M[é]{}nard, F., et al. 2017, , 597, L7 Bowler, B. P., Liu, M. C., Dupuy, T. J., & Cushing, M. C. 2010, , 723, 850 Bowler, B. P., Liu, M. C., Mawet, D., et al. 2017, , 153, 18 Buenzli, E., Apai, D., Morley, C. V., et al. 2012, , 760, L31 Buenzli, E., Apai, D., Radigan, J., Reid, I. N., & Flateau, D. 2014, , 782, 77 Burgasser, A. J., Marley, M. S., Ackerman, A. S., et al. 2002, , 571, L151 Burgasser, A. J., McElwain, M. W., Kirkpatrick, J. D., et al. 2004, , 127, 2856 Burgasser, A. J., Geballe, T. R., Leggett, S. K., Kirkpatrick, J. D., & Golimowski, D. A. 2006, , 637, 1067 Burgasser, A. J., Burrows, A., & Kirkpatrick, J. D. 2006, , 639, 1095 Burgasser, A. J., Liu, M. C., Ireland, M. J., Cruz, K. L., & Dupuy, T. J. 2008, , 681, 579-593 Burgasser, A. J., Cruz, K. L., Cushing, M., et al. 2010, , 710, 1142 Burgasser, A. J. 2014, Astronomical Society of India Conference Series, 11, Burgasser, A. J., Lopez, M. A., Mamajek, E. E., et al. 2016, , 820, 32 Burningham, B., Leggett, S. K., Homeier, D., et al. 2011, , 414, 3590 Burrows, A., Marley, M., Hubbard, W. B., et al. 1997, , 491, 856 Burrows, A., Heng, K., & Nampaisarn, T. 2011, , 736, 47 Carter, B. S. 1990, , 242, 1 Chauvin, G., Lagrange, A.-M., Dumas, C., et al. 2005, , 438, L25 Cohen, M., Wheaton, W. A., & Megeath, S. T. 2003, , 126, 1090 Cotten, T. H., & Song, I. 2016, , 225, 15 Currie, T., Burrows, A., Itoh, Y., et al. 2011, , 729, 128 Currie, T., Lisse, C. M., Kuchner, M., et al. 2015, , 807, L7 Cushing, M. C., Rayner, J. T., & Vacca, W. D. 2005, , 623, 1115 Cushing, M. C., Kirkpatrick, J. D., Gelino, C. R., et al. 2011, , 743, 50 Cutri, R. M., Skrutskie, M. F., van Dyk, S., et al. 2003, VizieR Online Data Catalog, 2246 Cutri, R. M., & et al. 2013, VizieR Online Data Catalog, 2328 De Rosa, R. J., Nielsen, E. L., Blunt, S. C., et al. 2015, , 814, L3 De Rosa, R. J., Rameau, J., Patience, J., et al. 2016, , 824, 121 Delorme, P., Gagn[é]{}, J., Malo, L., et al. 2012, , 548, A26 Delorme, P., Dupuy, T., Gagn[é]{}, J., et al. 2017, arXiv:1703.00843 Dupuy, T. J., Liu, M. C., & Ireland, M. J. 2009, , 699, 168 Dupuy, T. J., & Liu, M. C. 2012, , 201, 19 ESA 1997, ESA Special Publication, 1200, Faherty, J. K., Burgasser, A. J., West, A. A., et al. 2010, , 139, 176 Faherty, J. K., Riedel, A. R., Cruz, K. L., et al. 2016, , 225, 10 Feigelson, E. D., Lawson, W. A., Stark, M., Townsley, L., & Garmire, G. P. 2006, , 131, 1730 Filippazzo, J. C., Rice, E. L., Faherty, J., et al. 2015, , 810, 158 Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, , 125, 306 Fortney, J. J., Marley, M. S., Hubickyj, O., Bodenheimer, P., & Lissauer, J. J. 2005, Astronomische Nachrichten, 326, 925 Fortney, J. J., Marley, M. S., Saumon, D., & Lodders, K. 2008, , 683, 1104-1116 Freedman, R. S., Marley, M. S., & Lodders, K. 2008, , 174, 504-513 Gagn[é]{}, J., Lafreni[è]{}re, D., Doyon, R., Malo, L., & Artigau, [É]{}. 2014, , 783, 12 Gagn[é]{}, J., Faherty, J. K., Cruz, K. L., et al. 2015, , 219, 33 Gagn[é]{}, J., Burgasser, A. J., Faherty, J. K., et al. 2015, , 808, L20 Galicher, R., Marois, C., Macintosh, B., Barman, T., & Konopacky, Q. 2011, , 739, L41 Goldman, B., Marsat, S., Henning, T., Clemens, C., & Greiner, J. 2010, , 405, 1140 Greco, J. P., & Brandt, T. D. 2016, , 833, 134 Guarinos, J. 1992, European Southern Observatory Conference and Workshop Proceedings, 43, 301 H[ø]{}g, E., Fabricius, C., Makarov, V. V., et al. 2000, , 355, L27 Janson, M., Carson, J., Thalmann, C., et al. 2011, , 728, 85 Kalas, P. G., Rajan, A., Wang, J. J., et al. 2015, , 814, 32 Kasper, M., Apai, D., Janson, M., & Brandner, W. 2007, , 472, 321 Kasper, M., Apai, D., Wagner, K., & Robberto, M. 2015, , 812, L33 Kenyon, S. J., & Hartmann, L. 1995, , 101, 117 Kirkpatrick, J. D., Looper, D. L., Burgasser, A. J., et al. 2010, , 190, 100 Knapp, G. R., Leggett, S. K., Fan, X., et al. 2004, , 127, 3553 Koleva, M., & Vazdekis, A. 2012, , 538, A143 Konopacky, Q. M., Barman, T. S., Macintosh, B. A., & Marois, C. 2013, Science, 339, 1398 Konopacky, Q. M., Thomas, S. J., Macintosh, B. A., et al. 2014, Proc. SPIE, 9147, 84 Konopacky, Q. M., Rameau, J., Duch[ê]{}ne, G., et al. 2016, , 829, L4 Kraus, A. L., Ireland, M. J., Cieza, L. A., et al. 2014, , 781, 20 Kuzuhara, M., Tamura, M., Kudo, T., et al. 2013, , 774, 11 Lafreni[è]{}re, D., Marois, C., Doyon, R., Nadeau, D., & Artigau, [É]{}. 2007, , 660, 770 Leggett, S. K., Saumon, D., Marley, M. S., et al. 2007, , 655, 1079 Leggett, S. K., Saumon, D., Albert, L., et al. 2008, , 682, 1256-1263 Liu, M. C., Magnier, E. A., Deacon, N. R., et al. 2013, , 777, L20 Liu, M. C., Dupuy, T. J., & Allers, K. N. 2016, , 833, 96 Lodders, K. 2003, , 591, 1220 Looper, D. L., Kirkpatrick, J. D., & Burgasser, A. J. 2007, , 134, 1162 Luhman, K. L., Patten, B. M., Marengo, M., et al. 2007, , 654, 570 Macintosh, B., Graham, J. R., Ingraham, P., et al. 2014, Proceedings of the National Academy of Science, 111, 12661 Macintosh, B., Graham, J. R., Barman, T., et al. 2015, Science, 350, 64 Maire, J., Ingraham, P. J., De Rosa, R. J., et al. 2014, , 9147, 914785 Males, J. R., Close, L. M., Morzinski, K. M., et al. 2014, , 786, 32 Mamajek, E. E., & Bell, C. P. M. 2014, , 445, 2169 Marley, M. S., Saumon, D., Guillot, T., et al. 1996, Science, 272, 1919 Marley, M. S., Seager, S., Saumon, D., et al. 2002, , 568, 335 Marley, M. S., Fortney, J. J., Hubickyj, O., Bodenheimer, P., & Lissauer, J. J. 2007, , 655, 541 Marley, M. S., Saumon, D., & Goldblatt, C. 2010, , 723, L117 Marley, M. S., Saumon, D., Cushing, M., et al. 2012, , 754, 135 Marleau, G.-D., & Cumming, A. 2014, , 437, 1378 Marois, C., Lafreni[è]{}re, D., Doyon, R., Macintosh, B., & Nadeau, D. 2006, ApJ, 641, 556 Marois, C., Lafreni[è]{}re, D., Macintosh, B., & Doyon, R. 2006, , 647, 612 Marois, C., Macintosh, B., Barman, T., et al. 2008, Science, 322, 1348 Marois, C., Zuckerman, B., Konopacky, Q. M., Macintosh, B., & Barman, T. 2010, , 468, 1080 Marois, C., Macintosh, B., & V[é]{}ran, J.-P. 2010, , 7736, 77361J Milli, J., Hibon, P., Christiaens, V., et al. 2017, , 597, L2 Mordasini, C. 2013, , 558, A113 Morley, C. V., Fortney, J. J., Marley, M. S., et al. 2012, , 756, 172 Morley, C. V., Marley, M. S., Fortney, J. J., et al. 2014, , 787, 78 McCarthy, D. W., Burge, J. H., Angel, J. R. P., et al. 1998, , 3354, 750 McLean, I. S., & Sprayberry, D. 2003, , 4841, 1 Millar-Blanchaer, M. A., Wang, J. J., Kalas, P., et al. 2016, , 152, 128 Montet, B. T., Bowler, B. P., Shkolnik, E. L., et al. 2015, , 813, L11 Morzinski, K. M., Males, J. R., Skemer, A. J., et al. 2015, , 815, 108 Nakajima, T., Oppenheimer, B. R., Kulkarni, S. R., et al. 1995, , 378, 463 Naud, M.-E., Artigau, [É]{}., Malo, L., et al. 2014, , 787, 5 Nelder, J. A. Nelder and Mead, R. 1965, The Computer Journal, 7 (4): 308-313 Nielsen, E. L., De Rosa, R. J., Wang, J., et al. 2016, , 152, 175 berg, K. I., Murray-Clay, R., & Bergin, E. A. 2011, , 743, L16 Oppenheimer, B. R., Kulkarni, S. R., Matthews, K., & Nakajima, T. 1995, Science, 270, 1478 Patience, J., King, R. R., de Rosa, R. J., & Marois, C. 2010, , 517, A76 Perrin, M. D., Maire, J., Ingraham, P., et al. 2014, Proc. SPIE, 9147, 91473J Pickles, A. J. 1998, , 110, 863 Rameau, J., Chauvin, G., Lagrange, A.-M., et al. 2013, , 772, L15 Rajan, A., Barman, T., Soummer, R., et al. 2015, , 809, L33 Rayner, J. T., Cushing, M. C., & Vacca, W. D. 2009, , 185, 289 Rebull, L. M., Stapelfeldt, K. R., Werner, M. W., et al. 2008, , 681, 1484-1504 Riviere-Marichalar, P., Barrado, D., Montesinos, B., et al. 2014, , 565, A68 Robert, J., Gagn[é]{}, J., Artigau, [É]{}., et al. 2016, , 830, 144 Rufener, F., & Nicolet, B. 1988, , 206, 357 Samland, M., Molli[è]{}re, P., Bonnefoy, M., et al. 2017, arXiv:1704.02987 Saumon, D., & Marley, M. S. 2008, , 689, 1327-1344 Service, M., Lu, J. R., Campbell, R., et al. 2016, , 128, 095004 Sivaramakrishnan, A., & Oppenheimer, B. R. 2006, , 647, 620 Skemer, A. J., Morley, C. V., Zimmerman, N. T., et al. 2016, , 817, 166 Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, , 131, 1163 Smart, R. L., Tinney, C. G., Bucciarelli, B., et al. 2013, , 433, 2054 Soummer, R., Pueyo, L., & Larkin, J. 2012, , 755, L28 Spiegel, D. S., & Burrows, A. 2012, , 745, 174 Stephens, D. C., Leggett, S. K., Cushing, M. C., et al. 2009, , 702, 154 Tokunaga, A. T., Simons, D. A., & Vacca, W. D. 2002, , 114, van Leeuwen, F. 2007, A&A, 474, 653 Wagner, K., Apai, D., Kasper, M., et al. 2016, Science, 353, 673 Wahhaj, Z., Milli, J., Kennedy, G., et al. 2016, , 596, L4 Wang, J. J., Rajan, A., Graham, J. R., et al. 2014, Proc. SPIE, 9147, 55 Wang, J. J., Ruffio, J.-B., De Rosa, R. J., et al. 2015, Astrophysics Source Code Library, ascl:1506.001 Wolff, S. G., Perrin, M. D., Maire, J., et al. 2014, in Proc. SPIE, Vol. 9147, Ground-based and Airborne Instrumentation for Astronomy V, 91477H Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, , 140, 1868-1881 Yang, H., Apai, D., Marley, M. S., et al. 2016, , 826, 8 Yelda, S., Lu, J. R., Ghez, A. M., et al. 2010, , 725, 331-352 Yurchenko, S. N., Barber, R. J., & Tennyson, J. 2011, , 413, 1828 Yurchenko, S. N., & Tennyson, J. 2014, , 440, 1649 Zuckerman, B., Song, I., Bessell, M. S., & Webb, R. A. 2001, , 562, L87 Zurlo, A., Vigan, A., Galicher, R., et al. 2016, , 587, A57
[^1]: <https://bitbucket.org/pyKLIP/pyklip>
[^2]: <http://www.astro.caltech.edu/~jmc/2mass/v3/transformations/>
[^3]: <https://phoenix.ens-lyon.fr/Grids/BT-NextGen/SPECTRA/>
[^4]: <http://www.astro.princeton.edu/~burrows/>
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abstract: ' The third part of the present paper continues the investigation of the solution of the multivariable cubic algebraic equation for reparametrization invariance of the gravitational Lagrangian. The main result in this paper constitutes the fact that the earlier found parametrization functions of the cubic algebraic equation for reparametrization invariance of the gravitational Lagrangian can be considered also as uniformization functions. These functions are obtained as solutions of first - order nonlinear differential equations, as a result of which they depend only on the complex (uniformization) variable $z$. Further, it has been demonstrated that this uniformization can be extended to two complex variables, which is particularly important for investigating various physical metrics, for example the $ADS$ metric of constant negative curvature (Lobachevsky spaces).'
author:
- |
Bogdan G. Dimitrov [^1]\
Bogoliubov Laboratory for Theoretical Physics\
Joint Institute for Nuclear Research\
6 Joliot - Curie str.\
Dubna 141 980, Russia
title: 'Elliptic Curves and Algebraic Geometry Approach in Gravity Theory III. Uniformization Functions for a Multivariable Cubic Algebraic Equation'
---
INTRODUCTION
============
In previous papers \[1, 2\] the general approach for investigation of algebraic equations in gravity theory has been presented. The approach is based essentially on the important distinction between covariant and contravariant metric components in the framework of the *gravitational theories with covariant and contravariant metrics and connections (GTCCMC)*, which has been described in the review article \[3\]. In its essence, this distinction is related to the *affine geometry approach* \[4,5\], according to which the four-velocity tangent vector at each point of the observer’s worldline is *not normalized* and equal to one, i.e. $l_{a}l^{a}=l^{2}\neq 1$. Similarly, for a second-rank tensor one would have $g_{\mu \nu }g^{\nu \alpha }=l_{\mu }^{\alpha }$ $\neq \delta
_{\mu }^{\alpha }$.
In \[2\] a solution for the chosen variables $dX^{1}$, $dX^{2}$, $dX^{3}$ in terms of the elliptic Weierstrass function and its derivative has been found of the earlier proposed \[6\] *cubic algebraic equation of reparametrization invariance of the gravitational Lagrangian* $$dX^{i}dX^{l}\left( p\Gamma _{il}^{r}g_{kr}dX^{k}-\Gamma
_{ik}^{r}g_{lr}d^{2}X^{k}-\Gamma _{l(i}^{r}g_{k)r}d^{2}X^{k}\right)
-dX^{i}dX^{l}R_{il}=0\text{ \ \ \ \ .} \tag{1.1}$$Finding the solution enables one to find the dependence of the contravariant metric tensor components on the *elliptic Weierstrass function and its derivatives* and on variables, related to the covariant tensor and its derivatives.
However, it is much more important to find the dependence of the covariant tensor components. For some concrete cases - the Szafron-Szekeres inhomogeneous cosmological model and its subcase - the FLRW (Friedman - Lemaitre - Robertson - Walker) cosmology, such a representation has been found in \[7\]. The general solution of inhomogeneous relativistic cosmology was given in the form $$ds^{2}=dt^{2}-\frac{\left( \frac{M(z)}{2}\right) ^{2}}{(\rho (u+\epsilon
)-\rho (v_{0}))^{2}}e^{2\nu }(dx^{2}+dy^{2})-$$$$-h^{2}(z)(\Phi ^{^{\prime }}+\Phi \nu ^{^{\prime }})^{2}dz^{2}\text{ \ \ .}
\tag{1.2}$$The function $\rho (v_{0})$ in \[7\] is supposed to satisfy the elliptic curve $$\left( \rho ^{^{\prime }}(v_{0})\right) ^{2}=4\rho ^{3}(v_{0})-g_{2}\rho
(v_{0})-g_{3}\text{ \ \ \ , } \tag{1.3}$$where $g_{2}$ and $g_{3}$ are assumed to be the functions $$g_{2}=\frac{K^{2}(z)}{12}\text{ \ \ ; \ \ \ }g_{3}=\frac{1}{216}K^{3}(z)-\frac{1}{12}\Lambda M^{2}(z)\text{ \ \ .} \tag{1.7}$$Unfortunately, the presented in \[7\] solution cannot be claimed to be true due to the following simple reason: If $g_{2}$ and $g_{3}$ are functions and at the same time, they satisfy the elliptic curve equation (1.3), then these functions should be equal to the corresponding $\func{Ei}$*senstein series* (invariants) $g_{2}=60\sum\limits_{\omega \subset \Gamma }\frac{1}{\omega ^{4}}g_{3}=140\sum\limits_{\omega \subset \Gamma }\frac{1}{\omega ^{6}}$ . If this is taken into account, then it can easily be checked that the obtained solution (1.2) would not be of that form - for example, there would be no dependence on the $M(z)$ function in the second term.
Consequently, the problem about finding the correct solutions of the Einstein’s equations in terms of elliptic functions still remains open. But even if solutions for the metric tensor are found in the form $g_{ij}(z,\mathbf{x)}$ (here $z$ is the complex variable, on which the Weierstrass function depends, $\mathbf{x}$ are the coordinates in the chosen metric), then this dependence on additional complex variable would still complicate the solution.
In this third part of the paper, we shall consider an approach, when it will be possible to find the solution not in terms of the *parametrization functions* $g_{ij}(z,\mathbf{x)}$, but in terms of *uniformization functions* - these are the functions, which depend only on the complex variable $z$ (or, as we shall see, on two complex variables $z$ and $v$) and *on no other variables*. In fact, it will be shown that after solving a *first - order system of nonlinear differential equations* with respect to the generalized coordinates $X^{i}$, their dependence on the complex coordinate $z$ can be found. Thus, an important problem from the point of algebraic geometry will be solved - the uniformization functions $dX^{i}=dX^{i}(z)$ for the multivariable cubic algebraic equation (1.1) are found, as a result also the metric tensor $g_{ij}(z,X(z))=g_{ij}(z)$.
COMPLEX COORDINATE DEPENDENCE OF THE METRIC TENSOR COMPONENTS FROM THE UNIFORMIZATION OF A CUBIC ALGEBRAIC SURFACE
==================================================================================================================================
In this section it will be shown that the solutions (2. 16), (2. 21) and (2. 28) (see \[2\] - Part II) of the cubic algebraic equation (1.1) enable us to express not only the contravariant metric tensor components through the Weierstrass function and its derivatives, but the covariant components as well.
Let us write down for convenience the system of equations (2. 16), (2. 21) and (2. 28) for $dX^{1}$, $dX^{2}$ and $dX^{3}$ as ($l=1,2,3$) $$dX^{l}(X^{1},X^{2},X^{3})=F_{l}(g_{ij}(\mathbf{X}),\Gamma _{ij}^{k}(\mathbf{X}),\rho (z),\rho ^{^{\prime }}(z))=F_{l}(\mathbf{X},z)\text{ \ \ \ ,}
\tag{2.1}$$ where the appearence of the complex coordinate $z$ is a natural consequence of the uniformization procedure, applied with respect to each one of the cubic equations from the embedded sequence of equations.
Yet how the appearence of the additional complex coordinate $z$ on the R. H. S. of (2.1) can be reconciled with the dependence of the differentials on the L. H. S. only on the generalized coordinates $(X^{1},X^{2},X^{3})$ (and on the initial coordinates $x^{1},x^{2},x^{3}$ because of the mapping $X^{i}=X^{i}(x^{1},x^{2},x^{3})$)? The only reasonable assumption will be that *the initial coordinates depend also on the complex coordinate*, i.e. $$X^{l}\equiv X^{l}(x^{1}(z),x^{2}(z),x^{3}(z))=X^{l}(\mathbf{x,}\text{ }z)\text{ \ \ \ \ .} \tag{2.2}$$
Taking into account the important initial assumptions ($l=1,2,3$) $$d^{2}X^{l}=0=dF_{l}(\mathbf{X}(z),z)=\frac{dF_{l}}{dz}dz\text{ \ \ ,}
\tag{2.3}$$one easily gets the system of three inhomogeneous linear algebraic equations with respect to the functions $\frac{\partial X^{1}}{\partial z}$, $\frac{\partial X^{2}}{\partial z}$ and $\frac{\partial X^{3}}{\partial z}$ ($l=1,2,3$): $$\frac{\partial F_{l}}{\partial X^{1}}\frac{\partial X^{1}}{\partial z}+\frac{\partial F_{l}}{\partial X^{2}}\frac{\partial X^{2}}{\partial z}+\frac{\partial F_{l}}{\partial X^{3}}\frac{\partial X^{3}}{\partial z}+\frac{\partial F_{l}}{\partial z}=0\text{ \ \ \ ,} \tag{2.4}$$The solution of this algebraic system ($i,k,l=1,2,3$) $$\frac{\partial X^{l}}{\partial z}=G_{l}\left( \frac{\partial F_{i}}{\partial
X^{k}}\right) =G_{l}\left( X^{1},X^{2},X^{3},z\right) \text{ \ \ \ \ \ }
\tag{2.5}$$represents a system of *three first - order nonlinear differential equations*.** **A solution of this system can always be found in the form ** ** $$X^{1}=X^{1}(z)\text{ \ \ ; \ \ \ }X^{2}=X^{2}(z)\text{ \ \ ; \ \ \ \ }X^{3}=X^{3}(z)\text{ \ \ \ \ \ \ \ \ \ .} \tag{2.6}$$and therefore, the metric tensor components will also depend on the complex coordinate $z$, i.e. $g_{ij}=g_{ij}(\mathbf{X}(z))$. Note that since the functions $\frac{\partial F_{i}}{\partial X^{k}}$ in the R. H. S. of (2.5) depend on the Weierstrass function and its derivatives, it might seem natural to write that ** **the solution of the above system of nonlinear differential equations $g_{ij}$ will also depend on the Weierstrass function and its derivatives $$g_{ij}=g_{ij}(X^{1}(\rho (z),\rho ^{^{\prime }}(z),X^{2}(\rho (z),\rho
^{^{\prime }}(z),X^{3}(\rho (z),\rho ^{^{\prime }}(z))=g_{ij}(z)\text{ \ \ \
.} \tag{2.7}$$Note however that for the moment we do not have a theorem that the solution of the system (2.5) will also contain the Weierstrass function**.** But the dependence on the complex coordinate $z$ will be retained.
Now let us mention the other equations, which will further be taken into account.
The first set of equations simply means that the differentials $dF_{1}$, $dF_{2}$, $dF_{3}$, equal to the second differentials $d^{2}X^{1}$, $d^{2}X^{2}$, $d^{2}X^{3}$ can be taken with respect both to the generalized coordinates $X^{1}$, $X^{2}$, $X^{3}$ and the initial coordinates $x^{1}$, $x^{2}$, $x^{3}$ ($l=1,2,3$)$$d^{2}X^{l}=dF_{l}(\mathbf{X}(z),z)=dF_{l}(\mathbf{x}(z),z)\text{ \ \ \ \ .}
\tag{2.8}$$ Denoting further $\overset{.}{x}^{1}\equiv \frac{\partial x^{1}}{\partial z}$, $\overset{.}{x}^{2}\equiv \frac{\partial x^{2}}{\partial z}$ and $\overset{.}{x}^{3}\equiv \frac{\partial x^{3}}{\partial z}$, the above equalities result again in a system of three inhomogeneous algebraic equations with respect to $\overset{.}{X}^{1}\equiv \frac{\partial X^{1}}{\partial z}$, $\overset{.}{X}^{2}\equiv \frac{\partial X^{2}}{\partial z}$ and $\overset{.}{X}^{3}\equiv \frac{\partial X^{3}}{\partial z}$ $$\frac{\partial F_{l}}{\partial X^{1}}\frac{\partial X^{1}}{\partial z}+\frac{\partial F_{l}}{\partial X^{2}}\frac{\partial X^{2}}{\partial z}+\frac{\partial F_{l}}{\partial X^{3}}\frac{\partial X^{3}}{\partial z}=\frac{\partial F_{l}}{\partial x^{1}}\overset{.}{x}^{1}+\frac{\partial F_{l}}{\partial x^{2}}\overset{.}{x}^{2}+\frac{\partial F_{l}}{\partial x^{3}}\overset{.}{x}^{3}\text{.} \tag{2.9}$$Assuming for the moment that we know the functions $\overset{.}{x}^{1}$, $\overset{.}{x}^{2}$ and $\overset{.}{x}^{3}$, the solutions of this algebraic system will give again another system of three first - order nonlinear differential equations ($l=1,2,3$) $$\frac{\partial X^{l}}{\partial z}=H_{l}\left( X^{1},X^{2},X^{3},\text{ }z\text{ },\text{ }\overset{.}{x}^{1},\overset{.}{x}^{2},\overset{.}{x}^{3}\right) \text{ \ \ \ \ \ .} \tag{2.10}$$Again, a solution of this system like the one in (2.6) can be obtained but with account of the dependence on the additional variables $\overset{.}{x}^{1}$, $\overset{.}{x}^{2}$ and $\overset{.}{x}^{3}$. Let us also here note that the solution (2.6) of the nonlinear system of equations (2. 5) can be assumed to be dependent on some another complex variable $v$ $$X^{1}=X^{1}(z,v)\text{ \ \ \ ; \ \ }X^{1}=X^{1}(z,v)\text{ \ \ \ ; \ \ \ }X^{1}=X^{1}(z,v)\text{\ \ \ \ \ .} \tag{2.11}$$The system of equations (2.8) ($i=1,2,3$) $$d^{2}X^{i}=dF_{i}(\mathbf{X}(z,v),z)=dF_{1}(\mathbf{x}(z,v),z)\text{ \ \ \
,\ } \tag{2.12}$$with account of the expressions (2.10) now will be rewritten as $$\frac{\partial F_{i}}{\partial X^{1}}\frac{\partial X^{1}}{\partial v}+\frac{\partial F_{i}}{\partial X^{2}}\frac{\partial X^{2}}{\partial v}+\frac{\partial F_{i}}{\partial X^{3}}\frac{\partial X^{3}}{\partial v}=\frac{\partial F_{i}}{\partial x^{1}}\overset{.}{x}^{1}+\frac{\partial F_{i}}{\partial x^{2}}\overset{.}{x}^{2}+\frac{\partial F_{i}}{\partial x^{3}}\overset{.}{x}^{3}+$$ $$+\frac{\partial F_{i}}{\partial x^{1}}x^{^{\prime }1}+\frac{\partial F_{i}}{\partial x^{2}}x^{^{\prime }2}+\frac{\partial F_{i}}{\partial x^{3}}x^{^{\prime }3}-\frac{\partial F_{i}}{\partial X^{1}}H_{1}-\frac{\partial
F_{i}}{\partial X^{2}}H_{2}-\frac{\partial F_{i}}{\partial X^{3}}H_{3}=0\text{ \ ,} \tag{2.13}$$where $x^{^{\prime }1},x^{^{\prime }2},x^{^{\prime }3}$ denote the derivatives $\frac{\partial x^{1}}{\partial z},\frac{\partial x^{2}}{\partial z},\frac{\partial x^{3}}{\partial z}$. The same notation further will be used with respect to the variables $\frac{\partial X^{1}}{\partial v},\frac{\partial X^{2}}{\partial v},\frac{\partial X^{3}}{\partial v}$ . Similarly to (2. 10), the algebraic solution of this system of equations can be represented as $$\frac{\partial X^{i}}{\partial v}=K_{i}\left( \mathbf{X(}z,v\mathbf{),}\text{
}z\text{ },\overset{.}{\mathbf{x}}\text{ },\mathbf{x}^{^{\prime }}\right)
\text{ \ \ \ .} \tag{2.14}$$Note that instead of (2.12), we could have also written $$d^{2}X^{i}=dF_{i}(\mathbf{X}(z,v),z)=dF_{1}(\mathbf{x}(z),z,v)\text{ \ \ \
.\ } \tag{2.15}$$
Further in section 3 it shall be proved why this would be incorrect. The complete analysis of the system of equations, when both system of coordinates depend on the two pair of complex variables $z$ and $v$ will be given in the following sections. For the moment we give just the general qualitative motivations.
The other set of equations, which will further be used and which relates the generalized coordinates $X^{i}$ to the initial ones $x^{i}$ is $$d^{2}X^{i}=0=\frac{\partial ^{2}X^{i}}{\partial x^{k}\partial x^{r}}dx^{k}dx^{r}+\frac{\partial X^{i}}{\partial x^{k}}d^{2}x^{k}\text{ \ \ .}
\tag{2.16}$$For the moment we assume that the initial coordinates $x^{k}$ depend only on the $z$ coordinate, and therefore $$\frac{\partial ^{2}X^{i}}{\partial x^{k}\partial x^{r}}=\frac{\overset{..}{X}^{i}}{\overset{.}{x}^{k}\overset{.}{x}^{r}}-\overset{.}{X}^{i}\frac{\overset{..}{x}^{r}}{\overset{.}{x}^{k}\left( \overset{.}{x}^{r}\right) ^{2}}\text{ \
\ .} \tag{2.17}$$
Taking this into account, the system (2.16) in the $n-$dimensional case can be written as $$n^{2}\overset{..}{X}^{i}(dz)^{2}-(n-1)\overset{.}{X}^{i}\frac{\overset{..}{x}^{r}}{\overset{.}{x}^{r}}(dz)^{2}+n\overset{.}{X}^{i}d^{2}z=0\text{ \ \ \ .}
\tag{2.18}$$Introducing the notation $$y^{r}=\frac{\partial }{\partial z}\left( \ln \overset{.}{x}^{r}\right) =\frac{\overset{..}{x}^{r}}{\overset{.}{x}^{r}}\text{ \ \ \ \ \ } \tag{2.19}$$for the three-dimensional case, the system (2.18) can be written as $$2\overset{.}{X}^{i}(dz)^{2}(y^{1}+y^{2}+y^{3})=9\overset{..}{X}^{i}(dz)^{2}+3\overset{.}{X}^{i}d^{2}z\text{ \ \ \ \ .} \tag{2.20}$$ Dividing the L. H. S. and the R. H. S. of the $i$-th and the $j$-th equation of this system, it can easily be obtained $$\left( dz\right) ^{2}\left( \overset{..}{X}^{i}\overset{.}{X}^{j}-\overset{..}{X}^{j}\overset{.}{X}^{i}\right) =0\text{ \ \ \ \ ,} \tag{2.21}$$which can be written as $$(dz)^{2}\left( \overset{.}{X}^{j}\right) ^{2}\frac{\partial }{\partial z}\left( \frac{\overset{.}{X}^{i}}{\overset{.}{X}^{j}}\right) =0\text{ \ \ \ \
.} \tag{2.22}$$ Neglecting the case when $\overset{.}{X}^{j}=0$, the above relation simply means that $\overset{.}{X}^{2}$ and $\overset{.}{X}^{3}$should be proportional to $\overset{.}{X}^{1}$$$\overset{.}{X}^{2}=C_{2}\overset{.}{X}^{1}\text{ \ ; \ }\overset{.}{X}^{3}=C_{3}\overset{.}{X}^{1}\text{ \ \ ,} \tag{2.23}$$where $C_{2}$ and $C_{3}$ are constants. Indeed, it is easily seen that (2.23) holds since $$dX^{1}=\overset{.}{X}^{1}dz=F_{1}\text{ ; \ \ \ }dX^{2}=\overset{.}{X}^{2}dz=F_{2}\text{ ; \ \ \ }dX^{3}=\overset{.}{X}^{3}dz=F_{3} \tag{2.24}$$and consequently $$C_{2}=\frac{F_{2}}{F_{1}}\text{ \ \ \ \ ; \ \ \ \ }C_{3}=\frac{F_{3}}{F_{1}}\text{ \ \ \ .} \tag{2.25}$$
FIRST-ORDER NONLINEAR DIFFERENTIAL EQUATIONS FOR THE COMPLEX FUNCTIONS $x=x(z)$ AND $X=X(z)$
=====================================================================================================
For the purpose, the two systems of algebraic equations (2.4) and (2.9) will be used. If one substitutes the found expressions (2.23) for $\overset{.}{X}^{2}$and $\overset{.}{X}^{3}$into the system (2.4), it may be treated as an algebraic system of equations with respect to the variables $\overset{.}{X}^{1},C_{2}$ and $C_{3}$. Introducing the notation $$\{F_{i},F_{j}\}_{z,X^{k}}\equiv \frac{\partial F_{i}}{\partial z}\frac{\partial F_{j}}{\partial X^{k}}-\frac{\partial F_{i}}{\partial X^{k}}\frac{\partial F_{j}}{\partial z}\text{ \ \ } \tag{3.1}$$for the one-dimensional *Poisson bracket* $\{F_{i},F_{j}\}_{z,X^{k}}$ of the coordinates $z,X^{k}$ and also the notation $$\{F_{1},F_{2},F_{3}\}_{z,\left[ X^{i},X^{j}\right] }\equiv
\{F_{1},F_{2}\}_{z,X^{i}}\{F_{1},F_{3}\}_{z,X^{j}}-\{F_{1},F_{2}\}_{z,X^{j}}\{F_{1},F_{3}\}_{z,X^{i}}\text{ \ \ \ ,} \tag{3.2}$$one can show that the solution of the system of *linear algebraic equations* (2.4) with respect to $\overset{.}{X}^{1},C_{2}$ and $C_{3}$ $$\frac{\partial F_{i}}{\partial X^{1}}\overset{.}{X}^{1}+\frac{\partial F_{i}}{\partial X^{2}}\overset{.}{X}^{2}+\frac{\partial F_{i}}{\partial X^{3}}\overset{.}{X}^{3}+\frac{\partial F_{i}}{\partial z}=0 \tag{3.3}$$can be represented in the following compact form $$C_{2}=\frac{\{F_{1},F_{2},F_{3}\}_{z,\left[ X^{3},X^{1}\right] }}{\{F_{1},F_{2},F_{3}\}_{z,\left[ X^{2},X^{3}\right] }}\text{ \ \ \ ; \ \ \ \
\ }C_{3}=\frac{\{F_{1},F_{2},F_{3}\}_{z,\left[ X^{1},X^{2}\right] }}{\{F_{1},F_{2},F_{3}\}_{z,\left[ X^{2},X^{3}\right] }}\text{ \ \ \ ,\ }
\tag{3.4}$$$$\overset{.}{X}^{1}=-\frac{\frac{\partial F_{1}}{\partial z}\{F_{1},F_{2},F_{3}\}_{z,\left[ X^{2},X^{3}\right] }}{K_{1}}\text{ \ \ \ \ .}
\tag{3.5}$$In (3.5) the following notation has been introduced for $K_{i}$ ($i=1,2,3$) $$K_{i}\equiv \frac{\partial F_{i}}{\partial X^{1}}\{F_{1},F_{2},F_{3}\}_{z,\left[ X^{2},X^{3}\right] }+\frac{\partial F_{i}}{\partial X^{2}}\{F_{1},F_{2},F_{3}\}_{z,\left[ X^{3},X^{1}\right] }+\frac{\partial F_{i}}{\partial X^{3}}\{F_{1},F_{2},F_{3}\}_{z,\left[ X^{1},X^{2}\right] }\text{ \
\ \ \ .} \tag{3.6}$$The usefulness of introducing this notation will soon be understood.
Now let us rewrite the system of equations (2.9) in the form $$\frac{\partial F_{i}}{\partial x^{1}}\overset{.}{x}^{1}+\frac{\partial F_{i}}{\partial x^{2}}\overset{.}{x}^{2}+\frac{\partial F_{i}}{\partial x^{3}}\overset{.}{x}^{3}=M_{i}\text{ \ \ \ \ ,} \tag{3.7}$$where $M_{i}$ will be the notation for $$M_{i}\equiv \frac{\partial F_{i}}{\partial X^{1}}\overset{.}{X}^{1}+\frac{\partial F_{i}}{\partial X^{2}}\overset{.}{X}^{2}+\frac{\partial F_{i}}{\partial X^{3}}\overset{.}{X}^{3}\text{ \ \ .} \tag{3.8}$$Making use of the above formulaes (3.4 - 3.6) and also (2.23), $M_{i}$ can be calculated to be $$M_{i}=-\frac{\partial F_{1}}{\partial z}\frac{K_{i}}{K_{1}}\text{ \ \ \ .}
\tag{3.9}$$Further, the solutions of the linear algebraic system of equations (3.7) can be represented in the form $$\overset{.}{x}^{i}=S_{1}^{i}M_{1}+S_{2}^{i}M_{2}+S_{3}^{i}M_{3}\text{ \ \ \
\ ,} \tag{3.10}$$where the functions $S_{1}^{i},S_{2}^{i}$ and $S_{3}^{i}$ depend on $\frac{\partial F_{i}}{\partial x^{k}}$ ($i,k=1,2,3$). Since $M_{1}$, $M_{2}$, $M_{3}$ according to (3.9) and (3.6) are proportional to $\frac{\partial F_{1}}{\partial z}\frac{\{F_{1},F_{2},F_{3}\}_{z,\left[ X^{k},X^{j}\right] }}{K_{1}}$ (where $(k,j)=(2,3)$, $(3,1)$ or $(1,2)$), the resulting solution (3.10) will be of the kind $$\overset{.}{x}^{i}=\frac{\overline{S}_{1}^{i}\left( \frac{\partial F_{1}}{\partial z}\right) ^{2}+\overline{S}_{2}^{i}\frac{\partial F_{2}}{\partial z}\frac{\partial F_{1}}{\partial z}+\overline{S}_{3}^{i}\frac{\partial F_{1}}{\partial z}\frac{\partial F_{3}}{\partial z}}{\overline{S}_{4}^{i}\left(
\frac{\partial F_{1}}{\partial z}\right) +\overline{S}_{5}^{i}\frac{\partial
F_{2}}{\partial z}+\overline{S}_{6}^{i}\frac{\partial F_{3}}{\partial z}}\text{ \ \ \ \ ,} \tag{3.11}$$
where the functions $\overline{S}_{1}^{i},\overline{S}_{2}^{i},....,\overline{S}_{6}^{i}$ depend both on $\frac{\partial F_{i}}{\partial X^{k}}$ and $\frac{\partial F_{i}}{\partial x^{k}}$ and consequently on all the variables $x^{k},X^{k}$ and $z$. We have used also the following relation, obtained after simple algebra with account of (3.1) and (3.2) $$\{F_{1},F_{2},F_{3}\}_{z,\left[ X^{i},X^{j}\right] }=\left( \frac{\partial
F_{1}}{\partial z}\right) ^{2}\{F_{2},F_{3}\}_{X^{i},X^{j}}+$$$$+\frac{\partial F_{1}}{\partial z}\frac{\partial F_{2}}{\partial z}\{F_{3},F_{1}\}_{X^{i},X^{j}}+\frac{\partial F_{1}}{\partial z}\frac{\partial F_{3}}{\partial z}\{F_{1},F_{2}\}_{X^{i},X^{j}}\text{ \ \ .}
\tag{3.12}$$Thus we have obtained the system of *first order nonlinear differential equations* with respect to the initial coordinates $x^{i}=x^{i}(z)$**.** An analogous system of nonlinear differential equations is obtained for $X^{1}=X^{1}(z)$, $X^{2}=X^{2}(z)$ and $X^{3}=X^{3}(z)$ - for $X^{1}$ this is equation (3.5), and with account of (2.23) and expressions (3.4) for $C_{2}$ and $C_{3}$, the corresponding equations for $X^{2}(z)$ and $X^{3}(z)$ are $$\overset{.}{X}^{2}=-\frac{\frac{\partial F_{1}}{\partial z}\{F_{1},F_{2},F_{3}\}_{z,\left[ X^{3},X^{1}\right] }}{K_{1}}\text{ \ \ ; \ }\overset{.}{X}^{3}=-\frac{\frac{\partial F_{1}}{\partial z}\{F_{1},F_{2},F_{3}\}_{z,\left[ X^{1},X^{2}\right] }}{K_{1}}\text{.}
\tag{3.13}$$Therefore, if the generalized coordinates $X^{1},X^{2},X^{3}$ are determined as functions of the complex variable $z$ after solving the system (3.5), (3.13), the obtained functions $X^{1}=X^{1}(z)$, $X^{2}=X^{2}(z)$ and $X^{3}=X^{3}(z)$ can be substituted into the R. H. S. of the system (3.11) for $x^{1},x^{2}$ and $x^{3}$ and the corresponding solutions $x^{1}=x^{1}(z)
$, $x^{2}=x^{2}(z)$ and $x^{3}=x^{3}(z)$ can be found. Remember that we started from the assumption that only the generalized coordinates $X^{1},X^{2},X^{3}$ satisfy the original cubic algebraic equation and therefore equalities (2.24) are fulfilled. Nevertheless, the corresponding functions $x^{i}=x^{i}(z)$ is possible to be determined from the system (3.11), the R. H. S. of which also confirms that $dx^{i}\neq F_{i}$.
This conclusion is important since it shows that the two systems of coordinates should not be treated on an equal footing. This refers of course to the case of only one complex coordinate.
IS IT POSSIBLE TO HAVE A TWO COMPLEX COORDINATE DEPENDENCE OF THE GENERALIZED COORDINATES $X^{i}=X^{i}\left( \mathbf{x(}z\right) ,z,v)?$
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It will be proved below that such a case should be disregarded since it leads to an impossibility to determine the dependence $X^{i}$ on the $v$ coordinate.
Under the above assumption $X^{i}=X^{i}\left( \mathbf{x(}z\right) ,z,v)$, the first set of three equations $$dX^{i}=\frac{\partial X^{i}}{\partial x^{1}}dx^{1}+\frac{\partial X^{i}}{\partial x^{2}}dx^{2}+\frac{\partial X^{i}}{\partial x^{3}}dx^{3}\text{ \ \ }
\tag{4.1}$$can be represented as $$F_{i}=\overset{.}{X}^{i}\frac{\partial z}{\partial x^{1}}\overset{.}{x}^{1}dz+\overset{.}{X}^{i}\frac{\partial z}{\partial x^{2}}\overset{.}{x}^{2}dz+\overset{.}{X}^{i}\frac{\partial z}{\partial x^{3}}\overset{.}{x}^{3}dz=3\overset{.}{X}^{i}dz\text{ \ \ \ ,} \tag{4.2}$$so again relations (2.23) - (2.25) $\ \overset{.}{X}^{2}=\frac{F_{2}}{F_{1}}\overset{.}{X}^{1}$, $\overset{.}{X}^{3}=\frac{F_{3}}{F_{1}}\overset{.}{X}^{1}$will hold.
The second set of equations $$d^{2}X^{i}=dF_{i}(X,z)=dF_{i}(X(z,v),z)=dF_{i}(z,v)\text{ \ \ \ \ }
\tag{4.3}$$will express the equality of the differentials, expressed in terms of the two different sets of coordinates $(X,z)$ and $(z,v)$ $$d^{2}X^{i}=\frac{\partial F_{i}}{\partial X^{1}}dX^{1}+\frac{\partial F_{i}}{\partial X^{2}}dX^{2}+\frac{\partial F_{i}}{\partial X^{3}}dX^{3}+\frac{\partial F_{i}}{\partial z}dz=$$$$=\left[ \frac{\partial F_{i}}{\partial X^{1}}\overset{.}{X}^{1}+\frac{\partial F_{i}}{\partial X^{2}}\overset{.}{X}^{2}+\frac{\partial F_{i}}{\partial X^{3}}\overset{.}{X}^{3}+\frac{\partial F_{i}}{\partial z}\right]
dz+$$$$+\left[ \frac{\partial F_{i}}{\partial X^{1}}dX^{^{\prime }1}+\frac{\partial
F_{i}}{\partial X^{2}}dX^{^{\prime }2}+\frac{\partial F_{i}}{\partial X^{3}}dX^{^{\prime }3}\right] dv\text{ \ \ \ .} \tag{4.4}$$Taking into account that according to (2.25) $dX_{1}=F_{1}$, $dX_{2}=F_{2}$ and $dX_{3}=F_{3}$ and also the expressed from (2.25) differential $$dz=\frac{1}{3}\frac{F_{1}}{\overset{.}{X}^{1}}\text{ \ \ ,} \tag{4.5}$$one can obtain for (4.4) $$\left[ \frac{\partial F_{i}}{\partial X^{1}}dX^{^{\prime }1}+\frac{\partial
F_{i}}{\partial X^{2}}dX^{^{\prime }2}+\frac{\partial F_{i}}{\partial X^{3}}dX^{^{\prime }3}\right] dv=$$$$=\frac{2}{3}\left[ \frac{\partial F_{i}}{\partial X^{1}}F_{1}+\frac{\partial
F_{i}}{\partial X^{2}}F_{2}+\frac{\partial F_{i}}{\partial X^{3}}F_{3}\right]
\text{ \ \ \ \ \ .} \tag{4.6}$$Dividing the L. H. S. and the R. H. S. for different values of the indice $i=1,2,3$, one can obtain the following system of linear homogeneous algebraic equations with respect to $X^{^{\prime }1},X^{^{\prime }2}$ and $X^{^{\prime }3}$ (the indice $i$ takes values $1,2,3,1,2..,$i.e. if $i=3\,$, then $i+1$ would be $1$) $$\left( \frac{\partial F_{i}}{\partial X^{1}}Q_{i+1}-\frac{\partial F_{i+1}}{\partial X^{1}}Q_{i}\right) X^{^{\prime }1}+\left( \frac{\partial F_{i}}{\partial X^{2}}Q_{i+1}-\frac{\partial F_{i+1}}{\partial X^{2}}Q_{i}\right)
X^{^{\prime }2}+$$$$+\left( \frac{\partial F_{i}}{\partial X^{3}}Q_{i+1}-\frac{\partial F_{i+1}}{\partial X^{3}}Q_{i}\right) X^{^{\prime }3}=0\text{ \ \ ,} \tag{4.7}$$where $Q_{i}$ ($i=1,2,3$) denotes the expression $$Q_{i}\equiv \frac{\partial F_{i}}{\partial X_{1}}F_{1}+\frac{\partial F_{i}}{\partial X^{2}}F_{2}+\frac{\partial F_{i}}{\partial X^{3}}F_{3}\text{ \ \ .}
\tag{4.8}$$Note that for the moment we have not yet used the equations $d^{2}X^{i}=dF_{i}=0$, from where $Q_{i}=0$. Then the system of equations (4.7) would be identically satisfied for all $X^{^{\prime }1},X^{^{\prime }2}
$ and $X^{^{\prime }3}$ and it would be impossible to express them as solutions of the system. But even without making use of the equations $d^{2}X^{i}=dF_{i}=0$, the consistency (or inconsistency) of the system (4.7) is a necessary condition for the consistency (or inconsistency) of the assumption about $X^{i}=X^{i}\left( \mathbf{x(}z\right) ,z,v)$.
Making use of the notation (3.1), the determinant of the system can be written as $$\begin{vmatrix}
\dsum\limits_{l_{1}\neq 1}F_{l_{1}}\{F_{1},F_{2}\}_{1,l_{1}} &
\dsum\limits_{l_{2}\neq 2}F_{l_{2}}\{F_{1},F_{2}\}_{2,l_{2}} &
\dsum\limits_{l_{3}\neq 3}F_{l_{3}}\{F_{1},F_{3}\}_{3,l_{3}} \\
\dsum\limits_{m_{1}\neq 1}F_{m_{1}}\{F_{1},F_{3}\}_{1,m_{1}} &
\dsum\limits_{m_{2}\neq 2}F_{m_{2}}\{F_{1},F_{3}\}_{2,m_{2}} &
\dsum\limits_{m_{3}\neq 3}F_{m_{3}}\{F_{1},F_{3}\}_{3,m_{3}} \\
\dsum\limits_{n_{1}\neq 1}F_{n_{1}}\{F_{2},F_{3}\}_{1,n_{1}} &
\dsum\limits_{n_{2}\neq 2}F_{n_{2}}\{F_{2},F_{3}\}_{2,n_{2}} &
\dsum\limits_{n_{3}\neq 3}F_{n_{3}}\{F_{2},F_{3}\}_{3,n_{3}}\end{vmatrix}\text{ \ \ \ \ ,} \tag{4.9}$$where instead of $\{F_{i},F_{j}\}_{X^{k},X^{n_{k}}}$ we have written only $\{F_{i},F_{j}\}_{k,n_{k}}$ and each element in the determinant represents a sum either over $l_{1}$, $l_{2}$ or $l_{3}$.
The explicite calculation of the determinant (4.9) gives the non-zero expression $$\frac{\partial F_{2}}{\partial X^{1}}\{F_{1},F_{3}\}_{2,3}+\frac{\partial
F_{3}}{\partial X^{1}}\{F_{1},F_{2}\}_{1,2}\text{ \ \ .} \tag{4.10}$$Since the determinant is non-zero, the system of linear homogeneous algebraic equations does not have a solution and consequently the assumption that $X^{i}=X^{i}\left( \mathbf{x(}z\right) ,z,v)$ turns out to be incorrect.
COMPLEX STRUCTURE $X^{i}=X^{i}\left( \mathbf{x(}z,v\right) ,z) $ $\ $OF THE GENERALIZED COORDINATES AND OF THE METRIC TENSOR COMPONENTS
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****Now it shall be proved that the parametrization (2.1) of the initially given cubic algebraic curve (surface) can be extended to a parametrization in terms of a pair of complex coordinates $(z,v)$ and thus a *complex structure* can be introduced. Of particular interest in view of possible physical applications to theories with extra dimensions and relation to $ADS$ theories, which will be discussed in the conclusion, will be the case of $v=\overline{z}$, when a pair of holomorphic - antiholomorphic variables can be introduced.
In principle a manifold may admit a complex structure \[8\], if it can be covered with opened sets $U$ $_{1},V_{1},U_{2},V_{2}.....$, such that in any intersection $U_{i}\cap V_{i}$ the associated transformations $z^{k^{\prime
}}=z^{k^{\prime }}(z_{i},v_{i})$ are complex (analytical) functions.
The investigated problem may be formulated as follows. Let (again) the system of equations (2.1) is given, subjected to the additional constraining equation $d^{2}X^{i}=0$. Then the parametrization (2.1) of the initially given cubic algebraic surface can be extended to a parametrization by means of a pair of complex coordinates $(z,v)$ in the following way $$dX^{i}(\mathbf{X})=F_{i}(\mathbf{X}(\mathbf{x}(z,v)),z)\text{ \ \ .}
\tag{5.1}$$Therefore, it should be proved that the same system of equations, investigated in the previous sections, is not contradictable under the assumption $X^{i}=X^{i}\left( \mathbf{x(}z,v\right) ,z)$.
The *first set of equations* to be used is similar to (4.3), but this time expressing the equality of the differentials $$dF_{i}(\mathbf{X}(z,v),z)\text{ }=dF_{i}(\mathbf{x}(z,v),z)\text{ \ \ \ \ ,}
\tag{5.2}$$written in terms of the coordinates $(\mathbf{X,}z)$ and $(\mathbf{x},z)$ $$\left[ \frac{\partial F_{i}}{\partial X^{1}}\overset{.}{X}^{1}+\frac{\partial F_{i}}{\partial X^{2}}\overset{.}{X}^{2}+\frac{\partial F_{i}}{\partial X^{3}}\overset{.}{X}^{3}+\frac{\partial F_{i}}{\partial z}\right]
dz+$$$$+\left[ \frac{\partial F_{i}}{\partial X^{1}}X^{^{\prime }1}+\frac{\partial
F_{i}}{\partial X^{2}}X^{^{\prime }2}+\frac{\partial F_{i}}{\partial X^{3}}X^{^{\prime }3}\right] dv=$$$$=\left[ \frac{\partial F_{i}}{\partial x^{1}}\overset{.}{x}^{1}+\frac{\partial F_{i}}{\partial x^{2}}\overset{.}{x}^{2}+\frac{\partial F_{i}}{\partial x^{3}}\overset{.}{x}^{3}+\frac{\partial F_{i}}{\partial z}\right]
dz+$$$$+\left[ \frac{\partial F_{i}}{\partial x^{1}}x^{^{\prime }1}+\frac{\partial
F_{i}}{\partial x^{2}}x^{^{\prime }2}+\frac{\partial F_{i}}{\partial x^{3}}x^{^{\prime }3}\right] dv\text{ \ \ \ \ .} \tag{5.3}$$The *second set of equations* takes into account the fact that the second differential $d^{2}X^{i}$ is zero, or equivalently $$d^{2}X^{i}=dF_{i}(\mathbf{X}(z,v),z)=0\text{ \ \ \ \ \ ,} \tag{5.4}$$where $dF_{i}(\mathbf{X}(z,v),z)$ is given by the L. H. S. of equation (5.3).
The *third set of equations* is $$dX^{i}=F^{i}=\frac{\partial X_{i}}{\partial z}dz+\frac{\partial X_{i}}{\partial v}dv\text{ \ \ \ \ .} \tag{5.5}$$Let us now introduce the notations $$M_{i}(X,z)\equiv \frac{\partial F_{i}}{\partial X^{k}}\overset{.}{X}^{k}\text{ \ \ ; \ \ \ \ }M_{i}(x,z)\equiv \frac{\partial F_{i}}{\partial x^{k}}\overset{.}{x}^{k}\text{ \ \ ,} \tag{5.6}$$$$M_{i}(X,v)\equiv \frac{\partial F_{i}}{\partial X^{k}}X^{^{\prime }k}\text{
\ \ ; \ \ \ \ }M_{i}(x,v)\equiv \frac{\partial F_{i}}{\partial x^{k}}x^{^{\prime }k}\text{ \ \ ,} \tag{5.7}$$which will allow us to write down the first and the second set of equations (4.3) - (4.4) in the following compact form $$\left[ M_{i}(X,z)-M_{i}(x,z)\right] dz+\left[ M_{i}(X,v)-M_{i}(x,v)\right]
dv=0\text{ \ \ ,} \tag{5.8}$$ $$\left[ M_{i}(X,z)+\frac{\partial F_{i}}{\partial z}\right] dz+M_{i}(X,v)dv=0\text{ \ \ .} \tag{5.9}$$Expressing $\frac{\partial X^{i}}{\partial v}dv$ from (5.5), it can easily be proved that $$M_{i}(X,v)dv=\frac{\partial F_{i}}{\partial X^{k}}X^{^{\prime }k}dv=-\frac{\partial F_{i}}{\partial X^{k}}\overset{.}{X}^{k}dz+\frac{\partial F_{i}}{\partial X^{k}}F_{k}\text{ \ \ \ \ ,} \tag{5.10}$$where the last term is zero due to the fulfillment of the second set of equations (5.5). Consequently, from (5.10) it follows $$M_{i}(X,z)dz+M_{i}(X,v)dv=dF_{i}(z,v)=dF_{i}(\mathbf{X}(z,v),z)=0\text{ \ \
\ .} \tag{5.11}$$Additionally, if (5.11) is substracted from (5.8) and (5.9), one easily obtains $$M_{i}(x,z)dz+M_{i}(x,v)dv=dF_{i}(z,v)=dF_{i}(\mathbf{x}(z,v),z)=0\text{ \ \
\ ,} \tag{5.12}$$$$\frac{\partial F_{i}}{\partial z}dz=0\text{ \ \ \ }\Rightarrow \text{ \ \ }\frac{\partial F_{i}}{\partial v}dv=0\text{ \ \ \ \ .} \tag{5.13}$$In other words, if the differential $dF_{i}(\mathbf{X}(z,v),z)$ is zero in terms of the coordinates $(\mathbf{X,}z\mathbf{)}$, then it necessarily should be zero in the coordinates $(\mathbf{x},z)$. But in the spirit of the discussion at the end of section 2, this does not mean that if $dX^{i}=F_{i}$, then the same should hold also for the initial coordinates $x^{i}$, i.e. $dx^{i}=F_{i}$. Indeed, we can find $$M_{i}(x,z)\equiv \frac{\partial F_{i}}{\partial x^{k}}\overset{.}{x}^{k}=\frac{\partial F_{i}}{\partial X^{l}}\frac{\partial X^{l}}{\partial x^{k}}\overset{.}{x}^{k}=$$$$=\frac{\partial F_{i}}{\partial X^{l}}\left[ \frac{\overset{.}{X}^{l}}{\overset{.}{x}^{k}}+\frac{X^{^{\prime }l}}{x^{^{\prime }k}}\right] \overset{.}{x}^{k}=M_{i}(X,z)+M_{i}(X,v)\frac{\overset{.}{x}^{k}}{x^{^{\prime }k}}\text{ \ \ \ \ .} \tag{5.14}$$Similarly $$M_{i}(x,v)=M_{i}(X,v)+M_{i}(X,z)\frac{x^{^{\prime }k}}{\overset{.}{x}^{k}}\text{ \ \ .} \tag{5.15}$$If the above two expressions are substituted into (5.12), and (5.11) is taken into account, one can obtain $$M_{i}(X,v)\frac{\overset{.}{x}^{k}}{x^{^{\prime }k}}dz+M_{i}(X,z)\frac{x^{^{\prime }k}}{\overset{.}{x}^{k}}dv=0\text{ \ \ \ .} \tag{5.16}$$Additionally, we have $$M_{i}(X,v)=M_{i}(X,z)\frac{x^{^{\prime }k}}{\overset{.}{x}^{k}}\text{ \ \ \
; \ \ \ \ \ }dv=-\frac{M_{i}(x,z)}{M_{i}(x,v)}dz\text{ \ \ \ \ .} \tag{5.17}$$Therefore, the following equation in partial derivatives with respect to $F_{i}=F_{i}(x^{l})$ can be derived $$\frac{\partial F_{i}}{\partial x^{l}}x^{^{\prime }l}\frac{\overset{.}{x}^{k}}{x^{^{\prime }k}}\frac{x^{^{\prime }m}}{\overset{.}{x}^{m}}-\frac{\partial
F_{i}}{\partial x^{l}}\overset{.}{x}^{l}\frac{x^{^{\prime }k}}{\overset{.}{x}^{k}}=0\text{ \ \ \ \ .} \tag{5.18}$$A stronger statement may be proved, clearly showing that from $dF_{i}(X,z)=dF_{i}(x,z)=0$ and $dX_{i}=F_{i}$ it does not follow that $dx^{i}=F_{i}$. If expression (4.17) for $M_{i}(X,v)=M_{i}(X,z)\frac{x^{^{\prime }k}}{\overset{.}{x}^{k}}$ is substituted into (5.11), one obtains $$M_{i}(X,z)\left[ dz+\frac{x^{^{\prime }k}}{\overset{.}{x}^{k}}dv\right] =0
\tag{5.22}$$and since $M_{i}(X,z)\neq 0$ (and if $\frac{\partial F_{i}}{\partial X^{l}}\neq 0$), it follows $$dx^{k}=\overset{.}{x}^{k}dz+x^{^{\prime }k}dv=0\text{ \ \ \ .} \tag{5.23}$$
ANALYSIS OF THE FOURTH AND THE FIFTH SET OF EQUATIONS FOR THE PREVIOUS CASE $X^{i}=X^{i}(\mathbf{x}(z,v),z)$
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The *fourth set of equations*, which will be considered is $$dX^{k}=\frac{\partial X^{k}}{\partial x^{1}}dx^{1}+\frac{\partial X^{k}}{\partial x^{2}}dx^{2}+\frac{\partial X^{k}}{\partial x^{3}}dx^{3}=$$$$=3\overset{.}{X}^{k}dz+X^{^{\prime }k}\frac{\overset{.}{x}^{m}}{x^{^{\prime
}m}}dz+\overset{.}{X}^{k}\frac{x^{^{\prime }m}}{\overset{.}{x}^{m}}dv+X^{^{\prime }k}dv\text{ \ \ \ \ .} \tag{6.1}$$If multiplied by $\frac{\partial F_{i}}{\partial X^{k}}dzdv$ and also relation (5.11) $M_{i}(X,z)dz+M_{i}(X,v)dv=0$ is taken into account, the fourth set of equations can be written as $$\frac{\partial F_{i}}{\partial X^{k}}F_{k}dv=M_{i}(X,z)dz\left[ \frac{x^{^{\prime }m}}{\overset{.}{x}^{m}}\left( dv\right) ^{2}-\frac{\overset{.}{x}^{m}}{x^{^{\prime }m}}\left( dz\right) ^{2}\right] \text{ \ \ \ \ .}
\tag{6.2}$$The *fifth set* of equations is $$d^{2}X^{k}=0=\frac{\partial ^{2}X^{k}}{\partial x^{m}\partial x^{n}}dx^{m}dx^{n}+\frac{\partial X^{k}}{\partial x^{m}}d^{2}x^{m}\text{ \ \ ,}
\tag{6.3}$$where the expressions on the R.H.S. can easily be computed in terms of the coordinates $X$ and $x$ and their derivatives.
Our goal further will be to see *whether the fifth equation (6.3) constitutes a separate equation or whether it follows from the preceeding four ones.*
For the purpose, let us multiply both sides of the fifth equation by $\frac{\partial F_{i}}{\partial X^{k}}$ and see which are the terms, containing the second differentials $d^{2}z$ and $d^{2}v$ $$\left( \frac{\partial F_{i}}{\partial X^{k}}\frac{\overset{.}{X}^{k}}{\overset{.}{x}^{m}}+\frac{\partial F_{i}}{\partial X^{k}}\frac{X^{^{\prime
}k}}{x^{^{\prime }m}}\right) \left( \overset{.}{x}^{m}d^{2}z+x^{^{\prime
}m}d^{2}v\right) =$$$$=M_{i}(x,z)d^{2}z+M_{i}(x,v)d^{2}v\text{ \ \ .} \tag{6.4 }$$In (6.4) we have used relations (5.14) and (5.15) for $M_{i}(x,z)$ and $M_{i}(x,v)$. But we may note that the obtained term in (6.4) can be found from the relation (5.12) $M_{i}(x,z)dz+M_{i}(x,v)dv=0$, if it is differentiated by $z$ and $v$ and the resulting equations are summed up. Therefore $$M_{i}(x,z)d^{2}z+M_{i}(x,v)d^{2}v=-M_{i}(x,z)(dz)^{2}-M_{i}(x,v)(dv)^{2}-$$$$-(\overset{.}{M}_{i}(x,v)+M_{i}^{^{\prime }}(x,z))dxdv\text{ \ \ \ . }
\tag{6.5 }$$The derivatives $\overset{.}{M}_{i}(x,v)$ and $M_{i}^{^{\prime }}(x,z)$ can be found also from the already used expressions (5.14) and (5.15) $$M_{i}^{^{\prime }}(x,z)=M_{i}^{^{\prime }}(X,z)+\frac{\overset{.}{x}^{m}}{x^{^{\prime }m}}M_{i}^{^{\prime }}(X,v)+$$$$+M_{i}(X,v)\frac{\left[ \overset{.}{x}^{^{\prime }m}x^{^{\prime }m}-\overset{.}{x}^{m}x^{^{\prime \prime }m}\right] }{\left( x^{^{\prime }m}\right) ^{2}}\text{ \ \ \ \ ,} \tag{6.6 }$$$$\overset{.}{M}_{i}(x,v)=\overset{.}{M}_{i}(X,v)+\frac{x^{^{\prime }m}}{\overset{.}{x}^{m}}\overset{.}{M}_{i}(X,z)+$$$$+M_{i}(X,z)\frac{\left[ \overset{.}{x}^{^{\prime }m}\overset{.}{x}^{m}-x^{^{\prime }m}\overset{.}{x}^{m}\right] }{\left( x^{^{\prime
}m}\right) ^{2}}\text{ \ \ \ \ .} \tag{6.7}$$
Making use of all the expressions (6.4) - (6.7), the following expression for the fifth equation (6.3), multiplied by $\frac{\partial F_{i}}{\partial X^{k}}$, can be obtained: $$(dz)^{2}[-2M_{i}(X,z)+M_{i}(X,z)\frac{\overset{..}{x}^{m}}{\overset{.}{x}^{m}}+\frac{\partial F_{i}}{\partial X^{k}}\frac{\partial ^{2}X^{k}}{\partial
x^{m}\partial x^{n}}\overset{.}{x}^{m}\overset{.}{x}^{n}]+$$$$+(dv)^{2}[-2\frac{x^{^{\prime }m}}{\overset{.}{x}^{m}}M_{i}(X,z)+M_{i}(X,z)\frac{x^{^{\prime \prime }m}}{\overset{.}{x}^{m}}+\frac{\partial F_{i}}{\partial X^{k}}\frac{\partial ^{2}X^{k}}{\partial x^{m}\partial x^{n}}x^{^{\prime }m}x^{^{\prime }n}]+$$$$+dzdv[2M_{i}(X,z)\frac{\overset{.}{x}^{^{\prime }m}}{\overset{.}{x}^{m}}+2\frac{\partial F_{i}}{\partial X^{k}}\frac{\partial ^{2}X^{k}}{\partial
x^{m}\partial x^{n}}\overset{.}{x}^{m}x^{^{\prime }n}-2M_{i}^{^{\prime
}}(X,z)-$$$$-2\overset{.}{M_{i}}(X,z)\frac{x^{^{\prime }m}}{\overset{.}{x}^{m}}+2M_{i}(X,z)\frac{\left[ x^{^{\prime }m}\overset{..}{x}^{m}-\overset{.}{x}^{^{\prime }m}\overset{.}{x}^{m}\right] }{\left( \overset{.}{x}^{m}\right)
^{2}}]=0\text{ \ \ .} \tag{6.8 }$$The last two terms $\frac{\partial F_{i}}{\partial X^{k}}\frac{\partial
^{2}X^{k}}{\partial x^{m}\partial x^{n}}\overset{.}{x}^{m}\overset{.}{x}^{n}$and $\frac{\partial F_{i}}{\partial X^{k}}\frac{\partial ^{2}X^{k}}{\partial
x^{m}\partial x^{n}}x^{^{\prime }m}x^{^{\prime }n}$ in the first two square brackets can be found as follows: First, the derivatives $\overset{.}{M}_{i}(X,v)$ and $M_{i}^{^{\prime }}(X,v)$ can be expressed from the relation (4.17) $M_{i}(X,v)=M_{i}(X,z)\frac{x^{^{\prime }k}}{\overset{.}{x}^{k}}$: $$\overset{.}{M_{i}}(X,v)=\overset{.}{M_{i}}(X,z)\frac{x^{^{\prime }m}}{\overset{.}{x}^{m}}+M_{i}(X,z)\frac{\left[ \overset{.}{x}^{^{\prime }m}\overset{.}{x}^{m}-x^{^{\prime }m}\overset{..}{x}^{m}\right] }{\left(
\overset{.}{x}^{m}\right) ^{2}}\text{ \ \ \ ,} \tag{6.9 }$$$$M_{i}^{^{\prime }}(X,v)=M_{i}^{^{\prime }}(X,z)\frac{x^{^{\prime }m}}{\overset{.}{x}^{m}}+M_{i}(X,z)\frac{\left[ x^{^{\prime \prime }m}\overset{.}{x}^{m}-x^{^{\prime }m}\overset{.}{x}^{^{\prime }m}\right] }{\left( \overset{.}{x}^{m}\right) ^{2}}\text{ \ \ \ .} \tag{6.10 }$$But on the other hand, the same derivatives can be found by using the defining expressions (5.6 - 5.7) $$\overset{.}{M}_{i}(X,v)=\frac{\partial ^{2}F_{i}}{\partial X^{k}\partial
X^{l}}\frac{\partial X^{l}}{\partial x^{m}}\frac{\partial X^{k}}{\partial
x^{n}}\overset{.}{x}^{m}x^{^{\prime }n}+\frac{\partial F_{i}}{\partial X^{k}}\frac{\partial ^{2}X^{k}}{\partial x^{m}\partial x^{n}}\overset{.}{x}^{n}x^{^{\prime }m}+$$$$+M_{i}(X,z)\frac{x^{^{\prime }m}\overset{.}{x}^{n}}{\overset{.}{x}^{m}}+M_{i}(X,z)\overset{.}{x}^{^{\prime }n}\text{ \ \ ,} \tag{6.11 }$$$$M_{i}^{^{\prime }}(X,v)=\frac{\partial ^{2}F_{i}}{\partial X^{k}\partial
X^{l}}\frac{\partial X^{l}}{\partial x^{m}}\frac{\partial X^{k}}{\partial
x^{n}}x^{^{\prime }m}x^{^{\prime }n}+\frac{\partial F_{i}}{\partial X^{k}}\frac{\partial ^{2}X^{k}}{\partial x^{m}\partial x^{n}}x^{^{\prime
}n}x^{^{\prime }m}+$$$$+\frac{\partial F_{i}}{\partial X^{k}}\frac{\partial X^{k}}{\partial x^{n}}\frac{x^{^{\prime }m}\overset{.}{x}^{^{\prime }n}}{\overset{.}{x}^{m}}+\frac{\partial F_{i}}{\partial X^{k}}\frac{\partial X^{k}}{\partial x^{m}}x^{^{\prime \prime }n}\text{ \ \ .} \tag{6.12 }$$Therefore, the desired expressions can be found by setting up formulae (6.9) equal to (6.11) and also formulae (6.10) equal to (6.12). It can easily be derived how eq. (6.8) will transform, but unfortunately, this would not result in any simplification of the equation with respect to the generalized coordinates $X^{i}$.
It remained only to show whether the fourth equation (5.2) is independent from the preceeding ones (and thus can be treated separately) or it follows naturally from these equations. For the purpose, let us take the differential of (5.2) and use equations (5.10). After some lengthy, but straightforward calculations it can be obtained $$(dz)(dv)^{2}[\overset{.}{M_{i}}(X,z)\frac{x^{^{\prime }m}}{\overset{.}{x}^{m}}+2\frac{x^{^{\prime }m}}{\overset{.}{x}^{m}}M_{i}^{^{\prime
}}(X,z)-M_{i}(X,z)\frac{\left( \overset{.}{x}^{^{\prime }m}\overset{.}{x}^{m}-x^{^{\prime }m}\overset{..}{x}^{m}\right) }{\left( \overset{.}{x}^{m}\right) ^{2}}+$$$$+M_{i}(X,z)\frac{2\left( \overset{.}{x}^{^{\prime }m}\overset{.}{x}^{m}-x^{^{\prime }m}\overset{..}{x}^{m}+x^{^{\prime \prime }m}\overset{.}{\overset{.}{x}^{m}-\overset{.}{x}^{^{\prime }m}x^{^{\prime }m}}\right) }{\left( \overset{.}{x}^{m}\right) ^{2}}]+$$$$+(dz)^{2}(dv)[M_{i}^{^{\prime }}(X,z)\frac{\overset{.}{x}^{m}}{x^{^{\prime
}m}}+2\overset{.}{M}_{i}(X,z)+M_{i}(X,z)\frac{\left( \overset{.}{x}^{^{\prime }m}x^{^{\prime }m}-\overset{.}{x}^{m}x^{^{\prime \prime
}m}\right) }{\left( x^{^{\prime }m}\right) ^{2}}]+$$$$+(dz)^{3}[\overset{.}{M_{i}}(X,z)\frac{\overset{.}{x}^{m}}{x^{^{\prime }m}}+M_{i}(X,z)\frac{\left( \overset{..}{x}^{m}x^{^{\prime }m}-\overset{.}{x}^{m}\overset{.}{x}^{^{\prime }m}\right) }{\left( x^{^{\prime }m}\right) ^{2}}]+$$$$+(dv)^{3}[M_{i}^{^{\prime }}(X,z)\frac{x^{^{\prime }m}}{\overset{.}{x}^{m}}+M_{i}(X,z)\frac{\left( x^{^{\prime \prime }m}\overset{.}{x}^{m}-x^{^{\prime
}m}\overset{.}{x}^{^{\prime }m}\right) }{\left( \overset{.}{x}^{m}\right)
^{2}}]=0\text{ \ \ \ \ \ .} \tag{6.13 }$$This is an equation both for the initial coordinates $x^{i}$ and for the generalized ones $X^{i}$ and it is different from the fifth equation (6.8).
DISCUSSION
==========
In this third part of the paper it has been shown that from the expressions (2.1) a system of first - order nonlinear differential equations is obtained, for which always a solution $X^{1}=X^{1}(z)$, $X^{1}=X^{1}(z)$, $X^{1}=X^{1}(z)$ exists. Thus the dependence on the generalized coordinates $X^{1}$**,** $X^{2}$**,** $X^{3}$ in the uniformization functions (2.1 ) dissappears and only the dependence on the complex coordinate $z$ remains, as it should be for uniformization functions.
Moreover, the initial assumption $dX^{i}=0$ for obtaining the solutions (2.1) allows us to derive a system of nonlinear differential equations also for the initial variables $x^{1}$, $x^{2}$, $x^{3}$ and thus the corresponding solutions $x^{1}=x^{1}(z)$, $x^{2}=x^{2}(z)$, $x^{3}=x^{3}(z)$ in principle can be found. This analysis has been performed in section 3. In fact, it can easily be guessed that if we have the solutions $X^{1}=X^{1}(z)$, $X^{2}=X^{2}(z)$, $X^{3}=X^{3}(z)$ and the additional condition $d^{2}X^{i}=0$ (which in fact relates the generalized and the initial sets of coordinates), then the solutions $x^{1}=x^{1}(z)$, $x^{2}=x^{2}(z)$, $x^{3}=x^{3}(z)$ should also be coordinated with the previous ones. Indeed, this is evident from the dependence of the functions $\overline{S}_{1}^{(i)}$,$\overline{S}_{2}^{(i)}$,....,$\overline{S}_{6}^{(i)}$ in the system (3.11) for $\frac{dx^{i}}{dz}$ both on the functions $\frac{\partial F_{i}}{\partial X^{k}}$ and $\frac{\partial F_{i}}{\partial x^{k}}$ , i.e. on both system of coordinates. Of particular importance is the conclusion at the end of Section 3 that the two sets of coordinates $X^{1}$, $X^{2}$, $X^{3}$ and $\ x^{1}$, $x^{2}$, $x^{3}$ should not be treated on an equal footing. This means that if $dX^{1}$, $dX^{2}$, $dX^{3}$ satisfy the originally derived cubic algebraic equation, then it is not necessary to assume this for $\
dx^{1}$, $dx^{2}$, $dx^{3}$.
Much more interesting is the other investigated case in Sections 4 - 6, where a pair of complex coordinates $z,v$ has been introduced and thus through the generalized coordinates $X^{1}=X^{1}(z,v)$, $X^{2}=X^{2}(z,v)$, $X^{3}=X^{3}(z,v)$ a complex structure of the metric tensor components is introduced. For the investigated case under the assumption $d^{2}X^{i}=0$, there is only one way for introducing this complex structure - namely, through the dependence of the initial coordinates on $z$ and $v$, i. e. $X^{i}=X^{i}(\mathbf{x}(z,v),z)$. Otherwise, if some other possibility is assumed, for example $X^{i}=X^{i}(\mathbf{x}(z),z,v)$, then, as proved in section 4, the obtained system of equations is contradictory. Therefore, it remains to investigate the full system of equations for the only allowed case $X^{i}=X^{i}(\mathbf{x}(z,v),z)$, which has been performed in Sections 5 and 6. Remarkably, a nonlinear differential equation is obtained only for the initial coordinates. However, no such an equation only for the generalized coordinates can be obtained - the derived equation depends in a complicated manner on both system of coordinates. Since the existence of these noncontradictory systems of equations confirms that a complex structure can be introduced, one may express the line element $ds^{2}=g_{ij}(\mathbf{X})dX^{i}dX^{j}$ as $$ds^{2}=\widetilde{g}_{zz}(z,v)(dz)^{2}+\widetilde{g}_{zv}(z,v)dzdv+\widetilde{g}_{vv}(z,v)(dv)^{2}\text{ \ \ \ \ ,} \tag{7.1}$$where $$\widetilde{g}_{zz}(z,v)\equiv g_{ij}(\mathbf{X}(z,v))\overset{.}{X}^{i}\overset{.}{X}^{j}\text{ \ \ ; \ \ }\widetilde{g}_{vv}(z,v)\equiv g_{ij}(\mathbf{X}(z,v))X^{^{\prime }i}X^{^{\prime }j}\text{\ \ \ \ ,} \tag{7.2}$$$$\widetilde{g}_{zv}(z,v)\equiv g_{ij}(\mathbf{X}(z,v))\left[ \overset{.}{X}^{i}X^{^{\prime }j}+X^{^{\prime }i}\overset{.}{X}^{j}\right] \text{ \ \ .}
\tag{7.3}$$This result will be of particular importance in reference to possible physical applications, which will be considered in another paper. For example, the linear element of a unit surface in the *Lobachevsky space with a constant negative curvature* $-\frac{1}{R^{2}}$ can be represented as \[9\] $$ds^{2}=R^{2}\frac{(a^{2}-w^{2})du^{2}+2uwdudw+(a^{2}-u^{2})dw^{2}}{(a^{2}-u^{2}-w^{2})^{2}}\text{ \ \ \ ,} \tag{7.4}$$which by means of a suitable coordinate transformation can be brought to the form $$ds^{2}=d\rho ^{2}+e^{-\frac{2\rho }{R}}d\sigma ^{2}\text{ \ \ \ \ .}
\tag{7.5}$$The metric (7.5) is the standard form of the three - dimensional Lobachevsky metric \[10\] ($d\sigma ^{2}$ is a two - dimensional surface element), where the ratio $\frac{2\rho }{R}$ may or may not be identified with the Lobachevsky constant $k=\frac{1}{c}$ ($c$ is a natural unit length element in the Lobachevsky space). This is particularly important to be mentioned in reference to Randall - Sundrum models and theories with extra dimensions, which are based on the multi-dimensional analogue of the Lobachevsky metric (7.5). Some physical applications of the algebraic geometry formalism in these models will be considered in a separate paper.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author is grateful to Dr. L. K. Alexandrov, St. Mishev and especially to Prof. V. V. Nesterenko, Dr. O. Santillan (BLTP, JINR, Dubna) and to Prof. Sawa Manoff (INRNE, BAS, Sofia) for valuable comments, discussions and critical remarks.
This paper is written in memory of Prof. S. S. Manoff (1943 - 27.05.2005) - a specialist in classical gravitational theory and physics.
The author is grateful also to Dr. A. Zorin (LNP, JINR) and to J. Yanev (BLTP, JINR) for various helpful advises and to Dr.V. Gvaramadze (SAI, MSU, Moscow) and his family for their moral support and encouragement.
[99]{} B. G. Dimitrov 2006 Elliptic Curves and Algebraic Geometry Approach in Gravity Theory I. The General Approach* *(*Preprint* hep-th/0511136)
B. G. Dimitrov 2006 Elliptic Curves and Algebraic Geometry Approach in Gravity Theory II. Parametrization of Multivariable Cubic Equations (*Preprint* hep-th/0511136)
S. Manoff 1999 *Part. Nucl.* **30** 517 - 549 \[Rus. Edit. 1999 Fiz. Elem. Chast. Atomn.Yadra. **30** (5) 1211 - 1269\] (*Preprint* gr-qc/0006024)
A. P. Norden 1950 *Spaces of Affine Connection* (Moscow: Nauka Publ.House)
P. A. Shirokov, and A. P. Shirokov 1959 *Affine Differential Geometry* (Moscow: Fizmatgiz)
B. G. Dimitrov 2003 Cubic Algebraic Equations in Gravity Theory, Parametrization with the Weierstrass Function and Nonarithmetic Theory of Algebraic Equations *J. Math. Phys.* **44 (6)** 2542 - 2578 (*Preprint* hep-th/0107231)
G. V. Kraniotis, and S. B. Whitehouse 2002 General Relativity, the Cosmological Constant and Modular Forms *Class. Quant. Grav.* **19** 5073 - 5100 (*Preprint* gr-qc/0105022)
E. J. Flaherty 1976 *Hermitian and Kahlerian Geometry in Relativity (Lecture Notes in Physics 46)* (Berlin: Springer - Verlag)
E. Beltrami 1868 Saggio di interpretazione della geometria non-euclidea *Napoli* **6** 284 - 312
N. V. Yefimov 2003 *Higher Geometry (7th edition)* (Moscow: Fizmatlit)
[^1]: Electronic mail: [email protected]
|
---
abstract: |
We present a multiscale atomistic-to-continuum method for ionic crystals with defects. Defects often play a central role in ionic and electronic solids, not only to limit reliability, but more importantly to enable the functionalities that make these materials of critical importance. Examples include solid electrolytes that conduct current through the motion of charged point defects, and complex oxide ferroelectrics that display multifunctionality through the motion of domain wall defects. Therefore, it is important to understand the structure of defects and their response to electrical and mechanical fields. A central hurdle, however, is that interactions in ionic solids include both short-range atomic interactions as well as long-range electrostatic interactions. Existing atomistic-to-continuum multiscale methods, such as the Quasicontinuum method, are applicable only when the atomic interactions are short-range. In addition, empirical reductions of quantum mechanics to density functional models are unable to capture key phenomena of interest in these materials.
To address this open problem, we develop a multiscale atomistic method to coarse-grain the long-range electrical interactions in ionic crystals with defects. In these settings, the charge density is rapidly varying, but in an almost-periodic manner. The key idea is to use the polarization density field as a multiscale mediator that enables efficient coarse-graining by exploiting the almost-periodic nature of the variation. In regions far from the defect, where the crystal is close-to-perfect, the polarization field serves as a proxy that enables us to avoid accounting for the details of the charge variation. We combine this approach for long-range electrostatics with the standard Quasicontinuum method for short-range interactions to achieve an efficient multiscale atomistic-to-continuum method. As a side note, we examine an important issue that is critical to our method: namely, the dependence of the computed polarization field on the choice of unit cell. Potentially, this is fatal to our coarse-graining scheme; however, we show that consistently accounting for boundary charges leaves the continuum electrostatic fields invariant to choice of unit cell.
[**Keywords:**]{} electromechanics, multiscale modeling, atomistics, long-range interactions, Quasicontinuum method
author:
- |
Jason Marshall[^1] and Kaushik Dayal[^2]\
[Carnegie Mellon University]{}
bibliography:
- 'mybib.bib'
title: 'Atomistic-to-Continuum Multiscale Modeling with Long-Range Electrostatic Interactions in Ionic Solids'
---
****
Introduction
============
Ionic crystals such as solid electrolytes and complex oxides are central to modern technologies for energy storage, sensing, actuation, and other functional applications. Atomic-scale defects often play a central role in these materials, not only to limit reliability, but more importantly to enable the functionalities that make these materials of critical importance. E.g., in solid electrolytes, conduction is mediated by charged point defects [@solid-electrolytes]; and in complex oxide ferroelectrics, functionality is mediated by planar domain wall defects, and loss of functionality often occurs when domain walls are “pinned” by charged oxygen vacancy point defects [@ferroelectrics-scott-book]. A fundamental understanding of these materials therefore requires an accounting of the atomic-level structure of the defects. This poses a multiscale problem: atomic-level resolution is required at the defect, while complex geometries and boundary conditions require the modeling of a large specimen.
While defects play a critical role in determining properties, they occupy a tiny volume of the lattice; an exceedingly large fraction of the crystal is close-to-perfect. This feature is exploited in most leading atomistic multiscale methods such as the Quasicontinuum (QC) method [@QC-review1; @QC-review2]: typically, an adaptive coarse-graining is used with atomic resolution in the vicinity of the defect and a coarse-grained description further away as the crystal tends to a perfect lattice. Another important aspect of the coarse-graining is the use of sampling or quadrature to efficiently evaluate the energy in the coarse-grained region. It is essential that the atomic interactions are short-range to allow the evaluation of the energy at the quadrature points to be efficient. Therefore, existing multiscale methods cannot handle long-range electrostatic/ionic interactions that decay as $1/r$, where $r$ is the separation between charges.
A symptom of this difficulty with electrostatic interactions can be observed in standard proofs of the Cauchy-Born (CB) theorem that require the interactions to decay faster than $1/r^3$ [@friesecke-james; @blanc-lebris-lions]. Roughly, this implies that the standard CB theorem requires that the charge distribution in each unit cell of the lattice should not have net charge or net dipole character, but only higher-order multipoles. As shown in [@james-muller], it is not possible to define a meaningful energy density $W(\cdot)$ in such a setting, i.e., the standard decomposition of the energy used in elasticity $$E = \int_\Omega W(\cdot) \ d\Omega - \int_{\partial\Omega} \text{boundary working}$$ is not valid. When long-range electrostatic forces are involved, $W(\cdot)$ does not depend solely on the local value of a field (the strain field in elasticity or the polarization field in electrostatics). Rather, it depends on the electrostatic fields in a nonlocal manner as well as boundary conditions. While not always stated explicitly, some notion of the CB theorem is inherent in most atomistic multiscale formulations.
The restrictions described above on the nature of atomic-level interactions for conventional multiscale methods exclude an extremely large class of materials; essentially, all dielectrics, polarizable solids, and ionic solids. In dielectrics and polarizable solids, the non-vanishing dipole moment in the unit cell is central to the physics of dielectric response, spontaneous polarization. In ionic solids such as ionic conductors, the existence of charged defects is central to enabling conduction. Therefore, it is essential to develop methods that can handle long-range electrostatic interactions.
In this paper, we present a multiscale method that is tailored to allow both short-range atomic interactions as well as long-range electrostatic interactions. Some key features of these interactions are as follows. The short-range atomic interactions can be highly nonlinear and involve complex multibody interactions; however, they are typically restricted to 2nd or 3rd nearest-neighbors in a lattice. The long-range electrostatic interactions have the opposite features: the interactions between charges are entirely pairwise, but the interactions between [*every*]{} pair of charges in the system can be non-negligible. A further important feature is that the short-range and electrostatic contributions to the total energy combine additively. These features enable us to leverage much of the existing work for short-range interactions. In particular, we can use a standard version of the Quasicontinuum method [@Tadmor] for the short-range interactions in combination with a method that we develop for efficiently computing the electrostatic interactions.
While the presentation of our method in this paper is largely formal, key aspects build on – and are supported by – rigorous results of James and Müller [@james-muller] and others following their work, e.g. [@schlomerkemper-schmidt]. We note that seminal formal results in this topic were earlier obtained by [@toupin-elastic-dielectric]. While these works deal with point dipoles arranged in a lattice, our focus is on charges; however, due to the fact that the net charge in each unit cell of the lattice is $0$, many of the key results carry over largely unchanged, as also noticed previously by [@Xiao_thesis; @puri-bhatta]. The central idea that we exploit is that a charged lattice can be coarse-grained by introducing a polarization density field. I.e., electrostatic quantities that in principle require the solution of a Poisson problem with a rapidly and almost-periodically oscillating forcing term due to charge density can instead be computed with a much smoother forcing that is related to the polarization density field.
Efficient and accurate methods for interactions in charged systems have a long history in numerical methods. The key challenge is the long-range nature of interactions: in principle, a system of $N$ point charges requires $O(N^2)$ calculations. For $N$ of the order required for typical problems of interest today, this is completely infeasible. However, the seminal and beautiful Fast Multipole Method (FMM) of Greengard and Rokhlin [@greengard1987] provided a breakthrough in enabling this in $O(N)$ calculations with a controlled error. A key strength of the FMM is that the charge distribution can be completely arbitrary and non-uniform; however, this generality also means that the method does not exploit the structure in a given problem. As noted above, in the problems of relevance here, the crystalline structure is lost in the vicinity of the defect, but large parts of the crystal are almost perfect. The FMM, however, is unable to exploit this structure, whereas our method is more tailored (and thereby also less general) and appears to scale almost independent of $N$ asymptotically. Another leading method for atomic-level calculations with electrostatic interactions is the Ewald method (described in e.g. [@Tuckerman-MD-book]). It is restricted to periodic settings and therefore inapplicable to multiscale calculations with defects and complex geometries and boundary conditions.
As mentioned above, our coarse-graining of electrostatic interactions is based on the notion of a polarization density field. However, it is well-known, e.g. [@Resta-Vanderbilt], that the polarization density of a periodic solid depends on the choice of unit cell. At first sight, this is a disturbing observation and much work has been done in the materials physics community on using quantum mechanical notions such as the Berry phase to obtain a unique choice of polarization for a given periodic charge distribution [@Resta-Vanderbilt]. However, an important aspect of that approach is the insistence on starting from an infinite periodic solid. In both the formal calculations presented here, and the related rigorous calculations in the references above, the starting point is a finite periodic solid whose limit behavior is studied. From this “real-space” perspective, the boundaries of the crystal lattice enter naturally into the problem, in sharp contrast to starting from the infinite periodic solid where boundaries are ill-defined. The critical importance of the boundaries is that they, roughly speaking, compensate for the choice of unit cell. I.e., while different unit cell choices lead to different expressions for the polarization density, these also lead to different bound surface charges on the boundaries. When [*both*]{} the bulk bound charge and the surface bound charge are consistently accounted for in the calculations, the electric field and other quantities of relevance to the energy do not depend on the choice of unit cell up to an error that scales with size of the lattice and vanishes in the limit. Therefore, we take the view that a unique choice of unit cell to compute polarization density is unnecessary and any choice of unit cell is – in principle – equally valid. I.e., the polarization is an intermediate coarse-grained quantity, but there is no fundamental physical reason to have a specific choice. In practice, notions of crystal symmetry typically are most useful in selecting a unit cell. Heuristically, this perspective is comparable to the universally-accepted view in continuum mechanics that any reference configuration is – in principle – equally valid. While certain choices of reference configuration can lead to conceptual and algebraic simplifications, there is no fundamental physical preference for any specific choice. What is more relevant is that it is possible to go between different choices with appropriate transformations to the kinematic variables and the energy densities.
The paper is organized as follows.
- In Section \[sec:formulation\], we formulate the problem at the atomic level and briefly describe the well-developed QC approach to handle short-range interactions.
- In Section \[sec:electrostatics\], we describe our treatment of the long-range interactions. Formally, we show the appearance of the polarization as an intermediary multiscale quantity to link atomistic charge distributions with coarse-grained fields. We also examine the issue of the choice of unit cell for polarization; accounting consistently for the boundaries for a given unit cell does not affect the coarse-graining.
- In Section \[sec:implementation\], we outline the kinematic coarse-graining that follows the complex local QC method [@Tadmor] and other aspects of the numerical implementation.
- In Section \[sec:examples\], we outline the model material and its response to a variety of electrical and mechanical loadings.
- In Section \[sec:conclusion\], we discuss various aspects of the work including open problems for ongoing and future work.
Notation {#sec:notation}
--------
Throughout the paper, bold lowercase and uppercase letters denote vectors and tensors. The summation convention is [*not*]{} used in this paper. Sums will be explicitly written out to avoid confusion except where stated.
We define $L$ as a Bravais lattice with three independent lattice vectors that make up a unit cell. $$\label{eq:L_def}
L(\bfe_i,\bfzero) = \left( \bfx \in \R^3 \text{, }\bfx=\sum_i \nu^i\bfe_i \text{ where }\nu^i \in \Z \text{, }i=1,2,3\right)$$
-------------------------- --- ---------------------------------------------------------------------
$E_{total}$ = Total energy
$E$ = Electrostatic energy
$U$ = Interatomic potential energy (i.e. Lennard-Jones, Buckingham types)
$W$ = Short-range strain energy density (related to $U$)
$\Omega$ = Continuum body in the current configuration
$\Omega_0$ = Continuum body in the reference configuration
$Q^s$ = Charge of the atomic species indexed by $s$
$\rho$ = Charge density field in $\Omega$
$\epsilon_0$ = dielectric constant for vacuum
$\mathbb{K}$ = Dipole-dipole interaction electrostatic kernel
$\bfu$ = Displacement field
$\bfx$ = Slow variable representing position in current configuration
$\bfy$ = Fast variable representing position in current configuration
$\bfx_0$ = Position in reference configuration
$\bfzeta^s$ = Intra-unit cell position of species $s$, defined in the reference
$\bfp$ = Polarization density field
$\phi$ = Electric potential
$\grad_{\bfx} \phi$ = Electric field
$\bfF$ = Deformation gradient
$J$ = $\det \bfF$
$\square_i$ = $i$th unit cell
$\triangle_i$ = $i$th partial unit cell
$\epsilon$ = Continuum material point lengthscale
$l$ = Atomic lengthscale
$L$ = Lengthscale over which continuum fields vary
$\mathcal{B}_{\epsilon}$ = Ball of radius $\epsilon$
$\mathcal{D}_{\epsilon}$ = 2D disk of radius $\epsilon$
$\grad_{\bfx_0}$ = Gradient with respect to the reference configuration
$\grad_{\bfx}$ = Gradient with respect to the current configuration
$\sigma$ = Surface charge due to non-neutral partial unit cells.
-------------------------- --- ---------------------------------------------------------------------
$\Omega_{\#}$ and $\Omega_{\square}$ represent the decomposition of $\Omega$ into the partial unit cells on the boundary ($\Omega_{\#}$), and the remainder $\Omega_{\square} := \Omega \backslash \Omega_{\#}$; see Fig. \[fig:omega-decomposition\].
Problem Formulation {#sec:formulation}
===================
We consider a crystal occupying a region $\Omega$, composed of charged species indexed by $s$, each carrying a fixed charge $Q^s$. The notation of species is used broadly; it refers to ions and electrons, as well as electron “shells” as used in core-shell models [@core-shell-ref]. We assume that the charges are all point charges, e.g. nuclei, or that they can be represented through a center of charge as in electron shells. Therefore, the charge distribution $\rho(\bfx)$ is a collection of Dirac masses.
The total energy in the body can be written in the form below: $$\label{eq:energy_lattice}
E_{total} =
\underbrace{ \sum_{\substack{i \\ i \neq j}} U_i \left(\{\bfr_{ij} \} \right)}_\text{short-range}
+
\underbrace{\half \sum_{\substack{i,j \\ i \neq j}} \frac{Q^s_i Q^s_j}{4 \pi \epsilon_0\vert \bfr_{ij}\vert}}_\text{long-range}$$ $\bfr_{ij}$ is the vector between charges $i$ and $j$, and $Q^s_i$ is the charge carried by $i$th atom of species $s$. The function $U$ is the given short-range interatomic potential and can typically involve multibody interactions.
We restrict our attention to zero temperature. Our goal is to find local minimizers of $E_{total}$ to obtain the equilibrium structure subject to applied mechanical and electrostatic loadings. Brute force minimization is infeasible even for the short-range contributions for realistically large systems. This motivated the QC method and related approaches [@QC-review1; @QC-review2] for short-range interactions. We will use the so-called local QC multi-lattice method for the short-range energy largely following [@Tadmor]. As noted above, there are two ingredients to this multiscale approach: first, a kinematic condensation of the degrees of freedom using interpolations, and second, efficient calculation of the energy sum by using sampling or quadrature in relatively uniform regions. The term [*local*]{} refers to the fact that we use a sampling approximation [*everywhere*]{} in the specimen including at the defect core where it is likely to be quite inaccurate.
We begin with the species in the reference configuration arranged in a periodic multi-lattice. The unit cell is denoted $\square$ and the atomic length scale $l$ (Figure \[fig:unit\_cell\_omega\]). In a perfect lattice with short-range interactions, the energy converges to $\int_{\Omega} W(\grad_{\bfx_0} \bfu,\bfzeta^s) \ dV_{\bfx_0}$. Here, $W$ is the strain energy density, and $\grad_{\bfx_0} \bfu$ and $\bfzeta^s$ are the deformation gradient and the “shifts” or relative displacements between lattices [@blanc-lebris-lions; @friesecke-james]. While the expression for $W$ is algebraically involved, it is conceptually simple and comes directly from $U$. In a perfect multi-lattice, there is a well-defined notion of energy per atom since every atom of a given species is in the same environment. Therefore, it is possible to define an energy density by finding the energy of the atoms in a unit cell and dividing the cell volume, which is precisely $W$. The energy naturally depends on the shape of the unit cell and the positions of the different species within it, and this information is contained in $\grad_{\bfx_0} \bfu$ and $\bfzeta^s$ respectively. The QC method replaces the sum in (\[eq:energy\_lattice\]) by sampling the energy density $W(\grad_{\bfx_0} \bfu,\bfzeta^s)$ and using appropriate weights. In the more sophisticated formulations of QC, the energy is computed without this approximation in highly-distorted regions such as the vicinity of the defect [@knap-ortiz]. In the local QC, the approximation is used throughout the specimen, including at the defect. For further details, we refer the reader to recent reviews of the extensive literature on applying QC to materials with short-range interactions [@QC-review1; @QC-review2].
![Domain $\Omega$ showing the separation of scales with sample charge distribution.[]{data-label="fig:unit_cell_omega"}](Pictures/length_scales_good.png){width="125mm"}
Electrostatic Interactions {#sec:electrostatics}
==========================
In this section, we describe the formal calculations that enable us to efficiently account for the long-range electrostatic interactions. A key aspect is the appearance of the polarization density as a multiscale mediator. Because our treatment here is formal, we go between charge density fields and point charges as convenient by assuming that our calculations are valid even when the charge density field consists of Dirac masses. Rigorous treatments of many of the key aspects are available in the literature [@james-muller] and also are the focus of our ongoing work.
Why the Electrostatic Energy is Long-Range {#toy-example}
------------------------------------------
We construct and examine some simple examples to understand why the electrostatics is denoted “long-range”. E.g., the Lennard-Jones potential has interactions that decay as $r^{-6}$ and these interactions nominally extend to $\infty$. As we see, there are important differences when the interactions decay as $r^{-1}$ in electrostatics.
Consider a uniform lattice and 3 cases of charge arrangement within in each unit cell: (i) a charge, (ii) a pair of equal and opposite charges forming a dipole, and (iii) two pairs of equal and opposite charges that form a quadrupole with zero dipole moment (Fig. \[fig:toy-model\]).
![3 cases of charge arrangement: (i) net charge, (ii) net dipole but no net charge, and (iii) net quadrupole, but no net charge and no net dipole.[]{data-label="fig:toy-model"}](Pictures/lattice.png){width="170mm"}
We first consider the lattice of charges. As a rough measure of energy density, we compute the energy of the charge in the chosen unit cell due to its interaction with all the other unit cells in the body. This is the product between the magnitude of the charge in the chosen unit cell with the electrostatic potential created by the rest of the body. The potential due to a charge at a distance $r$ from the chosen unit cell scales as $r^{-1}$. Further, at a distance $r$ from the chosen unit cell, we consider a spatial region with the shape of a spherical shell with unit thickness. This shell has volume that scales as $r^2$ and therefore roughly contains $r^2$ charges. So the total potential at the chosen unit cell due to the rest of the system is $\sum_{r=1}^{\infty} \frac{1}{r} r^2 = \sum_{r=1}^{\infty} r \rightarrow \infty$. That is, the energy density of the body is unbounded in the large-body limit. The physical implication of this calculation is that large clusters of unbalanced charges have extremely high-energy and are thus unlikely to be observed in real materials.
Next consider the lattice of dipoles. As a rough measure of energy density, we compute the energy of the dipole in the chosen unit cell due to its interaction with all the other unit cells in the body. This is the product between the magnitude of the dipole in the chosen unit cell with the electrostatic field created by the rest of the body. The electric field due to a dipole at a distance $r$ from the chosen unit cell scales as $r^{-3}$. Further, the shell at a distance $r$ again contains roughly $r^2$ dipoles. So the total electric field at the chosen unit cell due to the rest of the body is $\sum_{r=1}^{\infty} \frac{1}{r^3} r^2 = \sum_{r=1}^{\infty} \frac{1}{r}$. This sum nominally also tends to infinity. However, the issue is more subtle. The full expression for the electric field due to a unit dipole oriented in the direction $\hat{\bfn}$ at a position $\bfx$ is $E_i = \sum_j\frac{\delta_{ij} - 3 \hat{\bfx}_i \hat{\bfx}_j}{4\pi |\bfx|^3} \hat{n}_j$. Certain components of the summation have alternating sign, and this leads to conditionally convergent sums, and in general this sum is at the border of convergence / divergence. The physical implication of this calculation is that the energy density of a large collection of dipoles is extremely sensitive to the precise boundary conditions that are imposed far away “at infinity”. Alternatively, in a finite body, this says that the energy [*density*]{} at a given point is extremely sensitive to the distribution of dipoles throughout the entire body. This physical implication is the reason to denote electrostatic interactions as “long-range”, namely, it is not possible to define a meaningful energy density that depends only on the local state of the crystal. The energy density at a given material point instead requires accounting for the state of the body at every other material point as well as boundary conditions. As we see below, this also poses practical difficulties for standard multiscale algorithms such as QC. This makes these methods inapplicable to an extremely broad class of solids: in all dielectrics, polarizable media, and ionic solids, the non-vanishing dipole moment in the unit cell is central to the physics of dielectric response, spontaneous polarization, and other key electrical properties.
Finally, consider the lattice of quadrupoles. As a rough measure of energy density, we compute the energy of the quadrupole in the chosen unit cell due to its interaction with all the other unit cells in the body. This is the product between the magnitude of the quadrupole in the chosen unit cell with the gradient of the electrostatic field created by the rest of the body. The electric field due to a quadrupole at a distance $r$ from the chosen unit cell scales as $r^{-5}$. Further, the shell at a distance $r$ contains again roughly contains $r^2$ quadrupoles. So the total electric field at the chosen unit cell due to the rest of the body is $\sum_{r=1}^{\infty} \frac{1}{r^5} r^2 = \sum_{r=1}^{\infty} \frac{1}{r^3}$. This sum converges rapidly. This setting corresponds to the case of metals and other systems with mobile electrons that allow the charge to redistribute itself to “shield” the dipole moment. This leads effectively to short-range interactions: though nominally the interactions are present at all values of $r$, the rapid convergence of the series allows truncation at finite cut-off without significant error. For this reason, among others, short-range potentials with interactions involving only nearest- and next-nearest-neighbors are sufficiently accurate to model metallic and related systems.
Existing Numerical Approaches to Compute Electrostatic Interactions
-------------------------------------------------------------------
The long-range nature of the electrical interactions described above leads to practical hurdles in atomic multiscale computations. Leading methods to handle these interactions are Ewald sums and the Fast Multipole Method (FMM). The Ewald method [@griebel-MD-book] assumes perfect periodicity. This is appropriate only for perfect crystals. Approximating defect calculations by periodic supercells has severe artifacts even with purely short-range interactions, a difficulty much more pronounced when interactions are long-range in nature. The fast multipole method (FMM) reduces the problem from $O(N^2)$ to $O(N)$, but is still extremely expensive with atomic multiscale calculations in crystals often as large as $N\sim 10^{21}$. In addition, the ability of FMM to deal with arbitrary charge distributions also implies that it does not exploit the close-to-uniform distortion away from the defect [@beatson-greengard-FMM-review].
For short-range interactions, multiscale atomistic methods such as the QC method borrow ideas of quadrature rules from FEM [@QC-review1; @QC-review2] to evaluate the energy at various sampling/quadrature atoms and then use quadrature weights. This idea depends critically on the energy evaluation at the quadrature point being a fast calculation. This is [*not*]{} a fast calculation if the quadrature charges interact directly with every other charge in the system. Therefore, these multiscale methods are applicable only to materials with short-range interactions. Multiscale QC-based methods for Orbital-Free Density Functional Theory – an empirical simplification of Density Functional Theory for metallic systems – use a continuous charge density rather than discrete point charges, but formally the issues are the same. Roughly, a [*predictor solution*]{} is patched together from the periodic solution in each “element”, and then a [*corrector solution*]{} due to the defect is superposed. The efficiency and accuracy of this approach requires that the corrector solution can be coarsely resolved away from the defect [@gavini-ortiz-bhatta]. However, in general there is a spatially-varying dipole moment in the specimen and zero dipole in the free space; therefore, the periodic calculation in any element will have large errors because it replaces this complex environment by a charge distribution with uniform dipole density. Consequently, the corrector can require fine resolution over much of the domain, except in settings such as metallic systems with net zero local dipole where they are currently applied.
An alternate approach to accounting for the large number of charge-charge interactions is to rewrite the problem as the electrostatic Poisson equation. However, this will lead to a highly-oscillatory forcing term that fluctuates at the atomic lengthscale while the problem is posed over the entire specimen. Therefore, this does not solve the essential difficulty. While numerical homogenization approaches may be feasible because the forcing close-to-periodic in many regions of the sample, it is not clear how to obtain full resolution in the vicinity of the defect. Further, the Poisson equation can be thought of as a nonlocal constraint that must be appended to (\[eq:energy\_lattice\]) in the minimization and therefore the essential non-local character remains.
Coarse-Graining the Electrostatic Field Energy {#sec:energy_derivation}
----------------------------------------------
We now consider the coarse-graining of the electrostatic energy. As noted above, we will work with a charge density field $\rho$ and assume that the coarse-graining is also valid for point charges by replacing $\rho$ with appropriate Dirac masses. Our starting point is to write $\rho$ following the ideas of 2-scale methods [@allaire-2scale; @debotton-bhattacharya]. I.e., we consider the setting where the charge density varies over 2 different lengthscales. There is a rapid almost-periodic variation of charge density at the lengthscale of the atomic unit cell (denoted $l$). In addition, there is a much slower variation over the characteristic continuum lengthscale denoted $L$. In the language of 2-scale methods, we can write the charge density field as $\rho(\bfx,\bfy)$ with $\bfy:=\bfx / l$ and $\rho$ periodic (with period of order one) in the second argument. A heuristic picture is that $\bfx$ specifies the location of the material point, and $\bfy$ specifies the location within the material point (Fig. \[fig:unit\_cell\_omega\]).
The electrostatic field energy can be written: $$\label{eqn:full-field-energy}
E
= \int_{\bfx,\bfx' \in\Omega} \frac{\rho(\bfx) \rho(\bfx')}{\left| \bfx - \bfx' \right|} dV_{\bfx} dV_{\bfx'}$$ We wish to examine the limit of the energy in the following setting. We introduce a lengthscale $\epsilon$ that, roughly speaking, denotes the size of the continuum material point. The limit of interest is then $l / \epsilon \rightarrow 0$ and $\epsilon / L \rightarrow 0$, or $l \ll \epsilon \ll L$. Essentially, the physical interpretation of this limit is that the atomic unit cell is much smaller than a material point, and a material point is much smaller than the lengthscale over which continuum fields vary.
We can now rewrite $E$ as $$\label{eqn:full-field-energy-2}
E
= \sum_{\bfx,\bfx' \in \Omega} \int_{(l \bfy) \in \mathcal{B}_\epsilon(\bfx), (l \bfy') \in\mathcal{B}_\epsilon(\bfx')} \frac{\rho(\bfx,\bfy) \rho(\bfx',\bfy')}{\left| \bfx + l \bfy - \bfx' - l\bfy' \right|} (l^3 \ dV_{\bfy}) (l^3 \ dV_{\bfy'})$$ The notation $\mathcal{B}_\epsilon(\bfx)$ denotes a ball of radius $\epsilon$ centered at $\bfx$. We note that $(l \bfy) \in \mathcal{B}_\epsilon(\bfx)$ implies $\bfy \in \mathcal{B}_{\epsilon/l}(\bfx)$.
We break up $E$ into 2 parts: a local term when $\bfx=\bfx'$, and a nonlocal term when $\bfx \neq \bfx'$:
[rCl]{} \[eqn:full-field-energy-3\] E & = & \_[ = ’]{}\
&& + \_[ ’]{}
In the limit that we will take, the nonlocal term represents the interactions between charges that are located at different material points $\bfx, \bfx'$, while the local term represents interactions between charges at the same material point.
We introduce some notation for what follows. We denote by $\square$ the rescaled atomic unit cell with characteristic dimension and volume of order $1$. The atomic unit cell with characteristic dimensions $l$ is denoted by $l\square$. We also use $\square_i$ and $l\square_i$ to denote the $i$-th atomic unit cell in a lattice.
### The Local Contribution of the Electrostatic Energy
For a fixed $\bfx$, the charge $\rho(\bfx,\bfy)$ is periodic in the second argument over $\square$. Therefore, we begin by rewriting the local term in (\[eqn:full-field-energy-3\]) in terms of integrals over $\square_i$: $$\label{eqn:local-energy-1}
\sum_{\bfx \in \Omega} \quad \sum_{\square_i, \square_{i'} \in \mathcal{B}_{\epsilon/l}(\bfx)} \int_{\bfy \in \square_i, \bfy' \in \square_{i'}} l^5 \frac{\rho(\bfx,\bfy) \rho(\bfx,\bfy')}{\left| \bfy - \bfy' \right|} dV_{\bfy} dV_{\bfy'}$$ The periodicity, and the fact that $\epsilon / l \rightarrow \infty$, together imply that every term in the sum relating the interaction between cells $i$ and $i'$ can be mapped to an interaction between cells $0$ and some $i''$. Therefore, the local term can now be written $$\label{eqn:local-energy-2}
\sum_{\bfx \in \Omega} \left( \frac{\epsilon}{l}\right)^3 \int_{\bfy \in \square_0, \bfy' \in \mathcal{B}_{\epsilon/l}(\bfx)} l^5 \frac{\rho(\bfx,\bfy) \rho(\bfx,\bfy')}{ \left| \bfy - \bfy' \right|} dV_{\bfy} dV_{\bfy'}$$ The factor $(\epsilon / l)^3$ is the number of terms in the sum that are replaced, obtained from dividing the volume of the ball of radius $\epsilon / l$ by the volume of $\square$.
This has the form of a Riemann sum: with $\epsilon \ll L$, the term $\epsilon^3$ is the volume measure. $$\label{eqn:local-energy-3}
\sum_{\bfx \in \Omega} \epsilon^3 \left(\int_{\bfy \in \square_0, \bfy' \in \mathcal{B}_{\epsilon/l}(\bfx)} l^2 \frac{\rho(\bfx,\bfy) \rho(\bfx,\bfy')}{\left| \bfy - \bfy' \right|} dV_{\bfy} dV_{\bfy'}\right)$$ The term in the brackets is the integrand and must be well-behaved, i.e. neither blow up nor go to $0$, in the limit $l \ll \epsilon$. The natural scaling is that the charge density must scale as $\rho(\bfx, \bfy) = \tilde{\rho}(\bfX,\bfy) / l$ where $\tilde{\rho}$ is the charge density on the rescaled unit cell $\square$ with characteristic dimension $1$, and $l \bfX = \bfx$. Note that $\tilde{\rho}$ has dimensions of charge per unit area. While this choice of scaling may appear arbitrary, we note that it can be recognized as the classical dipole scaling from elementary electrostatics. That is, in constructing the notion of a point dipole, one starts with charges that are separated by a finite distance and then takes the limit of the charges approaching each other. However, this limit leads to a finite dipole moment only when the charge magnitude is assumed to scale inversely with separation, thereby leaving the product of charge and separation distance finite. It is precisely this scaling which is required here for a finite local electrostatic energy. In our setting, if, for example, we assumed a fixed charge density and allowed the lattice spacing to go to $0$, charge neutrality would give us vanishing energy.
Using the charge scaling described above enables us to map the calculation of the integrand to a unit domain and gives the final form: $$\label{eqn:local-energy-4}
\int_{\bfx \in \Omega} \left(\int_{\bfy \in \square_0, \bfy' \in \mathcal{B}_{\epsilon/l}(\bfx)} \frac{\tilde{\rho}(\bfX,\bfy) \tilde{\rho}(\bfX,\bfy')}{\left| \bfy - \bfy' \right|} dV_{\bfy} dV_{\bfy'}\right) \ dV_{\bfx}$$ The energy in this form can be readily absorbed into standard energy densities that arise from applying the CB theorem to short-range interactions. This term has a number of different names: the Madelung energy in ionic solids [@kittel-book], the Lorentz local field, the weak-short contribution [@james-muller].
As an example, we replace the charge density with a set of Dirac masses representing point charges. The charge density in the unit cell is $\tilde{\rho}(\bfX,\bfy)= \sum_{\bfy\in\square_0} Q^s \delta_{\bfy_s}(\bfy)$ and extended periodically. The local energy has the form: $$\label{eq:local-energy-5}
E_{local} = \int_{\Omega} \left(\sum_{i,j \in \square_0} Q^i \bbS^{ij}Q^j + \frac{1}{3}\vert \bfp(\bfx)\vert^2 +\bfp(\bfx)\cdot \bfS \bfp(\bfx) \right) \ dV_{\bfx}$$ where $\bfp(\bfx) := \sum_{\bfy\in\square_0} Q^s \bfy \delta_{\bfy_s}(\bfy)$ is the polarization of the unit cell.
The quantities $\bbS, \bfS$ are defined as: $$\label{eq:SS_def}
\bbS^{ij} := \lim_{\omega \to \infty}\sum_{\bfy^i \in \square}\sum_{\substack{\bfz \in L_1 \setminus \bfy^i \\\cap \mathcal{B}_\omega}} \frac{1}{4 \pi \epsilon_0 \vert \bfy^i-\bfz^j \vert}
, \quad
\bfS := \lim_{\omega \to \infty}\sum_{\substack{\bfz \in L_1 \setminus \bfzero \\\cap \mathcal{B}_\omega}} \mathbb{K} (\bfz)$$ In the specific case that we have only 2 point charges in a unit cell, $\bbS$ vanishes and the local energy can be written in terms of $\bfp$ exclusively. The dipole kernel $\mathbb{K}$ is defined in (\[eqn:multipole-expansion\]).
An important point above is the presence of the limit in the definitions of $\bbS$ and $\bfS$. As noted previously, the full sums used above are conditionally convergent. The use of a limit is equivalent to enforcing a particular order of summation; in this case, it corresponds to using “neutral spheres” using the terminology of Ewald summation. Physically, it enforces that the far-field boundary conditions are set to $0$. The local contribution then is simply the energy of a uniform lattice of charges with vanishing far-field electric field; the lattice is uniform because the entire lattice is located at a single material point. In general, there can be a non-vanishing far-field electric field due to the other material points and continuum-scale boundary conditions, and this is introduced through the non-local contribution in the next section.
### Nonlocal Contribution of the Electrostatic Energy
We now focus on the nonlocal term in (\[eqn:full-field-energy-3\]), i.e., the interactions between charges at different material points. This contribution provides an energy that is very different from the standard local continuum energies. In particular, those energies are developed from the CB theorem that in the limit does not have any direct atomic interactions between different material points. Here, we have a clear nonlocal character to the energy.
We first introduce some notation regarding the multipole expansion. Consider the electrostatic interaction for charges located at $\bfx + l \bfy$ and $\bfx' + l\bfy'$: $$\label{eqn:multipole-expansion}
\begin{split}
\frac{1}{\left| \bfx + l\bfy - \bfx' - l\bfy' \right|}
=
& \frac{1}{\left| \bfx - \bfx' \right|}
+
\frac{\partial }{\partial \left(\bfx - \bfx' \right)} \left(\frac{1}{\left| \bfx - \bfx' \right|}\right) l \cdot \left( \bfy - \bfy' \right)
\\
&
+
\half
\underbrace{
\frac{\partial^2 }{\partial \left(\bfx - \bfx' \right)^2} \left(\frac{1}{\left| \bfx - \bfx' \right|}\right)
}_{\mathbb{K}} l^2 : \left( \bfy - \bfy' \right) \otimes \left( \bfy - \bfy' \right)
+
O(l^3)
\end{split}$$ The operator $\mathbb{K}$ is the dipole kernel.
In the nonlocal term in (\[eqn:full-field-energy-3\]), in anticipation of using the periodicity of $\rho(\bfx,\bfy)$ in $\bfy$ when $\bfx$ is held fixed, we reduce the integrations to unit cells $\square$: $$\label{eq:nonlocal-energy-1}
\sum_{\bfx,\bfx' \in \Omega; \bfx \neq \bfx'}
\quad
\sum_{\square_i \in \mathcal{B}_{\epsilon/l}(\bfx); \square_{i'} \in \mathcal{B}_{\epsilon/l}(\bfx')}
\quad
\int_{\bfy \in \square_i, \bfy' \in \square_{i'}} \frac{\rho(\bfx,\bfy) \rho(\bfx',\bfy')}{\left| \bfx + l \bfy - \bfx' - l\bfy' \right|} l^6 \ dV_{\bfy} dV_{\bfy'}$$ Assuming a separation of scales, i.e. $\epsilon \ll L$, the periodicity of $\rho$ implies that the interaction between charges contained in $\mathcal{B}_{\epsilon/l}(\bfx)$ and $\mathcal{B}_{\epsilon/l}(\bfx')$ can be replaced by interactions between charges in unit cells at $\bfx$ and $\bfx'$, and then multiplying by the number of unit cells in $\mathcal{B}_{\epsilon/l}(\bfx)$ and $\mathcal{B}_{\epsilon/l}(\bfx')$. $$\label{eq:nonlocal-energy-2}
\sum_{\bfx,\bfx' \in \Omega; \bfx \neq \bfx'}
\quad
\left(\frac{\epsilon}{l}\right)^3
\left(\frac{\epsilon}{l}\right)^3
\int_{\bfy \in \square, \bfy' \in \square} \frac{\rho(\bfx,\bfy) \rho(\bfx',\bfy')}{\left| \bfx + l \bfy - \bfx' - l\bfy' \right|} l^6 \ dV_{\bfy} dV_{\bfy'}$$ Canceling the factors of $l$ and using the notion of Riemann sums as above with $\epsilon^3$ as the volume measure, we can write this as a double integral: $$\label{eq:nonlocal-energy-3}
\int_{\bfx,\bfx' \in \Omega; \bfx \neq \bfx'}
\left(
\int_{\bfy \in \square, \bfy' \in \square} \frac{\rho(\bfx,\bfy) \rho(\bfx',\bfy')}{\left| \bfx + l \bfy - \bfx' - l\bfy' \right|} \ dV_{\bfy} dV_{\bfy'}
\right)
\ dV_{\bfx} \ dV_{\bfx'}$$ As in the local contribution, we require the integrand in brackets above to be well-defined when $l \rightarrow 0$. Recall the dipole scaling $\rho = \tilde{\rho} / l$: the integrand is therefore well-behaved if $\frac{1}{\left| \bfx + l \bfy - \bfx' - l\bfy' \right|}$ scales as $l^2$.
We substitute the multipole expansion from (\[eqn:multipole-expansion\]) and notice immediately that the first term scales independently of $l$ and the second term scales linearly in $l$. These would then potentially cause the integrand to diverge as $l\rightarrow 0$. However, we recall that each unit cell is charge-neutral, i.e. $\int_\square \tilde{\rho}(\bfx,\bfy) \ dV_{\bfy} = 0$. This causes both the first and second terms in the multipole expansion to vanish. Physically, this means that the energy is unbounded if every unit cell is not charge-neutral, e.g. recalling the example in Section \[toy-example\].
Next, we consider the term $\half l^2 \mathbb{K} : \left( \bfy - \bfy' \right) \otimes \left( \bfy - \bfy' \right)$. The terms containing $\bfy \otimes \bfy$ and $\bfy' \otimes \bfy'$ vanish from charge neutrality. The only remaining terms can be readily written as: $$\label{eq:nonlocal-energy-4}
\begin{split}
\int_{\bfx,\bfx' \in \Omega; \bfx \neq \bfx'}
\left(
\mathbb{K}(\bfx-\bfx') :
\underbrace{
\int_{\bfy \in \square} \tilde{\rho}(\bfX,\bfy) \bfy \ dV_{\bfy}
}_{\bfp(\bfx)}
\otimes
\underbrace{
\int_{\bfy' \in \square} \tilde{\rho}(\bfX',\bfy') \bfy' \ dV_{\bfy'}
}_{\bfp(\bfx')}
\right)
\ dV_{\bfx} \ dV_{\bfx'}
\\
=
\int_{\bfx,\bfx' \in \Omega; \bfx \neq \bfx'} \bfp(\bfx') \cdot \mathbb{K}(\bfx-\bfx') \bfp(\bfx) \ dV_{\bfx} \ dV_{\bfx'}
\end{split}$$ where the terms containing $\bfy \otimes \bfy'$ and $\bfy' \otimes \bfy$ have been combined using symmetry.
Consider finally the terms denoted $O(l^3)$. These will all go to $0$ as $l\rightarrow 0$. Physically, these terms represent the contributions from quadrupole and higher-order moments of the charge distribution, i.e. $\int_\square \tilde{\rho}(\bfx,\bfy) \bfy \otimes \bfy \ dV_{\bfy}$ and higher-order. We see that these terms vanish identically in the limit. Therefore, terms of higher-order than dipole do not appear in the nonlocal part of the continuum energy, recalling the example in Section \[toy-example\]. In general, all short-range forces – i.e. those that decay faster than dipolar interactions – do not contribute to the nonlocal term in the limit [@friesecke-james; @blanc-lebris-lions].
Finally, a long but straightforward calculation using the divergence theorem and integration-by-parts gives $$\label{eq:nonlocal-energy-5}
\begin{split}
& \int_{\bfx,\bfx' \in \Omega} \bfp(\bfx') \cdot \mathbb{K}(\bfx-\bfx') \bfp(\bfx) \ dV_{\bfx} \ dV_{\bfx'}
=
\\
& \int_{\bfx,\bfx' \in \Omega} \divergence \bfp(\bfx') G(\bfx-\bfx') \divergence \bfp(\bfx) \ dV_{\bfx} \ dV_{\bfx'}
+ \int_{\bfx,\bfx' \in \partial\Omega} \bfn \cdot \bfp(\bfx') G(\bfx-\bfx') \bfn \cdot \bfp(\bfx) \ dS_{\bfx} \ dS_{\bfx'}
\\
& - 2 \int_{\bfx \in \Omega,\bfx' \in \partial\Omega} \bfn \cdot \bfp(\bfx') G(\bfx-\bfx') \divergence \bfp(\bfx) \ dV_{\bfx} \ dS_{\bfx'}
\end{split}$$ where $G$ is the standard electrostatics Greens function and $\mathbb{K}:=\nabla^2 G$ from (\[eqn:multipole-expansion\]). Note that the condition $\bfx\neq\bfx'$ has not been written for brevity. An important conclusion from the above formula is that $- \divergence\bfp$ is equivalent to a bulk charge density (the so-called “bound bulk charge density”) and that $\bfp\cdot\bfn$ is equivalent to a surface charge density (the so-called “bound surface charge density”). In that perspective, the formula above simply gives the energy of this composite charge distribution using the usual Green’s function relation between charge density and energy. When $\bfp$ is discontinuous along interior surfaces, this formula gives bound surface charges along these surfaces.
While we have derived the equivalent bound charges using standard integration formulas after taking the limit of the Riemann sum, it is straightforward to derive these directly. Essentially, we manipulate the Riemann sum using the standard approach in the Riemann sum proof of the divergence theorem (e.g., [@tang-book]). In the interior, we find $\divergence\bfp$ appearing and the boundaries of the infinitesimal element canceling with it’s neighbors. On the boundary of $\partial\Omega$, there is no cancellation leading to the contribution $-\bfp\cdot\bfn$.
### Boundary Contributions due to Partial Unit Cells {#sec:boundary-charge}
We consider now the role of boundaries. As we have mentioned above and will discuss below in detail, the value of the polarization depends on the chosen unit cell. Boundary contributions are essential to ensure that the coarse-grained electric fields and other quantities do not depend on the arbitrary choice of unit cell. There are two contributions: first, due to polarization terminations $-\bfp\cdot\bfn$, and second, surface charges due to partial unit cells on the surface that are not charge-neutral. The polarization terminations have already been accounted for as shown in (\[eq:nonlocal-energy-5\]). In this section, we consider the case of the surface charges due to partial non-charge-neutral unit cells.
For simplicity, we do not compute the total energy which will have straightforward but tedious cross-terms between interior bound charges (due to $\divergence \bfp$) and surface charges as can be seen (\[eq:nonlocal-energy-5\]). Instead, we compute the electric potential field due to the surface charges where the calculations are more transparent. In a formal setting, one can readily go between these calculations.
Consider a point $\bfx \in \partial\Omega$. At a point $\bfx' \neq \bfx$, the electric potential due to the charges at $\bfx$ is given by: $$\label{eqn:boundaries-1}
\phi(\bfx') = \sum_{\bfx \in \partial\Omega} \int_{l\bfy \in \mathcal{D}_{\epsilon}(\bfx) \times C l \bfn} \frac{\rho(\bfx,\bfy)}{|\bfx + l\bfy - \bfx'|} l^3 \ dV_{\bfy}$$ Here $\mathcal{D}_{\epsilon}(\bfx)$ is a 2D disk of radius $\epsilon$ located at $\bfx$. Therefore, $\mathcal{D}_{\epsilon}(\bfx) \times l \bfn$ denotes a squat cylinder of height $C l$ oriented with axis $\bfn$ and cross section $\mathcal{D}_{\epsilon}(\bfx)$. The vector $\bfn$ is the unit outward normal to $\Omega$. It is implicit in the above formula that we are considering charges only in the partial unit cells.
We assume that the surface is a rational plane (see Appendix \[sec:appendix\] for the definition). The integration above in the directions along the surface (i.e., normal to $\bfn$) can then be reduced to an integration over a single unit cell because of periodicity in those directions[^3]. The integration in the direction along $\bfn$ reduces simply to the partial unit cells on the boundary and is therefore independent of $C l$. Following the ideas above for the volume contributions, we can then rewrite this as an integral over a unit cell by putting in the appropriate factor for the number of unit cells in the disk: $$\label{eqn:boundaries-2}
\phi(\bfx') = \sum_{\bfx \in \partial\Omega} \frac{\epsilon^2}{l^2} \int_{\bfy \in \triangle} \frac{\tilde{\rho}(\bfX,\bfy)/l}{|\bfx + l\bfy - \bfx'|} l^3 \ dV_{\bfy}$$ where we have used the notation $\triangle$ for partial non-neutral unit cells.
Therefore, we require that only the term independent of $l$ from (\[eqn:multipole-expansion\]) appear above. Upon taking the Riemann sum and defining the surface charge density $\sigma(\bfx)$, this gives the expected and simple result: $$\label{eqn:boundaries-3}
\phi(\bfx') = \int_{\bfx \in \partial\Omega} \frac{1}{|\bfx - \bfx'|} \underbrace{\left( \int_{\bfy \in \triangle} \tilde{\rho}(\bfX,\bfy) \ dV_{\bfy} \right)}_{\sigma} \ dS_{\bfx}$$ While we have assumed for simplicity that $\bfx \neq \bfx'$, considering the case $\bfx = \bfx'$ and examining the energy would give us a local contribution analogous to that in the case of the bulk.
Role of Boundaries in Compensating for the Non-uniqueness of Polarization
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It is well-known, e.g. [@Resta-Vanderbilt], that the value of $\bfp$ in a periodic solid depends on the choice of unit cell. This would appear to be a fatal difficulty in using $\bfp$ as a multiscale mediator between the atomic-scale-variation of $\rho$ and continuum-scale quantities. In the materials physics community, quantum mechanical notions are invoked to obtain a unique choice of polarization for a given periodic charge distribution [@Resta-Vanderbilt]. However, from the perspective of the calculations above, we are simply coarse-graining classical electrostatic interactions and there is no reason for quantum mechanics to play any role. As we describe in this section, the difficulties noticed by [@Resta-Vanderbilt] are entirely due to their starting-point of an infinite periodic solid. This makes the notion of boundaries ill-defined. If instead we begin from a finite solid and take the limit of lattice spacing being much smaller than the size of the body (the large-body limit), we see that the surface charges on the boundaries play a critical role. In short, while the polarization is itself not uniquely-defined, the electrostatic energy that comes from accounting for both the polarization and the surface charge is a unique quantity. While changing the unit cell changes the value of the polarization density, it also changes the boundary charge in the partial unit cells. These compensate to give the same value for the electrostatic energy.
### One-Dimensional Illustrative Example {#sec:unit_cell_example}
Consider a finite body $\Omega = (-L,L)\times (-1,1) \times (-1,1)$ with a one-dimensional charge distribution $\rho(\bfx) = \rho_0 \sin (2\pi \frac{x_1}{l})$ (Fig. \[fig:1D-example-boundaries\]). We assume that $L$ is an integer multiple of $l$, i.e. $n l = L$. Guided by the multipole expansion, we compute the dipole moment as the leading contributor to the behavior of the bar without using the fact that the charge distribution is in fact periodic in $\Omega$. We then compare this to the result obtained by using polarization density field that is defined on the unit cell.
![Charge distribution in the 1D illustrative example to show the effect of boundaries.[]{data-label="fig:1D-example-boundaries"}](Pictures/1D-example-boundaries.png){width="\textwidth"}
The total dipole moment of the bar $\bfP := \int_{\Omega} \rho(\bfr') \bfr' \ dV_{\bfr'}$ evaluates to $(1,0,0) \times \frac{-4\rho_0 L l}{ \pi}$.
The polarization density $\bfp$ in a single unit cell $l\square = (a,l+a)\times (-1,1) \times (-1,1)$ is defined as $\bfp := \frac{1}{\text{volume}(l \square)} \int_{\bfr' \in l \square} \rho(\bfr') \bfr' \ dV_{\bfr'}$. It evaluates to $(1,0,0) \times \frac{- \rho_0 l}{2 \pi} \cos(2\pi\frac{a}{l})$. This is a classic example showing that $\bfp$ depends on the chosen unit cell, here parametrized by $a$ [@Resta-Vanderbilt]. From (\[eq:nonlocal-energy-5\]) and the associated discussion, we have no bulk charge because $\bfp$ is identical in each unit cell, but there is a surface charge density given by $-\bfp\cdot\bfn$. Therefore, there is a total charge of $\frac{2 \rho_0 l}{\pi} \cos(2\pi\frac{a}{l})$ at $+L$, and $\frac{- 2 \rho_0 l}{\pi} \cos(2\pi\frac{a}{l})$ at $-L$. However, there are partial unit cells at each end: $(-L,-L+a)$ and $(L-l+a, L)$, and these are not charge neutral. The charge in these cells evaluates to $\frac{-2 \rho_0 l}{\pi} \left( 1 - \cos(2\pi\frac{a}{l})\right)$ and $\frac{2 \rho_0 l}{\pi} \left( 1 - \cos(2\pi\frac{a}{l})\right)$ respectively. Therefore the total charge at each end, both from the partial unit cells and from $-\bfp\cdot\bfn$, is $\pm \frac{2 \rho_0 l}{\pi}$. These equal and opposite charges are separated by a distance $2 L$. Therefore, this is a dipole of strength $\frac{-4 \rho_0 l L}{\pi} $. Note that we have errors up to order $l$ because the charges due to the polarization terminating on the surface are separated by $L-l$; however, the key point is that in the limit of $l \ll L$, we recover the dipole $\bfP$.
### The General Case
The key lesson from the example above is that it is critical to account for the charge in the partial unit cells on the boundary. This ensures that the coarse-graining that exploits the polarization as a multiscale mediator does not depend on the choice of unit cell. We now examine this in the 3D setting.
First, we decompose $\Omega$ into $\Omega_\square$, with only complete unit cells, and $\Omega_{\#}$ with the surface layer of incomplete unit cells, Fig. \[fig:omega-decomposition\].
![Decompose $\Omega$ into $\Omega_\square$ and $\Omega_{\#}$.[]{data-label="fig:omega-decomposition"}](Pictures/omega2.png){width="180mm"}
We now consider the unit cells adjacent to $\Omega_{\#}$. Our goal is to modify these unit cells in various ways, and show that the resulting changes in surface charge and polarization density balance each other.
An important element of our strategy is to transform the given unit cell to a unit cell that has a face parallel to the surface under consideration. Appendix \[sec:appendix\] shows that a unit cell with this property can always be found when the surface is rational. As in Section \[sec:boundary-charge\], we restrict attention to rational surfaces.
For this special choice of unit cell, we now consider the changes in surface charge and polarization density for various operations. We use the notation that the lattice vectors tangential to the surface are $\bfh_2$ and $\bfh_3$. We note that the volume of the unit cell can be written $\bfh_2 \times \bfh_3 \cdot \bfh_1 = \vert \bfh_2 \times \bfh_3 \vert \bfn \cdot \bfh_1$.
First, consider translations of the unit cell as shown in Fig. \[fig:unit-cell-change-1\].
Consider Fig. \[fig:unit-cell-change-1\]a. A translation along $\bfh_1$ causes an increase in the uncompensated surface charge area density $\Delta\sigma = \frac{1}{\vert \bfh_2 \times \bfh_3 \vert} \int_{\text{\textcircled{1}}} \rho$ in the partial unit cells. The increase in the polarization density in the translated unit cell is $\Delta\bfp = \frac{1}{\vert \bfh_2 \times \bfh_3 \vert \bfn \cdot \bfh_1} \left( \int_{\text{\textcircled{2}}} \rho \bfy - \int_{\text{\textcircled{1}}} \rho \bfy \right)$. From the periodicity of the charge distribution $\rho(\bfy+\bfh_1) = \rho(\bfy)$, we have $\Delta\bfp = \frac{1}{\vert \bfh_2 \times \bfh_3 \vert \bfn \cdot \bfh_1} \bfh_1 \int_{\text{\textcircled{2}}} \rho $. Therefore $\Delta\bfp\cdot\bfn = \Delta\sigma$.
Consider Fig. \[fig:unit-cell-change-1\]b. A translation along either $\bfh_2$ or $\bfh_3$ causes no change in $\sigma$. The change in polarization is $\frac{1}{\vert \bfh_2 \times \bfh_3 \vert \bfn \cdot \bfh_1} \bfh_2 \int_{\text{\textcircled{2}}} \rho $. Therefore $\Delta\bfp \cdot\bfn = 0$.
In general, one can have a translation along any direction. Such a translation can be decomposed into components along the lattice directions and the calculations above applied in succession to each direction.
![Change in charge and polarization due to a translation along $\bfh_1$ and $\bfh_2$ respectively in the special choice of unit cell.[]{data-label="fig:unit-cell-change-1"}](Pictures/1final.png){width="180mm"}
Second, consider distortions of the unit cell as shown in Fig. \[fig:unit-cell-change-2\].
From reasoning following very closely the previous case of translations of the unit cell, we find that the relation $\Delta\bfp\cdot\bfn = \Delta\sigma$ holds here too.
![Change in charge and polarization due a distortion of the unit cell in the special choice of unit cell.[]{data-label="fig:unit-cell-change-2"}](Pictures/2final.png){width="180mm"}
Third, consider changes in the unit cell due to remapping of the lattice vectors as shown in Fig. \[fig:unit-cell-change-3\].
As noted in Appendix \[sec:appendix\], the relation between $\{ \bff_1, \bff_2, \bff_3 \}$ and $\{ \bfh_1, \bfh_2, \bfh_3 \}$ must be of the form: $\bff_i = \sum_j \mu_i^j \bfh_j$ with $\mu_i^j$ a matrix of integers with determinant $\pm 1$. An example of such a remapping is in Fig. \[fig:unit-cell-change-3\]a.
As shown in the example in Fig. \[fig:unit-cell-change-2\]bcde, regions of the original unit cell are mapped to the new unit cell. For instance, maps to , maps to , and maps to . Each of these regions is translated by an integer linear combination of $\{ \bfh_1, \bfh_2, \bfh_3 \}$; however, each region may have a [*different*]{} integer combination translation. In addition, the uncompensated unit cell on the boundary also increases in extent (Fig. \[fig:unit-cell-change-2\]f). As above, for a periodic charge distribution, when a charged region is translated by an integer multiple of a lattice vector, the consequent change in polarization is simply the total charge in the region times the translation distance. Therefore $\Delta\bfp = \frac{1}{\vert \bfh_2 \times \bfh_3 \vert \bfn \cdot \bfh_1} \left(\sum_i \nu^1_i \bfh_1 \int_{A_i} \rho + \sum_i \nu^2_i \bfh_2 \int_{A_i} \rho + \sum_i \nu^3_i \bfh_3 \int_{A_i} \rho\right)$, where $\nu^1_i,\nu^2_i,\nu^3_i$ are integers.
The change in the uncompensated charge is simply $\Delta\sigma = \frac{1}{\vert \bfh_2 \times \bfh_3 \vert} \sum_i \nu^1_i \int_{A_i} \rho$ which is related to the extent of the translation along $\bfh_1$. This matches precisely with $\Delta \bfp\cdot\bfn$.
![Change in charge and polarization due a remapping of the unit cell from the special choice of unit cell.[]{data-label="fig:unit-cell-change-3"}](Pictures/3final.png){width="180mm"}
In the general case of modifying a given unit cell to another shape by any combination of the mechanisms studied above, we can conceptually consider mapping the given unit cell to the special unit cell with a surface-parallel face, conducting the modifications with the special unit cell, and then mapping to the desired final unit cell. Of course, in practice none of this need be done; as long as we are assured that changes in the unit cell are compensated by boundary charges appropriately, we can directly modify the unit cell as desired.
In the interior of the body, $\divergence\bfp$ changes by $O(l)$ when the unit cell is changed. This follows directly from the definition of $\bfp$ in (\[eq:nonlocal-energy-4\]) and the chain rule, using the fact that $\bfX / l = \bfx$. Therefore, the bound bulk charge density is the same in the limit of $l/L \ll 1$.
We note potential connections to ideas of Null-Lagrangians in the issue of a unique definition of the polarization [@ericksen-nilpotent]. Essentially, one can have different expressions for the free-energies, but these lead to the same Euler-Lagrange equation but with different boundary conditions. Similarly, different choices for the unit cell leave the bulk electrostatics unchanged, because the change in the bulk polarization is compensated by the boundary contributions.
Numerical Implementation {#sec:implementation}
========================
Our numerical implementation for the short-range interactions follows closely the standard QC, e.g. [@Tadmor]. In standard QC, there are two essential steps: first, a reduction of degrees of freedom by interpolation, typically using linear shape functions inspired by finite elements, with atomic resolution in critical regions and coarse-grained elsewhere (Figs. \[fig:mesh\], \[fig:mesh\_zoom\]); and second, a fast estimation of the energy (or derivative of the energy with respect to the retained degrees of freedom) using Cauchy-Born sampling. Among the many variants of QC, we use the local QC for multi-lattices, following [@Tadmor]. Local QC refers to the use of sampling [*everywhere*]{} in the specimen, not only in the coarse-grained region but also in regions with atomic resolution in the interpolation.
Dielectrics require a multi-lattice description because a dielectric response requires at least two charges in the unit cell that move independently to change the polarization in response to electromechanical fields. Essentially, the multi-lattice description uses $\bfF$ to track the deformation of the unit cell, and a set of vector-valued fields $\bfzeta^s$ that track the position of individual species $s$ within the unit cell. We use linear interpolation for the coarse-grained displacement field $\bfu$ implying that $\bfF$ is constant in a given element. To be consistent with this spatial variation of $\bfF$, we use the same “constant in each element” interpolation for $\bfzeta^s$. If the element were of infinite extent, this choice of interpolation would ensure that the energy density converges to the standard Cauchy-Born theorem. In the local QC approximation, we estimate the energy of a given element by finding the energy density of atomic unit cells at the selected quadrature / sampling points and multiply by the appropriate weight.
This interpolation of the kinematic variables $\bfF, \bfzeta^s$ implies that the polarization $\bfp$ is also constant in a given element. Therefore, in terms of effective bound charges, we have no bound bulk charges ($\divergence \bfp = 0$), and we have surface charge density $(\bfp_1 - \bfp_2)\cdot \bfn$ at the element faces. In the coarse-grained local QC approximation, the evaluation of electrostatic fields consists simply of finding the fields set up by the charge distributions. One element of electrostatics is that the electric potential and field are naturally posed in the current configuration. Therefore, for numerical updates, we compute the electrostatic fields in the current and then pull back to the reference using standard electromechanical transformations [@xiao-bhatta]. We assume in the remainder of the paper that we are dealing with point charges that can move around, but do not change their charge.
Energy Minimization through Gradient Descent
--------------------------------------------
Our interest is in finding energy minimizers to the coarse-grained problem. Under the local QC approximation, the coarse-grained problem can be written as the standard continuum energy for electromechanical solids [@shu-bhatta; @xiao-bhatta]: $$\label{eq:total_energy_again}
E = \underbrace{\int_{\Omega_0} W(\grad_{\bfx_0} \mathbf{u},\bfzeta^s) \ dV_{\bfx_0}}_\text{short-range energy}
+ \underbrace{ \frac{\epsilon_0}{2}\int_{\R^3}\vert \grad_{\bfx} \phi \vert^2 \ dV_{\bfx} }_\text{long-range (non-local) energy}
\quad , \quad \divergence_{\bfx} \grad_{\bfx} \phi = \divergence_{\bfx} \bfp$$ The nonlocal expression in (\[eq:nonlocal-energy-5\]) can be transformed to the form above by first noting that the right side of (\[eq:nonlocal-energy-5\]) is entirely in terms of the electrostatic Greens function $G$. Therefore, the electrostatic field $\phi$ can be defined as the solution of the electrostatic equation with charges given by $-\divergence\bfp$ and $\bfp\cdot\bfn$. The energy density is then simply $\vert \grad_{\bfx} \phi \vert^2$. The surface charges due to $\bfp\cdot\bfn$ as well as due to incomplete unit cells appear in the boundary conditions for the electrostatic equation. The local contribution of the electrostatic energy has been absorbed into the short-range term. Note that the non-local energy integral is posed in the current configuration.
We use a gradient-flow evolution to find the (local) minimizers following [@xiao-bhatta; @zhang-bhatta]. The independent variables that remain are $\bfu$ and $\bfzeta^s$. The polarization is completely defined in terms of unit cell geometry $\bfu$ and intra-cell positions of the charges $\bfzeta^s$, in turn defining the electrostatic potential $\phi$ through the electrostatic Poisson equation. Therefore, taking variations: $$\label{eq:deformation_variation}
\grad_{\bfx_0} \bfu \to \grad_{\bfx_0} \bfu +\eta \grad_{\bfx_0} \bfv
\quad , \quad
\bfzeta^s \to \bfzeta^s + \eta \bftheta^s$$ Additionally, the variation in the electric field is $\grad_{\bfx} \phi \to \grad_{\bfx} \phi +\eta \grad_{\bfx} \psi$ but this variation is constrained to variations in $\bfp \to \bfp + \eta \bfq$ by the Poisson equation, i.e. $\divergence_{\bfx} \grad_{\bfx} \psi = \divergence_{\bfx} \bfq$.
The definition of gradient flow results in the following statement: $$\label{eq:grad_flow}
\begin{split}
\int_{\Omega_0} \left (\dot{\bfu}\cdot \bfv + \sum_{s} \dot{\bfzeta}^s \cdot \bftheta^s \right ) \ dV_{\bfx_0}
= & - \left. \frac{\partial{}}{\partial{\eta}} E \left ( \grad_{\bfx_0} \bfu +\eta \grad_{\bfx_0} \bfv, \bfzeta^s + \eta \bftheta^s \right) \right|_{\eta=0} \\
= & \int_{\Omega} \left( \frac{\partial{W}}{\partial{(\grad_{\bfx_0} \bfu)}}: \grad_{\bfx_0}\bfv + \sum_{s} \frac{\partial{W}}{\partial{\boldsymbol{\zeta^s}}} \cdot \bftheta^s \right) \ dV_{\bfx_0} \\
& + \epsilon_0 \int_{\R^3} \grad_{\bfx} \phi \cdot \grad_{\bfx} \psi \ dV_{\bfx}
\end{split}$$ The nonlocal integral over $\R^3$ can be transformed by multiplying $\divergence_{\bfx} \grad_{\bfx} \psi = \divergence_{\bfx} \bfq$ by $\phi$ and integrating over $\R^3$. Using integration by parts on the left side and then pulling it back to the reference gives: $$\int_{\Omega_0}\grad_{\bfx} \phi \cdot \bfq J \ dV_{\bfx_0} = \int_{\R^3} \grad_{\bfx} \phi \cdot \grad_{\bfx} \psi \ dV_{\bfx}$$ where $J$ is the Jacobian of the deformation. This is substituted for the nonlocal integral in (\[eq:grad\_flow\]) to get: $$\label{eq:grad_flow-1}
\begin{split}
\int_{\Omega_0} \left (\dot{\bfu}\cdot \bfv + \sum_{s} \dot{\bfzeta}^s \cdot \bftheta^s \right ) \ dV_{\bfx_0}
=
&
\int_{\partial{\Omega_0}}\frac{\partial{W}}{\partial{(\grad_{\bfx_0} \bfu)}} : \bfn \bfv \ dS_{\bfx_0}
- \int_{\Omega_0} \left(\divergence_{\bfx_0} \frac{\partial{W}}{\partial{(\grad_{\bfx_0} \bfu)}}\right) \cdot \bfv \ dV_{\bfx_0}
\\
&
+\int_{\Omega_0}\sum_{s} \frac{\partial{W}}{\partial{\bfzeta^s}}\cdot \bftheta^s \ dV_{\bfx_0}
+ \epsilon_0 \int_{\Omega_0} \grad_{\bfx} \phi \cdot \bfq J \ dV_{\bfx_0}
\end{split}$$
We now examine the relation between $\bfq$ and the variations $\bfv$ and $\bftheta$. The polarization density $\bfp$ is defined by $$\label{eq:polar_def_2}
\bfp=\frac{1}{\det(\bfF) V_0} \sum_{s} Q^s \bfF \bfzeta^s$$ Note that $\bfzeta^s$ is the position of species $s$ in the reference configuration, while $\bfp$ is the polarization density in the current configuration. Similarly, $V_0$ is the volume of the unit cell in reference configuration. Hence, both $V_0$ and $\bfzeta^s$ are pushed forward to the current.
Taking the variation of $\bfp$ $$\label{eq:polar_def_q}
\bfp+ \eta \bfq= \frac{1}{\det(\bfF + \eta \bfG) V_0} \sum_{s} Q^s (\bfF+\eta \bfG) (\bfzeta^s+ \eta \bftheta^s)$$ where $\bfG:=\grad_{\bfx_0} \bfv$. Noting that $(\det(\bfF+\eta \bfG))^{-1} = \det(\bfF)^{-1} \det(\bfI+\eta \bfF^{-1} \bfG)^{-1} = \det(\bfF)^{-1} \left( 1-\eta \tr(\bfF^{-1}\bfG)- O(\eta^2) \right)$, we find: $$\label{eq:q_def}
\bfq = \frac{1}{\det(\bfF) V_0} \sum_{s} Q^s \left( \bfF \bftheta^s+ \bfzeta^s \cdot \grad_{\bfx_0} \bfv - \tr(\bfF^{-1} \grad_{\bfx_0} \bfv ) \bfF \bfzeta^s \right)$$ after ignoring terms of order higher than linear in $\eta$. Using this expression for $\bfq$, we obtain the following expression for $\int_{\Omega_0}\grad_{\bfx}\phi \cdot \bfq J \ dV_{\bfx_0}$ after using integration-by-parts and the divergence theorem: $$\label{eq:q_energy}
\begin{split}
\frac{1}{V_0} \sum_{s} Q^s \int_{\Omega} \grad_{\bfx}\phi \cdot \left(\bfF \bftheta^s \right) \ dV_{\bfx_0}
+
\frac{1}{V_0} \sum_{s} Q^s \left(\int_{\partial{\Omega_0}} \grad_{\bfx}\phi \bfzeta^s : \bfn \bfv \ dS_{\bfx_0}
-\int_{\Omega_0}\divergence_{\bfx_0}\left(\grad_{\bfx}\phi \bfzeta^s \right) \cdot \bfv \ dV_{\bfx_0} \right)
\\
-
\frac{1}{V_0} \sum_{s} Q^s \left( \int_{\partial{\Omega_0}}\grad_{\bfx}\phi \cdot \left( \bfF \bfzeta^s \right) \bfF^{-T} : \bfn \bfv \ dS_{\bfx_0}
- \int_{\Omega_0}\divergence_{\bfx_0}\left( \grad_{\bfx}\phi \cdot \left( \bfF \bfzeta^s \right) \bfF^{-T} \right) \cdot \bfv \ dV_{\bfx_0} \right)
\end{split}$$
Defining $\bfA^s := \grad_{\bfx}\phi \bfzeta^s - \grad_{\bfx}\phi \cdot \bfF\bfzeta^s \bfF^{-T}$, collecting terms in (\[eq:q\_energy\], \[eq:grad\_flow-1\]) and localizing using the arbitrariness of the variations gives us the compact form: $$\label{eq:u_gov}
\dot{\bfu} = \divergence_{\bfx_0} \left(\frac{\partial{W}}{\partial{\grad_{\bfx_0} \bfu}}+\epsilon_0\sum_{s} \frac{Q^s}{V_0}\bfA^s\right)
\quad , \quad
\dot{\bfzeta^s} = -\frac{\partial{W}}{\partial{\bfzeta^s}} -\epsilon_0\frac{Q^s}{V_0} \grad_{\bfx} \phi \cdot \bfF$$ with the boundary term: $$\label{eq:boundary_gov}
\left(\frac{\partial{W}}{\partial{\grad_{\bfx_0} \bfu}} + \epsilon_0\sum_{s}\frac{Q^s}{V_0}\bfA^s \right) \cdot \bfn =0$$ These equations provide the gradient descent equations for the fields $\bfu$ and $\bfzeta^s$. The equation for $\bfu$ involves a standard mechanical stress as well as a so-called Maxwell or electromechanical stress. The equation for $\bfzeta^s$ is a local equation (i.e., not a PDE), but is coupled through $W$ and the electric field to the nonlocal electrostatics and the PDE for momentum balance.
### Electromechanical Transformations to the Reference Configuration
For simplicity, we solve the electrostatic Poisson equation in the current configuration for the electric potential $\phi(\bfx)$ and compute the electric field $\grad_{\bfx}\phi(\bfx)$ and rewrite $\bfx=\bfx(\bfx_0)$ from the deformation map. In our setting, the energy (both short- and long-range) and electric fields are based entirely on the atomic position in the current configuration. Therefore, in common with much of continuum mechanics, the reference configuration can be considered simply a change of variables for bookkeeping convenience. It follows that the definition of electrical quantities in the reference configuration are not essential. This can also be readily observed from (\[eq:q\_energy\]); the physical quantity of interest there is $\grad_{\bfx}\phi(\bfx)$, and while it is certainly possible to define the electric field in the reference, it is not physically important.
This has led to a number of different proposals for referential electrical quantities in the literature. For instance, our (\[eq:polar\_def\_2\]) suggests that the reference polarization $\bfp_0(\bfx_0)$ is given by $\bfp(\bfx(\bfx_0)) = \det(\bfF)^{-1} \bfF \bfp_0(\bfx_0)$, i.e. the polarization transforms as material line elements that carry charges but with an additional factor accounting for volume changes.
In [@xiao-bhatta], they assume instead $\bfp(\bfx(\bfx_0)) = \det(\bfF)^{-1} \bfp_0(\bfx_0)$, and further assume that the reference electrostatic potential $\phi_0(\bfx_0) = \phi(\bfx(\bfx_0))$; this gives them that the electric field transforms as line elements using the identity $\grad_{\bfx} = \bfF^{-T}\grad_{\bfx_0}$. This is consistent with the differential geometric notion of the electric field as a $1$-form, i.e. it is a quantity that is integrated along lines.
In [@ponte-siboni], following [@dorfmann-ogden], they use yet another transformation; they work with electric displacement $\bfD$ and electric field $\bfE$ as primary variables rather than polarization. These variables are motivated by the fact that the continuum problem posed in polarization and electric potential leads to a saddle-point variational problem, whereas the minimization structure is preserved in $\bfD$ and $\bfE$. The electrostatic equations in these variables are $\divergence \bfD = 0$ and $\curl \bfE = 0$. For $\bfE$, these imply it should transform as a material line element and matches with [@xiao-bhatta] as well as the differential geometric notion of the electric field as a $1$-form. For $\bfD$ however, these imply a transformation $\bfD = \det(\bfF)^{-1} \bfF^{-T} \bfD_0$ following the differential geometric notion of the electric displacement as a $2$-form, i.e. a quantity that is integrated over surfaces; this is obviously identical to the standard stress transformation. This differential geometric notion is also implicitly exploited in Section 4 of [@kohn-shipman] in finding the appropriate averaging for the different field quantities. An essential practical advantage of this transformation that is exploited by [@ponte-siboni] in their homogenization analysis is that the equations retain their structure in the reference configuration, i.e. $\divergence_{\bfx_0} \bfD_0 = 0$ and $\curl_{\bfx_0} \bfE_0 = 0$.
While the transformations proposed by these other workers are physically appealing for the reasons mentioned above, the transformation implied by the microscopic model of the polarization is also physically motivated. These different transformations are not consistent with preserving the relation $\bfD = \epsilon_0 \bfE + \bfp$ between corresponding quantities in the reference.
Local Quasicontinuum for Multi-lattices with Short-Range Interactions
---------------------------------------------------------------------
We use standard finite element linear shape functions, $N_a$ defined at the nodes $a$ to approximate the displacements $\bfu$ of the lattice vectors: $$\label{eq:u_shape}
\bfu \approx \sum_{a} \bfu_a N_a \Rightarrow \grad_{\bfx_0} \bfu \approx \sum_{a} \bfu_a \grad_{\bfx_0} N_a$$ where $\bfu_a$ are the nodal displacements. Defining $\bfB$ as the Piola-Maxwell stress tensor, $$\label{eq:div_stress}
\divergence_{\bfx_0} \bfB = \divergence_{\bfx_0} \left(\frac{\partial{W}}{\partial{\grad_{\bfx_0} \bfu}}+\epsilon_0\sum_{s}\frac{Q^s}{V_0}\bfA^s\right)$$ we can write (\[eq:u\_gov\]) as ${\bf 0} = \divergence_{\bfx_0} \bfB$. Standard nonlinear finite element methods can now be used; the key difference is that we need to compute the electromechanical contribution to the stress at every iteration. At the same time, we also iterate with respect to $\bfzeta^s$ noting that our interpolation for these variables is piecewise constant, i.e. constant in a given element.
There are two short-range calculations: first, the Piola-Maxwell stress tensor, and second, the minimization over $\bfzeta^s$. Following the complex local QC method [@Tadmor], the deformation gradient and $\bfzeta^s$ are related to the atomic displacements through the Cauchy-Born rule. We assume that the energy of each element can be approximated by assuming a homogeneous deformation of the crystal through the deformation gradient. Given an interatomic potential $U$, we use $$\label{eq:pair_potential}
\frac{\partial{W}}{\partial{\grad_{\bfx_0} \bfu}} = \frac{1}{2V}\sum_{\text{atoms in cutoff}} \frac{\partial{U}}{\partial{\bfr}}\frac{\partial{\bfr}}{\partial{\grad_{\bfx_0} \bfu}}$$ where $\bfr$ are the positions of the atoms and are obtained from $\bfu$ and $\bfzeta^s$. Similarly, we can compute $$\label{eq:pair_potential2}
\frac{\partial{W}}{\partial{\boldsymbol{\zeta}^s}} = \frac{1}{2V}\sum_{\text{atoms in cutoff}} \frac{\partial{U}}{\partial{\bfr}}\frac{\partial{\bfr}}{\partial{\boldsymbol{\zeta}^s}}$$ The minimization over the nodal values of $\bfu$ and the element values of $\bfzeta^s$ are conducted in a coupled manner.
Coarse-graining of the Long-Range Electrostatic Interactions
------------------------------------------------------------
The energy minimization calculation in (\[eq:u\_gov\]) requires the electric field for a given distribution of charge, or equivalently for a given $\bfp$. As noted above, we have a constant $\bfF$ and $\bfzeta^s$ field in each element, i.e. they are discontinuous only along element boundaries. Consequently, $\bfp$ as defined in (\[eq:polar\_def\_2\]) is also constant in each element and discontinuous along the element boundaries. These discontinuities in polarization result in surface charge densities $(\bfp_1 - \bfp_2) \cdot \bfn$. We compute the fields due to these relatively simple charge distributions using direct Greens function integrations.
Crystal Free Surface Subject to Inhomogeneous Electric Fields {#sec:examples}
=============================================================
We apply the methodology described above to a simple setting of a crystal with a free surface subject to an inhomogeneous external electric field due to a point charge above the surface. We use short-range potentials based on the bi-species Lennard-Jones model used for Ni-Mn by [@Hildebrand], giving a tetragonal neutral lattice with a body-centered ion and a charged shell. The polarization is oriented along the tetragonal direction when external fields are absent, thereby providing a spontaneous polarization. This provides a simple model of many widely-used perovskite ferroelectrics such as barium titanate and lead titanate.
Given that this is a model material rather than numerically accurate, we aim to elucidate the physics of electromechanics. There are two independent ratios of interest. One is the strength of the ionic charges v. the strength of the external charge. The other is the strength of ionic interactions v. the strength of the bonded interactions. We explore the effect of the former by increasing external charge while holding ionic charges fixed. We explore the effect of the latter by scaling the electrostatic interactions. This enables us to test a class of materials that range from non-ionic to ionic.
We consider a specimen with the spontaneous polarization oriented tangential to the surface. A single point charge is placed near the surface. In all of these examples, atomic resolution is used in regions of interest – particularly beneath the external charges – while coarser resolution is provided everywhere else. A fine mesh is introduced near the point charge with coarsening throughout the rest of the body. Fig. \[fig:mesh\] shows the full mesh, while Fig. \[fig:mesh\_zoom\] shows the zoomed in atomistically resolved portion of the mesh with all atoms plotted in the current configuration. The black atoms are nodes while the green atoms are constrained by the interpolation.
We conduct three calculations with the electrostatic interactions scaled by factors of $1, 2, 4$ respectively, while holding the charge location and strength fixed. For the given charge strength, a scaling by $1$ produces a surface distortion near the point charge, while the scaling by $2$ and $4$ produce nucleation-like events[^4]. Fig. \[fig:scaling-1\] shows the stress and polarization fields near the point charge for electrostatic scaling of $1$. Similarly, Fig. \[fig:scaling-4\] shows the stress and polarization for the electrostatic scaling of $4$.
![The mesh of the entire specimen. The lengthscales are angstroms.[]{data-label="fig:mesh"}](Pictures/mesh_full.jpg){width="\textwidth"}
![Close up view of the atomically-resolved portion of the sample. Without applied fields and loads, the surface of the specimen is flat. The surface feature is caused by an applied electric field.[]{data-label="fig:mesh_zoom"}](Pictures/mesh_zoom.jpg){width="\textwidth"}
The next set of calculations replaces the single point charge by multiple point charges ($2$ and $4$ respectively), keeping the total charge the same as the case of the single point charge and with spacing between charges on the order of the height above the surface. In subsequent sets of simulations, these charges were moved closer to the surface until eventually a nucleation-like event occurred. Stress plots are provided for the $2$ and $4$ charges at various distances from the free surface in Figs. \[fig:stress-2chargefar\]-\[fig:stress-4chargeclose\].
An interesting feature is the presence of two stress lobes beneath the surface of the material. To further investigate this, we replace the point charge by a dipole (two charges with opposite signs). The resulting deformation as seen in Figure \[fig:stress-dipole\] is a depression instead of the up-down pattern seen in loadings with charges of the same sign. Additionally, the two stress lobes previously seen disappear and merge into a single stress lobe beneath the surface.
The stress lobes in non-dipole loadings do not appear to match continuum phase-field calculations, Fig. \[fig:Lun\] following [@Lun]. These continuum calculations are based on linearized electromechanics whereas the atomistic calculations in this paper are inherently large-deformation. The small-deformation approximation implies that electric fields are computed in the reference as well as potential large rotations being ignored. Further, the material model is also linear elastic. To examine if these assumptions are responsible for the inability of continuum models to capture this feature, we perform the following tests. First, we use precisely our approach except that the electric fields are computed in the reference as in linearized electromechanics. The resulting stress field still shows the double-lobe structure, though with a slightly reduced magnitude (Fig. \[fig:finite\_elect\]). Second, we examine the geometric linearization of the strains; Fig. \[fig:normal\_strain22\] shows a plot of the normalized difference between the non-linear and linearized $\epsilon_{22}$ strain measures which are far too small to be the cause. Third, we test against a fully atomic level description without any coarse-graining. While the system size is small due to computational limitations of the electrostatics, we see a 2-lobe structure Fig \[fig:full-atomic\]. Algorithmic reasons prevent us from computing the stress and deformation gradient in the fully-atomic setting, so we instead examine the change in energy of each atom from the perfect crystal. While qualitative, this calculation supports the view that the double-lobe structure is an atomic effect that we are able to capture despite the local QC approximation.
Regarding the lack of agreement with continuum phase-field, we conjecture that this may simply be due to the gradient penalty preventing the development of fine-scale features such as the double-lobe. Our approach can be viewed as a phase-field method with no gradient penalty. In typical phase-field modeling, the gradient penalty is taken to significantly larger than justified by domain wall widths, because the goal there is to predict microstructure and not defect structure. This may cause fine-scale features such as the double-lobe to be washed out. Of course, it is also debatable whether our fully local QC model is appropriate to model such fine-scale features.
Finally, we examine the effect of a mechanical indenter that is pressed into the surface. Plots of the stress and polarization are in Fig. \[fig:stress-indent\]. These are largely as expected.
We note that in these calculations, the electrostatics appears to induce a lengthscale. However, neither classical electrostatics not the local QC – which is essentially classical continuum mechanics with an atomically-informed constitutive model – has an intrinsic lengthscale. The induced lengthscale from the electrostatics is nonlocal though problem-dependent, i.e. it depends on sample size, boundary conditions, and so on.
Discussion {#sec:conclusion}
==========
We have presented a multiscale atomistic method for ionic solids and other materials where electrostatic interactions are long-range. Our approach is based on using the polarization field as a multiscale mediator between atomic-level rapidly-varying charge distributions and the continuum scale electrical quantities. Our coarse-graining strategy relies on ideas developed by [@james-muller] and others that followed them [@schlomerkemper-schmidt; @xiao-bhatta; @puri-bhatta]. The method that we have presented enables QC and other multiscale methods to go beyond purely short-range interactions, that are characteristic of many structural materials, to functional and electronic materials of interest today.
The method presented here is a first attempt towards the goal of understanding the multiscale electromechanics of defects in ionic solids. Consequently, there remain many important open questions.
1. The presentation here is based on energy minimization which implies zero temperature; recent methods based on extending this to finite temperature in the setting of short-range interactions can perhaps be adapted to our setting [@kulkarni]. An alternate approach is [@tadmor-EffHamil], where the powerful method of Effective Hamiltonians has been applied to study the finite temperature behavior of ferroelectrics in the local QC setting.
2. We have started from an atomic viewpoint in this paper whereas QC methods have been demonstrated for the orbital-free density functional theory – a version of (ground state) density functional theory that is restricted to metals with mobile electrons. However, variants of density functional theory have difficulty with charged defects as rigorously demonstrated in [@cances-ehrlacher]. In addition, density functional theory has important qualitative failings in computing bandgaps, van der Waals interactions, and so forth [@DFT-vdW-crap]. Therefore, it appears to be more useful to begin from the atomic level with well-calibrated and trustworthy potentials.
3. We have worked within the local QC setting which can alternately be considered a standard finite element approach with the constitutive relation being drawn from atomistics rather than a prescribed function. An important further step is to couple the coarse-grained model with a region with truly atomic resolution in the vicinity of the defect. In this regard, the key coarse-graining ideas presented here carry through to that setting. This open question is an area of our current research and we expect to report on this in the near future.
4. Our use of linear interpolation restricts us from capturing potentially important effects based on strain gradients, in particular the phenomenon of flexoelectricity that can be relevant at small scales [@pradeep-flexo]. However, this phenomenon is relevant at large strain gradients that typically can only occur in localized regions of the sample. Therefore, a QC method that fully resolves the defect region can potentially capture this phenomenon even with linear interpolations.
5. Our coarse-graining of the electrostatics in this paper assumed complete separation of scales. However, for a real calculation, this will naturally not be true. It is therefore important to study and quantify the errors when the separation of scales is large but not complete. This will enable the construction of an algorithm with controlled error tolerance. Directly related to this is the derivation of a rigorous limit as opposed to the formal presentation in this paper. These related open questions are an area of our current research, following techniques in [@schlomerkemper-schmidt].
The example that we have studied of the double-lobe structure beneath a point charge provides an important motivation for the further development of our method. Atomic-scale features such as the double-lobe are of importance to phenomena such as domain nucleation and cannot be captured by continuum models. On the other hand, the limited size of our admittedly-unoptimized brute force atomic calculation shows the need for coarse-graining efforts.
Finally, we have examined the issue of whether polarization is unique. As noted above, evaluating the classical definition of polarization provides a unit cell-dependent quantity. This is obviously a potential disaster for a coarse-graining scheme based on the polarization field. The materials physics orthodoxy appeals to quantum mechanical concepts to fix a unique value of the polarization [@Resta-Vanderbilt]. Continuum mechanicians have used variational notions to achieve similar ends [@puri-bhatta]. The view that we have taken in this paper is that there is simply no need to have a unique value for the polarization. When partial unit cells that provide boundary charges are accounted for consistently, the coarse-grained electric fields and other relevant quantities are independent of the choice of unit cell; the polarization is merely a multiscale intermediary. In continuum theories of electromechanical solids, e.g. [@xiao-bhatta; @shu-bhatta], the polarization appears in the standard local free energy density, not only in the electrostatics. In the local free energy density, the polarization can be considered as a quantity that tracks the atomic positions rather than relevant to electrostatics. In that perspective, the specific choice of unit cell is irrelevant. Broadly, the view advocated in this paper is in the spirit of classical continuum mechanics: the polarization field simply provides some information about the atomic-level, and one can take any choice as long as consistent transformations between energy, kinematics, and boundaries are respected. The immediate analog in continuum mechanics is the freedom in the choice of reference configuration and the corresponding value of the deformation field and strain energy density response function as long as care is taken to define suitable transformations between different choices.
Acknowledgments {#sec:acknowledgments .unnumbered}
===============
We thank ARO Numerical Analysis for financial support through a Young Investigator grant (W911NF-12-1-0156). Jason Marshall also acknowledges support from the Northrop Graduate Fellowship Award from Carnegie Mellon University. Kaushik Dayal also acknowledges support from AFOSR Computational Mathematics (FA9550-09-1-0393) and AFOSR Young Investigator Program (FA9550-12-1-0350). Kaushik Dayal thanks the Hausdorff Research Institute for Mathematics at the University of Bonn for hospitality. This research was also supported in part by the National Science Foundation through TeraGrid resources provided by Pittsburgh Supercomputing Center. We thank Richard D. James, Saurabh Puri, and Yu Xiao for useful discussions.
Transformation of Unit Cell in a Crystal Lattice to Obtain a Unit Cell with One Face Parallel to a Given Plane {#sec:appendix}
==============================================================================================================
We show in this section that any crystal lattice can be described by a unit cell with one face parallel to a given plane. This is an essential ingredient of our proof that the change in polarization due a change in the lattice unit cell is balanced by a corresponding change in surface charge density.
We first show the following proposition.
Consider a crystal described by lattice vectors $\{ \bff_1, \bff_2, \bff_3 \}$ and reciprocal lattice vectors $\{ \bff^1, \bff^2, \bff^3 \}$. Consider a rational plane[^5] with normal $\bfn$.
This lattice can also be described by lattice vectors $\{ \bfg_1, \bfg_2, \bfg_3 \}$, where $\bfg_2 \perp \bfn$, $\bfg_3 = \bff_3$, and $\bfg_1$ will be shown below to exist.
Result 3.1 of [@bhatta-book] states that a crystal described by lattice vectors $\{ \bff_1, \bff_2, \bff_3 \}$ can equivalently be described by lattice vectors $\bfg_1, \bfg_2, \bfg_3$ if and only if these satisfy $\bfg_i = \sum_j \mu_i^j \bff_j$ with $\mu_i^j$ being a $3 \times 3$ matrix of integers with determinant $\pm 1$.
We have $\bfg_3=\bff_3$, therefore $\mu_3^1 = \mu_3^2 = 0$ and $\mu_3^3=1$. Set $\bfg_2 = M_1 \bff_1 + M_2 \bff_2$ with $M_1 = -\frac{n_2}{\gcd(n_2,n_1)}$ and $M_2 = \frac{n1}{\gcd(n_2,n_1)}$. This choice ensures the following: $\bfg_2 \perp \bfn$, $M_1$ and $M_2$ are integers, and $\gcd(M_1, M_2)=1$. It implies that $\mu_2^1 = M_1, \mu_2^2 = M_2, \mu_2^3=0$.
The remaining step to complete the proof is to show that we can find $\bfg_1$ which satisfies $\bfg_i = \sum_j \mu_i^j \bff_j$ where $\mu_i^j$ has the restrictions mentioned above. Write $\bfg_1 = N_1 \bff_1 + N_2 \bff_2 + N_3 \bff_3$. Then $\det \mu = 1$ gives us the condition $N_1 M_2 - M_1 N_2 = 1$. The extended Euclidean algorithm provides the existence of integer solutions $N_1$ and $N_2$ to this equation [@euclidean-algorithm]. In general, this algorithm provides integer solutions $m,n$ to the equation $a m + b n = \gcd(a,b)$ where $a, b$ are given integers. While the algorithm is constructive and ensures existence of solutions, it does not provide explicit forms for the solutions.
Therefore, there exists $\bfg_1 = N_1 \bff_1 + N_2 \bff_2 + N_3 \bff_3$ where $N_1, N_2$ are obtained from the algorithm above and $N_3$ can be arbitrary.
We wish to show that given a crystal described by lattice vectors $\{ \bff_1, \bff_2, \bff_3 \}$, we can equivalently describe the crystal by lattice vectors $\{ \bfh_1, \bfh_2, \bfh_3 \}$ with $\bfh_2 \perp \bfn$ and $\bfh_3 \perp \bfn$. The vector $\bfn$ is the unit normal to a rational plane.
We simply apply the proposition above twice in succession. First, we use the proposition to transform from $\{ \bff_1, \bff_2, \bff_3 \}$ to $\{ \bfg_1, \bfg_2 \perp \bfn, \bfg_3 = \bff_3 \}$. We then use the proposition again to transform from $\{ \bfg_1, \bfg_2, \bfg_3 \}$ to $\{ \bfh_1, \bfh_2 = \bfg_2, \bfh_3 \perp \bfn\}$. Therefore, the final description has $\{ \bfh_1, \bfh_2 \perp \bfn, \bfh_3 \perp \bfn \}$. The condition $\det \mu = \pm 1$ ensures linear independence; in fact, it preserves the volume of the unit cell.
![Stress and polarization due to single point charge with an electrostatic scaling of $1$.[]{data-label="fig:scaling-1"}](Pictures/scaling-1.png){width="160mm"}
![Stress and polarization due to single point charge with an electrostatic scaling of $4$.[]{data-label="fig:scaling-4"}](Pictures/scaling-4.png){width="160mm"}
![Stress due to two point charges with an electrostatic scaling of $1$.[]{data-label="fig:stress-2chargefar"}](Pictures/stress-2chargesfar.jpg){width="160mm"}
![Stress due to two closely-spaced point charges with an electrostatic scaling of $1$.[]{data-label="fig:stress-2chargeclose"}](Pictures/stress-2chargesclose.jpg){width="130mm"}
![Stress due to four point charges with an electrostatic scaling of $1$.[]{data-label="fig:stress-4chargefar"}](Pictures/stress-4chargesfar.jpg){width="130mm"}
![Stress due to four closely-spaced point charges with an electrostatic scaling of $1$.[]{data-label="fig:stress-4chargeclose"}](Pictures/stress-4chargesclose.jpg){width="130mm"}
![Stress due to a dipole (two point charges of opposite sign) with an electrostatic scaling of $1$.[]{data-label="fig:stress-dipole"}](Pictures/stress-dipole.jpg){width="130mm"}
![Stress due to a point charge computed using a continuum phase-field model [@Lun].[]{data-label="fig:Lun"}](Pictures/Lun_stress.png){width="130mm"}
![Stress due to a point charge with electrostatic fields computed in the reference configuration.[]{data-label="fig:finite_elect"}](Pictures/finite_elect.jpg){width="130mm"}
![Normalized Difference between Nonlinear and Linearized Strain Measure ($\epsilon_{22}$ component).[]{data-label="fig:normal_strain22"}](Pictures/Normalized_Strain22.jpg){width="130mm"}
![Fully atomic calculation for a sample with a point charge. The plot shows the energy difference from the state without any external charge with the electrostatic field energy subtracted. The entire computational domain is shown.[]{data-label="fig:full-atomic"}](Pictures/two_lobes_different_color.png){width="130mm"}
![Stress and polarization due to mechanical indentation with no applied electric field.[]{data-label="fig:stress-indent"}](Pictures/StressPolarizationRed.png){width="130mm"}
[^1]: [email protected]
[^2]: [email protected]
[^3]: It is not clear to us how to proceed without assuming that the surfaces are rational planes. Irrational surfaces cause severe difficulties in defining surface energies even in simpler models of solids [@rosakis-surface-energy].
[^4]: A nucleation-like event refers to localized switching of polarization. We do not expect our method to be qualitatively accurate beyond this regime.
[^5]: A rational plane has a normal $\bfn$ that can be represented $\bfn = \sum_i n_i \bff^i$ with $n_i$ integers.
|
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abstract: 'Two dimensional hydrodynamical simulations of convective oxygen burning shell in the presupernova evolution of a star are extended to later times. We used the VULCAN code to simulate longer evolution times than previously possible. Our results confirm the previous work of [@ba98] over their time span (400 s). However, at 1200 s, we could identify a new steady state that is significantly different than the original one dimensional model. There is considerable overshooting at both the top and bottom boundaries of convection zone. Beyond the boundaries, the convective velocity falls off exponentially, with excitation of internal modes. The resulting mixing greatly affect the evolution of the simulations. Connections with other works of simulation of convection, in which such behavior is found in a different context, are discussed.'
author:
- 'S.M. Asida and David Arnett'
title: 'Further Adventures: Oxygen Burning in a Convective Shell'
---
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Introduction
============
There are many published studies of stellar convection using multidimensional hydrodynamical simulations, but few deal with convective nuclear burning occurring in stellar interior (see [@dpr98] for a study of core hydrogen convection). [@arn94] (A94) and [@ba94; @ba98] (BA94, BA98) have studied oxygen burning shell which is one of the last stages in a massive star presupernova evolution; we extend that work.
This is an important stage in presupernova evolution because in this convective region several phenomena take place. In or near this region: (1) most of the explosive nucleosynthesis and production of and occurs, (2) the “mass cut” between collapsed and ejected matter develops, and (3) mixing of different layers may happen. The standard model for treating convection in one dimensional (1D) stellar evolutionary codes, the mixing length theory (MLT), is usually used for modeling this convective region as well, even though the conditions of oxygen convective burning shell are more complicated than can be assumed for MLT to be valid (i.e., the flow is not strongly subsonic and there are both energy sources and sinks in the flow).
In [@arn94] this evolutionary stage was described, and the first two dimensional (2D) hydrodynamical simulation of this problem were presented. The 145 s time interval was simulated by these calculations, using the PROMETHEUS code, is much less than the duration of this stage ($10^3$ to $10^5\rm\ s$, see figures 10.5 and 10.6 in [@arn96]). During this time, convective flow was formed. [@ba94; @ba98] have modeled the evolution over a longer time interval of 300 to 400 s, and noticed a mixing of composition from the neighboring stable region near the end of the simulations. In addition,
- the flow velocities were up to $10\%-20\%$ of the sound speed,
- stellar structure was not altered significantly, and
- there were strong density and composition fluctuations in both space and time which did not reach a statistical steady state.
The main differences between this work and that of A94, BA94, and BA98 were (1) the use of a different hydrodynamic code, and (2) the simulation of a longer evolution time. The same initial model, equation of state, and nuclear reaction algorithms were used.
The numerical scheme
====================
For this study, we used a version of the hydrodynamical code VULCAN ([@lvn93]). This code uses an algorithm that begins with a hydrodynamic time step, and is completed by a relaxation of the numerical grid, thus giving an Arbitrary Lagrangian Eulerian (ALE) scheme. In the hydrodynamic time step, the Lagrangian hydrodynamic equations in two dimensions are solved explicitly or implicitly, allowing longer time steps to be taken if the flow is strongly subsonic. The mesh relaxation phase is necessary to eliminate distortion of the cells, especially for flow with vorticity. We used a quasi-1D Lagrangian relaxation, in which each radial row of cells kept a constant mass. The code was used by [@gl95] for computing convective novae outbursts, and by [@at97] and [@asd00] to simulate convection in a red giant envelope. One of the adaptation made by [@asd00] for simularions of convection was the inclusion of [@smg63] like sub grid scale mixing (SGSM) model, as was done in many two dimensional and three dimensional numerical studies of convection.
The equation of state and the thermonuclear reactions algorithms were essentially the same as in A94 and BA98. The equation of state was the sum of components for electrons, ions, and radiation. The nuclear reaction network used twelve species for helium, carbon, neon and oxygen burning. Neutrino cooling was included; see the above references for details. The initial model was the same as in A94 and BA98. The computational domain corresponds to a region containing the entire oxygen-burning shell in a star of , having an initial metallicity of 0.007 (about one third solar). This shell boundaries are at radii of $3\times 10^8\rm cm$ and $3\times 10^9\rm cm$ that are equivalent to mass coordinates of and .
We performed several simulations that were different in: the computational domain, spacial resolution, the initial temperature gradient and numerical parameters of the simulation. The computational domain typically includes all of the oxygen shell as well as the neighboring layers (the inner radius was located at $2\times 10^8\rm cm$ in some of the simulations, and the outer radius was located at $8\times 10^{10}\rm cm$ in other simulations). The angular extent of the wedge ran usually from 0.35 $\pi$ to 0.65 $\pi$ radians.
Typically. the oxygent shell was divided to $\sim $ 120 radial zones, and the spacing of zones was logarithmic in radius and linear in angular direction (i.e., $dr = r\ d\theta $) with 60 angular zones. Rotational symmetry was assumed. These characteristics are similar to the medium resolution simulations of BA98. Because of small differences in the equation of state in the initial one dimensional model and in the two dimensional code, as well as different radial zoning, we had two sets of simulations: in the first, the 1D model (with interpolation) was used as it is, and in the second it was slightly adapted, so that the temperature gradient would be superadiabatic in all zones of the convection shell.
The velocities on the inner boundary were set to be zero for the whole simulation, so that the inner core was a hard sphere. At the upper boundary there was no limitation on the velocities, but in order to eliminate mass flow out of the computational domain, an average radius was used to follow expansion or shrinking of the outer boundary. We used reflective boundary conditions on the sides (though such conditions enforce a downflow or upflow on the side boundaries, it was found in BA98 not to be important).
Results of the simulations
==========================
We present the results of one “standard” simulation with 172 radial zones and 60 angular zones. The inner boundary was at $2\times 10^8\rm cm$ and the outer was at $48\times 10^8\rm cm$. The temperature gradient was slightly super adiabatic in the oxygen shell, and no SGSM terms were used. Most of the results of the other simulations were similar and the differences are discussed mainly in the end of this section.
General Evolution of the Flow
-----------------------------
In our simulations, the initial velocities were zero, and the convective flow developed as a result of the instability from round off errors. Figure 1 presents the velocity field in the beginning of the simulations for times (a) 75 s and 150 s. As we can see, the convective flow starts at the bottom of the convection region (i.e., the burning layer), and then moves up with increasing eddy size. By time 150 seconds the convective flow penetrates the upper boundary of the convection region (seen as a thick line in panel a). As a result, there is a downflow of carbon-rich material. This can be seen in Figure 2, which presents contours of carbon nucleon fraction. Panels a-e represent times of 75, 150, 300, 600 and 1200 s. We can see that the downflow (panel b) penetrates the whole convective region, and results in mixing of carbon in this region. From comparison of panels c, d and e we can see that carbon abundance becomes more uniform, as the simulation evolves.
This penetration of carbon is almost identical to that seen by BA98; see their Figure 5. The two independent hydrocodes give consistent results over the whole time spanned (400 seconds) by BA98 simulations. The small difference in the timescale for the carbon penetration is due to the small differences in the extent of the super adiabatic gradient of the initial 1D model (when we used the initial 1D model as it is, without modifying the temperature gradient, we got very similar results with a longer timescale of the penetration).
Carbon Enrichment
-----------------
Carbon penetration, as well as other processes in the simulations, can be visualized by examining one dimensional averages of various parameters, which in principle would be equivalent to results from 1D simulations. In Figure 3 carbon abundance (nucleon fraction) is plotted as a function of stellar mass coordinate. From the initial profile (solid line) we can see the location of the oxygen shell. At 150 s (long dashed line), carbon rich material has entered the convection layer, and then mixed through the whole region. At later times, the fluctuations in the abundances of carbon decrease (compare the profiles at 300 s -short dashed line, 600 s - dotted line and 1200 s -dotted dashed line), and we have a carbon nucleon fraction of $\sim 5\times10^{-3} $.
A change in abundance in this plot does [*not*]{} necessarily corresponds to mixing, because this plot represents an one dimensional average of the compositions of all the material in each radial layer. This may be revealed by a closer look at the apparent widening of the jump in carbon abundance at the upper boundary of the oxygen shell at $m \approx $: this jump is much wider at 150 s than it is at later times. This “anti diffusion” is possible since the widening of the interface of the shell is the outcome of two processes: a mixture of material from the two sides of the interface [*and*]{} large scale motions that change the shape of the interface. Thus, at later times the interface is more spherical than it was at 150 s, so that averaging the abundances over radial layers yields a sharper jump (compare Fig 2 panels b and d).
To better understand the carbon enrichment we present Figure 4 which shows the RMS fluctuations of carbon nucleon fraction ($\sigma$) as a function of stellar mass coordinate at several different times. The tendency toward more uniform carbon mixing is clearly seen, as was indicated by Figures 3 and 4. From this figure, we can also see, that the previously mentioned widening of the oxygen shell interface is partially due to changes in its shape since the fluctuations in carbon fraction at the boundary are relatively high, and are even higher at time 150 s.
The mixing of carbon in the burning layer causes the reaction rate to change significantly. In Figure 5, the nuclear luminosity is presented as a function of time. In the beginning there is an adjustment phase in which the energy production rate decreases to about $5\times 10^{44}\rm\ erg\ s^{-1}$ . This transient phase is a thermal relaxation due to the fact that the initial model is inconsistent: it does not have a two dimensional velocity field which can carry the convective energy flux that the 1D model needed. As the carbon rich material penetrates to high temperature layers, it starts to react rapidly, yielding nuclear luminosities which increase to about hundred times the initial value. In this stage, the result is mainly carbon and neon consumption. Afterward, there is a smaller decrease in nuclear luminosity (compare time 600 and 1200 s). This decrease is related to the small decrease in the average carbon abundance seen between times 600 s and 1200 s in Figure 3.
A Quasi-Steady State
--------------------
The relaxation in the carbon abundances by time 300 s to a slowly varying state, and the small change between times 600 and 1200 s, suggest that the system may be close to a quasi - stationary state (a “steady” state on average). This one is significantly different from the steady state predicted by the one dimensional calculations used to definethe initial model. That the system is close to a steady state can be demonstrated by the following figures. Figure 6 shows the average rate of change in energy, as a function of mass coordinate (from time 400 to time 800 s). The energy produced by thermonuclear reactions (dashed line) is carried away by convection (solid line). The net change in energy is small, and balanced by changes in the gravitational energy (dotted line). This slow change is an adjustment of the structure driven by enhanced nuclear burning from the carbon ingestion. There is a net heating at the bottom, below , which will be addressed in the next section.
Figure 7 shows the average rate of change in carbon nucleon fraction, as a function of mass coordinate (from time 400 s to time 800 s, and from time 800 s to time 1200 s), comparing the changes due to the divergence of the convective abundance flux (solid line) with that due to nuclear burning (long dash line). The nuclear consumption of carbon at the bottom is almost completely balanced by convective inflow; This is the burning zone. Over most of the convection region the change in carbon abundances is small. There are larger fluctuations at the top (outer) boundary of the convection zone (where the carbon abundance increases by a factor of $\sim 20$, see Fig. 3).
The decrease in nuclear luminosity (seen in Figure 5, after the peak at 200 s) was caused by a decrease in the net flux of carbon into the burning layer. This is easily seen in Figure 7 in the comparison of the changes in the later time interval (800 s to 1200 s - short dashed line) to the changes in the previous time interval (400 s to 800 s - solid line). The total mass of carbon in the oxygen convection shell is decreasing as a result of carbon consumption and less mixing from the upper shells. Thus our convective burning is beginning to evolve on a secular time scale by the consumption of fuel, having approached a thermal “steady” state.
in Figure 8 we present the RMS fluctuations of density (divided by the density) as a function of stellar mass coordinate for the standard simulation for several times. As the composition becomes more uniform with time, the density fluctuation decreases throughout most of the oxygen shell to about $10^{-3}$ at time 1200 s. However at the edges of the convection zone, the fluctuations does not decrease and we have about $2\times 10^{-2}$ at the bottom and $7 \times 10^{-2}$ at the top, where there is a jump in the abundances. These values are similar to those obtained by BA98.
Interaction with Neighboring Shells
-----------------------------------
One important interaction of the convective shell with neighboring shells is the mixing of carbon from the upper carbon rich shells, as described before. Another important interaction is heating of the nearest shells below the bottom of the convection zone: in the initial model there is a sharp decrease in temperature just below the convection zone, as convective flow starts to grow inside the lower parts of the convection zone there is some penetration of the flow to these lower temperature shells. As a result of this flow, the lower temperature shells are being heated and the higher temperature shells are being cooled. This heating is easily seen in Figure 9 where we present the temperature profile in the initial model, and at later times. Most of the heating was done by time 300 s, but even at later times, some heating exsisted, as can be seen in the left end of Figure 6. Along with this “energy mixing” there is mixing of composition in those few zones as we can see from neutron excess $\eta$ plot presented in Figure 10. In the initial model, there is a jump in $\eta$ at the boundary of the oxygen shell (corresponds to the jump in composition). Due to mixing, this jump slightly moves towards lower mass coordinate and becomes slightly wider. From comparison of Fig. 9 and 10 we see that the heating penetrates to deeper shells than composition changes.
There are two important consequences of this interaction: (1) the heating and aditional oxygen and carbon in the few zones below the 1D oxygen convection shell cause these zones to be added to the oxygen shell and, together with few zones at the bottom of it, to be part of the burning zone (we can see a broader region with temperature above $2.2\times 10^9 \rm cm
$), (2) because of the cooling of the zones at the bottom of the oxygen shell, the temperature gradient becomes less than adiabatic in those zones and the convective flow exist as a result of a super adiabatic gradient at higher zones.
Thus we see that the neighboring stable shells greatly affect the oxygen convective shell. The same convective flow that penetrates to the neighboring shells and allow the mixing of elements and energy between the shells affects further more the stable shells. In Figure 11 we present the averaged convective velocity as a function of radius at the bottom of the oxygen shell. The dashed line corresponds to the fluctuations in radial component of the velocity, while the solid line corresponds to the amplitude of the fluctuating total velocity. The plus signs present points in which the pressure varies by a factor of ten. Three regions can be identified in this plot from right to left: a region of constant velocity, a region of exponential decrease, and another region of constant velocitis but with a noticeable difference between the total and the radial velocities. (i.e. the radial velocities are much smaller on average than the tangential velocities) which is indicative of g modes. This g modes can be identified in 2D presentation of the flow (Figure 12).
In another simulation we included a much larger shell above the oxygen shell. In this simulation we noticed similar flow characteristics of exponential decay (Figure 13) and tangential preference of the flow (Figure 14). From Fig. 11 and 13 we can also see that the exponential decrease at the bottom is on a scale which is about one third of the pressure scale height, while at the top it is over one pressure scale height.
Numerical Sensitivity
---------------------
We performed several simulations with different parameters as mentioned in §2. As our results are consistent with those of BA98, and since one of the main subjects in BA98 is the sensitivity of the results to numerical parameters, we would not present here all the results from our many simulations. Instead, we will focus on the conclusions from those simulations: the most sensitive feature in the results is the amount of heating below the oxygen shell. When there is more heating, the temperature gradient is less than adiabatic in a larger portion of the (bottom of) oxygen shell. As a result, the convective flow is damped, carbon can not reach the burning zone and we got less nuclear luminosity.
The opposite happened in simulations where we did not include the shell below the oxygen shell: the temperature gradient was super adiabatic so the velocities were higher (by a factor of five), more carbon entered the burning zone, and the nuclear luminosity was higher. From comparison of simulations with different resolution for this case (without the bottom shell), we found that the results are quite robust and not sensitive to the numerical resolution.
In all of our simulations the other features were essentially the same, namely: the evolution of the flow from the bottom, the penetration to neighboring shells and the enrichment of the oxygen shell by carbon.
In order to check extra mixing at scales shorter than numerical resolution, we performed a simulation with SGSM mixing terms. This simulation started from a 2D profile of the standard simulation at time 400 s and was carried out to time 800 s. The differences between this simulation and the standard simulation were: a more uniform composition at the oxygen shell - the RMS fluctuations of carbon abundance at time 800 s were similar to the results of the standard simulation at time 1200 s, and significantly more mixing at the stable bottom shell. This can be seen in Figure 15 were we plot the neutron excess $\eta$ for this simulation (SGSM) and the standard simulation. In this plot we can see a very small difference between the two profiles of the standard simulation (at time 400 s - solid line and at time 800 s - long dashed line). When SGSM terms were added, more mixing can be seen by time 600 s (short dashed line), and additional mixing occurs until the end of the simulation (time 800 s - dotted line).
Discussion
==========
When we compare our results to the results of BA98 we note that the evolution of the simulations is similar for times reached previously. In both, convective flow evolves from bottom to top, with velocities exceeding $10\%$ of the sound speed. The flow penetrates to the carbon rich region above the one dimensional edge of the convective region, and thus causes a significant enrichment of the convective region with carbon. These higher values of carbon abundances yield a much higher nuclear luminosity than did the original oxygen burning.
However, at later times, we could identify signs of a steady state configuration in which the average abundances changed very slowly, and turbulent mixing within the convective region was able to decrease the fluctuations in the oxygen shell. However, density fluctuations near the boundaries of the convection zone (which are also composition discontinuities) did [*not*]{} decrease at later times, and were equal to few percents.
The most prominent feature in our simulations is the interaction of the convective shell with the neighboring shells. This interaction is due to penetration of the convective flow to the stable shells and the resulting mixing of elements and energy. These effects were present in all of our simulations, and seemed to be physically resonable and valid. However, the exact amount of mixing (especially at the bottom) depends on numerical parameters of the simulations. Moreover, since the initial 1D model did not include such overshoot, we actually simulated a transient toward a more consistent model. For example, we mentioned that the heating of the very few neighboring zones ,at the bottom of the convection shell, lowers the temperature gradient below the adiabatic value. If the convective flow is damped because of that, then the heat generation in these zones would evantually increase the temperature gradient, and convection would be restored.
The properties of the internal modes excited in the stable shells are undoubtedly affected by the computational domain and the boundary condition. Reflective waves are probably the cause for the constant level of velocities we noticed at Figure 11. However, the volume below the oxygen shell in these stars is relatively small so a finite value of the velocity is quite plausible.
An interesting point is the validity of the results as we performed 2D simulations (and not 3D). This is not an easy question to answer, however, from various comparisons of 2D and 3D simulations (see for example [@krg99]) it seems that the main features of the interaction of the unstable layer and its neighboring layers are similar, and the overshoot and mixingin in both cases are comparable . As we do not claim to resolve this interaction quantitatively, we think that our results are valid.
Penetration of the convective flow to neighboring stable regions is a common feature that exists in many multidimensional simulations of convection, in a variety of stellar problems. [@hrl86] have studied such penetration beyond a shallow convective region in 2D, and found that it generated internal g modes in the stable region below the convection region, all the way to the bottom boundary. Two regions in this stable layer can be identified from their figure 13: close to the unstable layer there is a sharp decrease in the kinetic energy, but it does not go to zero, rather there is an almost constant level of kinetic energy throughout the rest of the stable layer. [@krg99] have tested penetration in both 2D and 3D, they found that the energy fluxes in the stable layer in the 3D are about two thirds of the 2D results and that the constant level is replaced with a continuas decrease when the viscosity coefficient is increased at the bottom to avoid reflection of waves.
In both these studies, ideal simplified physics was assumed. [@fls96] studied the structure and dynamics of shallow stellar surface convection zones, using two dimensional radiation hydrodynamic simulations with more “real” physics. They found convective motions extending “well beyond the boundary of convectively unstable region, with vertical velocities decaying exponentially with depth in the deeper parts of the lower overshoot region, as expected for linear $g^-$ modes.” They suggested to approximate the average velocity at the stable region as the convective velocity at the boundary of the unstable zone multiplied by an exponential decay factor with a lapse rate that is of order of the pressure scale height.
As explained by [@cox80] g$^-$ modes are unstable within convection regions, and generally decrease exponentially with increasing distance from the boundaries of the oscillatory region. As [@fls96] point out, [@ll59] have a relevant discussion in their §34, where they discuss some properties of potential flow. If compressibility and dissipation can be neglected, a steady state potential flow which is periodic in some plane, must be damped exponentially in a direction perpendicular to that plane. Also, the shortest wavelengths will be damped fastest. Consequently, most of the overshoot is given by the largest scales which are driven at the interface, that is, the largest convective scale.
We find a striking underlying unity in these results and ours, despite the significantly different astrophysical context involved: our simulations correspond to deep shell burning convection that is driven by nuclear burning at the bottom and cooled by neutrino radiation and in which spherical properties of the domain are important; while these studies deal with envelope convection driven by photon radiation at the surface in plane parallel geometry. In both cases The flow extand beyond the formal boundaries of the convection zone, with velocities that decay exponentially in the stable shells producing internal modes.
Near the boundaries of the convection zone, where the velocities are still large, quite efficient mixing is taking place. The amount of mixing beyond that region, is not entirely clear. [@fls96] have used trajectories of test particles to estimate the diffusion coefficient and conclude that the diffusion coefficient corresponds directly to the exponentially decaying velocities.
Some studies of stellar evolution that used these mixing terms for hydrogen core convection ([@hrw97]; Herwig 1998), found that the lapse rate of decay of the diffusion should be much smaller ($\approx$2 percents) than the pressure scale height. They speculate that this is due to the large stability of the layers that are near the core convection zone. Yet another explanation might be true, namely the flow characters at the stable shells and the many scales involved in mixing.
Mixing is a complex phenomenon which take place in a very small length scale comparing to the length scale of the star, and since stars have a lower effective viscosity than in the simulations, and since they evolve for longer times, an accurate prediction of mixing is very difficult. Further, we note that motion does not necessarily imply mixing! In particular, oscillatory potential flow, which is irrotational,would involve “stretching” and “contracting” motions that do not give mixing, at least at lowest order.
As we can see in Figures 11-14, The flow in the stable shells is different than the flow inside the convection region, even though it is [*not*]{} purely potential flow (as revealed when examining the vorticity). Due to these differences in the flow, it seems plausible that the effective mixing velocity may be less than the hydrodynamic velocity, and its lapse rate different as well.
As the various shells in our model have a different composition, mixing can be directly examined from the 2D profiles and the 1D average of composition. However, as the scale of mixing is much smaller than numerical resolution, the results must be carefully examined. One way to examine the results is to compare the mixing in the standard simulation to that in the simulation with sub grid mixing model. There are small differences in the average abundances in the oxygen shell between the two simulations, and the main difference is that with SGSM terms, the profile is more uniform because of the extra mixing. The difference between the simulations is much more prominent in the lower stable shell: the width of the abundance jump is much wider when SGSM terms were used, and penetrates to deeper zones, in which the “constant level” average velocity exists.
When we used SGSM we implicitly assumed that turbulent flow exists from the scale of the mesh resolution to the scale of molecular diffusion. Since this mixing occurs in a stable region, in which internal modes have been excited, the validity of this assumption is questionable. Thus it is fair to say that further work is needed to resolve this subject of mixing beyond the convective layers.
[@mdv99] have examined the effects of overshooting predicted by full spectrum turbulence model of convection (Canuto & Mazzitelli 1991, 1992) within the context of lithium production in AGB stars. This is an interesting example of the way stellar convection may be tested. A particular aspect of their work of interest here is the suggestion that [*symmetric*]{} overshoot may be constrained by existing observations. We note that a general feature of numerical simmulations is up-down asymmetry. Cooling flows are narrow, faster downdrafts than the corresponding updrafts, which are broader and slower. It is suggestive that in simulations of envelope convection driven by radiative cooling at the top, there is an overshoot at the bottom ([@hrl86; @fls96]), while we, who have simulated shell convection driven by burning at the bottom, have found exponential overshoot at the top (and the bottom). It may be that the causes of this asymmetry can be determined, so that a complete algorithm for the overshoot maybe devised for 1D stellar evolutionary calculations.
In summary, we found that stellar evolution models should take into account penetration and an exponential decay of the velocities, and the excitation of internal modes, when simulating convective burning shells. Direct hydrodynamic simulation of stellar evolution places heavy demands on computer resources, especially for carbon and neon burning stages, which are slower. The tendency toward steady state revealed in our simulations is encouraging in that simplified algorithms may allow the construction of an improved set of one dimensional models.
We used an initial model which was derived from results of a one dimensional stellar evolution code, in which earlear stages of evolution were modeled with standard MLT. Consequently, the profile we started with is unsatisfactory, and is not consistent with our conclusion that penetration and extra mixing should be taken into account. This should be kept in mind when using directly the results of our simulations.
Enlightening discussions with Grant Bazàn, Eli Livne and Falk Herwig, and the interaction with the anonymous referee, are gratefully acknowledged. This work was supported in part by DOE grant DE-FG03-98DP00214/A001.
Arnett, D. 1994 , 427, 932 Arnett, D. 1996, [*Supernovae and Nucleosynthesis*]{} (Princeton, New Jersy: Princeton University Press) Asida, S.M. 2000 , 528, 896 Asida, S.M., & Tuchman, Y. 1997 , 491, L47 Bazàn, G., & Arnett, D. 1994, , 433, L41 Bazàn, G., & Arnett, D. 1998, , 496, 316 Canuto, V. M., & Mazzitelli, I. 1991, , 370, 295 Canuto, V. M., & Mazzitelli, I. 1992, , 389, 724 Cox, J.P. 1980, [*Theory of Stellar Pulsation*]{} (Princeton, New Jersy: Princeton University Press) Deupree, R. G. 1998, , 499, 340 Freytag, B., Ludwig, H. -G, & Steffen, M. 1996, , 313, 497 Glasner, S. A., & Livne, E. 1995, , 445, L149 Herwig, F., Blöecker, T., Schöenberner, D., & El Eid, M. 1997, , 324, L81 Herwig, F., Schöenberner, D., & Blöecker, T. 1998, , 340, L43 Hurlburt, N.E., Toomre, J., & Massaguer, J.M. 1986, , 311, 563 Kiraga, M., Zahn, J.-P., Stȩpień, K., Jahn, K., Rózyczka, M. & Muthsam, H. J. 1999, in ASP Conf. Ser. 173, Theory and Test of Convection in Stellar Structure, ed. Á. Giménez, E. Guinan, B. Montesinos (San Francisco: ASP), 269 Landau, L. D., & Lifshitz, E. M., 1959, [*Fluid Mechanics*]{} (London: Pergamon Press) Livne, E. 1993, , 412, 634 Mazzitelli, I., D’Antona, F., & Ventura, P. 1999, , 348, 846
Smagorinsky, J. 1963, Mon. Weather Rev., 91(3) 99
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---
abstract: 'In the regression model $Y = b(X) +\varepsilon$, where $X$ has a density $f$, this paper deals with an oracle inequality for an estimator of $bf$, involving a kernel in the sense of Lerasle et al. (2016), selected via the PCO method. In addition to the bandwidth selection for kernel-based estimators already studied in Lacour, Massart and Rivoirard (2017) and Comte and Marie (2020), the dimension selection for anisotropic projection estimators of $f$ and $bf$ is covered.'
address:
- '\*LTCI, Télécom Paris, Palaiseau, France'
- '$^\dag$Laboratoire Modal’X, Université Paris Nanterre, Nanterre, France'
- '\*,$^\dag$ESME Sudria, Paris, France'
author:
- 'Hélène HALCONRUY\*'
- 'Nicolas MARIE$^\dag$'
title: Kernel Selection in Nonparametric Regression
---
**MSC2010:** 62G05 ; 62G08.
Introduction {#section_introduction}
============
Consider $n\in\mathbb N^*$ independent $\mathbb R^d\times\mathbb R$-valued ($d\in\mathbb N^*$) random variables $(X_1,Y_1),\dots,(X_n,Y_n)$, having the same probability distribution assumed to be absolutely continuous with respect to Lebesgue’s measure, and $$\widehat s_{K,\ell}(n;x) :=
\frac{1}{n}\sum_{i = 1}^{n}K(X_i,x)\ell(Y_i)
\textrm{ $;$ }x\in\mathbb R^d,$$ where $\ell :\mathbb R\rightarrow\mathbb R$ is a Borel function and $K$ is a symmetric continuous map from $\mathbb R^d\times\mathbb R^d$ into $\mathbb R$. This is an estimator of the function $s :\mathbb R^d\rightarrow\mathbb R$ defined by $$s(x) :=
\mathbb E(\ell(Y_1)|X_1 = x)f(x)
\textrm{ $;$ }
\forall x\in\mathbb R^d,$$ where $f$ is a density of $X_1$. For $\ell = 1$, $\widehat s_{K,\ell}(n;.)$ coincides with the estimator of $f$ studied in Lerasle et al. [@LMRB16], but for $\ell\not= 1$, it covers estimators involved in nonparametric regression. Assume that for every $i\in\{1,\dots,n\}$, $$\label{nonparametric_regression}
Y_i = b(X_i) +\varepsilon_i$$ where $\varepsilon_i$ is a centered random variable, independent of $X_i$, and $b :\mathbb R^d\rightarrow\mathbb R$ is a Borel function.
- If $\ell =\textrm{Id}_{\mathbb R}$, $k$ is a symmetric kernel and $$\label{PR_kernel}
K(x',x) =\prod_{q = 1}^{d}\frac{1}{h_q}k\left(\frac{x_q' - x_q}{h_q}\right)
\textrm{ with }
h_1,\dots,h_d > 0$$ for every $x,x'\in\mathbb R^d$, then $\widehat s_{K,\ell}(n;.)$ is the numerator of Nadaraya-Watson’s estimator of the regression function $b$. Precisely, $\widehat s_{K,\ell}(n;.)$ is an estimator of $s = bf$. If $\ell\not=\textrm{Id}_{\mathbb R}$, then $\widehat s_{K,\ell}(n;.)$ is the numerator of the estimator studied in Einmahl and Mason [@EM00; @EM05].
- If $\ell =\textrm{Id}_{\mathbb R}$, $\mathcal B_{m_q} =\{\varphi_{1}^{m_q},\dots,\varphi_{m_q}^{m_q}\}$ ($m_q\in\mathbb N^*$ and $q\in\{1,\dots,d\}$) is an orthonormal family of $\mathbb L^2(\mathbb R)$ and $$\label{projection_kernel}
K(x',x) =
\prod_{q = 1}^{d}
\sum_{j = 1}^{m_q}\varphi_{j}^{m_q}(x_q)\varphi_{j}^{m_q}(x_q')$$ for every $x,x'\in\mathbb R^d$, then $\widehat s_{K,\ell}(n;.)$ is the projection estimator on $\mathcal S =\textrm{span}(\mathcal B_{m_1}\otimes\dots\otimes\mathcal B_{m_d})$ of $s = bf$.
Now, assume that for every $i\in\{1,\dots,n\}$, $Y_i$ is defined by the heteroscedastic model $$\label{heteroscedastic_model}
Y_i =\sigma(X_i)\varepsilon_i,$$ where $\varepsilon_i$ is a centered random variable of variance $1$, independent of $X_i$, and $\sigma :\mathbb R^d\rightarrow\mathbb R$ is a Borel function. If $\ell(x) = x^2$ for every $x\in\mathbb R$, then $\widehat s_{K,\ell}(n;.)$ is an estimator of $s =\sigma^2f$.\
\
These ten last years, several data-driven procedures have been proposed in order to select the bandwidth of Parzen-Rosenblatt’s estimator ($\ell = 1$ and $K$ defined by (\[PR\_kernel\])). First, Goldenshluger-Lepski’s method, introduced in [@GL11], which reaches the adequate bias-variance compromise, but is not completely satisfactory on the numerical side (see Comte and Rebafka [@CR16]). More recently, in [@LMR17], Lacour, Massart and Rivoirard proposed the PCO (Penalized Comparison to Overfitting) method and proved an oracle inequality for the associated adaptative Parzen-Rosenblatt’s estimator by using a concentration inequality for the U-statistics due to Houdré and Reynaud-Bouret [@HRB03]. Together with Varet, they established the numerical efficiency of the PCO method in Varet et al. [@LMRV].\
Comte and Marie [@CM] deals with an oracle inequality and numerical experiments for an adaptative Nadaraya-Watson’s estimator with a numerator and a denominator having distinct bandwidths, both selected via the PCO method. Since the output variable in a regression model has no reason to be bounded, there were significant additional difficulties, bypassed in [@CM], to establish an oracle inequality for the numerator’s adaptative estimator. Via similar arguments, the present article deals with an oracle inequality for $\widehat s_{\widehat K,\ell}(n;.)$, where $\widehat K$ is selected via the PCO method in the spirit of Lerasle et al. [@LMRB16]. In addition to the bandwidth selection for kernel-based estimators already studied in [@LMR17; @CM], it covers the dimension selection for anisotropic projection estimators of $f$, $bf$ (when $Y_1,\dots,Y_n$ are defined by Model (\[nonparametric\_regression\])) and $\sigma^2f$ (when $Y_1,\dots,Y_n$ are defined by Model (\[heteroscedastic\_model\])). As for the bandwidth selection for kernel based estimators, for $d > 1$, the PCO method allows to bypass the numerical difficulties generated by the Goldenshluger-Lepski type method involved in the anisotropic model selection procedures (see Chagny [@CHAGNY13]).\
\
In Section \[section\_risk\_bound\], some examples of kernels sets are provided and a risk bound for $\widehat s_{K,\ell}(n;.)$ is established. Section \[section\_kernel\_selection\] deals with an oracle inequality for $\widehat s_{\widehat K,\ell}(n;.)$, where $\widehat K$ is selected via the PCO method. Finally, Section \[section\_simulations\] deals with a basic numerical study.
Risk bound {#section_risk_bound}
==========
Throughout the paper, $s\in\mathbb L^2(\mathbb R^d)$. Let $\mathcal K_n$ be a set of symmetric continuous maps from $\mathbb R^d\times\mathbb R^d$ into $\mathbb R$, of cardinal less or equal than $n$, fulfilling the following assumption.
\[assumption\_K\] There exists a deterministic constant $\mathfrak m_{\mathcal K,\ell} > 0$, not depending on $n$, such that
1. For every $K\in\mathcal K_n$, $$\sup_{x'\in\mathbb R^d}
\|K(x',.)\|_{2}^{2}
\leqslant
\mathfrak m_{\mathcal K,\ell}n.$$
2. For every $K\in\mathcal K_n$, $$\|s_{K,\ell}\|_{2}^{2}\leqslant\mathfrak m_{\mathcal K,\ell}$$ with $$s_{K,\ell} :=\mathbb E(\widehat s_{K,\ell}(n;.)) =\mathbb E(K(X_1,.)\ell(Y_1)).$$
3. For every $K,K'\in\mathcal K_n$, $$\mathbb E(\langle K(X_1,.),K'(X_2,.)\ell(Y_2)\rangle_{2}^{2})
\leqslant\mathfrak m_{\mathcal K,\ell}\mathfrak s_{K',\ell}$$ with $$\mathfrak s_{K',\ell} :=\mathbb E(\|K'(X_1,.)\ell(Y_1)\|_{2}^{2}).$$
4. For every $K\in\mathcal K_n$ and $\psi\in\mathbb L^2(\mathbb R^d)$, $$\mathbb E(\langle K(X_1,.),\psi\rangle_{2}^{2})\leqslant\mathfrak m_{\mathcal K,\ell}\|\psi\|_{2}^{2}.$$
The elements of $\mathcal K_n$ are called kernels. Let us provide two natural examples of kernels sets.
\[example\_kernels\_set\_1\] Consider $$\mathcal K_k(h_{\min}) :=
\left\{
(x',x)\mapsto
\prod_{q = 1}^{d}\frac{1}{h_q}k\left(\frac{x_q' - x_q}{h_q}\right)
\textrm{ $;$ }
h_1,\dots,h_d\in\{h_{\min},\dots,1\}\right\},$$ where $k$ is a symmetric kernel (in the usual sense) and $nh_{\min}^{d}\geqslant 1$. The kernels set $\mathcal K_k(h_{\min})$ fulfills Assumption \[assumption\_K\] and, for any $K\in\mathcal K_k(h_{\min})$ such that $$K(x',x) =
\prod_{q = 1}^{d}\frac{1}{h_q}k\left(\frac{x_q' - x_q}{h_q}\right)
\textrm{ $;$ }
\forall x,x'\in\mathbb R^d$$ with $h_1,\dots,h_d\in\{h_{\min},\dots,1\}$, $$\mathfrak s_{K,\ell} =\|k\|_{2}^{2d}\mathbb E(\ell(Y_1)^2)
\prod_{q = 1}^{d}\frac{1}{h_q}.$$
\[example\_kernels\_set\_2\] Consider $$\mathcal K_{\mathcal B_1,\dots,\mathcal B_n}(m_{\max}) :=
\left\{
(x',x)\mapsto
\prod_{q = 1}^{d}
\sum_{j = 1}^{m_q}\varphi_{j}^{m_q}(x_q)\varphi_{j}^{m_q}(x_q')
\textrm{ $;$ }
m_1,\dots,m_d\in\{1,\dots,m_{\max}\}\right\},$$ where $m_{\max}^{d}\in\{1,\dots,n\}$ and, for every $m\in\{1,\dots,n\}$, $\mathcal B_m =\{\varphi_{1}^{m},\dots,\varphi_{m}^{m}\}$ is an orthonormal family of $\mathbb L^2(\mathbb R)$ such that $$\sup_{x'\in\mathbb R}\sum_{j = 1}^{m}\varphi_{j}^{m}(x')^2
\leqslant\mathfrak m_{\mathcal B}m$$ with $\mathfrak m_{\mathcal B} > 0$ not depending on $m$ and $n$, and $$\label{projection_condition_1}
\mathcal B_m\subset\mathcal B_{m + 1}
\textrm{ $;$ }
\forall m\in\{1,\dots,n - 1\}$$ $$\label{projection_condition_2}
\overline{\mathfrak m}_{\mathcal B} :=
\sup\{
|\mathbb E(K(X_1,x))|\textrm{ $;$ }
K\in\mathcal K_{\mathcal B_1,\dots,\mathcal B_n}(m_{\max})
\textrm{ and }x\in\mathbb R^d\}
\textrm{ is finite and doesn't depend on $n$.}$$ The kernels set $\mathcal K_{\mathcal B_1,\dots,\mathcal B_n}(m_{\max})$ fulfills Assumption \[assumption\_K\] and, for any $K\in\mathcal K_{\mathcal B_1,\dots,\mathcal B_n}(m_{\max})$ such that $$K(x',x) =
\prod_{q = 1}^{d}
\sum_{j = 1}^{m_q}\varphi_{j}^{m_q}(x_q)\varphi_{j}^{m_q}(x_q')
\textrm{ $;$ }
\forall x,x'\in\mathbb R^d$$ with $m_1,\dots,m_n\in\{1,\dots,m_{\max}\}$, $$\mathfrak s_{K,\ell}\leqslant
\mathfrak m_{\mathcal B}^{d}\mathbb E(\ell(Y_1)^2)
\prod_{q = 1}^{d}m_q.$$
**Remark.** Note that Condition (\[projection\_condition\_1\]) (resp. (\[projection\_condition\_2\])) is close to (resp. the same that) Condition (19) (resp. (20)) of Lerasle et al. [@LMRB16], Proposition 3.2. See also Massart [@MASSART07], Chapter 7 on these conditions. For instance, the trigonometric basis and Hermite’s basis satisfy Condition (\[projection\_condition\_1\]). The regular histograms basis satisfy Condition (\[projection\_condition\_2\]). Indeed, by taking $\varphi_{j}^{m} =\psi_{j}^{m} :=\sqrt m\mathbf 1_{[(j - 1)/m,j/m[}$ for every $m\in\{1,\dots,n\}$ and $j\in\{1,\dots,m\}$, $$\begin{aligned}
\left|\mathbb E\left[
\prod_{q = 1}^{d}
\sum_{j = 1}^{m_q}\psi_{j}^{m_q}(X_{1,q})\psi_{j}^{m_q}(x_q)\right]\right| & = &
\sum_{j_1 = 1}^{m_1}\cdots
\sum_{j_d = 1}^{m_d}
\left(\prod_{q = 1}^{d}m_q\mathbf 1_{[(j_q - 1)/m_q,j_q/m_q[}(x_q)\right)\\
& &
\quad\quad\times
\int_{(j_1 - 1)/m_1}^{j_1/m_1}\cdots\int_{(j_d - 1)/m_d}^{j_d/m_d}f(x_1',\dots,x_d')dx_1'\cdots dx_d'\\
& \leqslant &
\|f\|_{\infty}
\prod_{q = 1}^{d}\sum_{j = 1}^{m_q}\mathbf 1_{[(j - 1)/m_q,j/m_q[}(x)\leqslant\|f\|_{\infty}\end{aligned}$$ for every $m_1,\dots,m_d\in\{1,\dots,n\}$ and $x\in\mathbb R^d$.\
\
The following proposition provides a suitable control of the variance of $\widehat s_{K,\ell}(n;.)$.
\[variance\_bound\_main\_estimator\] Under Assumption \[assumption\_K\].(1,2,3), if $s\in\mathbb L^2(\mathbb R^d)$ and if there exists $\alpha > 0$ such that $\mathbb E(\exp(\alpha|\ell(Y_1)|)) <\infty$, then there exists a deterministic constant $\mathfrak c_{\ref{variance_bound_main_estimator}} > 0$, not depending on $n$, such that for every $\theta\in ]0,1[$, $$\mathbb E\left(\sup_{K\in\mathcal K_n}\left\{
\left|\|\widehat s_{K,\ell}(n;.) - s_{K,\ell}\|_{2}^{2} -\frac{\mathfrak s_{K,\ell}}{n}\right|
-\frac{\theta}{n}\mathfrak s_{K,\ell}\right\}\right)
\leqslant
\mathfrak c_{\ref{variance_bound_main_estimator}}\frac{\log(n)^5}{\theta n}.$$
Finally, let us state the main result of this section.
\[risk\_bound\_main\_estimator\] Under Assumption \[assumption\_K\], if $s\in\mathbb L^2(\mathbb R^d)$ and if there exists $\alpha > 0$ such that $\mathbb E(\exp(\alpha|\ell(Y_1)|)) <\infty$, then there exists a deterministic constant $\mathfrak c_{\ref{risk_bound_main_estimator}},\overline{\mathfrak c}_{\ref{risk_bound_main_estimator}} > 0$, not depending on $n$, such that for every $\theta\in ]0,1[$, $$\mathbb E\left(\sup_{K\in\mathcal K_n}\left\{
\|\widehat s_{K,\ell}(n;.) - s\|_{2}^{2} - (1 +\theta)\left(\|s_{K,\ell} - s\|_{2}^{2} +\frac{\mathfrak s_{K,\ell}}{n}\right)\right\}\right)
\leqslant
\mathfrak c_{\ref{risk_bound_main_estimator}}\frac{\log(n)^5}{\theta n}$$ and $$\mathbb E\left(\sup_{K\in\mathcal K_n}\left\{
\|s_{K,\ell} - s\|_{2}^{2} +\frac{\mathfrak s_{K,\ell}}{n} -
\frac{1}{1 -\theta}\|\widehat s_{K,\ell}(n;.) - s\|_{2}^{2}\right\}\right)
\leqslant
\overline{\mathfrak c}_{\ref{risk_bound_main_estimator}}\frac{\log(n)^5}{\theta(1 -\theta) n}.$$
**Remark.** Note that the first inequality in Theorem \[risk\_bound\_main\_estimator\] gives a risk bound on the estimator $\widehat s_{K,\ell}(n;.)$: $$\mathbb E(\|\widehat s_{K,\ell}(n;.) - s\|_{2}^{2})
\leqslant
(1 +\theta)\left(\|s_{K,\ell} - s\|_{2}^{2} +\frac{\mathfrak s_{K,\ell}}{n}\right) +
\mathfrak c_{\ref{risk_bound_main_estimator}}
\frac{\log(n)^5}{\theta n}$$ for every $\theta\in ]0,1[$. The second inequality is useful in order to establish a risk bound on the adaptative estimator defined in the next section (see Theorem \[risk\_bound\_adaptative\_estimator\]).
Kernel selection {#section_kernel_selection}
================
This section deals with a risk bound on the adaptative estimator $\widehat s_{\widehat K,\ell}(n;.)$, where $$\widehat K\in\arg\min_{K\in\mathcal K_n}
\{\|\widehat s_{K,\ell}(n;\cdot) -\widehat s_{K_0,\ell}(n;\cdot)\|_{2}^{2} +\textrm{pen}(K)\},$$ $K_0$ is an overfitting proposal for $K$ and $$\label{penalty_proposal}
\textrm{pen}(K) :=
\frac{2}{n^2}
\sum_{i = 1}^{n}\langle K(.,X_i),K_0(.,X_i)\rangle_2\ell(Y_i)^2
\textrm{ $;$ }
\forall K\in\mathcal K_n.$$ **Example.** For $\mathcal K_n =\mathcal K_k(h_{\min})$, one should take $$K_0(x',x) =
\frac{1}{h_{\min}^{d}}\prod_{q = 1}^{d}k\left(\frac{x_q' - x_q}{h_{\min}}\right)
\textrm{ $;$ }
\forall x,x'\in\mathbb R^d,$$ and for $\mathcal K_n =\mathcal K_{\mathcal B_1,\dots,\mathcal B_n}(m_{\max})$, one should take $$K_0(x',x) =
\prod_{q = 1}^{d}
\sum_{j = 1}^{m_{\max}}
\varphi_{j}^{m_{\max}}(x_q)\varphi_{j}^{m_{\max}}(x_q')
\textrm{ $;$ }
\forall x,x'\in\mathbb R^d.$$ In the sequel, in addition to Assumption \[assumption\_K\], the kernels set $\mathcal K_n$ fulfills the following assumption.
\[additional\_assumption\_K\] There exists a deterministic constant $\overline{\mathfrak m}_{\mathcal K,\ell} > 0$, not depending on $n$, such that $$\mathbb E\left(\sup_{K,K'\in\mathcal K_n}
\langle K(X_1,.),s_{K',\ell}\rangle_{2}^{2}\right)
\leqslant\overline{\mathfrak m}_{\mathcal K,\ell}.$$
The following theorem provides an oracle inequality for the adaptative estimator $\widehat s_{\widehat K,\ell}(n;.)$.
\[risk\_bound\_adaptative\_estimator\] Under Assumptions \[assumption\_K\] and \[additional\_assumption\_K\], if $s\in\mathbb L^2(\mathbb R^d)$ and if there exists $\alpha > 0$ such that $\mathbb E(\exp(\alpha|\ell(Y_1)|)) <\infty$, then there exists a deterministic constant $\mathfrak c_{\ref{risk_bound_adaptative_estimator}} > 0$, not depending on $n$, such that for every $\vartheta\in ]0,1[$, $$\mathbb E(\|\widehat s_{\widehat K,\ell}(n;.) - s\|_{2}^{2})
\leqslant
(1 +\vartheta)\min_{K\in\mathcal K_n}
\mathbb E(\|\widehat s_{K,\ell}(n;.) - s\|_{2}^{2}) +
\frac{\mathfrak c_{\ref{risk_bound_adaptative_estimator}}}{\vartheta}
\left(\|s_{K_0,\ell} - s\|_{2}^{2} +\frac{\log(n)^5}{n}\right).$$
Finally, let us discuss about Assumption \[additional\_assumption\_K\]. Note that if $s$ is bounded and $$\mathfrak m_{\mathcal K} :=
\sup\{\|K(x',.)\|_{1}^{2}
\textrm{ $;$ }K\in\mathcal K_n\textrm{ and }x'\in\mathbb R^d\}$$ doesn’t depend on $n$, then $\mathcal K_n$ fulfills Assumption \[additional\_assumption\_K\]. Indeed,
$$\begin{aligned}
\mathbb E\left(\sup_{K,K'\in\mathcal K_n}
\langle K(X_1,.),s_{K',\ell}\rangle_{2}^{2}\right)
& \leqslant &
\left(\sup_{K'\in\mathcal K_n}\|s_{K',\ell}\|_{\infty}^{2}\right)\mathbb E\left(\sup_{K\in\mathcal K_n}\|K(X_1,.)\|_{1}^{2}\right)\\
& \leqslant &
\mathfrak m_{\mathcal K}\sup\left\{
\left(\int_{-\infty}^{\infty}|K'(x',x)s(x)|dx\right)^2
\textrm{ $;$ }
K'\in\mathcal K_n\textrm{ and }x'\in\mathbb R\right\}
\leqslant
\mathfrak m_{\mathcal K}^{2}\|s\|_{\infty}^{2}.\end{aligned}$$
In the nonparametric regression framework (see Model (\[nonparametric\_regression\])), to assume $s$ bounded means that $bf$ is bounded. For instance, this condition is fulfilled by the linear regression models with Gaussian inputs. The following examples focus on the condition on $\mathfrak m_{\mathcal K}$.\
\
**Examples:**
1. Consider $K\in\mathcal K_k(h_{\min})$. Then, there exists $h_1,\dots,h_d\in\{h_{\min},\dots,1\}$ such that $$K(x',x) =
\prod_{q = 1}^{d}
\frac{1}{h_q}k\left(\frac{x_q' - x_q}{h_q}\right)
\textrm{ $;$ }
\forall x,x'\in\mathbb R^d.$$ Clearly, $\|K(x',.)\|_1 =\|k\|_{1}^{d}$ for every $x'\in\mathbb R^d$. So, for $\mathcal K_n =\mathcal K_k(h_{\min})$, $\mathfrak m_{\mathcal K}\leqslant\|k\|_{1}^{2d}$.
2. For $\mathcal K_n =\mathcal K_{\mathcal B_1,\dots,\mathcal B_n}(m_{\max})$, the condition on $\mathfrak m_{\mathcal K}$ seems harder to check in general. Let us show that it is satisfied for the regular histograms basis defined in Section \[section\_risk\_bound\]. For every $m_1,\dots,m_d\in\{1,\dots,n\}$, $$\left\|\prod_{q = 1}^{d}\sum_{j = 1}^{m_q}\psi_{j}^{m_q}(x_q')\psi_{j}^{m_q}(._q)\right\|_1
\leqslant
\prod_{q = 1}^{d}\left(
m_q\sum_{j = 1}^{m_q}\mathbf 1_{[(j - 1)/m_q,j/m_q[}(x_q')\int_{(j - 1)/m_q}^{j/m_q}dx\right)
\leqslant 1.$$
The following proposition shows that $\mathcal K_{\mathcal B_1,\dots,\mathcal B_n}(m_{\max})$ fulfills Assumption \[additional\_assumption\_K\] for the trigonometric basis, even if the condition on $\mathfrak m_{\mathcal K}$ is not satisfied.
\[example\_kernels\_set+\] Consider $\chi_1 :=\mathbf 1_{[0,1]}$ and, for every $j\in\mathbb N^*$, the functions $\chi_{2j}$ and $\chi_{2j + 1}$ defined on $\mathbb R$ by $$\chi_{2j}(x) :=\sqrt 2\cos(2\pi jx)\mathbf 1_{[0,1]}(x)
\textrm{ and }
\chi_{2j + 1}(x) :=\sqrt 2\sin(2\pi j x)\mathbf 1_{[0,1]}(x)
\textrm{ $;$ }
\forall x\in\mathbb R.$$ If $s\in C^2(\mathbb R^d)$ and $\mathcal B_m =\{\chi_1,\dots,\chi_m\}$ for every $m\in\{1,\dots,n\}$, then $\mathcal K_{\mathcal B_1,\dots,\mathcal B_n}(m_{\max})$ fulfills Assumption \[additional\_assumption\_K\].
Basic numerical experiments {#section_simulations}
===========================
Throughout this section, $d = 1$, $\ell\in\{1,\textrm{Id}_{\mathbb R}\}$ and $Y_1,\dots,Y_n$ are defined by Model (\[nonparametric\_regression\]) with $\varepsilon_1,\dots,\varepsilon_n\rightsquigarrow\mathcal N(0,1)$. Some numerical experiments on $\widehat s_{K,1}(n;.)$ (resp. $\widehat s_{K,\textrm{Id}_{\mathbb R}}(n;.)$) for $K\in\mathcal K_k(h_{\min})$ have already been done in Varet et al. [@LMRV] (resp. Comte and Marie [@CM]). So, this section deals with basic numerical experiments on $\widehat s_{K,1}(n;.)$ and $\widehat s_{K,\textrm{Id}_{\mathbb R}}(n;.)$ for $K\in\mathcal K_{\mathcal B_1,\dots,\mathcal B_n}(m_{\max})$ and $\mathcal B_m =\{\psi_{1}^{m},\dots,\psi_{m}^{m}\}$ for every $m = 1,\dots,n$.\
\
In this case, $\widehat K =\textrm K_{\widehat m(\ell)}$ where $$\textrm K_m(x',x) :=
\sum_{j = 1}^{m}\psi_{j}^{m}(x')\psi_{j}^{m}(x)
\textrm{ $;$ }
\forall x,x'\in\mathbb R
\textrm{, }
\forall m\in\mathcal M =\{1,\dots,m_{\max}\},$$ $\widehat m(\ell)$ is a solution of the minimization problem $$\min_{m\in\mathcal M}
\{\|\widehat s_{\textrm K_m,\ell}(n;.) -\widehat s_{\textrm K_{m_{\max}},\ell}(n;.)\|_{2}^{2} +\textrm{pen}(m)\}$$ and $$\textrm{pen}(m) :=
\frac{2}{n^2}\sum_{i = 1}^{n}\langle\textrm K_m(.,X_i),\textrm K_{m_{\max}}(.,X_i)\rangle_2\ell(Y_i)^2
\textrm{ $;$ }
\forall m\in\mathcal M.$$ For $\ell\in\{1,\textrm{Id}_{\mathbb R}\}$, $n = 250$ and $m_{\max} = 30$, $m$ is selected in $\mathcal M$ for two basic densities and two nonlinear regression functions:
- $f = f_1$ the density of $\mathcal E(5)$.
- $f = f_2$ the density of $\mathcal N(1/2,(1/8)^2)$.
- $b(x) = b_1(x) := 10(x^2 - 1/2)$ for every $x\in [0,1]$.
- $b(x) = b_2(x) :=\cos(5\pi x)$ for every $x\in [0,1]$.
On the one hand, on the four following figures, one can see the beam of all possible estimations of $f$ and $bf$ (i.e. for each $m\in\mathcal M$) at left, the PCO criteria for $\widehat s_{K,1}(n;.)$ and $\widehat s_{K,\textrm{Id}_{\mathbb R}}(n;.)$ for each $m\in\mathcal M$ at the middle, and the PCO estimations of $f$ and $bf$ (i.e. for $m =\widehat m(1)$ and $m =\widehat m(\textrm{Id}_{\mathbb R})$) at right:
On the other hand, for $(f,b) = (f_1,b_2)$ and $(f,b) = (f_2,b_1)$, let us generate 10 datasets of $n = 250$ observations of $(X_1,Y_1)$ and, for each of these, select $m\in\mathcal M$ via the PCO criterion introduced previously. On the two following figures, the beam of all PCO estimations of $f$ (resp. $bf$) is plotted at left (resp. at right):
Details on kernels sets: proofs of Propositions \[example\_kernels\_set\_1\], \[example\_kernels\_set\_2\] and \[example\_kernels\_set+\]
=========================================================================================================================================
Proof of Proposition \[example\_kernels\_set\_1\]
-------------------------------------------------
Consider $K,K'\in\mathcal K_k(h_{\min})$. Then, there exist $h,h'\in\{h_{\min},\dots,1\}^d$ such that $$K(x',x) =
\prod_{q = 1}^{d}
\frac{1}{h_q}k\left(\frac{x_q' - x_q}{h_q}\right)
\textrm{ and }
K'(x',x) =
\prod_{q = 1}^{d}
\frac{1}{h_q'}k\left(\frac{x_q' - x_q}{h_q'}\right)$$ for every $x,x'\in\mathbb R^d$.
1. For every $x'\in\mathbb R^d$, $$\|K(x',.)\|_{2}^{2} =
\|k\|_{2}^{2d}
\prod_{q = 1}^{d}
\frac{1}{h_q}
\leqslant\|k\|_{2}^{2d}n.$$
2. Since $s_{K,\ell} = K\ast s$, $\|s_{K,\ell}\|_{2}^{2}\leqslant\|k\|_{1}^{2d}\|s\|_{2}^{2}$.
3. First, $$\mathfrak s_{K',\ell} =
\|k\|_{2}^{2d}\mathbb E(\ell(Y_1)^2)\prod_{q = 1}^{d}
\frac{1}{h_q'}.$$ Then, $$\begin{aligned}
\mathbb E(\langle K(X_1,.),K'(X_2,.)\ell(Y_2)\rangle_{2}^{2})
& = &
\mathbb E((K\ast K')(X_1 - X_2)^2\ell(Y_2)^2)\\
& \leqslant &
\|f\|_{\infty}\|K\ast K'\|_{2}^{2}
\mathbb E(\ell(Y_1)^2)\\
& \leqslant &
\|f\|_{\infty}\|k\|_{1}^{2d}\mathfrak s_{K',\ell}.
\end{aligned}$$
4. For every $\psi\in\mathbb L^2(\mathbb R^d)$, $$\begin{aligned}
\mathbb E(\langle K(X_1,.),\psi\rangle_{2}^{2})
& = &
\mathbb E((K\ast\psi)(X_1)^2)\\
& \leqslant &
\|f\|_{\infty}
\|K\ast\psi\|_{2}^{2}
\leqslant
\|f\|_{\infty}
\|k\|_{1}^{2d}\|\psi\|_{2}^{2}.
\end{aligned}$$
Proof of Proposition \[example\_kernels\_set\_2\]
-------------------------------------------------
Consider $K,K'\in\mathcal K_{\mathcal B_1,\dots,\mathcal B_n}(m_{\max})$. Then, there exist $m,m'\in\{1,\dots,m_{\max}\}^d$ such that $$K(x',x) =
\prod_{q = 1}^{d}
\sum_{j = 1}^{m_q}\varphi_{j}^{m_q}(x_q)\varphi_{j}^{m_q}(x_q')
\textrm{ and }
K'(x',x) =
\prod_{q = 1}^{d}
\sum_{j = 1}^{m_q'}\varphi_{j}^{m_q'}(x_q)\varphi_{j}^{m_q'}(x_q')$$ for every $x,x'\in\mathbb R^d$.
1. For every $x'\in\mathbb R^d$, $$\begin{aligned}
\|K(x',.)\|_{2}^{2} & = &
\prod_{q = 1}^{d}
\sum_{j,j' = 1}^{m_q}\varphi_{j'}^{m_q}(x_q')\varphi_{j}^{m_q}(x_q')
\int_{-\infty}^{\infty}\varphi_{j'}^{m_q}(x)\varphi_{j}^{m_q}(x)dx\\
& = &
\prod_{q = 1}^{d}
\sum_{j = 1}^{m_q}
\varphi_{j}^{m_q}(x_q')^2\leqslant
\mathfrak m_{\mathcal B}^{d}
\prod_{q = 1}^{d}m_q
\leqslant
\mathfrak m_{\mathcal B}^{d}n.
\end{aligned}$$
2. Since $$s_{K,\ell}(.) =
\sum_{j_1 = 1}^{m_1}\cdots\sum_{j_d = 1}^{m_d}
\langle s,\varphi_{j_1}^{m_1}\otimes\cdots\otimes\varphi_{j_d}^{m_d}\rangle_2
(\varphi_{j_1}^{m_1}\otimes\cdots\otimes\varphi_{j_d}^{m_d})(.),$$ by Pythagore’s theorem, $\|s_{K,\ell}\|_{2}^{2}\leqslant\|s\|_{2}^{2}$.
3. First, $$\mathfrak s_{K',\ell}
= \mathbb E\left[\ell(Y_1)^2
\prod_{q = 1}^{d}
\sum_{j = 1}^{m_q'}\varphi_{j}^{m_q'}(X_{1,q})^2\right]
\leqslant\mathfrak m_{\mathcal B}^{d}\mathbb E(\ell(Y_1)^2)
\prod_{q = 1}^{d}m_q'.$$ On the one hand, if $\mathcal B_1,\dots,\mathcal B_n$ satisfy Condition (\[projection\_condition\_1\]), then
$$\begin{aligned}
\mathbb E(\langle K(X_1,.),K'(X_2,.)\ell(Y_2)\rangle_{2}^{2})
& = &
\int_{\mathbb R^d}\mathbb E\left[
\left(\prod_{q = 1}^{d}
\sum_{j = 1}^{m_q\wedge m_q'}\varphi_{j}^{m_q'}(x_q')\varphi_{j}^{m_q'}(X_{2,q})\right)^2\ell(Y_2)^2\right]f(x')\lambda_d(dx')\\
& \leqslant &
\|f\|_{\infty}\mathbb E\left[\ell(Y_2)^2
\prod_{q = 1}^{d}
\sum_{j,j' = 1}^{m_q\wedge m_q'}\varphi_{j'}^{m_q'}(X_{2,q})\varphi_{j}^{m_q'}(X_{2,q})
\int_{-\infty}^{\infty}\varphi_{j'}^{m_q'}(x')\varphi_{j}^{m_q'}(x')dx'\right]\\
& &
\quad\quad\leqslant
\|f\|_{\infty}\mathfrak s_{K',\ell}.
\end{aligned}$$
On the other hand, if $\mathcal B_1,\dots,\mathcal B_n$ satisfy Condition (\[projection\_condition\_2\]), then $$\begin{aligned}
\mathbb E(\langle K(X_1,.),K'(X_2,.)\ell(Y_2)\rangle_{2}^{2})
& \leqslant &
\mathbb E(\|K(X_1,.)\|_{2}^{2}\|K'(X_2,.)\|_{2}^{2}\ell(Y_2)^2)\\
& = & \mathbb E(K(X_1,X_1))
\mathbb E(\|K'(X_2,.)\|_{2}^{2}\ell(Y_2)^2)
\leqslant
\overline{\mathfrak m}_{\mathcal B}\mathfrak s_{K',\ell}.
\end{aligned}$$
4. For every $\psi\in\mathbb L^2(\mathbb R^d)$, $$\begin{aligned}
\mathbb E(\langle K(X_1,.),\psi\rangle_{2}^{2})
& = &
\mathbb E\left[\left|\sum_{j_1 = 1}^{m_1}\cdots\sum_{j_d = 1}^{m_d}
\langle\psi,\varphi_{j_1}^{m_1}\otimes\cdots\otimes\varphi_{j_d}^{m_d}\rangle_2
(\varphi_{j_1}^{m_1}\otimes\cdots\otimes\varphi_{j_d}^{m_d})(X_1)\right|^2\right]\\
& \leqslant &
\|f\|_{\infty}\left\|\sum_{j_1 = 1}^{m_1}\cdots\sum_{j_d = 1}^{m_d}
\langle\psi,\varphi_{j_1}^{m_1}\otimes\cdots\otimes\varphi_{j_d}^{m_d}\rangle_2
(\varphi_{j_1}^{m_1}\otimes\cdots\otimes\varphi_{j_d}^{m_d})(.)\right\|_{2}^{2}
\leqslant\|f\|_{\infty}\|\psi\|_{2}^{2}.
\end{aligned}$$
Proof of Proposition \[example\_kernels\_set+\]
-----------------------------------------------
For the sake of readability, assume that $d = 1$. Consider $K,K'\in\mathcal K_{\mathcal B_1,\dots,\mathcal B_n}(m_{\max})$. Then, there exist $m,m'\in\{1,\dots,m_{\max}\}$ such that $$K(x',x) =
\sum_{j = 1}^{m}\chi_j(x)\chi_j(x')
\textrm{ and }
K'(x',x) =
\sum_{j = 1}^{m'}\chi_j(x)\chi_j(x')
\textrm{ $;$ }
\forall x,x'\in\mathbb R.$$ First, there exist $\mathfrak m_1(m,m')\in\{0,\dots,n\}$ and $\mathfrak c_1 > 0$, not depending on $n$, $K$ and $K'$, such that for any $x'\in [0,1]$, $$\begin{aligned}
|\langle K(x',.),s_{K',\ell}\rangle_2| & = &
\left|\sum_{j = 1}^{m\wedge m'}
\mathbb E(\ell(Y_1)\chi_j(X_1))\chi_j(x')\right|\\
& \leqslant &
\mathfrak c_1 + 2\left|
\sum_{j = 1}^{\mathfrak m_1(m,m')}
\mathbb E(\ell(Y_1)(\cos(2\pi jX_1)\cos(2\pi jx') +
\sin(2\pi jX_1)\sin(2\pi jx'))\mathbf 1_{[0,1]}(X_1))\right|\\
& = &
\mathfrak c_1 + 2\left|
\sum_{j = 1}^{\mathfrak m_1(m,m')}
\mathbb E(\ell(Y_1)\cos(2\pi j(X_1 - x'))\mathbf 1_{[0,1]}(X_1))\right|. \end{aligned}$$ Moreover, for any $j\in\{2,\dots,\mathfrak m_1(m,m')\}$, $$\begin{aligned}
\mathbb E(\ell(Y_1)\cos(2\pi j(X_1 - x'))\mathbf 1_{[0,1]}(X_1)) & = &
\int_{0}^{1}\cos(2\pi j(x - x'))s(x)dx\\
& = &
\frac{1}{j}\left[\frac{\sin(2\pi j(x - x'))}{2\pi}s(x)\right]_{0}^{1}\\
& &
+\frac{1}{j^2}\left[\frac{\cos(2\pi j(x - x'))}{4\pi^2}s'(x)\right]_{0}^{1}
-\frac{1}{j^2}\int_{0}^{1}\frac{\cos(2\pi j(x - x'))}{4\pi^2}s''(x)dx\\
& = &
\frac{s(0) - s(1)}{2\pi}\cdot\frac{\alpha_j(x')}{j} +\frac{\beta_j(x')}{j^2}\end{aligned}$$ where $\alpha_j(x') :=\sin(2\pi jx')$ and $$\beta_j(x') :=
\frac{1}{4\pi^2}\left(
(s'(1) - s'(0))\cos(2\pi jx') -\int_{0}^{1}\cos(2\pi j(x - x'))s''(x)dx\right).$$ Then, there exists a deterministic constant $\mathfrak c_2 > 0$, not depending on $n$, $K$, $K'$ and $x'$, such that $$\label{example_kernels_set+_1}
\langle K(x',.),s_{K',\ell}\rangle_{2}^{2}
\leqslant
\mathfrak c_2\left[1 +
\left(\sum_{j = 1}^{\mathfrak m_1(m,m')}\frac{\alpha_j(x')}{j}\right)^2 +
\left(\sum_{j = 1}^{\mathfrak m_1(m,m')}\frac{\beta_j(x')}{j^2}\right)^2\right].$$ Let us show that each term of the right-hand side of Inequality (\[example\_kernels\_set+\_1\]) are uniformly bounded in $x'$, $m$ and $m'$. On the one hand, $$\left|\sum_{j = 1}^{\mathfrak m_1(m,m')}\frac{\beta_j(x')}{j^2}\right|
\leqslant
\max_{j\in\{1,\dots,n\}}
\|\beta_j\|_{\infty}
\sum_{j = 1}^{n}\frac{1}{j^2}
\leqslant
\frac{1}{24}
(2\|s'\|_{\infty} +\|s''\|_{\infty}).$$ On the other hand, for every $x\in ]0,\pi[$ such that $[\pi/x] + 1\leqslant\mathfrak m_1(m,m')$ (without loss of generality), $$\begin{aligned}
\left|\sum_{j = 1}^{\mathfrak m_1(m,m')}\frac{\sin(jx)}{j}\right|
& \leqslant &
\left|\sum_{j = 1}^{[\pi/x]}\frac{\sin(jx)}{j}\right| +
\left|\sum_{j = [\pi/x] + 1}^{\mathfrak m_1(m,m')}\frac{\sin(jx)}{j}\right|
\nonumber\\
\label{example_kernels_set+_2}
& \leqslant & x\left[\frac{\pi}{x}\right] +\frac{2}{(1 + [\pi/x])\sin(x/2)}
\leqslant\pi + 2.\end{aligned}$$ Since $x\mapsto\sin(x)$ is continuous, odd and $2\pi$-periodic, Inequality (\[example\_kernels\_set+\_2\]) holds true for every $x\in\mathbb R$. So, $$\left|\sum_{j = 1}^{\mathfrak m_1(m,m')}\frac{\alpha_j(x')}{j}\right|
\leqslant
\pi + 2.$$ Therefore, $$\mathbb E\left[\sup_{K,K'\in\mathcal K_{\mathcal B_1,\dots,\mathcal B_n}(m_{\max})}
\langle K(X_1,.),s_{K',\ell}\rangle_{2}^{2}\right]
\leqslant
\mathfrak c_2\left(1 +(\pi + 2)^2 +
\frac{1}{24^2}
(2\|s'\|_{\infty} +\|s''\|_{\infty})^2\right).$$
Proofs of risk bounds
=====================
In this section, the proofs follow the same pattern as in Comte and Marie [@CM20; @CM].
Preliminary results
-------------------
This subsection provides three lemmas used several times in the sequel.
\[bound\_U\_statistics\] Consider $$U_{K,K',\ell}(n) :=\sum_{i\not= j}
\langle K(X_i,.)\ell(Y_i) - s_{K,\ell},
K'(X_j,.)\ell(Y_j) - s_{K',\ell}\rangle_2
\textrm{ $;$ }
\forall K,K'\in\mathcal K_n.$$ Under Assumption \[assumption\_K\].(1,2,3), if $s\in\mathbb L^2(\mathbb R^d)$ and if there exists $\alpha > 0$ such that $\mathbb E(\exp(\alpha|\ell(Y_1)|)) <\infty$, then there exists a deterministic constant $\mathfrak c_{\ref{bound_U_statistics}} > 0$, not depending on $n$, such that for every $\theta\in ]0,1[$, $$\mathbb E\left(\sup_{K,K'\in\mathcal K_n}
\left\{\frac{|U_{K,K',\ell}(n)|}{n^2}
-\frac{\theta}{n}\mathfrak s_{K',\ell}
\right\}\right)
\leqslant
\mathfrak c_{\ref{bound_U_statistics}}
\frac{\log(n)^5}{\theta n}.$$
\[bound\_trace\_term\] Consider $$V_{K,\ell}(n) :=\frac{1}{n}\sum_{i = 1}^{n}\|K(X_i,.)\ell(Y_i) - s_{K,\ell}\|_{2}^{2}
\textrm{ $;$ }
\forall K\in\mathcal K_n.$$ Under Assumption \[assumption\_K\].(1,2), if $s\in\mathbb L^2(\mathbb R^d)$ and if there exists $\alpha > 0$ such that $\mathbb E(\exp(\alpha|\ell(Y_1)|)) <\infty$, then there exists a deterministic constant $\mathfrak c_{\ref{bound_trace_term}} > 0$, not depending on $n$, such that for every $\theta\in ]0,1[$, $$\mathbb E\left(\sup_{K\in\mathcal K_n}\left\{
\frac{1}{n}|V_{K,\ell}(n) -\mathfrak s_{K,\ell}| -\frac{\theta}{n}\mathfrak s_{K,\ell}\right\}\right)
\leqslant\mathfrak c_{\ref{bound_trace_term}}\frac{\log(n)^3}{\theta n}.$$
\[bound\_crossed\_term\] Consider $$W_{K,K',\ell}(n) :=
\langle\widehat s_{K,\ell}(n;.) - s_{K,\ell},s_{K',\ell} - s\rangle_2
\textrm{ $;$ }
\forall K,K'\in\mathcal K_n.$$ Under Assumption \[assumption\_K\].(1,2,4), if $s\in\mathbb L^2(\mathbb R^d)$ and if there exists $\alpha > 0$ such that $\mathbb E(\exp(\alpha|\ell(Y_1)|)) <\infty$, then there exists a deterministic constant $\mathfrak c_{\ref{bound_crossed_term}} > 0$, not depending on $n$, such that for every $\theta\in ]0,1[$, $$\mathbb E\left(\sup_{K,K'\in\mathcal K_n}\{
|W_{K,K',\ell}(n)| -\theta\|s_{K',\ell} - s\|_{2}^{2}
\}\right)
\leqslant
\mathfrak c_{\ref{bound_crossed_term}}\frac{\log(n)^4}{\theta n}.$$
### Proof of Lemma \[bound\_U\_statistics\]
Consider $\mathfrak m(n) := 8\log(n)/\alpha$. For any $K,K'\in\mathcal K_n$, $$U_{K,K',\ell}(n)
= U_{K,K',\ell}^{1}(n) + U_{K,K',\ell}^{2}(n) + U_{K,K',\ell}^{3}(n) + U_{K,K',\ell}^{4}(n)$$ where $$U_{K,K',\ell}^{l}(n) :=
\sum_{i \not= j}
g_{K,K',\ell}^{l}(n;X_i,Y_i,X_j,Y_j)
\textrm{ $;$ }
l = 1,2,3,4$$ with, for every $(x',y),(x'',y')\in E =\mathbb R^d\times\mathbb R$, $$\begin{aligned}
g_{K,K',\ell}^{1}(n;x',y,x'',y') & := &
\langle K(x',.)\ell(y)\mathbf 1_{|\ell(y)|\leqslant\mathfrak m(n)} - s_{K,\ell}^{+}(n;.),
K'(x'',.)\ell(y')\mathbf 1_{|\ell(y)|\leqslant\mathfrak m(n)} - s_{K',\ell}^{+}(n;.)\rangle_2,\\
g_{K,K',\ell}^{2}(n;x',y,x'',y') & := &
\langle K(x',.)\ell(y)\mathbf 1_{|\ell(y)| >\mathfrak m(n)} - s_{K,\ell}^{-}(n;.),
K'(x'',.)\ell(y')\mathbf 1_{|\ell(y)|\leqslant\mathfrak m(n)} - s_{K',\ell}^{+}(n;.)\rangle_2,\\
g_{K,K',\ell}^{3}(n;x',y,x'',y') & := &
\langle K(x',.)\ell(y)\mathbf 1_{|\ell(y)|\leqslant\mathfrak m(n)} - s_{K,\ell}^{+}(n;.),
K'(x'',.)\ell(y')\mathbf 1_{|\ell(y)| >\mathfrak m(n)} - s_{K',\ell}^{-}(n;.)\rangle_2,\\
g_{K,K',\ell}^{4}(n;x',y,x'',y') & := &
\langle K(x',.)\ell(y)\mathbf 1_{|\ell(y)| >\mathfrak m(n)} - s_{K,\ell}^{-}(n;.),
K'(x'',.)\ell(y')\mathbf 1_{|\ell(y)| >\mathfrak m(n)} - s_{K',\ell}^{-}(n;.)\rangle_2\end{aligned}$$ and, for every $k\in\mathcal K_n$, $$s_{k,\ell}^{+}(n;.) :=
\mathbb E(k(X_1,.)\ell(Y_1)\mathbf 1_{|\ell(Y_1)|\leqslant\mathfrak m(n)})
\textrm{ and }
s_{k,\ell}^{-}(n;.) :=
\mathbb E(k(X_1,.)\ell(Y_1)\mathbf 1_{|\ell(Y_1)| >\mathfrak m(n)}).$$ On the one hand, since $\mathbb E(g_{K,K',\ell}^{1}(n;x',y,X_1,Y_1)) = 0$ for every $(x',y)\in E$, by Giné and Nickl [@GN15], Theorem 3.4.8, there exists a universal constant $\mathfrak m\geqslant 1$ such that for any $\lambda > 0$, with probability larger than $1 - 5.4e^{-\lambda}$, $$\frac{|U_{K,K',\ell}^{1}(n)|}{n^2}
\leqslant\frac{\mathfrak m}{n^2}(\mathfrak c_{K,K',\ell}(n)\lambda^{1/2} +\mathfrak d_{K,K',\ell}(n)\lambda +\mathfrak b_{K,K',\ell}(n)\lambda^{3/2} +\mathfrak a_{K,K',\ell}(n)\lambda^2)$$ where the constants $\mathfrak a_{K,K',\ell}(n)$, $\mathfrak b_{K,K',\ell}(n)$, $\mathfrak c_{K,K',\ell}(n)$ and $\mathfrak d_{K,K',\ell}(n)$ are defined and controlled later. First, note that $$\begin{aligned}
U_{K,K',\ell}^{1}(n) & = &
\sum_{i\not= j}
(\varphi_{K,K',\ell}(n;X_i,Y_i,X_j,Y_j)
\nonumber\\
\label{bound_U_statistics_1}
& &
\quad\quad\quad
-\psi_{K,K',\ell}(n;X_i,Y_i)
-\psi_{K',K,\ell}(n;X_j,Y_j)
+\mathbb E(\varphi_{K,K',\ell}(n;X_i,Y_i,X_j,Y_j))),\end{aligned}$$ where $$\varphi_{K,K',\ell}(n;x',y,x'',y'') :=
\langle K(x',.)\ell(y)\mathbf 1_{|\ell(y)|\leqslant\mathfrak m(n)},
K'(x'',.)\ell(y')\mathbf 1_{|\ell(y')|\leqslant\mathfrak m(n)}\rangle_2$$ and $$\begin{aligned}
\psi_{k,k',\ell}(n;x',y) :=
\langle k(x',.)\ell(y)\mathbf 1_{|\ell(y)|\leqslant\mathfrak m(n)},
s_{k',\ell}^{+}(n;.)\rangle_2 =
\mathbb E(\varphi_{k,k',\ell}(n;x',y,X_1,Y_1))\end{aligned}$$ for every $k,k'\in\mathcal K_n$ and $(x',y),(x'',y')\in E$. Let us now control $\mathfrak a_{K,K',\ell}(n)$, $\mathfrak b_{K,K',\ell}(n)$, $\mathfrak c_{K,K',\ell}(n)$ and $\mathfrak d_{K,K',\ell}(n)$:
- **The constant $\mathfrak a_{K,K',\ell}(n)$.** Consider $$\mathfrak a_{K,K',\ell}(n) :=
\sup_{(x',y),(x'',y')\in E}
|g_{K,K',\ell}^{1}(n;x',y,x'',y')|.$$ By (\[bound\_U\_statistics\_1\]), Cauchy-Schwarz’s inequality and Assumption \[assumption\_K\].(1), $$\begin{aligned}
\mathfrak a_{K,K',\ell}(n) & \leqslant &
4\sup_{(x',y),(x'',y')\in E}
|\langle K(x',.)\ell(y)\mathbf 1_{|\ell(y)|\leqslant\mathfrak m(n)},
K'(x'',.)\ell(y')\mathbf 1_{|\ell(y')|\leqslant\mathfrak m(n)}\rangle_2|\\
& \leqslant &
4\mathfrak m(n)^2
\left(\sup_{x'\in\mathbb R^d}\|K(x',.)\|_2\right)
\left(\sup_{x''\in\mathbb R^d}\|K'(x'',.)\|_2\right)
\leqslant 4\mathfrak m_{\mathcal K,\ell}\mathfrak m(n)^2n.
\end{aligned}$$ So, $$\frac{1}{n^2}\mathfrak a_{K,K',\ell}(n)\lambda^2
\leqslant
\frac{4}{n}\mathfrak m_{\mathcal K,\ell}\mathfrak m(n)^2\lambda^2.$$
- **The constant $\mathfrak b_{K,K',\ell}(n)$.** Consider $$\mathfrak b_{K,K',\ell}(n)^2 :=
n\sup_{(x',y)\in E}
\mathbb E(g_{K,K',\ell}^{1}(n;x',y,X_1,Y_1)^2).$$ By (\[bound\_U\_statistics\_1\]), Jensen’s inequality, Cauchy-Schwarz’s inequality and Assumption \[assumption\_K\].(1), $$\begin{aligned}
\mathfrak b_{K,K',\ell}(n)^2 & \leqslant &
16n\sup_{(x',y)\in E}
\mathbb E(\langle K(x',.)\ell(y)\mathbf 1_{|\ell(y)|\leqslant\mathfrak m(n)},
K'(X_1,.)\ell(Y_1)\mathbf 1_{|\ell(Y_1)|\leqslant\mathfrak m(n)}\rangle_{2}^{2})\\
& \leqslant &
16n\mathfrak m(n)^2\sup_{x'\in\mathbb R^d}
\|K(x',.)\|_{2}^{2}
\mathbb E(
\|K'(X_1,.)\ell(Y_1)\mathbf 1_{|\ell(Y_1)|\leqslant\mathfrak m(n)}\|_{2}^{2})
\leqslant
16\mathfrak m_{\mathcal K,\ell}n^2\mathfrak m(n)^2\mathfrak s_{K',\ell}.
\end{aligned}$$ So, for any $\theta\in]0,1[$, $$\begin{aligned}
\frac{1}{n^2}\mathfrak b_{K,K',\ell}(n)\lambda^{3/2}
& \leqslant &
2\left(\frac{3\mathfrak m}{\theta}\right)^{1/2}\frac{2}{n^{1/2}}\mathfrak m_{\mathcal K,\ell}^{1/2}\mathfrak m(n)\lambda^{3/2}
\times\left(\frac{\theta}{3\mathfrak m}\right)^{1/2}\frac{1}{n^{1/2}}\mathfrak s_{K',\ell}^{1/2}\\
& \leqslant &
\frac{\theta}{3\mathfrak mn}\mathfrak s_{K',\ell} +
\frac{12\mathfrak m\lambda^3}{\theta n}\mathfrak m_{\mathcal K,\ell}\mathfrak m(n)^2.
\end{aligned}$$
- **The constant $\mathfrak c_{K,K',\ell}(n)$.** Consider $$\mathfrak c_{K,K',\ell}(n)^2 :=
n^2\mathbb E(g_{K,K',\ell}^{1}(n;X_1,Y_1,X_2,Y_2)^2).$$ By (\[bound\_U\_statistics\_1\]), Jensen’s inequality and Assumption \[assumption\_K\].(3), $$\begin{aligned}
\mathfrak c_{K,K',\ell}(n)^2
& \leqslant &
16n^2\mathbb E(\langle K(X_1,.)\ell(Y_1)\mathbf 1_{|\ell(Y_1)|\leqslant\mathfrak m(n)},
K'(X_2,.)\ell(Y_2)\mathbf 1_{|\ell(Y_2)|\leqslant\mathfrak m(n)}\rangle_{2}^{2})\\
& \leqslant &
16n^2\mathfrak m(n)^2
\mathbb E(\langle K(X_1,.),K'(X_2,.)\ell (Y_2)\rangle_{2}^{2})
\leqslant 16\mathfrak m_{\mathcal K,\ell}n^2\mathfrak m(n)^2\mathfrak s_{K',\ell}.
\end{aligned}$$ So, $$\frac{1}{n^2}\mathfrak c_{K,K',\ell}(n)\lambda^{1/2}
\leqslant
\frac{\theta}{3\mathfrak mn}\mathfrak s_{K',\ell} +
\frac{12\mathfrak m\lambda}{\theta n}\mathfrak m_{\mathcal K,\ell}\mathfrak m(n)^2.$$
- **The constant $\mathfrak d_{K,K',\ell}(n)$.** Consider $$\mathfrak d_{K,K',\ell}(n) :=
\sup_{(a,b)\in\mathcal A}
\mathbb E\left[\sum_{i < j}a_i(X_i,Y_i)b_j(X_j,Y_j)g_{K,K',\ell}^{1}(n;X_i,Y_i,X_j,Y_j)\right],$$ where $$\mathcal A :=
\left\{(a,b) :
\sum_{i = 1}^{n - 1}\mathbb E(a_i(X_i,Y_i)^2)\leqslant 1
\textrm{ and }
\sum_{j = 2}^{n}\mathbb E(b_j(X_j,Y_j)^2)\leqslant 1\right\}.$$ By (\[bound\_U\_statistics\_1\]), Jensen’s inequality, Cauchy-Schwarz’s inequality and Assumption \[assumption\_K\].(3), $$\begin{aligned}
\mathfrak d_{K,K',\ell}(n) & \leqslant &
4\sup_{(a,b)\in\mathcal A}\mathbb E\left[
\sum_{i = 1}^{n - 1}
\sum_{j = i + 1}^{n}
|a_i(X_i,Y_i)b_j(X_j,Y_j)
\varphi_{K,K',\ell}(n;X_i,Y_i,X_j,Y_j)|\right]\\
& \leqslant &
4n\mathfrak m(n)
\mathbb E(\langle K(X_1,.),K'(X_2,.)\ell (Y_2)\rangle_{2}^{2})^{1/2}
\leqslant
4\mathfrak m_{\mathcal K,\ell}^{1/2}n\mathfrak m(n)\mathfrak s_{K',\ell}^{1/2}.
\end{aligned}$$ So, $$\frac{1}{n^2}\mathfrak d_{K,K',\ell}(n)\lambda
\leqslant
\frac{\theta}{3\mathfrak mn}\mathfrak s_{K',\ell} +
\frac{12\mathfrak m\lambda^2}{\theta n}\mathfrak m_{\mathcal K,\ell}\mathfrak m(n)^2.$$
Then, since $\mathfrak m\geqslant 1$ and $\lambda > 0$, with probability larger than $1 - 5.4e^{-\lambda}$, $$\frac{|U_{K,K',\ell}^{1}(n)|}{n^2}
\leqslant
\frac{\theta}{n}\mathfrak s_{K',\ell} +
\frac{40\mathfrak m^2}{\theta n}\mathfrak m_{\mathcal K,\ell}\mathfrak m(n)^2(1 +\lambda)^3.$$ So, with probability larger than $1 - 5.4|\mathcal K_n|e^{-\lambda}$, $$S_{\mathcal K,\ell}(n,\theta) :=
\sup_{K,K'\in\mathcal K_n}
\left\{\frac{|U_{K,K',\ell}^{1}(n)|}{n^2}
-\frac{\theta}{n}\mathfrak s_{K',\ell}
\right\}
\leqslant
\frac{40\mathfrak m^2}{\theta n}\mathfrak m_{\mathcal K,\ell}\mathfrak m(n)^2(1 +\lambda)^3.$$ For every $t\in\mathbb R_+$, consider $$\lambda_{\mathcal K,\ell}(n,\theta,t) :=
-1 +\left(\frac{t}{\mathfrak m_{\mathcal K,\ell}(n,\theta)}\right)^{1/3}
\textrm{ with }
\mathfrak m_{\mathcal K,\ell}(n,\theta) =
\frac{40\mathfrak m^2}{\theta n}\mathfrak m_{\mathcal K,\ell}\mathfrak m(n)^2.$$ Then, for any $T > 0$, $$\begin{aligned}
\mathbb E(S_{\mathcal K,\ell}(n,\theta))
& \leqslant &
T +\int_{T}^{\infty}\mathbb P(S_{\mathcal K,\ell}(n,\theta)\geqslant
(1 +\lambda_{\mathcal K,\ell}(n,\theta,t))^3\mathfrak m_{\mathcal K,\ell}(n,\theta))dt\\
& \leqslant &
2T + 5.4\mathfrak c_1|\mathcal K_n|\mathfrak m_{\mathcal K,\ell}(n,\theta)
\exp\left(-\frac{T^{1/3}}{2\mathfrak m_{\mathcal K,\ell}(n,\theta)^{1/3}}\right)
\textrm{ with }
\mathfrak c_1 =\int_{0}^{\infty}e^{1 - r^{1/3}/2}dr.\end{aligned}$$ Moreover, $$\mathfrak m_{\mathcal K,\ell}(n,\theta)
\leqslant\mathfrak c_2\frac{\log(n)^2}{\theta n}
\textrm{ with }
\mathfrak c_2 =
\frac{40\cdot 8^2\mathfrak m^2}{\alpha^2}\mathfrak m_{\mathcal K,\ell}.$$ So, by taking $$T = 2^3\mathfrak c_2\frac{\log(n)^5}{\theta n},$$ and since $|\mathcal K_n|\leqslant n$, $$\mathbb E(S_{\mathcal K,\ell}(n,\theta))
\leqslant
2^4\mathfrak c_2\frac{\log(n)^5}{\theta n} +
5.4\mathfrak c_1\mathfrak m_{\mathcal K,\ell}(n,\theta)
\frac{|\mathcal K_n|}{n}
\leqslant
(2^4 + 5.4\mathfrak c_1)
\mathfrak c_2\frac{\log(n)^5}{\theta n}.$$ On the other hand, by Assumption \[assumption\_K\].(1), Cauchy-Schwarz’s inequality and Markov’s inequality,
$$\begin{aligned}
\mathbb E\left(\sup_{K,K'\in\mathcal K_n}
|g_{K,K',\ell}^{2}(n ; X_1,Y_1,X_2,Y_2)|\right) & \leqslant &
4\mathfrak m(n)\sum_{K,K'\in\mathcal K_n}
\mathbb E(|\ell(Y_1)|\mathbf 1_{|\ell(Y_1)| >\mathfrak m(n)}
|\langle K(X_1,.),K'(X_2,.)\rangle_2|)\\
& \leqslant &
4\mathfrak m(n)\mathfrak m_{\mathcal K,\ell}n|\mathcal K_n|^2
\mathbb E(\ell(Y_1)^2)^{1/2}
\mathbb P(|\ell(Y_1)| >\mathfrak m(n))^{1/2}
\leqslant
\mathfrak c_3
\frac{\log(n)}{n}\end{aligned}$$
with $$\mathfrak c_3 =
\frac{32}{\alpha}\mathfrak m_{\mathcal K,\ell}
\mathbb E(\ell(Y_1)^2)^{1/2}
\mathbb E(\exp(\alpha|\ell(Y_1)|))^{1/2}.$$ So, $$\mathbb E\left(\sup_{K,K'\in\mathcal K_n}
\frac{|U_{K,K',\ell}^{2}(n)|}{n^2}\right)
\leqslant
\mathfrak c_3
\frac{\log(n)}{n}$$ and, symmetrically, $$\mathbb E\left(\sup_{K,K'\in\mathcal K_n}
\frac{|U_{K,K',\ell}^{3}(n)|}{n^2}\right)
\leqslant
\mathfrak c_3
\frac{\log(n)}{n}.$$ By Assumption \[assumption\_K\].(1), Cauchy-Schwarz’s inequality and Markov’s inequality,
$$\begin{aligned}
\mathbb E\left(\sup_{K,K'\in\mathcal K_n}
|g_{K,K',\ell}^{4}(n ; X_1,Y_1,X_2,Y_2)|\right) & \leqslant &
4\sum_{K,K'\in\mathcal K_n}
\mathbb E(|\ell(Y_1)\ell(Y_2)|\mathbf 1_{|\ell(Y_1)|,|\ell(Y_2)| >\mathfrak m(n)}
|\langle K(X_1,.),K'(X_2,.)\rangle_2|)\\
& \leqslant &
4\mathfrak m_{\mathcal K,\ell}n|\mathcal K_n|^2
\mathbb E(\ell(Y_1)^2)
\mathbb P(|\ell(Y_1)| >\mathfrak m(n))
\leqslant
\frac{\mathfrak c_4}{n^5}\end{aligned}$$
with $$\mathfrak c_4 =
4\mathfrak m_{\mathcal K,\ell}
\mathbb E(\ell(Y_1)^2)
\mathbb E(\exp(\alpha|\ell(Y_1)|)).$$ So, $$\mathbb E\left(\sup_{K,K'\in\mathcal K_n}
\frac{|U_{K,K',\ell}^{4}(n)|}{n^2}\right)
\leqslant
\frac{\mathfrak c_4}{n^5}.$$ Therefore, $$\mathbb E\left(\sup_{K,K'\in\mathcal K_n}
\left\{\frac{|U_{K,K',\ell}(n)|}{n^2}
-\frac{\theta}{n}\mathfrak s_{K',\ell}
\right\}\right)
\leqslant
(2^4 + 5.4\mathfrak c_1)
\mathfrak c_2\frac{\log(n)^5}{\theta n} +
2\mathfrak c_3\frac{\log(n)}{n} +
\frac{\mathfrak c_4}{n^5}.$$
### Proof of Lemma \[bound\_trace\_term\]
First, the two following results are used several times in the sequel: $$\begin{aligned}
\|s_{K,\ell}\|_{2}^{2} & \leqslant &
\mathbb E(\ell(Y_1)^2)
\int_{\mathbb R^d}f(x')\int_{\mathbb R^d}
K(x',x)^2\lambda_d(dx)\lambda_d(dx')
\nonumber\\
\label{bound_trace_term_1}
& \leqslant &
\mathbb E(\ell(Y_1)^2)\mathfrak m_{\mathcal K,\ell}n\end{aligned}$$ and $$\begin{aligned}
\mathbb E(V_{K,\ell}(n)) & = &
\mathbb E(\|K(X_1,.)\ell(Y_1) - s_{K,\ell}\|_{2}^{2})
\nonumber\\
\label{bound_trace_term_2}
& = &
\mathbb E(\|K(X_1,.)\ell(Y_1)\|_{2}^{2}) +
\|s_{K,\ell}\|_{2}^{2}
- 2\int_{\mathbb R^d}s_{K,\ell}(x)\mathbb E(K(X_1,x)\ell(Y_1))\lambda_d(dx)
=\mathfrak s_{K,\ell} -\|s_{K,\ell}\|_{2}^{2}.\end{aligned}$$ Consider $\mathfrak m(n) := 2\log(n)/\alpha$ and $$v_{K,\ell}(n) := V_{K,\ell}(n) -\mathbb E(V_{K,\ell}(n))
= v_{K,\ell}^{1}(n) + v_{K,\ell}^{2}(n),$$ where $$v_{K,\ell}^{j}(n) =
\frac{1}{n}\sum_{i = 1}^{n}
(g_{K,\ell}^{j}(n;X_i,Y_i) -\mathbb E(g_{K,\ell}^{j}(n;X_i,Y_i)))
\textrm{ $;$ }
j = 1,2$$ with, for every $(x',y)\in E$, $$g_{K,\ell}^{1}(n;x',y) :=
\|K(x',.)\ell(y) - s_{K,\ell}\|_{2}^{2}\mathbf 1_{|\ell(y)|\leqslant\mathfrak m(n)}$$ and $$g_{K,\ell}^{2}(n;x',y) :=
\|K(x',.)\ell(y) - s_{K,\ell}\|_{2}^{2}\mathbf 1_{|\ell(y)| >\mathfrak m(n)}.$$ On the one hand, by Bernstein’s inequality, for any $\lambda > 0$, with probability larger than $1 - 2e^{-\lambda}$, $$|v_{K,\ell}^{1}(n)|
\leqslant
\sqrt{\frac{2\lambda}{n}\mathfrak v_{K,\ell}(n)}
+\frac{\lambda}{n}\mathfrak c_{K,\ell}(n)$$ where $$\mathfrak c_{K,\ell}(n) =\frac{\|g_{K,\ell}^{1}(n;.)\|_{\infty}}{3}
\textrm{ and }
\mathfrak v_{K,\ell}(n) =
\mathbb E(g_{K,\ell}^{1}(n;X_1,Y_1)^2).$$ Moreover, $$\begin{aligned}
\mathfrak c_{K,\ell}(n) & = &
\frac{1}{3}\sup_{(x',y)\in E}
\|K(x',.)\ell(y) - s_{K,\ell}\|_{2}^{2}
\mathbf 1_{|\ell(y)|\leqslant\mathfrak m(n)}\\
& \leqslant &
\frac{2}{3}\left(
\mathfrak m(n)^2\sup_{x'\in\mathbb R^d}\|K(x',.)\|_{2}^{2} +\|s_{K,\ell}\|_{2}^{2}
\right)
\leqslant
\frac{2}{3}(\mathfrak m(n)^2 +\mathbb E(\ell(Y_1)^2))\mathfrak m_{\mathcal K,\ell}n\end{aligned}$$ by Inequality (\[bound\_trace\_term\_1\]), and $$\begin{aligned}
\mathfrak v_{K,\ell}(n) & \leqslant &
\|g_{K,\ell}^{1}(n;.)\|_{\infty}\mathbb E(V_{K,\ell}(n))\\
& \leqslant &
2(\mathfrak m(n)^2 +\mathbb E(\ell(Y_1)^2))\mathfrak m_{\mathcal K,\ell}n
(\mathfrak s_{K,\ell} -\|s_{K,\ell}\|_{2}^{2})\end{aligned}$$ by Inequality (\[bound\_trace\_term\_1\]) and Equality (\[bound\_trace\_term\_2\]). Then, for any $\theta\in ]0,1[$, $$\begin{aligned}
|v_{K,\ell}^{1}(n)|
& \leqslant &
2\sqrt{\lambda
(\mathfrak m(n)^2 +\mathbb E(\ell(Y_1)^2))\mathfrak m_{\mathcal K,\ell}
(\mathfrak s_{K,\ell} -\|s_{K,\ell}\|_{2}^{2})}
+\frac{2\lambda}{3}
(\mathfrak m(n)^2 +\mathbb E(\ell(Y_1)^2))\mathfrak m_{\mathcal K,\ell}\\
& \leqslant &
\theta\mathfrak s_{K,\ell} +
\frac{5\lambda}{3\theta}
(1 +\mathbb E(\ell(Y_1)^2))\mathfrak m_{\mathcal K,\ell}\mathfrak m(n)^2\end{aligned}$$ with probability larger than $1 - 2e^{-\lambda}$. So, with probability larger than $1 - 2|\mathcal K_n|e^{-\lambda}$, $$S_{\mathcal K,\ell}(n,\theta) :=
\sup_{K\in\mathcal K_n}
\left\{\frac{|v_{K,\ell}^{1}(n)|}{n}
-\frac{\theta}{n}\mathfrak s_{K,\ell}
\right\}
\leqslant
\frac{5\lambda}{3\theta n}
(1 +\mathbb E(\ell(Y_1)^2))\mathfrak m_{\mathcal K,\ell}\mathfrak m(n)^2.$$ For every $t\in\mathbb R_+$, consider $$\lambda_{\mathcal K,\ell}(n,\theta,t) :=
\frac{t}{\mathfrak m_{\mathcal K,\ell}(n,\theta)}
\textrm{ with }
\mathfrak m_{\mathcal K,\ell}(n,\theta) =
\frac{5}{3\theta n}
(1 +\mathbb E(\ell(Y_1)^2))\mathfrak m_{\mathcal K,\ell}\mathfrak m(n)^2.$$ Then, for any $T > 0$, $$\begin{aligned}
\mathbb E(S_{\mathcal K,\ell}(n,\theta))
& \leqslant &
T +\int_{T}^{\infty}\mathbb P(S_{\mathcal K,\ell}(n,\theta)\geqslant
\lambda_{\mathcal K,\ell}(n,\theta,t)\mathfrak m_{\mathcal K,\ell}(n,\theta))dt\\
& \leqslant &
2T + 2\mathfrak c_1|\mathcal K_n|\mathfrak m_{\mathcal K,\ell}(n,\theta)
\exp\left(-\frac{T}{2\mathfrak m_{\mathcal K,\ell}(n,\theta)}\right)
\textrm{ with }
\mathfrak c_1 =
\int_{0}^{\infty}e^{-r/2}dr = 2.\end{aligned}$$ Moreover, $$\mathfrak m_{\mathcal K,\ell}(n,\theta)
\leqslant\mathfrak c_2\frac{\log(n)^2}{\theta n}
\textrm{ with }
\mathfrak c_2 =
\frac{10}{3\alpha^2}(1 +\mathbb E(\ell(Y_1)^2))\mathfrak m_{\mathcal K,\ell}.$$ So, by taking $$T = 2\mathfrak c_2\frac{\log(n)^3}{\theta n},$$ and since $|\mathcal K_n|\leqslant n$, $$\mathbb E(S_{\mathcal K,\ell}(n,\theta))
\leqslant
4\mathfrak c_2\frac{\log(n)^3}{\theta n} +
4\mathfrak m_{\mathcal K,\ell}(n,\theta)
\frac{|\mathcal K_n|}{n}
\leqslant
8\mathfrak c_2\frac{\log(n)^3}{\theta n}.$$ On the other hand, by Inequality (\[bound\_trace\_term\_1\]) and Markov’s inequality, $$\begin{aligned}
\mathbb E\left[\sup_{K\in\mathcal K_n}\frac{|v_{K,\ell}^{2}(n)|}{n}\right]
& \leqslant &
\frac{2}{n}\mathbb E\left(
\sup_{K\in\mathcal K_n}\|K(X_1,.)\ell(Y_1) - s_{K,\ell}\|_{2}^{2}\mathbf 1_{|\ell(Y_1)| >\mathfrak m(n)}\right)\\
& \leqslant &
\frac{4}{n}\mathbb E\left[\left|
\ell(Y_1)^2
\sup_{K\in\mathcal K_n}\|K(X_1,.)\|_{2}^{2}
+\sup_{K\in\mathcal K_n}\|s_{K,\ell}\|_{2}^{2}\right|^2\right]^{1/2}
\mathbb P(|\ell(Y_1)| >\mathfrak m(n))^{1/2}
\leqslant
\frac{\mathfrak c_3}{n}\end{aligned}$$ with $$\mathfrak c_3 = 8\mathfrak m_{\mathcal K,\ell}\mathbb E(\ell(Y_1)^4)^{1/2}\mathbb E(\exp(\alpha|\ell(Y_1)|))^{1/2}.$$ Therefore, $$\mathbb E\left(\sup_{K\in\mathcal K_n}\left\{
\frac{|v_{K,\ell}(n)|}{n} -\frac{\theta}{n}\mathfrak s_{K,\ell})\right\}\right)
\leqslant 8\mathfrak c_2\frac{\log(n)^3}{\theta n} +\frac{\mathfrak c_3}{n}$$ and, by Equality (\[bound\_trace\_term\_2\]), the definition of $v_{K,\ell}(n)$ and Assumption \[assumption\_K\].(2), $$\mathbb E\left(\sup_{K\in\mathcal K_n}\left\{
\frac{1}{n}|V_{K,\ell}(n) -\mathfrak s_{K,\ell}| -\frac{\theta}{n}\mathfrak s_{K,\ell}\right\}\right)
\leqslant 8\mathfrak c_2\frac{\log(n)^3}{\theta n} +\frac{\mathfrak c_3 +\mathfrak m_{\mathcal K,\ell}}{n}.$$
### Proof of Lemma \[bound\_crossed\_term\]
Consider $\mathfrak m(n) = 12\log(n)/\alpha$. For any $K,K'\in\mathcal K_n$, $$W_{K,K',\ell}(n) = W_{K,K',\ell}^{1}(n) + W_{K,K',\ell}^{2}(n)$$ where $$W_{K,K',\ell}^{j}(n) :=
\frac{1}{n}\sum_{i = 1}^{n}
(g_{K,K',\ell}^{j}(n;X_i,Y_i) -\mathbb E(g_{K,K',\ell}^{j}(n;X_i,Y_i)))
\textrm{ $;$ }
j = 1,2$$ with, for every $(x',y)\in E$, $$g_{K,K',\ell}^{1}(n;x',y) :=
\langle K(x',.)\ell(y),s_{K',\ell} - s\rangle_2
\mathbf 1_{|\ell(y)|\leqslant\mathfrak m(n)}$$ and $$g_{K,K',\ell}^{2}(n;x',y) :=
\langle K(x',.)\ell(y),s_{K',\ell} - s\rangle_2
\mathbf 1_{|\ell(y)| >\mathfrak m(n)}.$$ On the one hand, by Bernstein’s inequality, for any $\lambda > 0$, with probability larger than $1 - 2e^{-\lambda}$, $$|W_{K,K',\ell}^{1}(n)|
\leqslant
\sqrt{\frac{2\lambda}{n}\mathfrak v_{K,K',\ell}(n)}
+\frac{\lambda}{n}\mathfrak c_{K,K',\ell}(n)$$ where $$\mathfrak c_{K,K',\ell}(n) =\frac{\|g_{K,K',\ell}^{1}(n;.)\|_{\infty}}{3}
\textrm{ and }
\mathfrak v_{K,K',\ell}(n) =
\mathbb E(g_{K,K',\ell}^{1}(n;X_1,Y_1)^2).$$ Moreover, $$\begin{aligned}
\mathfrak c_{K,K',\ell}(n) & = &
\frac{1}{3}\sup_{(x',y)\in E}
|\langle K(x',.)\ell(y),s_{K',\ell} - s\rangle_2|
\mathbf 1_{|\ell(y)|\leqslant\mathfrak m(n)}\\
& \leqslant &
\frac{1}{3}\mathfrak m(n)\|s_{K',\ell} - s\|_2
\sup_{x'\in\mathbb R^d}\|K(x',.)\|_2
\leqslant
\frac{1}{3}\mathfrak m_{\mathcal K,\ell}^{1/2}n^{1/2}\mathfrak m(n)\|s_{K',\ell} - s\|_2\end{aligned}$$ by Assumption \[assumption\_K\].(1), and $$\mathfrak v_{K,\ell}(n)\leqslant
\mathbb E(\langle K(X_1,.)\ell(Y_1),s_{K',\ell} - s\rangle_{2}^{2}
\mathbf 1_{|\ell(Y_1)|\leqslant\mathfrak m(n)})\\
\leqslant
\mathfrak m(n)^2
\mathfrak m_{\mathcal K,\ell}\|s_{K',\ell} - s\|_{2}^{2}$$ by Assumption \[assumption\_K\].(4). Then, since $\lambda > 0$, for any $\theta\in ]0,1[$, $$\begin{aligned}
|W_{K,K',\ell}^{1}(n)|
& \leqslant &
\sqrt{\frac{2\lambda}{n}
\mathfrak m(n)^2
\mathfrak m_{\mathcal K,\ell}\|s_{K',\ell} - s\|_{2}^{2}}
+\frac{\lambda}{3n^{1/2}}\mathfrak m_{\mathcal K,\ell}^{1/2}\mathfrak m(n)\|s_{K',\ell} - s\|_2\\
& \leqslant &
\theta\|s_{K',\ell} - s\|_{2}^{2} +
\frac{\mathfrak m_{\mathcal K,\ell}}{2\theta n}\mathfrak m(n)^2(1 +\lambda)^2\end{aligned}$$ with probability larger than $1 - 2e^{-\lambda}$. So, with probability larger than $1 - 2|\mathcal K_n|e^{-\lambda}$, $$S_{\mathcal K,\ell}(n,\theta) :=
\sup_{K,K'\in\mathcal K_n}
\{|W_{K,K',\ell}^{1}(n)|
-\theta\|s_{K',\ell} - s\|_{2}^{2}\}
\leqslant
\frac{\mathfrak m_{\mathcal K,\ell}}{2\theta n}\mathfrak m(n)^2(1 +\lambda)^2.$$ For every $t\in\mathbb R_+$, consider $$\lambda_{\mathcal K,\ell}(n,\theta,t) :=
-1 +
\left(\frac{t}{\mathfrak m_{\mathcal K,\ell}(n,\theta)}\right)^{1/2}
\textrm{ with }
\mathfrak m_{\mathcal K,\ell}(n,\theta) =
\frac{\mathfrak m_{\mathcal K,\ell}}{2\theta n}\mathfrak m(n)^2.$$ Then, for any $T > 0$, $$\begin{aligned}
\mathbb E(S_{\mathcal K,\ell}(n,\theta))
& \leqslant &
T +\int_{T}^{\infty}\mathbb P(S_{\mathcal K,\ell}(n,\theta)\geqslant
(1 +\lambda_{\mathcal K,\ell}(n,\theta,t))^2\mathfrak m_{\mathcal K,\ell}(n,\theta))dt \\
& \leqslant &
2T + 2\mathfrak c_1|\mathcal K_n|\mathfrak m_{\mathcal K,\ell}(n,\theta)
\exp\left(-\frac{T^{1/2}}{2\mathfrak m_{\mathcal K,\ell}(n,\theta)^{1/2}}\right)
\textrm{ with }
\mathfrak c_1 =
\int_{0}^{\infty}e^{1 - r^{1/2}/2}dr.\end{aligned}$$ Moreover, $$\mathfrak m_{\mathcal K,\ell}(n,\theta)
\leqslant\mathfrak c_2\frac{\log(n)^2}{\theta n}
\textrm{ with }
\mathfrak c_2 =
\frac{12^2}{2\alpha^2}\mathfrak m_{\mathcal K,\ell}.$$ So, by taking $$T = 2^2\mathfrak c_2\frac{\log(n)^4}{\theta n},$$ and since $|\mathcal K_n|\leqslant n$, $$\mathbb E(S_{\mathcal K,\ell}(n,\theta))
\leqslant
2^3\mathfrak c_2\frac{\log(n)^4}{\theta n} +
2\mathfrak c_1\mathfrak m_{\mathcal K,\ell}(n,\theta)
\frac{|\mathcal K_n|}{n}
\leqslant
(2^3 + 2\mathfrak c_1)\mathfrak c_2\frac{\log(n)^4}{\theta n}.$$ On the other hand, by Assumption \[assumption\_K\].(2,4), Cauchy-Schwarz’s inequality and Markov’s inequality, $$\begin{aligned}
\mathbb E\left(\sup_{K,K'\in\mathcal K_n}|W_{K,K',\ell}^{2}(n)|\right)
& \leqslant &
2\mathbb E(\ell(Y_1)^2\mathbf 1_{|\ell(Y_1)| >\mathfrak m(n)})^{1/2}
\sum_{K,K'\in\mathcal K_n}
\mathbb E(
\langle K(X_1,.),s_{K',\ell} - s\rangle_{2}^{2})^{1/2}\\
& \leqslant &
2\mathfrak m_{\mathcal K,\ell}^{1/2}\|s_{K',\ell} - s\|_2
\mathbb E(\ell(Y_1)^4)^{1/4}|\mathcal K_n|^2\mathbb P(|\ell(Y_1)| >\mathfrak m(n))^{1/4}
\leqslant
\frac{\mathfrak c_3}{n}\end{aligned}$$ with $$\mathfrak c_3 = 2\mathfrak m_{\mathcal K,\ell}^{1/2}
(\mathfrak m_{\mathcal K,\ell}^{1/2} +\|s\|_2)
\mathbb E(\ell(Y_1)^4)^{1/4}
\mathbb E(\exp(\alpha|\ell(Y_1)|))^{1/4}.$$ Therefore, $$\mathbb E\left(\sup_{K,K'\in\mathcal K_n}\{
|W_{K,K',\ell}(n)| -\theta\|s_{K',\ell} - s\|_{2}^{2}\}\right)
\leqslant (2^3 + 2\mathfrak c_1)\mathfrak c_2\frac{\log(n)^4}{\theta n} +\frac{\mathfrak c_3}{n}
\leqslant
\mathfrak c_4\frac{\log(n)^4}{\theta n}$$ with $\mathfrak c_4 = (2^3 + 2\mathfrak c_1)\mathfrak c_2 +\mathfrak c_3$.
Proof of Proposition \[variance\_bound\_main\_estimator\]
---------------------------------------------------------
For any $K\in\mathcal K_n$, $$\label{variance_bound_main_estimator_1}
\|\widehat s_{K,\ell}(n;.) - s_{K,\ell}\|_{2}^{2} =
\frac{U_{K,\ell}(n)}{n^2} +\frac{V_{K,\ell}(n)}{n}$$ with $U_{K,\ell}(n) = U_{K,K,\ell}(n)$ and $V_{K,\ell}(n) = V_{K,K,\ell}(n)$. Then, by Lemmas \[bound\_U\_statistics\] and \[bound\_trace\_term\], $$\mathbb E\left(\sup_{K\in\mathcal K_n}\left\{
\left|\|\widehat s_{K,\ell}(n;.) - s_{K,\ell}\|_{2}^{2} -\frac{\mathfrak s_{K,\ell}}{n}\right|
-\frac{\theta}{n}\mathfrak s_{K,\ell}\right\}\right)
\leqslant
\mathfrak c_{\ref{variance_bound_main_estimator}}\frac{\log(n)^5}{\theta n}$$ with $\mathfrak c_{\ref{variance_bound_main_estimator}} =\mathfrak c_{\ref{bound_U_statistics}} +\mathfrak c_{\ref{bound_trace_term}}$.
Proof of Theorem \[risk\_bound\_main\_estimator\]
-------------------------------------------------
On the one hand, for every $K\in\mathcal K_n$, $$\|\widehat s_{K,\ell}(n;.) - s\|_{2}^{2} - (1 +\theta)\left(\|s_{K,\ell} - s\|_{2}^{2} +\frac{\mathfrak s_{K,\ell}}{n}\right)$$ can be written $$\|\widehat s_{K,\ell}(n;.) - s_{K,\ell}\|_{2}^{2} - (1 +\theta)\frac{\mathfrak s_{K,\ell}}{n} +
W_{K,K,\ell}(n) -\theta\|s_{K,\ell} - s\|_{2}^{2}.$$ Then, by Proposition \[variance\_bound\_main\_estimator\] and Lemma \[bound\_crossed\_term\], $$\mathbb E\left(\sup_{K\in\mathcal K_n}\left\{
\|\widehat s_{K,\ell}(n;.) - s\|_{2}^{2} - (1 +\theta)\left(\|s_{K,\ell} - s\|_{2}^{2} +\frac{\mathfrak s_{K,\ell}}{n}\right)\right\}\right)
\leqslant
\mathfrak c_{\ref{risk_bound_main_estimator}}\frac{\log(n)^5}{\theta n}$$ with $\mathfrak c_{\ref{risk_bound_main_estimator}} =\mathfrak c_{\ref{variance_bound_main_estimator}} +\mathfrak c_{\ref{bound_crossed_term}}$. On the other hand, for any $K\in\mathcal K_n$, $$\|s_{K,\ell} - s\|_{2}^{2} =
\|\widehat s_{K,\ell}(n;.) - s\|_{2}^{2}
-\|\widehat s_{K,\ell}(n;.) - s_{K,\ell}\|_{2}^{2}
- W_{K,\ell}(n).$$ Then, $$(1 -\theta)\left(\|s_{K,\ell} - s\|_{2}^{2} +\frac{\mathfrak s_{K,\ell}}{n}\right)
-\|\widehat s_{K,\ell}(n;.) - s\|_{2}^{2}
\leqslant
|W_{K,\ell}(n)| -\theta\|s_{K,\ell} - s\|_{2}^{2} +
\Lambda_{K,\ell}(n) -\theta\frac{\mathfrak s_{K,\ell}}{n}$$ where $$\Lambda_{K,\ell}(n) :=
\left|
\|\widehat s_{K,\ell} - s_{K,\ell}\|_{2}^{2} -\frac{\mathfrak s_{K,\ell}}{n}\right|.$$ By Equalities (\[variance\_bound\_main\_estimator\_1\]) and (\[bound\_trace\_term\_2\]), $$\Lambda_{K,\ell}(n) =
\left|\frac{U_{K,\ell}(n)}{n^2} +\frac{v_{K,\ell}(n)}{n} -\frac{\|s_{K,\ell}\|_{2}^{2}}{n}\right|.$$ By Lemmas \[bound\_trace\_term\] and \[bound\_U\_statistics\], there exists a deterministic constant $\mathfrak c_1 > 0$, not depending $n$ and $\theta$, such that $$\mathbb E\left(\sup_{K\in\mathcal K_n}\left\{
\Lambda_{K,\ell}(n) -\theta\frac{\mathfrak s_{K,\ell}}{n}\right\}\right)
\leqslant
\mathfrak c_1\frac{\log(n)^5}{\theta n}.$$ By Lemma \[bound\_crossed\_term\], $$\mathbb E\left(\sup_{K\in\mathcal K_n}\{
|W_{K,\ell}(n)| -\theta\|s_{K,\ell} - s\|_{2}^{2}
\}\right)
\leqslant
\mathfrak c_{\ref{bound_crossed_term}}\frac{\log(n)^3}{\theta n}.$$ Therefore, $$\mathbb E\left(\sup_{K\in\mathcal K_n}\left\{
\|s_{K,\ell} - s\|_{2}^{2} +\frac{\mathfrak s_{K,\ell}}{n} -
\frac{1}{1 -\theta}\|\widehat s_{K,\ell}(n;.) - s\|_{2}^{2}\right\}\right)
\leqslant
\overline{\mathfrak c}_{\ref{risk_bound_main_estimator}}\frac{\log(n)^5}{\theta(1 -\theta) n}$$ with $\overline{\mathfrak c}_{\ref{risk_bound_main_estimator}} =\mathfrak c_{\ref{bound_crossed_term}} +\mathfrak c_1$.
Proof of Theorem \[risk\_bound\_adaptative\_estimator\]
-------------------------------------------------------
The proof of Theorem \[risk\_bound\_adaptative\_estimator\] is dissected in three steps.\
\
**Step 1.** This first step is devoted to provide a suitable decomposition of $$\|\widehat s_{\widehat K,\ell}(n;\cdot) - s\|_2^2.$$ First, $$\|\widehat s_{\widehat K,\ell}(n;\cdot) - s\|_2^2 = \|\widehat s_{\widehat K,\ell}(n;\cdot) - \widehat s_{K_0,\ell}(n;\cdot)\|_2^2 + \|\widehat s_{K_0,\ell}(n;\cdot) - s\|_2^2 - 2\langle \widehat s_{K_0,\ell}(n;\cdot) - \widehat s_{\widehat K,\ell}(n;\cdot) , \widehat s_{K_0,\ell}(n;\cdot) - s\rangle_2$$ From , it follows that for any $K\in\mathcal K_n$, $$\begin{aligned}
\label{oracle_1}
\nonumber
\|\widehat s_{\widehat K,\ell}(n;\cdot) - s\|_2^2
& \leqslant & \|\widehat s_{K,\ell}(n;\cdot) - s\|_2^2 + \textrm{pen}(K) - \textrm{pen}(\widehat K) + \|\widehat s_{K_0,\ell}(n;\cdot) - s\|_2^2 \\
\nonumber
&&- 2 \langle \widehat s_{K_0,\ell}(n;\cdot) - \widehat s_{\widehat K,\ell}(n\;\cdot) , \widehat s_{K_0,\ell}(n;\cdot) - s \rangle_2\\
&=& \|\widehat s_{K,\ell}(n;\cdot) - s\|_2^2 +\psi_n(K) -\psi_n(\widehat K)
\end{aligned}$$ where $$\psi_n(K) := 2 \langle \widehat s_{K,\ell}(n;\cdot)-s , \widehat s_{K_0,\ell}(n;\cdot) - s \rangle_2- \textrm{pen}(K).\vspace{10pt}\\$$ Let’s complete the decomposition of $\|\widehat s_{\widehat K,\ell}(n;\cdot) - s\|_2^2$ by writing $$\psi_n(K) = 2(\psi_{1,n}(K) + \psi_{2,n}(K) + \psi_{3,n}(K)),$$ where $$\begin{aligned}
\psi_{1,n}(K) &:=& \dfrac{U_{K,K_0,\ell}(n)}{n^2},\\
\psi_{2,n}(K) &:=& -\dfrac{1}{n^2}\left(\displaystyle\sum_{i=1}^n\ell(Y_i)\langle K_0(X_i,.),s_{K,\ell}\rangle_2+ \sum_{i=1}^n\ell(Y_i)
\langle K(X_i,.), s_{K_0,\ell}\rangle_2 \right)+\dfrac{1}{n}\langle s_{K_0,\ell},s_{K,\ell} \rangle_2
\textrm{ and}\\
\psi_{3,n}(K) &:=& W_{K,K_0,\ell}(n)+ W_{K_0,K,\ell}(n) + \langle s_{K,\ell}- s, s_{K_0,\ell}- s \rangle_2.\\
\end{aligned}$$ **Step 2.** In this step, we give controls of the quantities $$\mathbb E(\psi_{i,n}(K)) \; \text{and} \; \mathbb E(\psi_{i,n}(\widehat K)) \; ; \; i=1,2,3.$$
- By Lemma \[bound\_U\_statistics\], for any $\theta\in ]0,1[$, $$\mathbb E(|\psi_{1,n}(K)|)
\leqslant \frac{\theta}{n}\mathfrak s_{K,\ell} +
\mathfrak c_{\ref{bound_U_statistics}}\frac{\log(n)^5}{\theta n}$$ and $$\mathbb E(|\psi_{1,n}(\widehat K)|)
\leqslant \frac{\theta}{n}\mathbb E(\mathfrak s_{\widehat K,\ell}) +
\mathfrak c_{\ref{bound_U_statistics}}\frac{\log(n)^5}{\theta n}.$$
- On the one hand, for any $K,K'\in\mathcal K_n$, consider $$\Psi_{2,n}(K,K') := \dfrac{1}{n}\displaystyle\sum_{i=1}^n\ell(Y_i)\langle K(X_i,.),s_{K',\ell}\rangle_2.$$ Then, by Assumption \[additional\_assumption\_K\], $$\begin{aligned}
\mathbb E\left(\sup_{K,K'\in\mathcal K_n}|\Psi_{2,n}(K,K')|\right)
&\leqslant&
\mathbb E(\ell(Y_1)^2)^{1/2}
\mathbb E\left(\sup_{K,K'\in\mathcal K_n}
\langle K(X_1,.),s_{K',\ell}\rangle_{2}^{2}\right)^{1/2}\\
&\leqslant&
\overline{\mathfrak m}_{\mathcal K,\ell}^{1/2}\mathbb E(\ell(Y_1)^2)^{1/2}.
\end{aligned}$$ On the other hand, by Assumption \[assumption\_K\].(2), $$|\langle s_{K,\ell},s_{K_0,\ell} \rangle_2|
\leqslant \mathfrak{m}_{\mathcal K,\ell}.$$ Then, there exists a deterministic constant $\mathfrak c_1 > 0$, not depending on $n$ and $K$, such that $$\mathbb E(|\psi_{2,n}(K)|)\leqslant \frac{\mathfrak c_1}{n}
\textrm{ and }
\mathbb E(|\psi_{2,n}(\widehat K)|)\leqslant \frac{\mathfrak c_1}{n}.$$
- By Lemma \[bound\_crossed\_term\], $$\begin{aligned}
\mathbb E(|\psi_{3,n}(K)|)
&\leqslant & \dfrac{\theta}{4}(\|s_{K,\ell} - s\|_{2}^{2}+\|s_{K_0,\ell} - s\|_{2}^{2})+8\mathfrak c_{\ref{bound_crossed_term}}\frac{\log(n)^4}{\theta n}\\
&&+\left(\dfrac{\theta}{2}\right)^{1/2}\|s_{K,\ell} - s\|_{2}\times \left( \dfrac{2}{\theta}\right)^{1/2} \|s_{K_0,\ell} - s\|_{2}\\
&\leqslant & \dfrac{\theta}{2}\|s_{K,\ell} - s\|_{2}^{2} + \left(\dfrac{\theta}{4}+\dfrac{1}{\theta}\right)\|s_{K_0,\ell} - s\|_{2}^{2}+8\mathfrak c_{\ref{bound_crossed_term}}\frac{\log(n)^4}{\theta n}
\end{aligned}$$ and $$\mathbb E(|\psi_{3,n}(\widehat K)|)\leqslant \frac{\theta}{2}\mathbb E(\|s_{\widehat K,\ell} - s\|_{2}^{2}) + \left(\dfrac{\theta}{4}+\dfrac{1}{\theta}\right)\|s_{K_0,\ell} - s\|_{2}^{2}+8\mathfrak c_{\ref{bound_crossed_term}}\frac{\log(n)^4}{\theta n}.\vspace{3pt}\\$$
**Step 3.** By the previous step, there exists a deterministic constant $\mathfrak c_2 > 0$, not depending on $n$, $\theta$, $K$ and $K_0$, such that $$\mathbb E(|\psi_n(K)|)
\leqslant \theta \left(\|s_{K,\ell} - s\|_{2}^{2} +\dfrac{\mathfrak s_{K,\ell}}{n}\right) +\left(\dfrac{\theta}{2}+\dfrac{2}{\theta}\right)\|s_{K_0,\ell} - s\|_{2}^{2} +\mathfrak c_2\dfrac{\log(n)^5}{\theta n}$$ and $$\mathbb E(|\psi_n(\widehat K)|)
\leqslant \theta\mathbb E\left(\|s_{\widehat K,\ell} - s\|_{2}^{2} +\dfrac{\mathfrak s_{\widehat K,\ell}}{n}\right) +\left(\dfrac{\theta}{2}+\dfrac{2}{\theta}\right)\|s_{K_0,\ell} - s\|_{2}^{2} +\mathfrak c_2\dfrac{\log(n)^5}{\theta n}.$$ Then, by Theorem \[risk\_bound\_main\_estimator\], $$\begin{aligned}
\mathbb E(|\psi_n(K)|)
&\leqslant& \dfrac{\theta}{1-\theta}
\mathbb E(\|\widehat s_{K,\ell}(n;.) - s\|_{2}^{2}) +\left(\dfrac{\theta}{2}+\dfrac{2}{\theta}\right)\|s_{K_0,\ell} - s\|_{2}^{2}+\left(\dfrac{\mathfrak c_2}{\theta} +\dfrac{\mathfrak c_{\ref{risk_bound_main_estimator}}}{1 -\theta}\right)\dfrac{\log(n)^5}{n}
\end{aligned}$$ and $$\begin{aligned}
\mathbb E(|\psi_n(\widehat K)|)
&\leqslant& \dfrac{\theta}{1-\theta}
\mathbb E(\|\widehat s_{\widehat K,\ell}(n;.) - s\|_{2}^{2}) +\left(\dfrac{\theta}{2}+\dfrac{2}{\theta}\right)\|s_{K_0,\ell} - s\|_{2}^{2}+\left(\dfrac{\mathfrak c_2}{\theta} +\dfrac{\mathfrak c_{\ref{risk_bound_main_estimator}}}{1 -\theta}\right)\dfrac{\log(n)^5}{n}.
\end{aligned}$$ By decomposition , there exist two deterministic constants $\mathfrak c_3,\mathfrak c_4 >0$, not depending on $n$, $\theta$, $K$ and $K_0$, such that $$\begin{aligned}
\mathbb E(\|\widehat s_{\widehat K,\ell}(n;\cdot)-s\|_2^2)
&\leqslant & \mathbb E(\|\widehat s_{K,\ell}(n;\cdot)-s\|_2^2) +\mathbb E(| \psi_n(K)|) + \mathbb E(|\psi_n(\widehat K)|)\\
&\leqslant & \left(1 + \dfrac{\theta}{1-\theta}\right) \mathbb E(\|\widehat s_{K,\ell}(n;\cdot)-s\|_2^2) + \dfrac{\theta}{1-\theta}\mathbb E(\|\widehat s_{\widehat K,\ell}(n;.) - s\|_{2}^{2})\\
&&+\dfrac{\mathfrak c_3}{\theta}\|s_{K_0,\ell} - s\|_{2}^{2}+\dfrac{\mathfrak c_4}{\theta(1-\theta)}\cdot\dfrac{\log(n)^5}{n}.
\end{aligned}$$ This concludes the proof.\
\
**Acknowledgments.** Many thanks to Fabienne Comte for her careful reading and advices.
[99]{} G. Chagny. *Warped Bases for Conditional Density Estimation.* Math. Methods Statist. 22, 253-282, 2013. F. Comte and N. Marie. *Bandwidth Selection for the Wolverton-Wagner Estimator.* Journal of Statistical Planning and Inference 207, 198-214, 2020. F. Comte and N. Marie. *On a Nadaraya-Watson Estimator with Two Bandwidths.* Submitted, 2020. F. Comte and T. Rebafka. *Nonparametric Weighted Estimators for Biased Data.* Journal of Statistical Planning and Inference 174, 104-128, 2016. U. Einmahl and D.M. Mason. *An Empirical Process Approach to the Uniform Consistency of Kernel-Type Function Estimators.* Journal of Theoretical Probability 13, 1-37, 2000. U. Einmahl and D.M. Mason. *Uniform in Bandwidth Consistency of Kernel-Type Function Estimators.* Annals of Statistics 33, 1380-1403, 2005. E. Giné and R. Nickl. *Mathematical Foundations of Infinite-Dimensional Statistical Models.* Cambridge university press, 2015. A. Goldenshluger and O. Lepski. *Bandwidth Selection in Kernel Density Estimation: Oracle Inequalities and Adaptive Minimax Optimality.* The Annals of Statistics 39, 1608-1632, 2011. C. Houdré and P. Reynaud-Bouret. *Exponential Inequalities, with Constants, for $U$-Statistics of Order Two.* Stochastic Inequalities and Applications, vol. 56 of Progr. Proba., 55-69, Birkhauser, 2003. C. Lacour, P. Massart and V. Rivoirard. *Estimator Selection: a New Method with Applications to Kernel Density Estimation.* Sankhya A 79, 2, 298-335, 2017. M. Lerasle, N.M. Magalhaes and P. Reynaud-Bouret. *Optimal Kernel Selection for Density Estimation.* High dimensional probabilities VII: The Cargese Volume, vol. 71 of Prog. Proba., 435–460, Birkhauser, 2016. P. Massart. *Concentration Inequalities and Model Selection.* Lecture Notes in Mathematics 1896, Springer, 2007. S. Varet, C. Lacour, P. Massart and V. Rivoirard. *Numerical Performance of Penalized Comparison to Overfitting for Multivariate Density Estimation.* Preprint, 2020.
|
---
author:
- |
Xin Wang\
Cerebras Systems\
`[email protected]`\
bibliography:
- 'references.bib'
title: 'The curious case of developmental BERTology\'
---
|
---
abstract: 'We define a $K$-theoretic analogue of Fomin’s dual graded graphs, which we call dual filtered graphs. The key formula in the definition is $DU-UD= D + I$. Our major examples are $K$-theoretic analogues of Young’s lattice, of shifted Young’s lattice, and of the Young-Fibonacci lattice. We suggest notions of tableaux, insertion algorithms, and growth rules whenever such objects are not already present in the literature. (See the table below.) We also provide a large number of other examples. Most of our examples arise via two constructions, which we call the Pieri construction and the Möbius construction. The Pieri construction is closely related to the construction of dual graded graphs from a graded Hopf algebra, as described in [@BLL; @HN; @LS]. The Möbius construction is more mysterious but also potentially more important, as it corresponds to natural insertion algorithms.'
address:
- ' Department of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church Street, Minneapolis, MN 55455, USA'
- 'Department of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church Street, Minneapolis, MN 55455, USA'
author:
- Rebecca Patrias
- Pavlo Pylyavskyy
title: Dual filtered graphs
---
[^1]
[^2]
[^3]
--------------------------------- ------------------------------------------------------------------- --------------------------------------------------- -------------------------------
$ $ Tableaux Insertion Growth
\[.5ex\] Young Standard Young tableaux [@Young] RSK insertion [@Robinson; @Schensted; @Knuth] [@Fomin]
\[.5ex\] Shifted Young Standard shifted Young tableaux with and without circles [@Sagan] Shifted Robinson-Schensted insertion [@Sagan; @W] [@FSchen]
\[.5ex\] Young-Fibonacci Young-Fibonacci tableaux [@Fomin; @F] Young-Fibonacci insertion [@Fomin] [@FSchen]
\[.5ex\] Möbius Young Increasing and set-valued tableaux [@ThY; @B] Hecke insertion [@BKSTY] Section \[sec:HeckeGrowth\]
\[.5ex\] Möbius shifted Young Definitions \[def:increasingshifted\] and \[def:shiftedset\] Section \[sec:shiftedinsertion\] Section \[sec:shiftedgrowth\]
\[.5ex\] Möbius Young-Fibonacci Definitions \[def:KYFtableau\] and \[def:KYFsetvalued\] Section \[sec:KYFinsertion\] Section \[sec:KYFgrowth\]
\[.5ex\]
--------------------------------- ------------------------------------------------------------------- --------------------------------------------------- -------------------------------
Introduction
============
Fomin’s [*[dual graded graphs]{}*]{} [@F], as well as their predecessors - Stanley’s [*[differential posets]{}*]{} [@St], were invented as a tool to better understand the Robinson-Schensted insertion algorithm. Dual graded graphs are significant in the areas where the Robinson-Schensted correspondence appears, for example in Schubert calculus or in the study of representations of towers of algebras [@BLL; @BS]. A recent work of Lam and Shimozono [@LSh] associates a dual graded graph to any Kac-Moody algebra, bringing their study to a new level of generality.
Weyl algebra and its deformations
---------------------------------
One way to view the theory is as a study of certain [*[combinatorial representations]{}*]{} of the [*[first Weyl algebra]{}*]{}. Let us briefly recall the definitions. The [*[Weyl algebra]{}*]{}, or the first Weyl algebra, is an algebra over some field $K$ (usually $K = {\mathbb {R}}$) generated by two generators $U$ and $D$ with a single relation $DU-UD=1$. It was originally introduced by Hermann Weyl in his study of quantum mechanics. We refer the reader to [@Bjo], for example, for more background on the Weyl algebra.
A [*[graded graph]{}*]{} is a triple $G=(P,\rho,E)$, where $P$ is a discrete set of vertices, $\rho:P\to \mathbb{Z}$ is a rank function, and $E$ is a multiset of edges/arcs $(x,y)$, where $\rho(y)=\rho(x)+1$. In other words, each vertex is assigned a rank, and edges can only join vertices in successive ranks. For the set of vertices $P$, let $P_n$ denote the subset of vertices of rank $n$. For any field $K$ of characteristic zero, the formal linear combinations of vertices of $G$ form the vector space $KP$.
Let $G_1=(P,\rho,E_1)$ and $G_2=(P,\rho,E_2)$ be a pair of graded graphs with a common vertex set and rank function. From this pair, define an *oriented graded graph* $G=(P,\rho, E_1,E_2)$ by orienting the edges of $G_1$ in the direction of increasing rank and the edges of $G_2$ in the direction of decreasing rank. Let $a_i(x,y)$ denote the number of edges of $E_i$ joining $x$ and $y$ or the multiplicity of edge $(x,y)$ in $E_i$. We define the *up* and *down operators* $U,D\in End(KP)$ associated with graph $G$ by $$Ux=\sum_{y} a_1(x,y)y$$ and $$Dy=\sum_x a_2(x,y) x.$$
Graded graphs $G_1$ and $G_2$ with a common vertex set and rank function are said to be *dual* if $$DU-UD=I,$$ where $I$ is an identity operator acting on $KP$. Thus, we see that $KP$ is a representation of the Weyl algebra. Furthermore, it is a representation of a very special kind, where $U$ and $D$ act in a particularly nice combinatorial way on a fixed basis.
One would then expect that variations of the Weyl algebra would correspond to some variations of the theory of dual graded graphs. One such variation is the [*[$q$-Weyl algebra]{}*]{} defined by the relation $$DU-qUD=1.$$ The corresponding theory of [*[quantum dual graded graphs]{}*]{} was pursued by Lam in [@L].
In this paper, we shall study the theory arising from an analogue $W$ of the Weyl algebra with defining relation $$DU-UD=D+1.$$ We shall refer to it as the Ore algebra [@O], and we shall see that it is a very natural object. In particular, the corresponding theory of [*[dual filtered graphs]{}*]{} fits naturally with the existing body of work on the $K$-theory of Grassmannians.
\[rem:rescale\] It is easy to see that by rescaling $U$ and $D$, we can pass from arbitrary $DU-UD= \alpha D + \beta$ to $DU-UD= D + I$. We thus suffer no loss in generality by writing down the relation in the latter form.
Differential vs difference operators {#sec:differential}
------------------------------------
The following provides some intuitive explanation for the relation between representations of Weyl and Ore algebras. The simplest representation of the Weyl algebra is that acting on a polynomial ring. Indeed, consider polynomial ring $R = \mathbb R[x]$, and for $f \in R$ let $$U(f) = xf, \;\;\; D(f) = \frac{\partial f}{\partial x}.$$ It is an easy exercise to verify that this indeed produces a representation of the Weyl algebra. In fact, this is how it is often defined.
Let us now redefine the down operator to be the [*[difference operator]{}*]{}: $$U(f) = xf, \;\;\; D(f) = f(x+1) - f(x).$$
This gives a representation of the Ore algebra $W$. In other words, those operators satisfy $$DU-UD = D+I,$$ where $I$ is the identity map on $R$.
We have $$(DU-UD)(f) = ((x+1)f(x+1)-xf(x)) - x(f(x+1)-f(x))$$ $$= f(x+1) = f(x) + (f(x+1)-f(x)) = (I+D)(f).$$
It can also be noted that differential and difference operators can be related as follows: $$D = e^{\frac{\partial}{\partial x}}-1.$$ We omit the easy proof for brevity. As we shall see in Example \[ex:polys\], this representation of $W$ corresponds to a very basic dual filtered graph.
Pieri and Möbius constructions
------------------------------
We make use of two conceptual ways to build examples of dual filtered graphs. The first construction starts with an algebra $A$, a derivation $D$ on $A$, and an element $f \in A$ such that $D(f)=f+1$. We often build the desired derivation $D$ using a [*[bialgebra]{}*]{} structure on $A$. This construction is very close to an existing one in the literature [@BLL; @HN; @LS], where dual graded graphs are constructed from graded Hopf algebras. In fact, if we were to require $D(f)=1$, we would get dual graded graphs instead of dual filtered graphs in our construction. We refer to this method as the [*[Pieri construction]{}*]{} or sometimes as the [*[Pieri deformation]{}*]{}.
Instead, we can also start with an existing dual graded graph $G=(P, \rho, E_1, E_2)$ (see definition below), composed of graphs $G_1=(P,\rho, E_1)$ and $G_2=(P,\rho,e_2)$, and adjust $E_1$ and $E_2$ in the following manner. To obtain $G_1'$, we add $\#\{x|y\text{ covers }x\text{ in }G_1\}$ loops at each vertex $y \in P$ to $E_1$. As for $G_2'$, we create a new edge set $E'_2$ by forming $$a'_2(x,y) = |\mu(x,y)|$$ edges between vertices $x$ and $y$, where $\mu$ denotes the Möbius function in $G_2=(P, \rho, E_2)$. We refer to this construction as the [*[Möbius construction]{}*]{} or [*[Möbius deformation]{}*]{}. Note that this does not always produce a pair of dual filtered graphs, and it is mysterious to determine when it does. In some major examples, however, it is the result of this construction that relates to Robinson-Schensted-like algorithms, for example to [*[Hecke insertion]{}*]{} of [@BKSTY].
The following observation seems remarkable, and we call it the [*[Möbius via Pieri phenomenon]{}*]{}. Let $A$ be a graded Hopf algebra, and let bialgebra $\tilde A$ be its [*[$K$-theoretic deformation]{}*]{} in some appropriate sense which we do not know how to formalize. Let $G$ be a natural dual graded graph associated with $A$. What we observe is that applying the Möbius construction to $G$ yields [*[the same result]{}*]{} as a natural Pieri construction applied to $\tilde A$. In other words, the following diagram commutes.
\(A) at (-2,2) [$A$]{}; (tildeA) at (2,2) [$\tilde A$]{}; (G) at (-2,0) [$G$]{}; (tildeG) at (2,0) [$\tilde G$]{}; (G) edge node \[above\] (tildeG); (A) edge node \[left\] [$\substack{\text{Pieri}\\\text{construction}}$]{} (G); (tildeA) edge node \[right\] [$\substack{\text{Pieri}\\\text{construction}}$]{} (tildeG); (A) edge node \[above\] (tildeA);
This happens for Young’s lattice, see Section \[ex:G\], and for the binary tree dual graded graph, see Section \[sec:bin\]. We expect this phenomenon to also occur for the shifted Young’s lattice.
The crucial condition necessary for this phenomenon to be observed is for $A$ to be the associated graded algebra of $\tilde A$, but this is not always the case. For example, this is not the case for $K$-theoretic analogues of the Poirier-Reutenauer and Malvenuto-Reutenauer Hopf algebras, as described in Sections \[sec:PR\], \[sec:MR\]. To put it simply, the numbers of basis elements for the filtered components of $A$ and $\tilde A$ are distinct in those cases, thus there is no hope to obtain corresponding graphs one from the other via Möbius construction. Interestingly, those are exactly the examples we found where the Möbius construction fails to produce a dual filtered graph.
Synopsis of the paper
---------------------
In Section \[sec:dgg\], we recall the definition of dual graded graphs from [@F] as well as three major examples: Young’s lattice, Young-Fibonacci lattice, and shifted Young’s lattice. We then remind the reader of the definition of the Robinson-Schensted algorithm and how one can obtain it locally via the machinary of [*[growth diagrams]{}*]{}, also introduced by Fomin in [@F; @FSchen].
In Section \[sec:dfg\], we formulate our definition of dual filtered graphs. We then introduce the trivial, Pieri and Möbius constructions.
In Section \[sec:Y\], we build the Pieri and Möbius deformations of Young’s lattice. We recall the definition of Hecke insertion and observe it is a map into a pair of paths in the Möbius deformation of Young’s lattice. We provide growth rules that realize Hecke insertion. We also show how the Pieri construction applied to the ring generated by [*[Grothendieck polynomials]{}*]{} yields the same result as the Möbius construction applied to Young’s lattice. Thus we demonstrate an instance of the Möbius via Pieri phenomenon.
In Section \[sec:sY\], we build Pieri and Möbius deformations of the shifted Young’s lattice. We introduce [*[shifted Hecke insertion]{}*]{} and remark that its result always coincides with that of $K$-theoretic jeu de taquin rectification as defined in [@ThYshift]. We also provide the corresponding growth rules.
In Section \[sec:YF\], we build Pieri and Möbius deformations of the Young-Fibonacci lattice. We define $K$-Young-Fibonacci tableaux and suggest the corresponding insertion algorithm and growth rules.
In Section \[sec:other\], we consider some other examples of dual filtered graphs. Of special interest are the Pieri constructions associated to the quasisymmetric functions, to the Poirier-Reutenauer and Malvenuto-Reutenauer Hopf algebras, as well as to their $K$-theoretic analogues, which we draw from [@LP; @PP]. The Hopf algebra of quasisymmetric functions and its $K$-theoretic analogue provide another instance of the Möbius via Pieri phenomenon.
In Section \[sec:enum\], we use the calculus of up and down operators to prove some identities similar to the ones known for dual graded graphs. In particular, we formulate and prove a $K$-theoretic analogue of the Frobenius-Young identity.
Acknowledgements
----------------
We thank Vic Reiner, Alexander Garver, and Thomas Lam for helpful discussions.
Dual graded graphs {#sec:dgg}
==================
Dual graded graphs {#dual-graded-graphs}
------------------
This section follows [@F], and we refer the reader to this source for further reading on dual graded graphs.
A graded graph is a triple $G=(P,\rho,E)$, where $P$ is a discrete set of vertices, $\rho:P\to \mathbb{Z}$ is a rank function, and $E$ is a multiset of edges/arcs $(x,y)$, where $\rho(y)=\rho(x)+1$. In other words, each vertex is assigned a rank, and edges can only join vertices in successive ranks. For the set of vertices $P$, let $P_n$ denote the subset of vertices of rank $n$. For any field $K$ of characteristic zero, the formal linear combinations of vertices of $G$ form the vector space $KP$.
In future examples, the idea of *inner corners* and *outer corners* of certain configurations of boxes or cells will be necessary. We will call a cell an inner corner if it can be removed from the configuration and the resulting configuration is still valid. A cell is an outer corner of a configuration if it can be added to the configuration, and the resulting configuration is still valid.
Young’s lattice is an example of a graded graph where for any partition $\lambda$, $\rho(\lambda)=|\lambda|$, and there is an edge from $\lambda$ to $\mu$ if $\mu$ can be obtained from $\lambda$ by adding one box. Ranks zero through five are shown below. We see that $(3,31)\in E$ and $\rho(3)+1=3+1=\rho(31)=4$.
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,1)
$ $
; (2) at (1,2)
$ $ & $ $
; (11) at (-1,2)
$ $\
$ $
; (111) at (-2,3)
$ $\
$ $\
$ $
; (21) at (0,3)
$ $ & $ $\
$ $
; (3) at (2,3)
$ $ & $ $ & $ $
; (1111) at (-3,4)
$ $\
$ $\
$ $\
$ $
; (211) at (-1,4)
$ $ & $ $\
$ $\
$ $
; (22) at (0,4)
$ $ & $ $\
$ $ & $ $
; (31) at (1,4)
$ $ & $ $ & $ $\
$ $
; (4) at (3,4)
$ $ & $ $ & $ $ & $ $
; (5) at (4,5)
$ $ & $ $ & & &
; (41) at (2,5)
$ $ & $ $ & &\
$ $
; (32) at (1,5)
$ $ & $ $ &\
$ $ & $ $
; (311) at (0,5)
$ $ & $ $ &\
$ $\
$ $
; (221) at (-1,5)
$ $ & $ $\
$ $ & $ $\
$ $
; (2111) at (-2,5)
$ $ & $ $\
$ $\
$ $\
$ $
; (11111) at (-4,5)
$ $\
$ $\
$ $\
$ $\
$ $
; (empty) – (1) (1) – (2)–(3)–(4)–(5) (1)–(11)–(111)–(1111)–(11111) (11)–(21)–(31)–(41) (111)–(211)–(2111) (1111)–(2111) (4)–(41) (22)–(221) (211)–(221) (31)–(311) (3)–(31)–(32) (21)–(22)–(32) (2)–(21)–(211)–(311);
If we look at the partition (4,4,2,1), visually represented below, we see that the inner corners are in positions (2,4), (3,2), and (4,1), and the outer corners are in positions (1,5), (3,3), (4,2), and (5,1).
$ $ & $ $ & $ $ & $ $\
$ $ & $ $ & $ $ & $ $\
$ $ & $ $\
$ $
Let $G_1=(P,\rho,E_1)$ and $G_2=(P,\rho,E_2)$ be a pair of graded graphs with a common vertex set and rank function. From this pair, define an *oriented graded graph* $G=(P,\rho, E_1,E_2)$ by orienting the edges of $G_1$ in the direction of increasing rank and the edges of $G_2$ in the direction of decreasing rank. Let $a_i(x,y)$ denote the number of edges of $E_i$ joining $x$ and $y$ or the multiplicity of edge $(x,y)$ in $E_i$. We define the *up* and *down operators* $U,D\in End(KP)$ associated with graph $G$ by $$Ux=\sum_{y} a_1(x,y)y$$ and $$Dy=\sum_x a_2(x,y) x.$$
For example, in Young’s lattice shown above, $U(21)=31+22+211$ and $D(21)=2+11$.
The restrictions of $U$ and $D$ to “homogeneous” subspaces $KP_n$ are denoted by $U_n$ and $D_n$, respectively. Graded graphs $G_1$ and $G_2$ with a common vertex set and rank function are said to be *r-dual* if $$D_{n+1}U_n=U_{n-1}D_n+rI_n$$ for some $r\in\mathbb{R}$ and simply *dual* if $$D_{n+1}U_n=U_{n-1}D_n+I_n.$$ We focus on the latter.
Young’s lattice is an example of a self-dual graded graph.
Another well-known example of a self-dual graded graph is the *Young-Fibonacci lattice*, $\mathbb{YF}$. The first six ranks are shown below. The vertices of the graph are all finite words in the alphabet $\{1,2\}$, where the rank of a word is the sum of its entries. The covering relations are as follows: a word $w'$ covers $w$ if and only if either $w'=1w$ or $w'=2v$ for some $v$ covered by $w$. For example, the word $121$ covers the word $21$ since $121$ is obtained by concatentating $1$ and $21$, and $121$ is covered by $221$ because $121$ covers $21$. The words in the alphabet $\{1,2\}$ are sometimes called snakes and can be represented as a collection of boxes whose heights correspond to the entries of the word called snakeshapes. For example, the word $122112$ can be pictured as
& $ $ & $ $ & & & $ $\
$ $ & $ $ & $ $ & $ $ & $ $ & $ $
.
Thought of this way, the rank of a word is the number of boxes in its corresponding snakeshape.
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,1)
$ $
; (11) at (1,2)
$ $ & $ $
; (2) at (-1,2)
$ $\
$ $
; (12) at (-2,3)
& $ $\
$ $ &
; (21) at (0,3)
$ $ &\
$ $ & $ $
; (111) at (2,3)
$ $ & &
; (112) at (-3,4)
& & $ $\
$ $ & &
; (22) at (-1,4)
$ $ &\
$ $ &
; (121) at (0,4)
& $ $ &\
$ $ & &
; (211) at (1,4)
$ $ & &\
$ $ & &
; (1111) at (3,4)
$ $ & & &
; (11111) at (4,5)
$ $ & & & &
; (2111) at (2.92,5)
$ $ & & &\
$ $ & & &
; (1211) at (1.78,5)
& $ $ & &\
$ $ & & &
; (221) at (.64,5)
$ $ & &\
$ $ & &
; (1121) at (-.64,5)
& & $ $ &\
$ $ & & &
; (212) at (-2.92,5)
$ $ & & $ $\
$ $ & &
; (122) at(-1.78,5)
& $ $ &\
$ $ & &
; (1112) at (-4,5)
& & & $ $\
$ $ & & &
; (empty) – (1) –(11)–(111)–(1111)–(11111) (1)–(2)–(12)–(112)–(1112) (11)–(21)–(22)–(212) (2)–(21)–(211)–(2111) (111)–(211)–(221) (21)–(121)–(1121) (12)–(22)–(212) (112)–(212) (22)–(122) (22)–(221) (121)–(221) (211)–(1211) (1111)–(2111);
Young’s lattice and the Young-Fibonacci lattice discussed above are the most interesting examples of self-dual graded graphs. We close this section by describing the graph of shifted shapes, $\mathbb{SY}$, and its dual, which together form a dual graded graph.
\[ex:shifted\] Given a strict partition $\lambda$ (i.e. $\lambda = (\lambda_1>\lambda_2>\ldots>\lambda_k)$), the corresponding shifted shape $\lambda$ is an arrangement of cells in $k$ rows where row $i$ contains $\lambda_i$ cells and is indented $i-1$ spaces. For example, the strict partition $(5,4,3,1)$ corresponds to the shifted shape shown below.
$ $ & $ $ & $ $ & $ $ & $ $\
& $ $ & $ $ & $ $ & $ $\
& & $ $ & $ $ & $ $\
& & & $ $
We say that cells in row $i$ and column $i$ are *diagonal* cells and all other cells are *off-diagonal*.
The graph of shifted shapes, $\mathbb{SY}$, has shifted shapes as vertices, the rank function counts the number of cells in a shape, and shifted shape $\lambda$ covers $\mu$ if $\lambda$ is obtained from $\mu$ by adding one cell. The first six ranks of the graph $\mathbb{SY}$ are shown below on the left. The graph shown on the right is its dual. In the dual graph, there is one edge between $\lambda$ and $\mu$ if $\lambda$ is obtained from $\mu$ by adding a diagonal cell, and there are two edges between $\lambda$ and $\mu$ if $\lambda$ is obtained from $\mu$ by adding an off-diagonal cell. To form the dual graded graph, the edges of $\mathbb{SY}$ are oriented upward and those of its dual are oriented downward.
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,1)
$ $
; (2) at (0,2)
$ $ & $ $
; (3) at (1,3)
$ $ & $ $ & $ $
; (21) at (-1,3)
$ $ & $ $\
& $ $
; (31) at (0,4)
$ $ & $ $ & $ $\
& $ $
; (4) at (2,4)
$ $ & $ $ & $ $ & $ $
; (32) at (-1,5)
$ $ & $ $ & $ $\
& $ $ & $ $
; (41) at (1,5)
$ $ & $ $ & $ $ & $ $\
& $ $
; (5) at (3,5)
$ $ & $ $ & $ $ & $ $ & $ $
; (empty)–(1)–(2)–(3)–(4)–(5) (2)–(21)–(31)–(32) (3)–(31)–(41) (4)–(41);
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,1)
$ $
; (2) at (0,2)
$ $ & $ $
; (3) at (1,3)
$ $ & $ $ & $ $
; (21) at (-1,3)
$ $ & $ $\
& $ $
; (31) at (0,4)
$ $ & $ $ & $ $\
& $ $
; (4) at (2,4)
$ $ & $ $ & $ $ & $ $
; (32) at (-1,5)
$ $ & $ $ & $ $\
& $ $ & $ $
; (41) at (1,5)
$ $ & $ $ & $ $ & $ $\
& $ $
; (5) at (3,5)
$ $ & $ $ & $ $ & $ $ & $ $
; (empty) – (1) (2) – (21) (3) – (31) (4) – (41); (1)–(2) (2)–(3) (3)–(4) (4)–(5) (21)–(31) (31)–(32) (31)–(41);
Growth rules as generalized RSK
-------------------------------
There is a well-known combinatorial correspondence between words in the alphabet of positive integers and pairs consisting of a semistandard Young tableau and a standard Young tableau of the same shape called the Robinson-Schensted-Knuth (RSK) correspondence. We briefly review the correspondence here and refer the reader to [@EC2] for more information.
Given a word $w$, the RSK correspondence maps $w$ to a pair of tableaux via a row insertion algorithm consisting of inserting a positive integer into a tableau. The algorithm for inserting positive integer $k$ into a row of a semistandard tableau is as follows. If $k$ is greater than or equal to all entries in the row, add a box labeled $k$ to the end of the row. Otherwise, find the first $y$ in the row with $y>k$. Replace $y$ with $k$ in this box, and proceed to insert $y$ into the next row. To insert $k$ into semistandard tableau $P$, we start by inserting $k$ into the first row of $P$. To create the *insertion tableau* of a word $w=w_1w_2\cdots w_r$, we first insert $w_1$ into the empty tableau, insert $w_2$ into the result of the previous insertion, insert $w_3$ into the result of the previous insertion, and so on until we have inserted all letters of $w$. We denote the resulting insertion tableau by $P(w)$. The insertion tableau will always be a semistandard tableau.
To obtain a standard Young tableau from $w$, we define the *recording tableau*, $Q(w)$, of $w$ by labeling the box of $P(w_1\cdots w_s)/P(w_1\cdots w_{s-1})$ by $s$. For example, $w=14252$ has insertion and recording tableau
$P(w)=$
1 & 2 & 2\
4 & 5
$Q(w)=$
1 & 2 & 4\
3 & 5
.
The RSK correspondence described above is a bijection between words consisting of positive integers and pairs $(P,Q)$, where $P$ is a semistandard Young tableau and $Q$ is a standard Young tableau of the same shape.
If $w=w_1w_2\cdots w_k$ is a permutation, we can also obtain $(P(w),Q(w))$ from $w$ by using growth diagrams. First, we create a $k \times k$ array with an $X$ in the $w_i^{th}$ square from the bottom of column $i$. For example, if $w=14253$, we have
$ $ & $ $ & $ $ & X & $ $\
$ $ & X & $ $ & $ $ & $ $\
$ $ & $ $ & $ $ & $ $ & X\
$ $ & $ $ & X & $ $ & $ $\
X & $ $ & $ $ & $ $ & $ $
.
We will label the corners of each square with a partition. We begin by labeling all corners along the bottom row and left side of the diagram with the empty shape, $\varnothing$.
To complete the labeling of the corners, suppose the corners $\mu$, $\lambda$, and $\nu$ are labeled, where $\mu$, $\lambda$, and $\nu$ are as in the picture below. We label $\gamma$ according to the following rules.
\(A) at (-1,-1) [$\lambda$]{}; (B) at (1,-1) [$\nu$]{}; (C) at (-1,1) [$\mu$]{}; (D) at (1,1) [$\gamma$]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
- If the square does not contain an $X$ and $\lambda=\mu=\nu$, then $\gamma=\lambda$.
- If the square does not contain an $X$ and $\lambda\subset\mu=\nu$, then $\mu/\lambda$ is one box in some row $i$. In this case, $\gamma$ is obtained from $\mu$ by adding one box in row $i=1$.
- If the square does not contain an $X$ and $\mu\neq\nu$, define $\gamma=\mu\cup\nu$.
- If the square contains an $X$, then $\lambda=\nu=\mu$ and $\gamma$ is obtained from $\mu$ by adding one box in the first row.
Following these rules, there is a unique way to label the corners of the diagram. The resulting array is called the *growth diagram* of $w$, and rules (L1)-(L4) are called *growth rules*. For the remainder of this paper, we will use the word “square” when referring to a square in the growth diagram and the word “box” or “cell” when referring to a box in a partition, shifted partition, or snakeshape.
The chains of partitions read across the top of the diagram and up the right side of the diagram determine $Q(w)$ and $P(w)$, respectively. To obtain a tableau of shape $\lambda$ from a chain of partitions $\varnothing \subset \lambda^1 \subset \lambda^2 \subset \ldots \subset \lambda^k=\lambda$, label the box of $\lambda^i/\lambda^{i-1}$ by $i$.
The growth diagram for the word $14253$ is shown below.
\(00) at (0,0) [$\varnothing$]{}; (10) at (1,0) [$\varnothing$]{}; (20) at (2,0) [$\varnothing$]{}; (30) at (3,0) [$\varnothing$]{}; (40) at (4,0) [$\varnothing$]{}; (50) at (5,0) [$\varnothing$]{}; (01) at (0,1) [$\varnothing$]{}; (.5.5) at (.5,.5) [$X$]{}; (11) at (1,1) [$1$]{}; (21) at (2,1) [$1$]{}; (31) at (3,1) [$1$]{}; (41) at (4,1) [$1$]{}; (51) at (5,1) [$1$]{}; (02) at (0,2) [$\varnothing$]{}; (12) at (1,2) [$1$]{}; (22) at (2,2) [$1$]{}; (32) at (3,2) [$2$]{}; (2.51.5) at (2.5,1.5) [$X$]{}; (42) at (4,2) [$2$]{}; (52) at (5,2) [$2$]{}; (03) at (0,3) [$\varnothing$]{}; (13) at (1,3) [$1$]{}; (23) at (2,3) [$1$]{}; (33) at (3,3) [$2$]{}; (43) at (4,3) [$2$]{}; (53) at (5,3) [$3$]{}; (4.52.5) at (4.5,2.5) [$X$]{}; (04) at (0,4) [$\varnothing$]{}; (14) at (1,4) [$1$]{}; (24) at (2,4) [$2$]{}; (34) at (3,4) [$21$]{}; (44) at (4,4) [$21$]{}; (54) at (5,4) [$31$]{}; (1.53.5) at (1.5,3.5) [$X$]{}; (05) at (0,5) [$\varnothing$]{}; (15) at (1,5) [$1$]{}; (25) at (2,5) [$2$]{}; (35) at (3,5) [$21$]{}; (45) at (4,5) [$31$]{}; (55) at (5,5) [$32$]{}; (3.54.5) at (3.5,4.5) [$X$]{};
(00)–(10)–(20)–(30)–(40)–(50) (01)–(11)–(21)–(31)–(41)–(51) (02)–(12)–(22)–(32)–(42)–(52) (03)–(13)–(23)–(33)–(43)–(53) (04)–(14)–(24)–(34)–(44)–(54) (05)–(15)–(25)–(35)–(45)–(55) (00)–(01)–(02)–(03)–(04)–(05) (10)–(11)–(12)–(13)–(14)–(15) (20)–(21)–(22)–(23)–(24)–(25) (30)–(31)–(32)–(33)–(34)–(35) (40)–(41)–(42)–(43)–(44)–(45) (50)–(51)–(52)–(53)–(54)–(55);
From the chains of partitions on the rightmost edge and uppermost edge of the diagram, we can read the insertion tableau and recording tableau, respectively.
$P(14253)=$
1 & 2 & 3\
4 & 5
$Q(14253)=$
1 & 2 & 4\
3 & 5
We next draw a connection between the growth rules described and an explicit cancellation in Young’s lattice that shows that $DU-UD=I$. We shall start by giving explicit cancellation rules to pair all down-up paths from $\mu$ to $\nu$ with up-down paths from $\mu$ to $\nu$, which will leave one up-down path unpaired in $\mu=\nu$. (Note that if $\rho(\mu)\neq \rho (\nu)$, then there are no up-down or down-up paths from $\mu$ to $\nu$.)
- If $\mu\neq \nu$, there is only one down-up path from $\mu$ to $\nu$, which passes through $\mu\cap\nu$. Pair this with the only up-down path from $\mu$ to $\nu$, which passes through $\mu\cup\nu$.
- If $\mu=\nu$, then any down-up path can be represented by the inner corner of $\mu$ that is deleted in the downward step, and any up-down path can be represented by the outer corner of $\mu$ that is added in the upward step. Pair each down-up path with the up-down path that adds the first outer corner of $\mu$ strictly below the inner corner of the down-up path.
Now, the only up-down path that has not been paired with a down-up path is the path corresponding to adding the outer corner of $\mu$ in the first row. This gives $(DU-UD)\mu=\mu$.
It is easy to see that this cancellation yields exactly the growth rules described above. Cancellation rule (C1) determines growth rule (L3), cancellation rule (C2) determines growth rule (L2), and the up-down path left unmatched determines growth rule (L4). If we started with a different explicit cancellation, we could modify (L2), (L3), and (L4) accordingly to obtain new growth rules. Note that growth rule (L1) is simply a way to transfer information and will remain the same for any explicit cancellation.
Starting with $\mu=(3,1)$, cancellation rule (C2) says that the down-up path from $\mu$ to itself passing through $(2,1)$ is paired with the up-down path passing through $(3,2)$. The corresponding growth rule in this situation, (L4), is illustrated on the right.
\(A) at (0,-1) [21]{}; (B) at (0,0) [31]{}; (D) at (0,1) [32]{}; (A) – (B) – (D);
\(A) at (-1,-1) [21]{}; (B) at (1,-1) [31]{}; (C) at (-1,1) [31]{}; (D) at (1,1) [32]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
Dual filtered graphs {#sec:dfg}
====================
The definition
--------------
A weak filtered graph is a triple $G=(P,\rho,E)$, where $P$ is a discrete set of vertices, $\rho:P\to \mathbb{Z}$ is a rank function, and $E$ is a multiset of edges/arcs $(x,y)$, where $\rho(y) \geq \rho(x)$. A strict filtered graph is a triple $G=(P,\rho,E)$, where $P$ is a discrete set of vertices, $\rho:P\to \mathbb{Z}$ is a rank function, and $E$ is a multiset of edges/arcs $(x,y)$, where $\rho(y) > \rho(x)$. Let $P_n$ as before denote the subset of vertices of rank $n$.
Let $G_1=(P,\rho,E_1)$ and $G_2=(P,\rho,E_2)$ be a pair of filtered graphs with a common vertex set, $G_1$ weak while $G_2$ strict. From this pair, define an *oriented filtered graph* $G=(P, E_1,E_2)$ by orienting the edges of $G_1$ in the positive filtration direction and the edges of $G_2$ in the negative filtration direction. Recall that $a_i(x,y)$ denotes the number of edges of $E_i$ joining $x$ and $y$ or the multiplicity of edge $(x,y)$ in $E_i$. Using the *up* and *down operators* associated with graph $G$ as previously defined, we say that $G_1$ and $G_2$ are *dual filtered graphs* if for any $x\in G$, $$(DU-UD)x=\alpha x+\beta Dx$$ for some $\alpha,\beta\in\mathbb{R}$. We focus on examples where $DU-UD=D+I$, see Remark \[rem:rescale\].
In the next sections, we describe three constructions of dual filtered graphs.
Trivial construction
--------------------
Every dual graded graph has an easy deformation that makes it a dual filtered graph. To construct it, simply add $\rho(x)$ “upward” loops to each element $x$ of the dual graded graph, where $\rho$ is the rank function. We will call this the *trivial construction*.
For any dual graded graph $G$, the trivial construction gives a dual filtered graph $G'$.
It suffices to show that if $q$ covers $p$ in $G$, then $[p](DU-UD)(q)=[p](D)(q)$. If we restrict to all paths that do not contain a loop, the coefficient of $p$ in $(DU-UD)(q)$ is $0$ since without loops, we have a pair of dual graded graphs. Hence we may restrict our attention to up-down paths and down-up paths that contain a loop. The up-down paths beginning at $q$ and ending at $p$ are exactly the paths consisting of a loop at $q$ followed by an edge from $q$ to $p$. There are $\rho(q)e(q\to p)$ such paths. The down-up paths containing a loop are the paths consisting of an edge from $q$ to $p$ followed by a loop at $p$. There are $e(q\to p)\rho(p)$ such paths. Thus $$[p](DU-UD)(q)=e(q \to p)(\rho(q)-\rho(p))=e(q \to p) = [p]D(q).$$
The trivial construction for the first four ranks of Young’s lattice is shown below. The edges in the graph on the left are oriented upward, and the edges in the graph on the right are oriented downward.
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; (empty) – (1) (1) – (2)–(3) (1)–(11)–(111) (11)–(21) (2)–(21); (1) to \[out=60,in=100,distance=.5cm\] (1); (2) to \[out=60,in=100,distance=.5cm\] (2); (2) to \[out=60,in=100,distance=.7cm\] (2); (11) to \[out=60,in=100,distance=.5cm\] (11); (11) to \[out=60,in=100,distance=.7cm\] (11); (111) to \[out=60,in=100,distance=.5cm\] (111); (111) to \[out=60,in=100,distance=.7cm\] (111); (111) to \[out=60,in=100,distance=.9cm\] (111); (21) to \[out=60,in=100,distance=.5cm\] (21); (21) to \[out=60,in=100,distance=.7cm\] (21); (21) to \[out=60,in=100,distance=.9cm\] (21); (3) to \[out=60,in=100,distance=.5cm\] (3); (3) to \[out=60,in=100,distance=.7cm\] (3); (3) to \[out=60,in=100,distance=.9cm\] (3);
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Pieri construction
------------------
The following construction is close to the one for dual graded graphs described in the works of Bergeron-Lam-Li [@BLL], Nzeutchap [@HN], and Lam-Shimozono [@LS].
Let $$A = \oplus_{i \geq 0} A_i$$ be a filtered, commutative algebra over a field $\mathbb F$ with $A_0 = \mathbb F$ and a linear basis $a_{\lambda} \in A_{|\lambda|}$. Here, the $\lambda$’s belong to some infinite indexing set $P$, each graded component of which $$P_i = \{\lambda \mid |\lambda|=i\}$$ is finite. The property of being filtered means that $a_\lambda a_\mu=\sum c_\nu a_\nu$, where $|\lambda|+|\mu|\leq|\nu|.$
Let $f = f_1 +f_2 + \ldots \in \hat A$ be an element of the completion $\hat A$ of $A$ such that $f_i \in A_i$. Assume $D \in End(A)$ is a derivation on $A$ satisfying $D(f) = f+1$.
To form a graph, we define $E_1$ by defining $a_1(\lambda,\mu)$ to be the coefficient of $a_{\lambda} \text{ in } D (a_{\mu}).$ We also define $E_2$ by defining $a_2(\lambda,\mu)$ to be the coefficient of $a_{\mu} \text{ in } a_{\lambda} f.$
\[thm:pieri\] The resulting graph is a dual filtered graph.
It follows from $$D(f a_{\lambda}) = D(f) a_{\lambda} + f D(a_{\lambda}),$$ or $$D(f a_{\lambda}) - f D(a_{\lambda}) = a_{\lambda} + D(a_{\lambda}).$$
Assume $A_1$ is one-dimensional, and let $g$ be its generator. We shall often find a derivation $D$ from a [*[bialgebra]{}*]{} structure on $A$. Indeed, assume $\Delta$ is a coproduct on $A$ such that $\Delta(a) = 1 \otimes a + a \otimes 1 + \dotsc$, where the rest of the terms lie in $(A_1 \oplus A_2 \oplus \ldots ) \otimes (A_1 \oplus A_2 \oplus \ldots )$. Assume also $\xi$ is a map $A \otimes A \longrightarrow A$ such that $\xi(p \otimes g) = p$ and for any element $q \in A_i$, $i \not = 1$, we have $\xi(p \otimes q) = 0$.
\[lem:derivation\] The map $D(a) = \xi(\Delta(a))$ is a derivation.
We have $\xi(\Delta(ab)) = \xi(\Delta(a) \Delta(b)))$. Pick a linear basis for $A$ that contains $1$ and $g$. Express $\Delta(a)$ and $\Delta(b)$ so that the right side of tensors consists of the basis elements. The only terms that are not killed by $\xi$ are the ones of the form $p \otimes g$. The only way we can get such terms is from $(p_1 \otimes 1) \cdot (p_2 \otimes g)$ or $(p_1 \otimes g) \cdot (p_2 \otimes 1)$. Those are exactly the terms that contribute to $$\xi(\Delta(a))\cdot b + a \cdot \xi(\Delta(b)).$$
Möbius construction
-------------------
Let $G=(P, \rho, E_1, E_2)$ be a dual graded graph. From $G$, form a pair of filtered graphs as follows. Let $G_1' = (P, \rho, E_1')$ have the set of edges $E_1'$ consisting of the same edges as $E_1$ plus $\#\{x|y\text{ covers }x\text{ in }G_1\}$ loops at each vertex $y \in P$. Let $G_2' = (P, \rho, E_2')$ have the set of edges $E_2'$ consisting of $$a'_2(x,y) = |\mu(x,y)|$$ edges between vertices $x$ and $y$, where $\mu$ denotes the Möbius function in $(P, \rho, E_2)$. Compose $G_1'$ and $G_2'$ into an oriented filtered graph $\tilde G$.
As we shall see, in all three of our major examples, the Möbius deformation forms a dual filtered graph. Unlike with the Pieri construction, we do not have the algebraic machinary explaining why the construction yields dual filtered graphs. Instead, we provide the proofs on a case-by-case basis. Nevertheless, the Möbius construction is the most important one, as it is this construction that relates to insertion algorithms and thus to $K$-theory of Grassmannians and Lagrangian Grassmannians.
It is not the case that every dual graded graph’s Möbius deformation makes it a dual filtered graph. For example, it is easy to see that the Möbius deformation of the graphs in Example \[ex:mMR\] and Example \[ex:SYTtree\] are not dual filtered graphs.
Determine for which dual graded graphs $G$ the Möbius construction $\tilde G$ yields a dual filtered graph.
Major examples: Young’s lattice {#sec:Y}
===============================
Pieri deformations of Young’s lattice
-------------------------------------
Let $A = \Lambda$ be the ring of symmetric functions. Let $s_{\lambda}$, $p_{\lambda}$, and $h_{\lambda}$ be its bases of Schur functions, power sum symmetric functions, and complete homogeneous symmetric functions. Let $$f = h_1 + h_2 + \ldots.$$ Define up and down edges of a filtered graph $G = (P, \rho, E_1, E_2)$ by letting $a_1(\mu, \nu)$ be the coefficient of $s_{\nu} \text{ in } p_1 s_{\mu}, \text{ and } a_2(\mu, \nu)$ be the coefficient of $s_{\nu} \text{ in } f s_{\mu}.$
Recall that given two partitions, $\mu$ and $\nu$, $\mu/\nu$ forms a *horizontal strip* if no two boxes of $\mu/\nu$ lie in the same column. For example, $(4,2,1)/(2,1)$ forms a horizontal strip while $(4,3,1)/(2,1)$ does not.
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The up edges $E_1$ coincide with those of Young’s lattice, while the down edges $E_2$ connect shapes that differ by a horizontal strip.
The statement follows from the Pieri rule and the fact that $p_1=h_1$, see [@EC2].
In the figure below, the first six ranks are shown. The edges in the graph on the left are upward-oriented, and the edges on the graph on the right are downward-oriented.
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; (empty) – (1) (1) – (2)–(3)–(4) (1)–(11)–(111)–(1111) (11)–(21)–(31) (111)–(211) (3)–(31) (21)–(22) (2)–(21)–(211) (3) to \[bend left=20\] (empty) (4) to \[bend left=25\] (empty) (2) to \[bend left=15\] (empty) (3) to \[bend left =15\] (1) (4) to \[bend left =15\](2) (4) to \[bend left =20\] (1) (21)–(1) (211)–(11) (22) to \[bend left =10\] (2) (31) to \[bend left =15\] (11) (31) to \[bend left =10\] (1) (31)–(2);
\[thm:young\] The Pieri deformation of Young’s lattice is a dual filtered graph with $$DU-UD=D+I.$$
Recall the following facts regarding the ring of symmetric functions.
[@EC2; @Mac]
- $\Lambda$ is a free polynomial ring in $p_1, p_2, \ldots$.
- $\Lambda$ can be endowed with a standard bilinear inner product determined by $\langle s_{\lambda}, s_{\mu} \rangle = \delta_{\lambda, \mu}$.
Because of the first property, we can differentiate elements of $\Lambda$ with respect to $p_1$ by expressing them first as a polynomial in the $p_i$’s. We shall need the following two properties of $f$ and $p_1$.
\[lem:young\]
- For any $h,g \in \Lambda$ we have $$\langle g, p_1 h \rangle = \langle \frac{d}{d p_1} g, h \rangle.$$
- We have $$\frac{d}{d p_1} f = f+1.$$
By bilinearity, it suffices to prove the first claim for $g,h$ in some basis of $\Lambda$. Let us choose the power sum basis. Then we want to argue that $$\langle p_{\lambda}, p_1 p_{\mu} \rangle = \langle \frac{d}{d p_1} p_{\lambda}, p_{\mu} \rangle.$$ This follows from the fact that $p_{\lambda}$’s form an orthogonal basis and [@EC2 Proposition 7.9.3].
For the second claim, using [@EC2 Proposition 7.7.4] we have $$1 + f = e^{p_1 + \frac{p_2}{2} + \ldots},$$ which of course implies $\frac{d}{d p_1} (1+f) = 1+f$.
Now we are ready for the proof of Theorem \[thm:young\].
Applying Lemma \[lem:young\] we see that $A = \Lambda$, $a_{\lambda} = s_{\lambda}$, $D = \frac{d}{d p_1}$ and $f = h_1 + h_2 + \ldots$ satisfy the conditions of Theorem \[thm:pieri\]. The claim follows.
Alternatively, we can give the following direct combinatorial argument.
Suppose shape $\mu$ covers shape $\lambda$. Marking the boxes of $\mu/\lambda$ in $\mu$, we see that up-down paths ending at $\lambda$ correspond to ways to add a box to $\mu$ so that this new box and the boxes in $\mu/\lambda$ form a horizontal strip. Down-up paths ending at $\lambda$ correspond to ways to remove a box from $\lambda$ so that this box and the boxes in $\mu/\lambda$ form a horizontal strip.
To see there is one more up-down path than down-up path, black out each column of $\mu$ that contains a box of $\mu/\lambda$. This will partition the columns of $\mu$ which do not have a box in $\mu/\lambda$. In each non-empty grouping of columns, there is exactly one way to form an up-down path, by adding the leftmost box in the grouping to $\mu$, and one way to form a down-up path, by deleting the rightmost box in the grouping from $\lambda$. There is, in addition, one way to form an up-down path by adding a box to the first row of $\mu$. Thus $\lambda$ appears once in $(DU-UD)\mu$.
Take $\mu=(6,4,3,1)$ and $\lambda = (5,3,3)$. There are $3$ up-down paths and $2$ down-up paths between $\mu$ and $\lambda$.\
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Blacking out the columns of $\mu$ that contain boxes in $\mu/\lambda$, we have partitioned the remaining columns in the groups $\{2,3\}$ and $5$. In columns $2$ and $3$, we get an up-down path by adding $(4,2)$ to $\mu$ and a down-up path by removing $(3,3)$ from $\lambda$. In column $5$, we get an up-down path by adding $(2,5)$ to $\mu$ and a down-up path by removing $(1,5)$ from $\lambda$. We get an additional up-down path by adding the box $(7,1)$ to $\mu$.
We can make a similar argument for the analogous construction where the edges in $E_1$ are again those of Young’s lattice and the edges in $E_2$ connect shapes that differ by a vertical strip. In this case we use $f = e_1 + e_2 + \ldots,$ where the $e_i$ are the elementary symmetric functions.
Möbius deformation of Young’s lattice {#sec:MobiusYoung}
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Given two partitions, $\lambda$ and $\nu$, $\lambda/\nu$ forms a *rook strip* if no two boxes of $\lambda/\nu$ lie in the same row and no two boxes of $\lambda/\nu$ lie in the same column. In other words, $\lambda/\nu$ is a rook strip if it is a disconnected union of boxes. We state the following well-known result about the Möbius function on Young’s lattice.
We have $$|\mu(\lambda, \nu)| =
\begin{cases}
1 & \text{if $\lambda/\nu$ is a rook strip;}\\
0 & \text{otherwise.}
\end{cases}$$
The statement follows from the fact that Young’s lattice is a distributive lattice, and the interval $[\lambda,\mu]$ is isomorphic to a Boolean algebra if and only if $\mu/\lambda$ is a rook strip. See [@EC1 Example 3.9.6].
The Möbius deformation of Young’s lattice is shown below with upward-oriented edges shown on the left and downward-oriented edges shown on the right. Notice that loops at a partition $\lambda$ may be indexed by inner corners of $\lambda$.
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; (empty) – (1) (1) – (2)–(3)–(4) (1)–(11)–(111)–(1111) (11)–(21)–(31) (111)–(211) (3)–(31) (21)–(22) (2)–(21)–(211) (1) to \[out=60,in=100,distance=.5cm\] (1) (2) to \[out=60,in=100,distance=.5cm\] (2) (11) to \[out=60,in=100,distance=.5cm\] (11) (3) to \[out=60,in=100,distance=.5cm\] (3) (4) to \[out=60,in=100,distance=.5cm\] (4) (111) to \[out=60,in=100,distance=.5cm\] (111) (1111) to \[out=60,in=100,distance=.5cm\] (1111) (21) to \[out=-60,in=-100,distance=.5cm\] (21) (21) to \[out=-60,in=-100,distance=.7cm\] (21) (211) to \[out=60,in=100,distance=.5cm\] (211) (211) to \[out=60,in=100,distance=.7cm\] (211) (31) to \[out=60,in=100,distance=.5cm\] (31) (31) to \[out=60,in=100,distance=.7cm\] (31) (22) to \[out=60,in=100,distance=.5cm\] (22) ;
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The Möbius deformation of Young’s lattice forms a dual filtered graph with $$DU-UD=D+I.$$
It suffices to show that $[\lambda](DU-UD)\mu=[\lambda]D\mu$ when $\mu$ covers $\lambda$ since we started with a dual graded graph.
Suppose shape $\lambda$ is covered by shape $\mu$. Each inner corner of $\mu$ contributes one to the coefficient of $\lambda$ in $DU\mu$ via taking the loop corresponding to that inner corner of $\mu$ as an upward move and then going down to $\lambda$.
Next, consider shape $\mu$ and mark the outer corners of $\lambda$ contained in $\mu$. The marked boxes will be inner corners of $\mu$. The shapes with an upward arrow from $\mu$ and a downward arrow to $\lambda$ are exactly those obtained by adding one box to an outer corner of $\mu$ which is not adjacent to one of the marked boxes. There are $\#\{\text{outer corners of $\lambda$}\}-|\mu/\lambda|$ possible places to add such a box. Thus the coefficient of $\lambda$ in $DU\mu$ is $$\#\{\text{inner corners of $\mu$}\}+\#\{\text{outer corners of $\lambda$}\}-|\mu/\lambda|.$$
For down-up paths that involve three shapes, consider the inner corners of $\mu$ that are also inner corners of $\lambda$. Every shape obtained by removing one of these inner corners determines one down-up path from $\mu$ to $\lambda$, and there are $\#\{\text{inner corners of $\mu$}\}-|\mu/\lambda|$ such shapes. The remaining down-up paths are given by going from $\mu$ to $\lambda$ and then taking a loop at $\lambda$. Hence the coefficient of $\lambda$ in $UD\mu$ is $$\#\{\text{inner corners of $\mu$}\}-|\mu/\lambda|+\#\{\text{inner corners of }\lambda\}.$$ Since all shapes have one more outer corner than inner corner, the coefficient of $\lambda$ in $(DU-UD)\mu$ is 1.
Hecke insertion
---------------
We next describe an insertion and reverse insertion procedure that correspond to the Möbius deformation of Young’s lattice in the same way that RSK corresponds to Young’s lattice, which is called *Hecke insertion*. This procedure was introduced in [@BKSTY]. In other words, if we insert a word $w$ of length $n$ using Hecke insertion, the construction of the insertion tableau will be represented as a path of length $n$ downward that ends at $\varnothing$ in the Möbius deformation of Young’s lattice, and the construction of the recording tableau is represented as a path of length $n$ upward starting at $\varnothing$.
An *increasing tableau* is a filling of a Young diagram with positive integers such that the entries in rows are strictly increasing from left to right and the entries in columns are strictly increasing from top to bottom.
The tableau shown on the left is an increasing tableau. The tableau on the right is not an increasing tableau because the entries in the first row are not strictly increasing.
1 & 2 & 4 & 5\
2 & 3 & 5 & 7\
6 & 7\
8
1 & 2 & 2 & 4\
3 & 4\
5
We follow [@BKSTY] to give a description of Hecke (row) insertion of a positive integer $x$ into an increasing tableau $Y$ resulting in an increasing tableau $Z$. The shape of $Z$ is obtained from the shape of $Y$ by adding at most one box. If a box is added in position $(i,j)$, then we set $c=(i,j)$. In the case where no box is added, then $c=(i,j)$, where $(i,j)$ is a special corner indicating where the insertion process terminated. We will use a parameter $\alpha\in\{0,1\}$ to keep track of whether or not a box is added to $Y$ after inserting $x$ by setting $\alpha=0$ if $c\in Y$ and $\alpha=1$ if $c\notin Y$. We use the notation $Z=(Y {\overset{H}{\longleftarrow}} x)$ to denote the resulting tableau, and we denote the outcome of the insertion by $(Z,c,\alpha)$.
We now describe how to insert $x$ into increasing tableau $Y$ by describing how to insert $x$ into $R$, a row of $Y$. This insertion may modify the row and may produce an output integer, which we will insert into the next row. To begin the insertion process, insert $x$ into the first row of $Y$. The process stops when there is no output integer. The rules for insertion of $x$ into $R$ are as follows:
- If $x$ is weakly larger than all integers in $R$ and adjoining $x$ to the end of row $R$ results in an increasing tableau, then $Z$ is the resulting tableau and $c$ is the new corner where $x$ was added.
- If $x$ is weakly larger than all integers in $R$ and adjoining $x$ to the end of row $R$ does not result in an increasing tableau, then $Z=Y$, and $c$ is the box at the bottom of the column of $Z$ containing the rightmost box of the row $R$.
For the next two rules, assume $R$ contains boxes strictly larger than $x$, and let $y$ be the smallest such box.
- If replacing $y$ with $x$ results in an increasing tableau, then replace $y$ with $x$. In this case, $y$ is the output integer to be inserted into the next row
- If replacing $y$ with $x$ does not result in an increasing tableau, then do not change row $R$. In this case, $y$ is the output integer to be inserted into the next row.
\
$1$ & $2$ & $3$ & $5$\
$2$ & $3$ & $4$ & $6$\
$6$\
$7$
${\overset{H}{\longleftarrow}} 3$ $=$
$1$ & $2$ & $3$ & $5$\
$2$ & $3$ & $4$ & $6$\
$6$\
$7$
We use rule (H4) in the first row to obtain output integer $5$. Notice that the $5$ cannot replace the $6$ in the second row since it would be directly below the $5$ in the first row. Thus we use (H4) again and get output integer $6$. Since we cannot add this $6$ to the end of the third row, we use (H2) and get $c=(4,1)$. Notice that the shape did not change in this insertion, so $\alpha=0$.
\[ex:ins2\]\
$2$ & $4$ & $6$\
$3$ & $6$ & $8$\
$7$
${\overset{H}{\longleftarrow}} 5$ $=$
$2$ & $4$ & $5$\
$3$ & $6$ & $8$\
$7$ & $8$
The integer $5$ bumps the $6$ from the first row using (H3). The $6$ is inserted into the second row, which already contains a $6$. Using (H4), the second row remains unchanged and we insert $8$ into the third row. Since $8$ is larger than everything in the third row, we use (H1) to adjoin it to the end of the row. Thus $\alpha=1$.
Using this insertion algorithm, we define the *Hecke insertion tableau* of a word $w=w_1w_2\cdots w_n$ to be $$P_H(w)=(\ldots((\varnothing\overset{H}{\longleftarrow} w_1)\overset{H}{\longleftarrow} w_2)\ldots) \overset{H}{\longleftarrow} w_n.$$
In [@BKSTY], Buch, Kresch, Shimozono, Tamvakis, and Yong give the following algorithm for reverse Hecke insertion starting with the triple $(Z,c,\alpha)$ as described above and ending with a pair $(Y,x)$ consisting of an increasing tableau and a postive integer.
- If $y$ is the cell in square $c$ of $Z$ and $\alpha=1$, then remove $y$ and reverse insert $y$ into the row above.
- If $\alpha=0$, do not remove $y$, but still reverse insert it into the row above.
In the row above, let $x$ be the largest integer such that $x<y$.
- If replacing $x$ with $y$ results in an increasing tableau, then we replace $x$ with $y$ and reverse insert $x$ into the row above.
- If replacing $x$ with $y$ does not result in an increasing tableau, leave the row unchanged and reverse insert $x$ into the row above.
- If $R$ is the first row of the tableau, the final output consists of $x$ and the modified tableau.
[@BKSTY Theorem 4]\[thm:insbijection\] Hecke insertion $(Y,x)\mapsto (Z,c,\alpha)$ and reverse Hecke insertion $(Z,c,\alpha)\mapsto (Y,x)$ define mutually inverse bijections between the set of pairs consisting of an increasing tableau and a positive integer and the set of triples consisting of an increasing tableau, a corner cell of the increasing tableau, and $\alpha\in\{0,1\}$.
We next describe the tableaux that will act as recording tableau for Hecke insertion.
A *set-valued tableau* $T$ of shape $\lambda$ is a filling of the boxes with finite, non-empty subsets of positive integers so that
1. the smallest number in each box is greater than or equal to the largest number in the box directly to the left of it (if that box is present), and
2. the smallest number in each box is strictly greater than the largest number in the box directly above it (if that box is present).
We call of set-valued tableau *standard* if it contains exactly the integers $[n]$ for some $n$.
Given a word $w=w_1w_2\ldots w_l$, we can associate a pair of tableaux $(P_H(w), Q_H(w))$, where $P_H(w)$ is the Hecke insertion tableau described previously and $Q_H(w)$ is a standard set-valued tableau called the *Hecke recording tableau* obtained as follows. Start with $Q_H(\varnothing)=\varnothing$. At each step of the insertion of $w$, let $Q_H(w_1\ldots w_k)$ be obtained from $Q_H(w_1\ldots w_{k-1})$ by labeling the special corner, $c$, in the insertion of $w_k$ into $P_H(w_1\ldots w_{k-1})$ with the positive integer $k$. Then $Q_H(w)=Q_H(w_1 w_2\ldots w_l)$ is the resulting standard set-valued tableau.
Let $w$ be $15133$. We obtain $(P_H(w),Q_H(w))$ with the following sequence, where in column $k$, $Q_H(w_1 \ldots w_k)$ is shown below $P_H(w_1 \ldots w_k)$.
1
1 & 5
1 & 5\
5
1 & 3\
5
1 & 3\
5
$=P_H(w)$
1
1 & 2
1 & 2\
3
1 & 2\
34
1 & 25\
34
$=Q_H(w)$
Call a word $w$ [*[initial]{}*]{} if the letters appearing in it are exactly the numbers in $[k]$ for some positive integer $k$.
The word $13422$ is initial since the letters appearing in it are the numbers from $1$ to $4$. On the other hand, the word $1422$ is not initial because the letters appearing in it are $1$, $2$, and $4$ and do not form a set $[k]$ for any $k$.
As with RSK, Hecke insertion gives a useful bijection.
[@PP] The map sending $w=w_1w_2\cdots w_n$ to $(P_H(w),Q_H(w))$ is a bijection between words and ordered pairs of tableaux of the same shape $(P,Q)$, where $P$ is an increasing tableau and $Q$ is a set-valued tableau with entries $\{1,2,\ldots,n\}$. It is also a bijection if there is an extra condition of being initial imposed both on $w$ and $P$.
Hecke insertion is closely related to the $K$-theoretic jeu de taquin algorithm introduced in [@ThY]. This relationship is explored in [@BuchSam; @ThY2]
Hecke growth and decay {#sec:HeckeGrowth}
----------------------
Given any word, $h=h_1h_2\ldots h_k$ containing $n \leq k$ distinct numbers, we can create an $n \times k$ array with an $X$ in the $h_i^{th}$ square from the bottom of column $i$. Note that there can be multiple $X$’s in the same row but is at most one $X$ per column. For example, if $h=121331$, we have
$ $ & $ $ & $ $ & X & X & $ $\
$ $ & X & $ $ & $ $ & $ $ & $ $\
X & $ $ & X & $ $ & $ $ & X
We will label each of the four corners of each square with a partition and some of the horizontal edges of the squares with a specific inner corner of the partition on the left vertex of the edge, which will indicate the box where the insertion terminates. We do this by recording the row of the inner corner at which the insertion terminates. We begin by labeling all corners along the bottom row and left side of the diagram with the partition $\varnothing$.
\(00) at (0,0) [$\varnothing$]{}; (10) at (1,0) [$\varnothing$]{}; (20) at (2,0) [$\varnothing$]{}; (30) at (3,0) [$\varnothing$]{}; (40) at (4,0) [$\varnothing$]{}; (50) at (5,0) [$\varnothing$]{}; (01) at (0,1) [$\varnothing$]{}; (.5.5) at (.5,.5) [$X$]{}; (11) at (1,1) [$ $]{}; (21) at (2,1) [$ $]{}; (31) at (3,1) [$ $]{}; (41) at (4,1) [$ $]{}; (51) at (5,1) [$ $]{}; (02) at (0,2) [$\varnothing$]{}; (12) at (1,2) [$ $]{}; (22) at (2,2) [$ $]{}; (32) at (3,2) [$ $]{}; (1.51.5) at (1.5,1.5) [$X$]{}; (42) at (4,2) [$ $]{}; (52) at (5,2) [$ $]{}; (03) at (0,3) [$\varnothing$]{}; (13) at (1,3) [$ $]{}; (23) at (2,3) [$ $]{}; (33) at (3,3) [$ $]{}; (43) at (4,3) [$ $]{}; (53) at (5,3) [$ $]{}; (2.5.5) at (2.5,.5) [$X$]{}; (3.52.5) at (3.5,2.5) [$X$]{}; (4.53.5) at (4.5,2.5) [$X$]{}; (5,51.5) at (5.5,.5) [$X$]{}; (60) at (6,0) [$\varnothing$]{}; (61) at (6,1) [$ $]{}; (62) at (6,2) [$ $]{}; (63) at (6,3) [$ $]{};
(00)–(10)–(20)–(30)–(40)–(50)–(60) (01)–(11)–(21)–(31)–(41)–(51)–(61) (02)–(12)–(22)–(32)–(42)–(52)–(62) (03)–(13)–(23)–(33)–(43)–(53)–(63)
(00)–(01)–(02)–(03) (10)–(11)–(12)–(13) (20)–(21)–(22)–(23) (30)–(31)–(32)–(33) (40)–(41)–(42)–(43) (50)–(51)–(52)–(53) (60)–(61)–(62)–(63);
To complete the labeling of the corners, suppose the corners $\mu$, $\lambda$, and $\nu$ are labeled, where $\mu$, $\lambda$, and $\nu$ are as in the picture below. We label $\gamma$ according to the following rules.
\(A) at (-1,-1) [$\lambda$]{}; (B) at (1,-1) [$\nu$]{}; (C) at (-1,1) [$\mu$]{}; (D) at (1,1) [$\gamma$]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
**If the square contains an X:**
- If $\mu_1=\nu_1$, then $\gamma/\mu$ consists of one box in row 1.
- If $\mu_1 \neq \nu_1$, then $\gamma=\mu$ and the edge between then is labeled by the row of the highest inner corner of $\mu$.
**If the square does not contain an X and if either $\mu=\lambda$ or $\nu=\lambda$ with no label between $\lambda$ and $\nu$**:
- If $\mu=\lambda$, then set $\gamma = \nu$. If $\nu=\lambda$, then $\gamma = \mu$.
**If $\nu \nsubseteq \mu$ and the square does not contain an X:**
- In the case where $\nu \nsubseteq \mu$, $\gamma = \nu \cup \mu$.
**If $\nu \subseteq \mu$ and the square does not contain an X:**
- If $\nu/\lambda$ is one box in row $i$ and $\mu/\nu$ has no boxes in row $i+1$, then $\gamma/\mu$ is one box in row $i+1$.
- If $\nu/\lambda$ is one box in row $i$ and $\mu/\nu$ has a box in row $i+1$, then $\gamma=\mu$ and the edge between then is labeled $i+1$.
- If $\nu=\lambda$, the edge between them is labeld $i$, and there are no boxes of $\mu/\nu$ immediately to the right or immediately below this inner corner of $\nu$ in row $i$, then $\gamma=\mu$ with the edge between them labeled $i$.
- If $\nu=\lambda$, the edge between them is labeled $i$, and there is a box of $\mu/\nu$ directly below this inner corner of $\nu$ in row $i$, then $\gamma=\mu$ with the edge between them labeled $i+1$.
- If $\nu=\lambda$, the edge between them is labeled $i$, and there is a box of $\mu/\nu$ immediately to the right of this inner corner of $\nu$ in row $i$ but no box of $\mu/\nu$ in row $i+1$, then $\gamma/\mu$ is one box in row $i+1$.
- If $\nu=\lambda$, the edge between them is labeled $i$, and there is a box of $\mu/\nu$ immediately to the right of this inner corner of $\nu$ in row $i$ and a box of $\mu/\nu$ in row $i+1$, then $\gamma=\mu$ with the edge between them labeled $i+1$.
We call the resulting array the *Hecke growth diagram* of $h$. In our previous example with $h=121342$, we would have the diagram below.
\(00) at (0,0) [$\varnothing$]{}; (10) at (1,0) [$\varnothing$]{}; (20) at (2,0) [$\varnothing$]{}; (30) at (3,0) [$\varnothing$]{}; (40) at (4,0) [$\varnothing$]{}; (50) at (5,0) [$\varnothing$]{}; (01) at (0,1) [$\varnothing$]{}; (.5.5) at (.5,.5) [$X$]{}; (11) at (1,1) [$1$]{}; (21) at (2,1) [$1$]{}; (31) at (3,1) [$1$]{}; (41) at (4,1) [$1$]{}; (51) at (5,1) [$1$]{}; (02) at (0,2) [$\varnothing$]{}; (12) at (1,2) [$1$]{}; (22) at (2,2) [$2$]{}; (32) at (3,2) [$21$]{}; (1.51.5) at (1.5,1.5) [$X$]{}; (42) at (4,2) [$21$]{}; (52) at (5,2) [$21$]{}; (03) at (0,3) [$\varnothing$]{}; (13) at (1,3) [$1$]{}; (23) at (2,3) [$2$]{}; (33) at (3,3) [$21$]{}; (43) at (4,3) [$31$]{}; (53) at (5,3) [$31$]{}; (2.5.5) at (2.5,.5) [$X$]{}; (3.52.5) at (3.5,2.5) [$X$]{}; (4.53.5) at (4.5,2.5) [$X$]{}; (5,51.5) at (5.5,.5) [$X$]{}; (60) at (6,0) [$\varnothing$]{}; (61) at (6,1) [$1$]{}; (62) at (6,2) [$21$]{}; (63) at (6,3) [$31$]{};
(2.51.1) at (2.5,1.15) [1]{}; (5.51.1) at (5.5,1.15) [1]{}; (5.52.1) at (5.5,2.15) [2]{}; (4.53.1) at (4.5,3.15) [1]{}; (5.53.1) at (5.5,3.15) [2]{};
(00)–(10)–(20)–(30)–(40)–(50)–(60) (01)–(11)–(21)–(31)–(41)–(51)–(61) (02)–(12)–(22)–(32)–(42)–(52)–(62) (03)–(13)–(23)–(33)–(43)–(53)–(63)
(00)–(01)–(02)–(03) (10)–(11)–(12)–(13) (20)–(21)–(22)–(23) (30)–(31)–(32)–(33) (40)–(41)–(42)–(43) (50)–(51)–(52)–(53) (60)–(61)–(62)–(63);
Let $\mu_0 = \varnothing \subseteq \mu_1 \subseteq \ldots \mu_k$ be the seqence of partitions across the top of the Hecke growth diagram, and let $\nu_0=\varnothing \subseteq \nu_1 \subseteq \ldots \nu_n$ be the sequence of partitions on the right side of the Hecke growth diagram. These sequences correspond to increasing tableaux $Q(h)$ and $P(h)$, respectively. If the edge between $\mu_i$ and $\mu_{i+1}$ is labeled $j$, then $\mu_i=\mu_{i+1}$, and the label $i+1$ of $Q(h_1\cdots h_{i+1})$ is placed in the box at the end of row $j$ of $Q(h_1\cdots h_i)$. In the example above, we have
$P(h)=$
1 & 2 & 3\
2
$Q(h)=$
1 & 2 & 45\
36
For any word $h$, the increasing tableau $P(h)$ and set-valued tableau $Q(h)$ obtained from the sequence of partitions across the right side of the Hecke growth diagram for $h$ and across the top of the Hecke growth diagram for $h$, respectively, are $P_H(h)$ and $Q_H(h)$, the Hecke insertion and recording tableau for $h$.
For the following proof, again note that the loops at shape $\lambda$ may be indexed by inner corners of $\lambda$.
Suppose that the square described in rules 1 through 10 is in row $t$ and column $s$. We will argue the result by induction.
Oriented as in the square above, $|\nu/\lambda|\leq 1$.
By the induction hypothesis, $\lambda$ is the recording tableau after inserting the numbers in columns $1$ through $s-1$, and $\nu$ is the insertion tableau after inserting the number in column $s$. Since we are only inserting one number, $|\nu/\lambda|$ is at most 1.
Oriented as in the square above, $\mu/\lambda$ is a rook strip, that is, no two boxes in $\mu/\lambda$ are in the same row or column.
By induction, $\mu/\lambda$ represents the positions of boxes filled with $t$ at some point in the insertion. Since insertion results in an increasing tableau, $\mu/\lambda$ must be a rook strip.
- First note that in any square with an X, $\lambda=\nu$ since there is exactly one square in each column and that an X in the square we are considering corresponds to inserting a $t$ into the tableau of shape $\lambda$. Since, by the induction hypothesis, shapes along columns represent the insertion tableau, $\mu_1=\nu_1=\lambda_1$ means that before adding the $t$, there are no $t$’s in the first row of the insertion tableau. Since $t$ is weakly the largest number inserted to this point, inserting the $t$ will result in a $t$ being added to the first row of the insertion tableau. Thus $\gamma=(\mu_1+1,\mu_2,\mu_3,\ldots)$.
- If $\mu_1\neq\nu_1=\lambda_1$, then after inserting the numbers in columns $1$ through $s-1$, there is a $t$ at the end of the first row of the insertion tableau. It follows that inserting another $t$ will result in this $t$ merging with the $t$ at the end of the first row, and the special corner in this case becomes the bottom box of the last column, or in other words, the highest inner corner of $\mu$.
- If $\mu=\lambda$, then there is no X in row $t$ in columns $1$ to $s$. Thus there is no $t$ to add, and so the insertion tableau $\nu$ should not change. Similarly for if $\nu=\lambda$.
- Suppose that $\nu/\lambda$ is one box in row i and $\mu/\lambda$ is a rook strip containing boxes in rows $j_1,j_2,\ldots ,j_k$. Then $\nu \nsubseteq \mu$ implies that $i\notin \{j_1,j_2,\ldots ,j_k\}$. Since $\nu/\lambda$ is one box, the last action in the insertion sequence of $r$ into the insertion tableau of shape $\lambda$ is a box being added in row $i$. Since there is not a $t$ in row $i$, the bumping sequence when inserting $r$ into the insertion tableau of shape $\mu$ will not disturb the $t$’s.
- Suppose $\nu$ is $\lambda$ plus one box in position $(i,j)$. Then since $\nu \subseteq \mu$, $\mu/\lambda$ contains the box at $(i,j)$. It follows that inserting $r$ into the insertion tableau of shape $\mu$ will result in bumping the $t$ in position $(i,j)$ since the bumping path of $r$ inserting in the insertion tableau of shape $\lambda$ ends by adding a box at $(i,j)$. Since there is no box in row $i+1$ of $\mu/\lambda$, everything in row $i+1$ of the insertion tableau of shape $\mu$ is strictly less than $t$. It follows that the $t$ bumped from $(i,j)$ can be added to the end of row $i+1$.
- This is almost identical to the proof of Rule 4. Since there is now a box in row $i+1$ of $\mu/\lambda$, there is a $t$ at the end of row $i+1$ of the insertion tableau of shape $\mu$. Inserting $r$ will bump the $t$ in row $i$ as above, but now the $t$ will merge with the $t$ in row $i+1$. Thus the special corner is now at the inner corner of $\mu$ in row $i+1$.
- There are two ways that inserting $r$ into the insertion tableau of shape $\mu$ will involve the $t$’s. Assume $\nu$ is obtained from $\lambda$ by taking a loop in row $i$ and column $j$. Assume rows $k$ through $i$ all have length $j$.
Case 1: A box containing $y$ was bumped from row $k-1$, merged with itself in row $k$, and there was a $t$ to the right of the $y$. Note that $k\neq i$. Then the $y$ duplicates and bumps the $t$ in row $k$, but this $t$ cannot be added to row $k+1$ as it would be directly below the original $t$. Thus, by the rules of Hecke insertion, the special corner is the box at the bottom of column $j$, box $(i,j)$. See the left-hand figure below where the dot indicates the box $(i,j)$.
Case 2: Consider the situation in the right-hand figure below where the dot indicates the box $(i,j)$. Assume that during the insertion of $r$, some number merged with itself to the left of $z$ in row $k-1$ and $z$ is duplicated bumped to the next row. Following the rules of Hecke insertion, $z$ bumps the $t$ in row $k$, but as it cannot replace the $t$, the $t$ remains at the end of row $k$. The $t$ that was bumped cannot be added to end of row $k+1$, so no new boxes are created in the insertion and the special corner is the box at the bottom of column $j$, box $(i,j)$.
- There are three ways that inserting $r$ into the insertion tableau of shape $\mu$ will involve the $t$’s. Assume $\nu$ is obtained from $\lambda$ by taking a loop in row $i$ and column $j$. Assume rows $k$ through $i$ all have length $j$.
Case 1: If there is a $t$ in square $(i,j+1)$, we are in the situation described by Rule 9.
Cases 2 and 3 are described in the proof of Rule 7. The difference is that the bottom of column $j$ after we insert $r$ is now the box $(i+1,j)$, so the special corner in each case will be $(i+1,j)$, as desired.
- Note that a loop in row $i$ with a box of $\mu/\nu$ to its right implies that $\nu_{i-1}>\nu_i>\nu_{i+1}$. There are two ways for the special corner of the insertion of $r$ into the insertion tableau of shape $\lambda$ could have been $(i,j)$.
Case 1: Some $y$ was bumped from row $i-1$ and merged with itself in box $(i,j)$. Since there is a $t$ to the right of box $(i,j)$ in $\mu$, this will result in duplicating and bumping $t$. Since there is nothing in row $i+1$ of $\mu/\nu$, everything in row $i+1$ of $\mu$ is strictly less than $t$. Hence the $t$ bumped from row $i$ can be added to the end of row $i+1$.
Case 2: As in the figure below, $z$ was duplicated and bumped from box $(i-1,j+1)$. This $z$ cannot replace the $t$ in box $(i,j+1)$ since it would be directly below the orginal $z$, but it bumps the $t$ to row $i+1$. Since there is nothing in row $i+1$ of $\mu/\nu$, everything in row $i+1$ of $\mu$ is strictly less than $t$. Hence the $t$ bumped from row $i$ can be added to the end of row $i+1$.
- This is similar to the situation described in the proof of Rule 9. The difference is that a box in row $i+1$ of $\mu/\nu$ means there is already a $t$ at the end of row $i+1$ of $\mu$, so each insertion will end with the bumped $t$ merging with itself at the end of row $i+1$.
Möbius via Pieri {#ex:G}
----------------
The following shows that we have an instance of the Möbius via Pieri phenomenon in the case of Young’s lattice. Namely, the Möbius deformation of Young’s lattice is obtained by the Pieri construction from an appropriate $K$-theoretic deformation of the underlying Hopf algebra.
Let $A$ be the subring of the completion $\hat\Lambda$ of the ring of symmetric functions with basis $\{G_\lambda\}$, the stable Grothendieck polynomials. For details, see [@B; @LP]. Define the *signless stable Grothendieck polynomials* $\tilde G_\lambda$ by omitting the sign. Note that these coincide with the $\tilde K_\lambda$ defined in [@LP]. The structure constants will coincide with those of the stable Grothendieck polynomials up to sign. We recall these structure constants here.
Let $S_\mu$ be the superstandard tableau of shape $\mu$, that is, the first row of $S$ is filled with $1,2,\ldots,\mu_1$, the second row with $\mu_1+1,m_1+2,\ldots,\mu_1+\mu_2$, etc. This space has a bialgebra product structure given by $$\tilde G_\lambda \tilde G_\mu=\displaystyle\sum c_{\lambda,\mu}^\nu \tilde G_\nu,$$ where $c_{\lambda,\mu}^\nu$ is the number of increasing tableaux $R$ of skew shape $\nu/\lambda$ such that $P_H(\r(R))=S_\mu$. The coproduct structure is given by $$\Delta(\tilde G_\nu)=\displaystyle\sum_{\lambda,\mu} d_{\lambda,\mu}^\nu \tilde G_\lambda \otimes \tilde G_\mu,$$ where $d_{\lambda,\mu}^\nu$ is the number of increasing tableaux $R$ of skew shape $\lambda \oplus \mu$ such that $P_H(\r(R))=S_\nu$, see [@BuchSam; @PP; @ThY3]. Here $\lambda\oplus\mu$ denotes the skew shape obtained by joining the rightmost top corner of $\lambda$ to the leftmost bottom corner of $\mu$. For example, $(2,1)\oplus (2,2)=(4,4,2,1)/(2,2)$ is shown below.
& & $ $ & $ $\
& & $ $ & $ $\
$ $ & $ $\
$ $
We consider the following Pieri construction in $A$. Let $g=\tilde G_1$, and define an operator $D$ by $D(\tilde G_\nu)=\xi(\Delta(\tilde G_\nu))$. We define a graph $G$, where elements are partitions, $a_1(\mu,\lambda)$ is the coefficient of $\tilde G_\mu$ in $D(\tilde G_\lambda)$, and $a_2(\mu,\lambda)$ is the coefficient of $\tilde G_\lambda$ in $\tilde G_\mu \tilde G_1$.
\[prop:mobpieriyoung\] The resulting graph coincides with the one obtained via Möbius construction in Section \[sec:MobiusYoung\].
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,1)
$ $
; (2) at (1,2)
$ $ & $ $
; (11) at (-1,2)
$ $\
$ $
; (111) at (-2,3)
$ $\
$ $\
$ $
; (21) at (0,3)
$ $ & $ $\
$ $
; (3) at (2,3)
$ $ & $ $ & $ $
; (1111) at (-3,4)
$ $\
$ $\
$ $\
$ $
; (211) at (-1,4)
$ $ & $ $\
$ $\
$ $
; (22) at (0,4)
$ $ & $ $\
$ $ & $ $
; (31) at (1,4)
$ $ & $ $ & $ $\
$ $
; (4) at (3,4)
$ $ & $ $ & $ $ & $ $
; (empty) – (1) (1) – (2)–(3)–(4) (1)–(11)–(111)–(1111) (11)–(21)–(31) (111)–(211) (3)–(31) (21)–(22) (2)–(21)–(211) (1) to \[out=60,in=100,distance=.5cm\] (1) (2) to \[out=60,in=100,distance=.5cm\] (2) (11) to \[out=60,in=100,distance=.5cm\] (11) (3) to \[out=60,in=100,distance=.5cm\] (3) (4) to \[out=60,in=100,distance=.5cm\] (4) (111) to \[out=60,in=100,distance=.5cm\] (111) (1111) to \[out=60,in=100,distance=.5cm\] (1111) (21) to \[out=-60,in=-100,distance=.5cm\] (21) (21) to \[out=-60,in=-100,distance=.7cm\] (21) (211) to \[out=60,in=100,distance=.5cm\] (211) (211) to \[out=60,in=100,distance=.7cm\] (211) (31) to \[out=60,in=100,distance=.5cm\] (31) (31) to \[out=60,in=100,distance=.7cm\] (31) (22) to \[out=60,in=100,distance=.5cm\] (22) ;
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,1)
$ $
; (2) at (1,2)
$ $ & $ $
; (11) at (-1,2)
$ $\
$ $
; (111) at (-2,3)
$ $\
$ $\
$ $
; (21) at (0,3)
$ $ & $ $\
$ $
; (3) at (2,3)
$ $ & $ $ & $ $
; (1111) at (-3,4)
$ $\
$ $\
$ $\
$ $
; (211) at (-1,4)
$ $ & $ $\
$ $\
$ $
; (22) at (0,4)
$ $ & $ $\
$ $ & $ $
; (31) at (1,4)
$ $ & $ $ & $ $\
$ $
; (4) at (3,4)
$ $ & $ $ & $ $ & $ $
; (empty) – (1) (1) – (2)–(3)–(4) (1)–(11)–(111)–(1111) (11)–(21)–(31) (111)–(211) (3)–(31) (21)–(22) (2)–(21)–(211) (21)–(1) (31)–(2) (211)–(11);
To verify this claim, we first show why the coefficient of $\tilde G_\nu$ in $\tilde G_\lambda \tilde G_1$ is 1 if $\nu/\lambda$ is a rook strip and is 0 otherwise. This follows from that fact that the only words that Hecke insert into $S_1$ are words consisting only of the letter $1$. Thus, $\r(R)=11\cdots 1$ for any $R$ contributing to $c_{\lambda,1}^\nu$. Increasing tableaux of shape $\nu/\lambda$ with row word $11\cdots 1$ are precisely rook strips.
Secondly, it must be true that $d_{\lambda,1}^\lambda$ is the number of inner corners of $\lambda$ and if $\nu\neq\lambda$, $d_{\lambda,1}^\nu$ is 1 if $\nu$ is obtained from $\lambda$ by adding one box and 0 otherwise. Suppose $\nu\neq\lambda$ and consider all skew tableaux $R$ of shape $\lambda \oplus 1$ such that $P_H(\r(R))=S_\nu$. To obtain such $\lambda$, simply reverse insert an entry of $S_\nu$ with $\alpha=1$ and use the output integer to fill the single box in $\lambda\oplus 1$. The resulting shapes are the shapes obtained from $\nu$ by deleting one inner corner of $\mu$. Similarly, if $\nu=\lambda$, we reverse insert each inner corner of $S_\lambda$ using $\alpha=0$ and then let $R$ be $S_\lambda\oplus x$, where $x$ is the result of the reverse insertion.
Major examples: shifted Young’s lattice {#sec:sY}
=======================================
Pieri deformation of shifted Young’s lattice
--------------------------------------------
Let $A = \Lambda'$ be the subring of the ring of symmetric functions generated by the odd power sums $p_1, p_3, \ldots$. Let $Q_{\lambda}$ and $P_{\lambda}$ be the bases of Schur $Q$ and Schur $P$ functions for this ring. Here $\lambda$ varies over the set $P$ of partitions with distinct parts. We refer the reader to [@Mac] for background. Let $q_i$ be defined by $$q_i = \sum_{0 \leq j \leq i} e_j h_{i-j},$$ and let $f = q_1 + q_2 + \ldots.$
Define up and down edges of a filtered graph $G = (P, \rho, E_1, E_2)$ by letting $a_2(\mu, \nu)$ be the coefficient of $Q_\mu$ in $fQ_\nu$ and $a_1(\mu,\nu)$ be the coefficient of $P_\mu$ in $p_1P_\nu$.
We will fill a shifted partition $\lambda$ with ordered alphabet $1'<1<2'<2<3'<3\ldots$ according to the usual rules:
- The filling must be weakly increasing in rows and columns.
- There can be at most one instance of $k'$ in any row.
- There can be at most one instance of $k$ in any column.
- There can be no primed entries on the main diagonal.
For example, the shifted partition below is filled according to the rules stated above.
1 & 2’ & 2 & 2 & 4’\
& 2 & 3’ & 5’ & 5\
& & 4 & 5’\
& & & 6
Given two shifted shapes, $\mu$ and $\nu$, we say that $\mu/\nu$ forms a *border strip* if it contains no 2-by-2 square.
The Pieri deformation of the shifted Young’s lattice is formed by adding downward-oriented edges from $\mu$ to $\nu$ whenever $\mu/\nu$ forms a border strip. The number of such edges added is the same as the number of ways to fill the boxes of the border strip with $k$ and $k'$ according to the usual rules.
Follows from formula (8.15) in [@Mac], which is an analogue of the Pieri rule for Schur $P$ functions, and the fact that $q_1 = 2 p_1$.
The first six ranks of the Pieri deformation are shown below.
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,1)
$ $
; (2) at (0,2)
$ $ & $ $
; (3) at (1,3)
$ $ & $ $ & $ $
; (21) at (-1,3)
$ $ & $ $\
& $ $
; (31) at (0,4)
$ $ & $ $ & $ $\
& $ $
; (4) at (2,4)
$ $ & $ $ & $ $ & $ $
; (32) at (-1,5)
$ $ & $ $ & $ $\
& $ $ & $ $
; (41) at (1,5)
$ $ & $ $ & $ $ & $ $\
& $ $
; (5) at (3,5)
$ $ & $ $ & $ $ & $ $ & $ $
; (empty)–(1)–(2)–(3)–(4)–(5) (2)–(21)–(31)–(32) (3)–(31)–(41) (4)–(41);
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,1)
$ $
; (2) at (0,2)
$ $ & $ $
; (3) at (1,3)
$ $ & $ $ & $ $
; (21) at (-1,3)
$ $ & $ $\
& $ $
; (31) at (0,4)
$ $ & $ $ & $ $\
& $ $
; (4) at (2,4)
$ $ & $ $ & $ $ & $ $
; (32) at (-1,5)
$ $ & $ $ & $ $\
& $ $ & $ $
; (41) at (1,5)
$ $ & $ $ & $ $ & $ $\
& $ $
; (5) at (3,5)
$ $ & $ $ & $ $ & $ $ & $ $
; (empty) – (1) to \[bend right =5\] (2) (2) to \[bend left=5\] (3) (3) to \[bend left =5\] (2) (2) – (21) to \[bend left =5\] (31) (21) to \[bend right =5\] (31) (1) to \[bend left=5\] (2) (3) to \[bend left=5\] (4) (3) to \[bend right=5\] (4) (3) – (31) to \[bend left=5\] (41) (31) to \[bend right =5\] (41) (31) to \[bend left =5\] (32) (31) to \[bend right=5\] (32) (4) to \[bend right=5\] (5) (4) to \[bend left=5\] (5) (2) to \[bend left=15\] (empty) (3) to \[bend left =15\] (1) (3) to \[bend left =25\] (1) (3) to \[bend left =15\] (empty) (4) to \[bend left =15\] (2) (4) to \[bend left =25\] (2) (4) to \[bend left =15\] (1) (4) to \[bend left =25\] (1) (4) to \[bend left = 15\] (empty) (5) to \[bend left =15\] (3) (5) to \[bend left =25\] (3) (5) to \[bend left =15\] (2) (5) to \[bend left =25\] (2) (5) to \[bend left =15\] (1) (5) to \[bend left =25\] (1) (5) to \[bend left=15\] (empty) (31) to \[bend left=5\] (2) (31) to \[bend right =5\] (2) (31) to \[bend right = 15\] (1) (21) to \[bend right = 15\] (1) (32) to \[bend right =20\] (3) (41) to \[bend left = 5\] (3) (41) to \[bend right =5\] (3) (41) to \[bend right = 15\] (21) (41) to \[bend right =20\] (21) (41) to \[bend left = 5\] (2) (41) to \[bend right = 5\] (2) (32) – (2) (4) – (41);
Notice that there are two edges from $41$ to $2$ corresponding to the following fillings of the border strip $(4,1)/(2)$.
& & $k$’ & $k$\
& $k$
& & $k$ & $k$\
& $k$
\[thm:syoung\] The Pieri deformation of the dual graded graph of shifted shapes satisfies $$DU-UD=D+I.$$
We shall need the following two properties of $\Lambda'$.
[@Mac]
- $\Lambda'$ is a free polynomial ring in odd power sums $p_1, p_3, \ldots$.
- $\Lambda'$ inherits the standard bilinear inner product from $\Lambda$, which satisfies $\langle Q_{\lambda}, P_{\mu} \rangle = 2^{l(\mu)} \delta_{\lambda, \mu}$.
Because of the first property, we can again differentiate elements of $\Lambda'$ with respect to $p_1$, by expressing them first as a polynomial in the $p_i$’s. We shall need the following property of $f$.
\[lem:syoung\] We have $$\frac{d}{d p_1} f = 2f+2.$$
From [@Mac Ex. 6 (a), III, 8] we have $$1 + f = e^{2p_1 + \frac{2p_3}{3} + \ldots},$$ which of course implies $\frac{d}{d p_1} (1+f) = 2+2f$.
Now we are ready for the proof of Theorem \[thm:syoung\].
Applying Lemma \[lem:young\] we see that $A = \Lambda'$, $a_{\lambda} = Q_{\lambda}$, $D = \frac{1}{2} \frac{d}{d p_1}$ and $f = q_1 + q_2 + \ldots$ satisfy the conditions of Theorem \[thm:pieri\]. The claim follows.
If we instead take $a_1(\mu, \nu)$ to be 2 times the coefficient of $Q_{\nu} \text{ in } p_1 Q_{\mu}$, and $a_2(\mu, \nu)$ to be $\frac{1}{2}$ times the coefficient of $P_{\mu} \text{ in } f P_{\nu},$ we get a dual filtered graph with $$DU-UD=2D+I.$$ This choice corresponds to first swapping $E_1$ and $E_2$ in the original dual graded graph of $\mathbb{SY}$ and then adding downward edges to $E_2$ as described above.
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,1)
$ $
; (2) at (0,2)
$ $ & $ $
; (3) at (1,3)
$ $ & $ $ & $ $
; (21) at (-1,3)
$ $ & $ $\
& $ $
; (31) at (0,4)
$ $ & $ $ & $ $\
& $ $
; (4) at (2,4)
$ $ & $ $ & $ $ & $ $
; (32) at (-1,5)
$ $ & $ $ & $ $\
& $ $ & $ $
; (41) at (1,5)
$ $ & $ $ & $ $ & $ $\
& $ $
; (5) at (3,5)
$ $ & $ $ & $ $ & $ $ & $ $
; (empty)–(1) to \[bend right =5\] (2) (2) to \[bend left=5\] (3) (3) to \[bend left =5\] (2) (2) – (21) to \[bend left =5\] (31) (21) to \[bend right =5\] (31) (1) to \[bend left=5\] (2) (3) to \[bend left=5\] (4) (3) to \[bend right=5\] (4) (3) – (31) to \[bend left=5\] (41) (31) to \[bend right =5\] (41) (31) to \[bend left =5\] (32) (31) to \[bend right=5\] (32) (4) to \[bend right=5\] (5) (4) – (41) (4) to \[bend left=5\] (5);
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,1)
$ $
; (2) at (0,2)
$ $ & $ $
; (3) at (1,3)
$ $ & $ $ & $ $
; (21) at (-1,3)
$ $ & $ $\
& $ $
; (31) at (0,4)
$ $ & $ $ & $ $\
& $ $
; (4) at (2,4)
$ $ & $ $ & $ $ & $ $
; (32) at (-1,5)
$ $ & $ $ & $ $\
& $ $ & $ $
; (41) at (1,5)
$ $ & $ $ & $ $ & $ $\
& $ $
; (5) at (3,5)
$ $ & $ $ & $ $ & $ $ & $ $
; (empty)–(1)–(2)–(3)–(4)–(5) (2)–(21)–(31)–(32) (3)–(31)–(41) (4)–(41) (2) to \[bend left=15\] (empty) (3) to \[bend left =15\] (1) (3) to \[bend left =25\] (1) (3) to \[bend left =15\] (empty) (4) to \[bend left =15\] (2) (4) to \[bend left =25\] (2) (4) to \[bend left =15\] (1) (4) to \[bend left =25\] (1) (4) to \[bend left = 15\] (empty) (5) to \[bend left =15\] (3) (5) to \[bend left =25\] (3) (5) to \[bend left =15\] (2) (5) to \[bend left =25\] (2) (5) to \[bend left =15\] (1) (5) to \[bend left =25\] (1) (5) to \[bend left=15\] (empty) (31) to \[bend left=5\] (2) (31) to \[bend right =5\] (2) (31) to \[bend right = 15\] (1) (21) to \[bend right = 15\] (1) (32) to \[bend right =20\] (3) (41) to \[bend left = 5\] (3) (41) to \[bend right =5\] (3) (41) to \[bend right = 15\] (21) (41) to \[bend right =20\] (21) (41) to \[bend left = 5\] (2) (41) to \[bend right = 5\] (2) (32) – (2) (4) – (41);
Möbius deformation of shifted Young’s lattice
---------------------------------------------
The Mobius deformation of the shifted Young’s lattice is shown below with upward-oriented edges shown on the left and downward-oriented edges on the right. Note that we swapped the $E_1$ and $E_2$ of Example \[ex:shifted\].
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,1)
$ $
; (2) at (0,2)
$ $ & $ $
; (3) at (1,3)
$ $ & $ $ & $ $
; (21) at (-1,3)
$ $ & $ $\
& $ $
; (31) at (0,4)
$ $ & $ $ & $ $\
& $ $
; (4) at (2,4)
$ $ & $ $ & $ $ & $ $
; (32) at (-1,5)
$ $ & $ $ & $ $\
& $ $ & $ $
; (41) at (1,5)
$ $ & $ $ & $ $ & $ $\
& $ $
; (5) at (3,5)
$ $ & $ $ & $ $ & $ $ & $ $
; (empty) – (1) to \[bend right =5\] (2) (2) to \[bend left=5\] (3) (3) to \[bend left =5\] (2) (2) – (21) to \[bend left =5\] (31) (21) to \[bend right =5\] (31) (1) to \[bend left=5\] (2) (3) to \[bend left=5\] (4) (3) to \[bend right=5\] (4) (3) – (31) to \[bend left=5\] (41) (31) to \[bend right =5\] (41) (31) to \[bend left =5\] (32) (31) to \[bend right=5\] (32) (4) to \[bend right=5\] (5) (4) to \[bend left=5\] (5) (4) – (41) (1) to \[out=160,in=100,distance=.5cm\] (1) (2) to \[out=60, in=100, distance=.5cm\] (2) (2) to \[out=60, in=100, distance=.7cm\] (2) (21) to \[out=60, in=100, distance=.5cm\] (21) (3) to \[out=60, in=100, distance=.5cm\] (3) (3) to \[out=60, in=100, distance=.7cm\] (3) (4) to \[out=60, in=100, distance=.5cm\] (4) (4) to \[out=60, in=100, distance=.7cm\] (4) (5) to \[out=60, in=100, distance=.5cm\] (5) (5) to \[out=60, in=100, distance=.7cm\] (5) (31) to \[out=60, in=100, distance=.5cm\] (31) (31) to \[out=60, in=100, distance=.7cm\] (31) (31) to \[out=60, in=100, distance=.9cm\] (31) (41) to \[out=60, in=100, distance=.5cm\] (41) (41) to \[out=60, in=100, distance=.7cm\] (41) (41) to \[out=60, in=100, distance=.9cm\] (41) (32) to \[out=60, in=100, distance=.5cm\] (32) (32) to \[out=60, in=100, distance=.7cm\] (32);
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,1)
$ $
; (2) at (0,2)
$ $ & $ $
; (3) at (1,3)
$ $ & $ $ & $ $
; (21) at (-1,3)
$ $ & $ $\
& $ $
; (31) at (0,4)
$ $ & $ $ & $ $\
& $ $
; (4) at (2,4)
$ $ & $ $ & $ $ & $ $
; (32) at (-1,5)
$ $ & $ $ & $ $\
& $ $ & $ $
; (41) at (1,5)
$ $ & $ $ & $ $ & $ $\
& $ $
; (5) at (3,5)
$ $ & $ $ & $ $ & $ $ & $ $
; (empty)–(1)–(2)–(3)–(4)–(5) (31)–(2) (41)–(3) (2)–(21)–(31)–(32) (3)–(31)–(41) (4)–(41);
For the lattice of shifted shapes, the Mobius function is given by the following formula: $$\mu(p,q)=\left\{ \begin{array}{ll}
(-1)^{|p|-|q|} & \text{if }q/p\text{ is a disjoint union of boxes} \\
0 & \text{otherwise}
\end{array}
\right.$$
It is known that $\mathbb{SY}$ is a distributive lattice [@F Example 2.2.8], and so it follows that $\mu(p,q)$ is nonzero if and only if the interval $[p,q]$ is a Boolean lattice, in which case $\mu(p,q)$ is as described above, [@EC2 Example 3.9.6]. And $[p,q]$ is a Boolean lattice exactly when $q/p$ is a disjoint union of boxes, i.e. no two boxes of $q/p$ share an edge.
The Möbius deformation of the shifted Young’s lattice forms a dual filtered graph with $$DU-UD=D+I.$$
Given that the non-deformed pair form a pair of dual graded graphs, it suffices to show that $$[\lambda](DU-UD)(\mu)=[\lambda](D)(\mu)$$ when $\mu$ covers $\lambda$.
For a shifted shape $x$, let $I_{diag}(x)$ denote the size of the set of inner corners of $x$ that are on the diagonal, $I_{odiag}(x)$ denote the size of the set of inner corners of $x$ that are not on the diagonal, $O_{diag}(x)$ denote the size of the set of outer corners of $x$ that are on the diagonal, and $O_{odiag}(x)$ denote the size of the set of outer corners of $x$ that are not on the diagonal.
\[lem:shiftededges\] For any shifted shape $\lambda$, $$O_{diag}(\lambda) + 2O_{odiag}(\lambda)-I_{diag}(\lambda)-2I_{odiag}(\lambda)=1.$$
Consider some element $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_k)$. Suppose $\lambda_k=1$. We can add a box at some subset of rows $\{i_1=1,i_2,\ldots,i_t\}$ and fill each with either $k$ or $k'$. Thus $O_{diag}(\lambda) + 2O_{odiag}(\lambda)=2t$. We can delete a box in rows $i_2-1,i_3-1,\ldots,i_t-1,k\}$, where each can be filled with $k$ or $k'$ except the box in row $k$, which is on the diagonal. Thus $I_{diag}(\lambda)-2I_{odiag}(\lambda)=2(t-1)+1$.
Now suppose $\lambda_k\geq 2$. We can add a box at some subset of rows $\{i_1=1,i_2,\ldots,i_t\,k+1\}$ and fill each with either $k$ or $k'$ except for the box in row $k+1$, which is on the diagonal. This gives $O_{diag}(\lambda) + 2O_{odiag}(\lambda)=2t+1$. We can delete boxes in rows $\{i_2-1,\ldots,i_t-1\,k\}$, where each can be filled with $k$ or $k'$, giving $I_{diag}(\lambda)-2I_{odiag}(\lambda)=2t$.
First, consider all up-down paths from $\mu$ to $\lambda$ that begin with a loop at $\mu$. There are $$2I_{odiag}(\mu)+I_{diag}(\mu)$$ such paths since there are the same number of loops at $\mu$. Up-down paths that do not start with a loop can be counted by the number of outer corners of $\lambda$ that are not inner corners of $q$ counted with multiplicity two if they are off the main diagonal since each of these corners may be added to $q$ for the step up and then removed in addition to the boxes of $\mu/\lambda$ for the step down. There are $$2(O_{odiag}(\lambda)-I_{odiag}(\mu/\lambda))+(O_{diag}(\lambda)-I_{diag}(\mu/\lambda))$$ such corners. Thus there are $$2I_{odiag}(\mu)+I_{diag}(\mu)+2(O_{odiag}(\lambda)-I_{odiag}(\mu/\lambda))+(O_{diag}(\lambda)-I_{diag}(\mu/\lambda))$$ up-down paths from $\mu$ to $\lambda$.
Next, consider all down-up paths from $\mu$ to $\lambda$ that end with a loop at $\lambda$. There are exactly $$2I_{odiag}(\lambda)+I_{diag}(\lambda)$$ such paths. Down-up paths that do not end with a loop can be counted by the inner corners of $q$ that are also inner corners of $\lambda$ counted twice if they are off the main diagonal since removing this type of inner corner in addition to the boxes of $\lambda/\mu$ will result in a shifted shape covered by $\lambda$. There are $$2(I_{odiag}(\mu)-I_{odiag}(\mu/\lambda))+(I_{diag}(\mu)-I_{diag}(\mu/\lambda))$$ such corners. Thus there are $$2I_{odiag}(\lambda)+I_{diag}(\lambda)+2(I_{odiag}(\mu)-I_{odiag}(\mu/\lambda))+(I_{diag}(\mu)-I_{diag}(\mu/\lambda))$$ down-up paths from $\mu$ to $\lambda$.
Hence $$\begin{aligned}
[\lambda](DU-UD)(\mu) &=& 2I_{odiag}(\mu)+I_{diag}(\mu)+2(O_{odiag}(\lambda)-I_{odiag}(\mu/\lambda))+(O_{diag}(\lambda)-I_{diag}(\mu/\lambda)) \nonumber \\
&-&(2I_{odiag}(\lambda)+ I_{diag}(\lambda)+2(I_{odiag}(\mu)-I_{odiag}(\mu/\lambda))+(I_{diag}(\mu)-I_{diag}(\mu/\lambda)))) \nonumber \\
&=& 2O_{0diag}(\lambda) + O_{diag}(\lambda)-2I_{odiag}(\lambda)-I_{diag}(\lambda) \nonumber \\
&=& 1 \nonumber\end{aligned}$$ by Lemma \[lem:shiftededges\].
Shifted Hecke insertion {#sec:shiftedinsertion}
-----------------------
\[def:increasingshifted\] An *increasing shifted tableau* is a filling of a shifted shape with non-negative integers such that entries are strictly increasing across rows and columns.
The following are examples of increasing shifted tableaux.
1 & 2 & 4\
& 4
2 & 3 & 5 & 7 & 8\
& 5 & 8\
& & 9
We now describe an insertion procedure for increasing shifted shapes that is similar to Hecke insertion, which we call *shifted Hecke insertion*. This procedure is a natural analogue of that of Sagan and Worley, [@Sagan; @W]. As before, any insertion tableau will correspond to a walk downward ending at $\varnothing$ in the dual filtered graph of shifted shapes from the Möbius deformation and any recording tableau corresponds to a walk upward starting at $\varnothing$.
We start by describing how to insert $x$ into an increasing shifted tableau $Y$ to obtain increasing shifted tableau $Z$. We begin by inserting $x=y_1$ into the first row of $Y$. This insertion may modify the first row of $Y$ and may produce an output integer $y_2$. Following the rules below, we then insert $y_2$ into the second row of $Y$ and continue in this manner until one of two things happens. Either at some stage the insertion will not create an output integer, in which case the insertion of $x$ into $Y$ terminates, or $y_k$ will replace some diagonal element of $Y$. In the latter case, we continue the insertion process along the columns: $y_{k+1}$ is inserted into the column to the right of its original position and so on until some step in this process does not create an output integer.
For each insertion, we designate a specific box of the resulting tableau, $Z$, as being the box where the insertion terminated. We will later use this notion to define recording tableaux.
The rules for inserting any positive integer $x$ into a row or column are as follows:
- If $x$ is weakly larger than all integers in the row (resp. column) and adjoining $x$ to the end of the row (resp. column) results in an increasing tableau, then $Z$ is the resulting tableau. We say that the insertion terminated at this new box.
Inserting $4$ into the first row of the tableau on the left gives the tableau on the right. This insertion terminated in position $(1,3)$.
1 & 2\
& 4
1 & 2 & 4\
& 4
Inserting $5$ into the third column of the resulting tableau gives the tableau shown below. This insertion terminated in position $(2,3)$.
1 & 2 & 4\
& 4 & 5
- If $x$ is weakly larger than all integers in the row (resp. column) and adjoining $x$ to the end of the row (resp. column) does not result in an increasing tableau, then $Z=Y$. If $x$ was row inserted into a nonempty row, we say that the insertion terminated at the box at the bottom of the column containing the rightmost box of this row. If $x$ was row inserted into an empty row, we say the insertion terminated at the rightmost box of the previous row. If $x$ was column inserted, we say the insertion terminated at the rightmost box of the row containing the bottom box of the column $x$ could not be added to.
Inserting $4$ into the first row of the tableau on the left does not change the row and does not give an output integer, and so the insertion does not change the tableau. The insertion terminated in position $(2,3)$
1 & 2 & 4\
& 3 & 5
Inserting $4$ into the third column of the tableau below does not change the column and does not produce an output integer, and so the insertion does not change the tableau. The insertion terminated in position $(1,4)$.
1 & 2 & 4 & 5\
& 3
Inserting 2 into the (empty) second row of the tableau below does not change the row. The insertion terminated in position (1,2)
1 & 2
For the last two rules, suppose the row (resp. column) contains a box with label strictly larger than $x$, and let $y$ be the smallest such box.
- If replacing $y$ with $x$ results in an increasing tableau, then replace $y$ with $x$. In this case, $y$ is the output integer. If $x$ was inserted into a column or if $y$ was on the main diagonal, proceed to insert all future output integers into the next column to the right. If $x$ was inserted into a row and $y$ was not on the main diagonal, then insert $y$ into the row below.
Inserting $6$ into the second row of the tableau on the left results in the tableau on the right with output integer $7$ to be inserted into the third row.
1 & 2 & 3 & 4\
& 4 & 7 & 8\
& & 8
1 & 2 & 3 & 4\
& 4 & 6 & 8\
& & 8
Inserting $6$ into the third column of the tableau on the left also results in the tableau on the right with output integer $7$, but this time, $7$ is to be inserted into the fourth column.
Inserting $3$ into the second row of the tableau on the left results in the tableau shown below with output integer $4$ to be inserted into the third column.
1 & 2 & 3 & 4\
& 3 & 7 & 8\
& & 8
- If replacing $y$ with $x$ does not result in an increasing tableau, then do not change the row (resp. column). In this case, $y$ is the output integer. If $x$ was inserted into a column or if $y$ was on the main diagonal, proceed to insert all future output integers into the next column to the right. If $x$ was inserted into a row, then insert $y$ into the row below.
Inserting $2$ into the first row of the tableau on the left does not change the tableau and produces output integer $5$ to be inserted into the second row.
1 & 2 & 5 & 8\
& 3 & 6
Inserting $5$ into the third column does not change the tableau and gives output integer $6$ to be inserted into the fourth column.
Inserting $2$ into the second row does not change the tableau. In this case, the output integer is 3 and is to be inserted into the third column.
Using this insertion algorithm, we define the *shifted Hecke insertion tableau* of a word $w=w_1w_2\cdots w_n$ to be $$P_S(w)=(\ldots((\varnothing\leftarrow w_1)\leftarrow w_2)\ldots) \leftarrow w_n.$$
We show the sequence of shifted tableaux obtained while computing $P_S(4211232)$. We start by inserting $4$ into the first row.
4
2 & 4
1 & 2 & 4
1 & 2 & 4
1 & 2 & 4\
& 4
1 & 2 & 3\
& 4
1 & 2 & 3\
& 3 & 4
The result of such insertion agrees with that of $K$-theoretic jeu de taquin rectification described in [@ThYshift]. See [@REU2015].
In this setting, recording tableaux will be set-valued shifted tableaux.
\[def:shiftedset\] A *set-valued shifted tableau* $T$ of shifted shape $\lambda$ is a filling of the boxes with finite, nonempty subsets of primed and unprimed positive integers so that
1. the smallest number in each box is greater than or equal to the largest number in the box directly to the left of it (if that box is present),
2. the smallest number in each box is greater than or equal to the largest number in the box directly to the above it (if that box is present),
3. any positive integer appears at most once, either primed or unprimed, but not both, and
4. there are no primed entries on the main diagonal.
A set-valued shifted tableau is called *standard* if the set of labels consists of $1,2,\ldots,n$, each appearing either primed or unprimed exactly once, for some $n$.
A recording tableau for a word $w=w_1w_2\ldots w_n$ is a standard shifted set-valued tableau and is obtained as follows. Begin with $Q_S(\varnothing)=\varnothing$. If the insertion of $w_k$ into $P_S(w_1\cdots w_{k-1})$ resulted in adding a new box to $P_S(w_1\cdots w_{k-1})$, add this same box with label $k$ if the box was added via row insertion and $k'$ if the box was added via column insertion to $Q_S(w_1\cdots w_{k-1})$ to obtain $Q_S(w_1\cdots w_k)$. If the insertion of $k$ into $P_S(w_1\cdots w_{k-1})$ did not change the shape of $P_S(w_1\cdots w_{k-1})$, obtain $Q_S(w_1\cdots w_k)$ from $Q_S(w_1\cdots w_{k-1})$ by adding the label $k$ to the box where the insertion terminated if the last move was a row insertion into a nonempty row and $k'$ if the last move was a column insertion. If the last move was row insertion into an empty row, label the box where the insertion terminated $k'$.
The top row of tableaux shows the sequence of tableaux obtained from inserting $w=4211232$ as in the previous example, and the bottom row shows the corresponding steps to form $Q_S(w)$.
4
2 & 4
1 & 2 & 4
1 & 2 & 4
1 & 2 & 4\
& 4
1 & 2 & 3\
& 4
1 & 2 & 3\
& 3 & 4
$=P_S(w)$
1
1 & 2’
1 & 2’ & 3’
1 & 2’ & 3’4’
1 & 2’ & 3’4’\
& 5
1 & 2’ & 3’4’\
& 56
1 & 2’ & 3’4’\
& 56 & 7’
$=Q_S(w)$
We next define a reverse insertion procedure so that given a pair $(P_S(w),Q_S(w))$, we can recover $w$.
First locate the box containing the largest label of $Q_S(w)$, call the label $n$ and the position of the box $(i_n,j_n)$, and find the corresponding box in $P_S(w)$. Say the integer in positioin $(i_n,j_n)$ of $P_S(w)$ is $y_n$. We then perform reverse insertion on cell $(i_n,j_n)$ of $P_S(w)$ by following the rules below.
- If $n$ is the only label in cell $(i_n,j_n)$ of $Q(w)$, remove box $(i_n,j_n)$ from $P_S(w)$ and reverse insert $y_n$ into the row above if $n$ is unprimed and into the column to the left if $n$ is primed.
- If $n$ is not the only label in cell $(i_n,j_n)$ of $Q_S(w)$, do not remove box $(i_n,j_n)$ from $P_S(w)$, but still reverse insert $y_n$ into the row above if $n$ is unprimed and into the column to the left if $n$ is primed.
In the row above if $y_n$ is reverse inserted into a row or the column to the left if it is reverse inserted into a column, let $x$ be the largest label with $x<y_n$.
- If replacing $x$ with $y_n$ results in an increasing shifted tableau, replace $x$ with $y_n$. If $y_n$ was reverse column inserted and $x$ was not on the main diagonal, reverse insert $x$ into the column to the left. Otherwise, reverse insert $x$ into the row above.
- If replacing $x$ with $y_n$ does not result in an increasing shifted tableau, leave the row or column unchanged. If $y_n$ was reverse column inserted and $x$ was not on the main diagonal, reverse insert $x$ into the column to the left. Otherwise, reverse insert $x$ into the row above.
If we are in the first row of the tableau and the last step was a reverse row insertion or we are in the first column and the last step was a reverse column insertion, then $x$ and the modified tableau, which we will call $P_{S,1}(w)$, are the final output value. Define $x_1=x$.
Repeat this process using the pair $(P_{S,1}(w),Q_{S,n-1}(w))$, where $Q_{S,n-1}(w)$ is the result of removing the entry $n$ or $n'$ from $Q_S(w)$. Define $x_2$ to be the output value and $P_{S,2}(w)$ to be the modified tableau. Continue this process with pairs $(P_{S,2}(w),Q_{S,n-2}(w)),\ldots,(P_{S,n-1}(w),Q_{S,1}(w))$. Then we claim $w=x_1x_2\cdots x_n$.
There is a bijection between pairs consisting of an increasing shifted tableau and a standard set-valued shifted tableau of the same shape, $(P,Q)$, and words, where the word $w$ corresponds to the pair $(P_S(w),Q_S(w))$.
We show that given a pair of tableaux of the same shape, $(P,Q)$, where $P$ is an increasing shifted tableau on some $[n]$ and $Q$ is standard shifted set-valued tableau on $1<2'<2<3'<\ldots<n'<n$, we can recover $w$ so that $P=P_S(w)$ and $Q=Q_S(w)$. Assume first that $(P,Q)$ has been obtained by inserting some integer $h$ into a pair $(Y,Z)$. We must show that the reverse insertion procedure defined above recovers $h$ and $(Y,Z)$ from $(P,Q)$.
It is clear that if the insertion of $h$ ended by adding a new box to $Y$ and $Z$ using (S1), then reverse insertion steps (rS1), (rS2), (rS3), and (rS4) undo insertion steps (S1), (S2), (S3), and (S4), respectively. Therefore, it suffices to show that if the insertion of $h$ does not result in adding a new box to $Y$ and $Z$ using (S2), then the reverse insertion procedure can recover $h$ from $(Y,Z)$.
Suppose the last step of the insertion of $h$ into $Y$ is inserting some $x$ into nonempty row $i$ using (S2) and the insertion terminates in row ${i+j}$. We first use $(rS2)$ and reverse insert the entry at the end of row ${i+j}$ into row ${i+j-1}$, which will not change row ${i+j-1}$ because the entry being reverse inserted is strictly larger than all entries in row $i+j-1$ but cannot replace the entry at the end of row $i+j-1$ since this would put it directly above the occurence of the entry in row $i+j$. Next, we reverse insert the entry at the end of row ${i+j-1}$ into ${i+j-2}$, and in the same manner, it will not change row ${i+j-2}$. This process continues until we reverse insert the entry at the end of row ${i+1}$ into $i$ and obtain output integer $x$. From this point onward, the reverse insertion rules will exactly undo the corresponding insertion rules to recover $h$. We can use the transpose of this argument if $x$ is column inserted into column $j$ using (S2) and the insertion terminates in column $j+i$.
For example, inserting 4 into the first row of the tableau below does not change the tableau, and the insertion terminates at $(3,4)$. Starting reverse insertion at $(3,4)$, we reverse insert 6 into the second row. It cannot replace the 5, so the third row doesn’t change, and 5 is reverse inserted into the first row. The 5 cannot replace the 4, so the first row is unchanged, and we end with the original tableau and the integer 4.
1 & 2 & 3 & 4\
& 3 & 4 & 5\
& & 5 & 6
If $x$ is inserted into empty row $i$ using (S2). Then our original tableau looked like the one shown below, where the bottom row shown is row $i-1$ and the dot indicates where the insertion terminated. In this case, $x$ must have been bumped by a row insertion of $a$ into row $i-1$. We show reverse insertion starting from the box where the insertion terminated recovers the step of inserting $a$ into row $i-1$.
$ $ & $ $ & $ $ & $ $ &$ $ & $ $ & $ $ & $ $\
& a & x & $ $ & $ $ & $ $ &
Using (rS4), we reverse insert the entry at the end of row $i-1$ into the column to the left. The column will not change since the entry being reverse inserted is strictly larger than all entries in the column and cannot replace the entry at the end of the column. We then reverse insert the second rightmost entry of row $i-1$ into the column to its left. Continuing in this manner, we eventually reverse column insert $x$ into the main diagonal, which results in reverse row inserting $a$ into row $i-2$. From this point on, the reverse insertion rules will exactly undo the corresponding insertion rules to recover $h$.
For example, suppose some insertion ends with inserting 2 into the empty second row of the tableau below. The insertion terminates in position $(1,4)$, and we see that the 2 must have been bumped by a row insertion of 1 into the first row. We start reverse column insertion in position $(1,4)$ and reverse column insert 4 into the third column. The integer 4 cannot replace the 3 in the third column since it would be directly to the left of the 4 in the fourth column, so the third column is left unchanged. We reverse column insert 3 into the second column. Again, the column remains unchanged, and we reverse insert 2 into the first column. The 2 cannot replace the 1, so the column is unchanged. Since the 1 was on the main diagonal, we next reverse row insert 1 into the row above.
1 & 2 & 3 & 4
Shifted Hecke growth and decay {#sec:shiftedgrowth}
------------------------------
As before, given any word $w=w_1w_2\cdots w_k$, containing $n \leq k$ distinct numbers, we can create an $n \times k$ array with an $X$ in the $w_i^{th}$ square from the bottom of column $i$. Note that there can be multiple $X$’s in the same row but is at most one $X$ per column.
We will label the corners of each square with a shifted partition and label some of the horizontal edges of the squares with a specific inner corner where the insertion terminated and/or a ‘c’ to designate column insertion. To denote a specific inner corner, we will give either its row or column. For example, the edge label 2 denotes the inner corner at the end of the second row of the shifted diagram, and an edge labeled 2c denotes the inner corner at the end of the second column of the shifted diagram. We begin by labeling all corners along the bottom row and left side of the diagram with the empty shape, $\varnothing$, as before.
To complete the labeling of the corners, suppose the corners $\mu$, $\lambda$, and $\nu$ are labeled, where $\mu$, $\lambda$, and $\nu$ are as in the picture below. We label $\gamma$ according to the following rules.
\(A) at (-1,-1) [$\lambda$]{}; (B) at (1,-1) [$\nu$]{}; (C) at (-1,1) [$\mu$]{}; (D) at (1,1) [$\gamma$]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
**If the square contains an X:**
- If $\lambda_1=\mu_1$, then $\gamma/\mu$ consists of one box in the first row.
- If $\lambda_1+1=\mu_1$, then $\gamma=\mu$ and the edge between them is labeled with a $1$ to signify the inner corner in the first row of $\gamma$.
\(A) at (-1,-1) [1]{}; (B) at (1,-1) [1]{}; (C) at (-1,1) [1]{}; (D) at (1,1) [2]{}; (E) at (0,0) [$X$]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
\(A) at (-1,-1) [1]{}; (B) at (1,-1) [1]{}; (C) at (-1,1) [2]{}; (D) at (1,1) [2]{}; (E) at (0,0) [$X$]{}; (F) at (0,1.25) [1]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
**If the square does not contain an X and $\mu=\lambda$ or if $\nu=\lambda$ with no edge label on the bottom edge**
- If $\mu=\lambda$, then set $\gamma = \nu$ and label the top edge label as the bottom edge if one exists. If $\nu=\lambda$, then $\gamma = \mu$.
\(A) at (-1,-1) [1]{}; (B) at (1,-1) [2]{}; (C) at (-1,1) [1]{}; (D) at (1,1) [2]{}; (E) at (0,1.25) [c]{}; (F) at (0,-.75)[c]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
\(A) at (-1,-1) [1]{}; (B) at (1,-1) [1]{}; (C) at (-1,1) [2]{}; (D) at (1,1) [2]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
**If $\nu \nsubseteq \mu$ and the square does not contain an X:**
- In the case where $\nu \nsubseteq \mu$, $\gamma = \nu \cup \mu$, and the top edge label is the same as the bottom edge label.
\(A) at (-1,-1) [2]{}; (B) at (1,-1) [3]{}; (C) at (-1,1) [21]{}; (D) at (1,1) [31]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
**If $\nu \subseteq \mu$ and the square does not contain an X:**
Suppose the square of $\nu/\lambda$ is one box in position $(i,j)$. If $(i,j)$ is not on the diagonal and the edge between $\lambda$ and $\nu$ is not labeled $c$:
- If there is no box of $\mu/\lambda$ in row $i+1$ of $\mu$, then $\gamma$ is obtained from $\mu$ by adding a box to the end of row $i+1$.
- If there is a box of $\mu/\lambda$ in row $i+1$ of $\mu$, then $\gamma=\mu$ and the top edge is labeled $i+1$.
\(A) at (-1,-1) [1]{}; (B) at (1,-1) [2]{}; (C) at (-1,1) [2]{}; (D) at (1,1) [21]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
\(A) at (-1,-1) [2]{}; (B) at (1,-1) [3]{}; (C) at (-1,1) [31]{}; (D) at (1,1) [31]{}; (E) at (0,1.25)[2]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
If $(i,j)$ is on the main diagonal or the bottom edge is labeled $c$:
- If there is no box of $\mu/\lambda$ in column $j+1$ of $\mu$, then $\gamma$ is obtained from $\mu$ by adding a box to the end of column $j+1$. The top edge is labeled $c$.
- If there is a box of $\mu/\lambda$ in column $j+1$ of $\mu$, then $\gamma=\mu$. The top edge is labeled with $c$ and $j+1$.
\(A) at (-1,-1) [1]{}; (B) at (1,-1) [2]{}; (C) at (-1,1) [2]{}; (D) at (1,1) [3]{}; (E) at (0,1.25)[c]{}; (F) at (0,-.75)[c]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
\(A) at (-1,-1) [2]{}; (B) at (1,-1) [21]{}; (C) at (-1,1) [31]{}; (D) at (1,1) [31]{}; (E) at (0,1.25)[3c]{}; (F) at (0,-.75)[c]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
If $\lambda=\nu$ and the bottom edge is labeled with $i$ but not with $c$:
- If there is no box of $\mu/\lambda$ immediately to the right of or immediately below the box at the end of row $i$ of $\nu$, then $\gamma=\mu$ and the top edge is labeled by $i$.
- If there is a box of $\mu/\lambda$ directly below the box at the end of row $i$ of $\nu$, then $\gamma=\mu$ and the edge between them is labeled by $i+1$.
- If there is a box of $\mu/\lambda$ immediately to the right of the box at the end of row $i$ of $\nu$, say in position $(i,j+1)$, no box of $\mu/\lambda$ in row $i+1$, and the outer corner of row $i+1$ is not in column $j$, then $\gamma/\mu$ is one box in row $i+1$.
- If there is a box of $\mu/\lambda$ immediately to the right of the box at the end of row $i$ of $\nu$, say in position $(i,j+1)$, no box of $\mu/\lambda$ in row $i+1$, and the outer corner of row $i+1$ is in column $j+1$, then the outer corner of row $i+1$ is on the main diagonal. In this case, $\gamma=\mu$, and the top edge is labeled with $j+1$ and $c$.
\(A) at (-1,-1) [3]{}; (B) at (1,-1) [3]{}; (C) at (-1,1) [31]{}; (D) at (1,1) [31]{}; (E) at (0,1.25)[1]{}; (F) at (0,-.75)[1]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
\(A) at (-1,-1) [2]{}; (B) at (1,-1) [2]{}; (C) at (-1,1) [21]{}; (D) at (1,1) [21]{}; (E) at (0,1.25)[2]{}; (F) at (0,-.75)[1]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
\(A) at (-1,-1) [2]{}; (B) at (1,-1) [2]{}; (C) at (-1,1) [3]{}; (D) at (1,1) [31]{}; (F) at (0,-.75)[1]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
\(A) at (-1,-1) [1]{}; (B) at (1,-1) [1]{}; (C) at (-1,1) [2]{}; (D) at (1,1) [2]{}; (E) at (0,1.25)[2c]{}; (F) at (0,-.75)[1]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
If the $\lambda=\nu$ and the bottom edge is labeled with $j$ and with $c$:
- If there is no box of $\mu/\lambda$ immediately to the right of or immediately below the box at the end of column $j$ of $\nu$, then $\gamma=\mu$ and the edge between them is labeled by $j$ and $c$.
- If there is a box of $\mu/\lambda$ immediately to the right of the box at the end of column $j$ of $\nu$, then $\gamma=\mu$ and the top edge is labeled by $j+1$ and $c$.
- If there is a box of $\mu/\lambda$ directly below the box at the end of column $j$ of $\nu$ but no box of $\mu/\lambda$ in column $j+1$, then $\gamma/\mu$ is one box in column $j+1$. The top edge is labeled $c$.
\(A) at (-1,-1) [41]{}; (B) at (1,-1) [41]{}; (C) at (-1,1) [42]{}; (D) at (1,1) [42]{}; (E) at (0,1.25)[4c]{}; (F) at (0,-.75)[4c]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
\(A) at (-1,-1) [31]{}; (B) at (1,-1) [31]{}; (C) at (-1,1) [41]{}; (D) at (1,1) [41]{}; (E) at (0,1.25)[4c]{}; (F) at (0,-.75)[3c]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
\(A) at (-1,-1) [31]{}; (B) at (1,-1) [31]{}; (C) at (-1,1) [32]{}; (D) at (1,1) [42]{}; (F) at (0,-.75)[3c]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
We call the resulting array the *shifted Hecke growth diagram* of $w$. In our previous example with $w=4211232$, we would have the diagram below.
\(00) at (0,0) [$\varnothing$]{}; (10) at (1,0) [$\varnothing$]{}; (20) at (2,0) [$\varnothing$]{}; (30) at (3,0) [$\varnothing$]{}; (40) at (4,0) [$\varnothing$]{}; (50) at (5,0) [$\varnothing$]{}; (01) at (0,1) [$\varnothing$]{}; (.53.5) at (.5,3.5) [$X$]{}; (11) at (1,1) [$\varnothing$]{}; (21) at (2,1) [$\varnothing$]{}; (31) at (3,1) [$1$]{}; (41) at (4,1) [$1$]{}; (51) at (5,1) [$1$]{}; (02) at (0,2) [$\varnothing$]{}; (12) at (1,2) [$\varnothing$]{}; (22) at (2,2) [$1$]{}; (32) at (3,2) [$2$]{}; (1.51.5) at (1.5,1.5) [$X$]{}; (42) at (4,2) [$2$]{}; (52) at (5,2) [$2$]{}; (03) at (0,3) [$\varnothing$]{}; (13) at (1,3) [$\varnothing$]{}; (23) at (2,3) [$1$]{}; (33) at (3,3) [$2$]{}; (43) at (4,3) [$2$]{}; (53) at (5,3) [$2$]{}; (2.5.5) at (2.5,.5) [$X$]{}; (04) at (0,4) [$\varnothing$]{}; (14) at (1,4) [$1$]{}; (24) at (2,4) [$2$]{}; (34) at (3,4) [$3$]{}; (44) at (4,4) [$3$]{}; (54) at (5,4) [$31$]{}; (3.5.5) at (3.5,.5) [$X$]{}; (4.51.5) at (4.5,1.5) [$X$]{}; (5,52.5) at (5.5,2.5) [$X$]{}; (60) at (6,0) [$\varnothing$]{}; (61) at (6,1) [$1$]{}; (62) at (6,2) [$2$]{}; (63) at (6,3) [$3$]{}; (64) at (6,4) [$31$]{}; (70) at (7,0) [$\varnothing$]{}; (71) at (7,1) [$1$]{}; (72) at (7,2) [$2$]{}; (73) at (7,3) [$31$]{}; (74) at (7,4) [$32$]{}; (6.51.5) at (6.5,1.5) [$X$]{}; (3.51.15) at (3.5,1.15) [1]{}; (3.52.15) at (3.5,2.15) [2c]{}; (3.53.15) at (3.5,3.15) [2c]{}; (3.54.15) at (3.5,4.15) [3c]{}; (2.52.15) at (2.5,2.15) [c]{}; (2.53.15) at (2.5,3.15) [c]{}; (2.54.15) at (2.5,4.15) [c]{}; (1.54.15) at (1.5,4.15) [c]{}; (4.52.15) at (4.5,2.15) [1]{}; (4.53.15) at (4.5,3.15) [1]{}; (5.54.15) at (5.5,4.15) [2]{}; (6.52.15) at (6.5,2.15) [1]{}; (6.54.15) at (6.5,4.15) [c]{};
(00)–(10)–(20)–(30)–(40)–(50)–(60)–(70) (01)–(11)–(21)–(31)–(41)–(51)–(61)–(71) (02)–(12)–(22)–(32)–(42)–(52)–(62)–(72) (03)–(13)–(23)–(33)–(43)–(53)–(63)–(73) (04)–(14)–(24)–(34)–(44)–(54)–(64)–(74) (00)–(01)–(02)–(03)–(04) (10)–(11)–(12)–(13)–(14) (20)–(21)–(22)–(23)–(24) (30)–(31)–(32)–(33)–(34) (40)–(41)–(42)–(43)–(44) (50)–(51)–(52)–(53)–(54) (60)–(61)–(62)–(63)–(64) (70)–(71)–(72)–(73)–(74);
Let $\mu_0 = \varnothing \subseteq \mu_1 \subseteq \ldots \mu_k$ be the seqence of shifted partitions across the top of the growth diagram, and let $\nu_0=\varnothing \subseteq \nu_1 \subseteq \ldots \nu_n$ be the sequence of shifted partitions on the right side of the growth diagram. These sequences correspond to increasing shifted tableaux $Q(w)$ and $P(w)$, respectively. If the edge between $\mu_i$ and $\mu_{i+1}$ is labeled $j$, then $\mu_i=\mu_{i+1}$, and the label $i+1$ of $Q(w_1\cdots w_{i+1})$ is placed in the box at the end of row $j$ of $Q(w_1\cdots w_i)$. If the edge between $\mu_i$ and $\mu_{i+1}$ is labeled $jc$, then $\mu_i=\mu_{i+1}$, and the label $i+1'$ of $Q(w_1\cdots w_{i+1})$ is placed in the box at the end of column $j$ of $Q(w_1\cdots w_i)$. For the shifted Hecke growth diagram above, we have:
$P(w)=$
1 & 2 & 3\
& 3 & 4
$Q(w)=$
1 & 2’ & 3’4’\
& 56 & 7’
.
We see that this $P$ and $Q$ agree with $P_S(w)$ and $Q_S(w)$, which is not a coincidence.
For any word $w$, the increasing shifted tableaux $P$ and $Q$ obtained from the sequence of shifted partitions across the right side of the shifted Hecke growth diagram for $w$ and across the top of the shifted Hecke growth diagram for $w$, respectively, are $P_S(w)$ and $Q_S(w)$, the shifted Hecke insertion and recording tableau for $w$.
Suppose the square we are examining is in row $s$ and column $t$. We argue by induction. We will use the same two lemmas from the Hecke growth diagram proof, which have analogous proofs.
Oriented as in the square above, $|\nu/\lambda|\leq 1$.
Oriented as in the square above, $\mu/\lambda$ is a rook strip, that is, no two boxes in $\mu/\lambda$ are in the same row or column.
(1-2) Note that if the square contains an $X$, then $\lambda=\nu$ because there are no other $X$’s in column $t$. If $\lambda_1=\mu_1$, then there are no $s$’s in the first row of insertion tableau $\mu$. Thus the inserted $s$ will be added to the end of the first row, creating $\gamma$. If $\lambda_1=\mu_1$, then there is already an $s$ at the end ofthe first row of insertion tableau $\mu$. The inserted $s$ cannot be added to the first row, so the $\mu=\gamma$. In this case, the insertion terminated at the end of the first row. We know this is an inner corner because $s$ is the largest number inserted so far, so there can not be anything directly below the box labeled $s$ in the first row.
\(3) If $\mu=\lambda$, there is no $X$ to the left of square $(s,t)$ in row $s$. Thus nothing changes between $\nu$ and $\gamma$ as we consider occurences of entry $s$. If $\lambda=\nu$, there is no $X$ below square $(s,t)$ in column $t$ and so no new insertion.Thus nothing changes between $\mu$ and $\gamma$.
\(4) Suppose $\nu/\lambda$ is one box in row $n$ and $\mu/\lambda$ contains boxes in rows exactly $\{j_1,j_2,\ldots,j_k\}$. Suppose the box in $\nu/\lambda$ corresponds to inserting $r<s$. Then $\nu \nsubseteq \mu$ implies that $n\notin \{j_1,j_2,\ldots,j_k\}$. Since $\nu/\lambda$ is one box, the last action in the insertion sequence of $r$ into the insertion tableau of shape $\lambda$ is a box being inserted in row $n$. Since there is no $s$ in row $n$, the bumping sequence when inserting $r$ into the insertion tableau of shape $\mu$ will not disturb the $s$’s. The edge between $\lambda$ and $\nu$ is either labeled $c$ or not, corresponding to whether or not inserting the number in column $t$ involves column insertions. This remains unchanged.
(5-6) Since $\nu/\lambda$ is one box in position $(i,j)$, there is some $X$ in position $(r,t)$ for $r<s$. It follows that inserting $r$ into the insertion tableau $\mu$ will result in bumping the $s$ in position $(i,j)$ of $\mu$. If there is no box in row $i+1$ of $\mu/\lambda$, there is not an $s$ in row $i+1$ of insertion tableau $\mu$. Thus the $s$ in position $(i,j)$ is bumped from row $i$ by $r$ and added to the end of row $i+1$. If there is a box in row $i+1$ of $\mu/\lambda$, there is already an $s$ in row $i+1$ of insertion tableau $\mu$. It follows that the $s$ bumped from row $i$ cannot be added to row $i+1$, and so $\gamma=\mu$. The insertion terminates at the end of row $i+1$ since there cannot be any boxes directly below the $s$ in row $i+1$.
(7-8) If $(i,j)$ is on the main diagonal or the edge between $\lambda$ and $\nu$ is labeled $c$, any bumped entries will be column inserted into the column $j$. The rules are the transpose of rules 5 and 6, and the new edge is labeled $c$ since all subsequent bumping will be column inserted after the first column insertion.
\(9) Since there is no box directly to the right of $(i,j)$, nothing is bumped when inserting $r<s$ into insertion tableau $\mu$. Since there is no box directly below $(i,j)$, the insertion terminates at inner corner $(i,j)$.
\(10) In the situation described above, if there is a box directly below $(i,j)$, the insertion now terminates at the inner corner in row $i+1$.
\(11) When inserting $r$ into $\mu$, the $s$ in position $(i,j+1)$ is bumped but not replaced and inserted into row $i+1$. Since there is no $s$ in row $i+1$, this $s$ can be added to the row as long as it is not directly below the $s$ in position $(i,j)$.
\(12) When inserting $r$ into $\mu$, the $s$ in position $(i,j+1)$ is bumped but not replaced and inserted into row $i+1$. Since there is no $s$ in row $i+1$, we attempt to add this $s$ to the end of row $i+1$. However, this $s$ would end directly below the $s$ in row $i$, and so cannot be added. If the outer corner of row $i+1$ is not on the main diagonal, the insertion terminates at the end of row $i+1$. If it is on the diagonal, the insertion terminates at the end of row $i$, and we know the last box in row $i$ must be in column $j$ since there are no entries larger than $s$. This edge is labeled with $c$ since this action may bump an entry from the main diagonal in column $j$ in future steps.
(13-15) Rules 13-15 are the tranpose of rules 10-12, and may be explained in an analogous way.
We can also formulate the rules for the reverse insertion, as follows. The proof is omitted for brevity.
[**[Reverse Rules]{}**]{}
- If $\gamma/\mu$ is one box in the first row, then the square has a $X$ and $\lambda=\nu$.
- If $\gamma=\mu$ and the edge between them is labeled 1, then the square has an $X$ and $\lambda=\nu$.
- If $\gamma=\mu$ with no edge label, then $\lambda=\mu$. If $\gamma=\nu$, then $\lambda=\mu$ and the edge label between $\lambda$ and $\nu$ is the same as the label between $\gamma$ and $\mu$.
- If $\nu\not\subseteq\mu$, then $\lambda=\mu \cap \nu$.
- If $\gamma/\mu$ is one box in row $i+1$ for some $i>0$, the edge between $\gamma$ and $\mu$ has no label, and $\mu/\nu$ has no box in row $i$, then $\nu/\lambda$ is one box in row $i$.
- If $\gamma/\mu$ is one box in row $i+1$ for some $i>0$, the edge between $\gamma$ and $\mu$ has no label, and $\mu/\nu$ has a box in row $i$, then $\lambda=\nu$ and the edge between them is labeled $i$.
- If $\gamma/\mu$ is one box in column $j+1$ for some $j>0$, the edge between $\gamma$ and $\mu$ is labeled $c$, and $\mu/\nu$ has no box in column $j$, then $\nu/\lambda$ is one box in column $j$ and the edge between $\nu$ and $\lambda$ is labeled $c$.
- If $\gamma/\mu$ is one box in column $j+1$ for some $j>0$, the edge between $\gamma$ and $\mu$ is labeled $c$, and $\mu/\nu$ has a box in column $j$, then $\nu=\lambda$ and the edge between them is labeled $j$ and $c$.
- If $\gamma=\mu$, the edge between them is labeled $i+1$ for some $i>0$, $\mu/\nu$ has a box in row $i+1$, and $\mu_i\neq \mu_{i+1}+1$, then $\nu/\lambda$ is one box in row $i$.
- If $\gamma=\mu$, the edge between them is labeled $i+1$ for some $i>0$, $\mu/\nu$ has no box in row $i+1$, then $\lambda=\nu$ and the edge between them is labeled $i+1$.
- If $\gamma=\mu$, the edge between them is labeled $i+1$ for some $i>0$, $\mu/\nu$ has a box in row $i+1$, and $\mu_i=\mu_{i+1}+1$, then $\lambda=\nu$ and the edge between them is labeled $i$.
- If $\gamma=\mu$, the edge between them is labeled $j+1$ and $c$ for some $j>0$, $\mu/\nu$ has a box in column $j+1$, and the $j^{th}$ and $j+1^{th}$ columns of $\mu$ are not the same size, then $\nu/\lambda$ is one box in column $j$.
- If $\gamma=\mu$, the edge between them is labeled $j+1$ and $c$ for some $j>0$, and the bottom box of column $j+1$ is in the last row of $\mu$, then $\lambda=\nu$ and the edge between them is labeled with the bottom row of $\nu$.
- If $\gamma=\mu$, the edge between them is labeled $j+1$ and $c$ for some $j>0$, $\mu/\mu$ has no box in column $j+1$, then $\lambda=\nu$ and the edge between them is labeled $j+1$ and $c$.
- If $\gamma=\mu$, the edge between them is labeled $j+1$ and $c$ for some $j>0$, $\mu/\mu$ has a box in column $j+1$, and and the $j^{th}$ and $j+1^{th}$ columns of $\mu$ are the same size, then $\lambda=\nu$ and the edge between them is labeled $j$ and $c$.
Major examples: Young-Fibonacci lattice {#sec:YF}
=======================================
Möbius deformation of the Young-Fibonacci lattice
-------------------------------------------------
In order to describe the Möbius deformation of the Young-Fibonacci lattice, we first need to determine its Möbius function.
Let $X>Y$ be elements in the Young-Fibonacci lattice and suppose $X$ has $n$ edges below it. Then $$\mu(Y,X)=\left\{ \begin{array}{ll}
n-1 & \text{if }2Y=X \\
-1 & \text{if $X$ covers $Y$} \\
0 & \text{otherwise}
\end{array}
\right.$$
Suppose $X>Y$. If $\rho(X)=\rho(Y)+1$, then clearly, $\mu(Y,X)=-1$.
If $\rho(X)=\rho(Y)+2$ and $X=2Y$, then $\mu(Y,X)=n-1$ since $2Y$ covers exactly the elements that cover $Y$. If $X\neq 2Y$, then it can be checked that $X$ covers exactly one element that covers $Y$. Therefore $\mu(Y,X)=0$.
Assume now $\rho(X) \geq \rho(Y)+2$. We will argue that $\mu(Y,X)=0$ by induction on $\rho(X)$. The base case $\rho(X)=\rho(Y)+2$ was just proved. Let us argue the step of induction. There are two cases.
- We have $2Y < X$. In this case, consider $\displaystyle\sum_{Y\leq W < X} \mu(Y,W)$, we argue it is $0$. Indeed, $\displaystyle\sum_{Y\leq W \leq 2Y} \mu(Y,W) =0$. For the $2Y \not \leq W < X$ in the interval, we use induction assumption to conclude $\mu(Y,W)=0$.
- We have $2Y \not < X$. We claim there is only one $Z$ which covers $Y$ such that $Z < X$. Indeed, since Young-Fibonacci lattice is a lattice, and since join of any two distinct such $Z$’s is $2Y$, we would have a contradiction. Then $\mu(Y,Y)+\mu(Y,Z)=1 + (-1)=0$. For the rest of $Y < W < X$ we have $\mu(Y,W)=0$ by induction assumption.
By the previous result, for each element $X$ in the Young-Fibonacci lattice, the Möbius deformation is formed by adding an upward-oriented loop for each of the $n$ elements $X$ covers and adding $n-1$ downward-oriented edges from $X$ to $Y$ if $X=2Y$. For example, there are 2 edges from 221 to 21. The first few ranks are shown below.
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,1)
$ $
; (11) at (1,2)
$ $ & $ $
; (2) at (-1,2)
$ $\
$ $
; (12) at (-2,3)
& $ $\
$ $ &
; (21) at (0,3)
$ $ &\
$ $ & $ $
; (111) at (2,3)
$ $ & &
; (112) at (-3,4)
& & $ $\
$ $ & &
; (22) at (-1,4)
$ $ &\
$ $ &
; (121) at (0,4)
& $ $ &\
$ $ & &
; (211) at (1,4)
$ $ & &\
$ $ & &
; (1111) at (3,4)
$ $ & & &
; (empty) – (1) –(11)–(111)–(1111) (1)–(2)–(12)–(112) (11)–(21)–(22) (1) to \[out=60, in=100, distance=.5cm\] (1) (2) to \[out=60, in=100, distance=.5cm\] (2) (11) to \[out=60, in=100, distance=.5cm\] (11) (12) to \[out=60, in=100, distance=.5cm\] (12) (21) to \[out=-60, in=-100, distance=.5cm\] (21) (21) to \[out=-60, in=-100, distance=.7cm\] (21) (111) to \[out=60, in=100, distance=.5cm\] (111) (112) to \[out=60, in=100, distance=.5cm\] (112) (22) to \[out=60, in=100, distance=.5cm\] (22) (22) to \[out=60, in=100, distance=.7cm\] (22) (121) to \[out=60, in=100, distance=.5cm\] (121) (211) to \[out=60, in=100, distance=.5cm\] (211) (211) to \[out=60, in=100, distance=.7cm\] (211) (1111) to \[out=60, in=100, distance=.5cm\] (1111) (2)–(21)–(211) (111)–(211) (21)–(121) (12)–(22);
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,1)
$ $
; (11) at (1,2)
$ $ & $ $
; (2) at (-1,2)
$ $\
$ $
; (12) at (-2,3)
& $ $\
$ $ &
; (21) at (0,3)
$ $ &\
$ $ & $ $
; (111) at (2,3)
$ $ & &
; (112) at (-3,4)
& & $ $\
$ $ & &
; (22) at (-1,4)
$ $ &\
$ $ &
; (121) at (0,4)
& $ $ &\
$ $ & &
; (211) at (1,4)
$ $ & &\
$ $ & &
; (1111) at (3,4)
$ $ & & &
; (empty) – (1) –(11)–(111)–(1111) (1)–(2)–(12)–(112) (11)–(21)–(22) (2)–(21)–(211) (111)–(211) (21)–(121) (12)–(22) (21)–(1) (22)–(2) (211)–(11);
The Möbius deformation of the Young-Fibonacci lattice forms a dual graded graph with $$(DU-UD)=D+I.$$
Suppose that shape $X$ covers shape $Y$, and first assume that $\rho(Y)=\rho(X)-1$. The up-down paths from $X$ to $Y$ that consist of a loop at $X$ followed by following the edge from $X$ to $Y$ can be counted by the number of loops at $X$, i.e. the number of edges below $X$ in the Young-Fibonacci lattice. In the second type of up-down path, we start at $X$, move up to a distinct shape $W$, and then move down to $Y$. The only $W$ with $\rho(W)=\rho(Y)+2$ that cover $Y$ is the $W$ defined by $W=2Y$. This $W$ covers $X$ by definition of the Young-Fibonacci lattice. To count the number of paths through $W$, we need to count the number of edges from $W$ to $Y$. This number is the same as one fewer than the number edges below $W$ in $\mathbb{YF}$, which is the same as one fewer than the number of edges above $Y$ in $\mathbb{YF}$.
To count down-up paths, we first count the paths that consist of the edge from $X$ to $Y$ followed by a loop at $Y$. These can be counted by the number of loops at $Y$ or, in other words, the one fewer than the number of edges above $Y$ in $\mathbb{YF}$. The second type of down-up paths involve traveling from $X$ down to some $Z$ with $\rho(Z)+2=\rho(X)$ and then going up to $Y$. There is a unique such $Z$ defined by $2Z=X$, and $Y$ covers this $Z$. We thus need to count the number of ways to get from $X$ to $Z$, which is the same as one fewer than the number of edges below $X$ in $\mathbb{YF}$.
Putting the previous two arguments together, $[Y](DU-UD)(X)=1$ in this case, as desired.
Now suppose $\rho(Y)=\rho(X)-2$. The only up-down paths from $X$ to $Y$ consist of a loop at $X$ followed by an edge from $X$ to $Y$. There are $$(\#\{\text{edges below $X$ in $\mathbb{YF}$}\})(\#\{\text{edges below $X$ in $\mathbb{YF}$}\}-1)$$ such paths.
To count down-up paths, we must count the number of ways to choose an edge from $X$ to $Y$ and then choose a loop of $Y$. There are $$(\#\{\text{edges below $X$ in $\mathbb{YF}$}\}-1)(\#\{\text{edges above $Y$ in $\mathbb{YF}$}\}-1)$$ such paths. Since $$\#\{\text{edges down from $X$ in $\mathbb{YF}$}\}=\#\{\text{edges above $Y$ in $\mathbb{YF}$}\},$$ we have that $$[Y](DU-UD)(X)=\#\{\text{edges below $X$ in $\mathbb{YF}$}\}-1=[Y]D(X).$$
K-Young-Fibonacci insertion {#sec:KYFinsertion}
---------------------------
As with the previous examples of Möbius construction, we describe an insertion procedure that corresponds to the Möbius deformation of the Young-Fibonacci lattice, which is an analogue of the Young-Fibonacci insertion of Fomin. We recommend [@F] for details about Young-Fibonacci insertion. In this analogue, any walk upward from $\varnothing$ will correspond to a recording tableau and any walk downward to $\varnothing$ will correspond to an insertion tableau. We begin by describing the insertion tableaux in this setting.
\[def:KYFtableau\] A *K-Young-Fibonacci (KYF) tableau* is a filling of a snakeshape with positive integers such that
- for any pair
B\
A
the inequality $A\leq B$ holds;
- to the right of any
B\
A
, there are no numbers from the interval $[A,B]$ in either the upper or lower rows;
- if the position above
A
is not occupied yet, then to the right of
A
there are no numbers greater than or equal to $A$ in either the lower or upper rows;
- any pair
A\
A
must come before any single box
B
and must not be in rightmost column of the snakeshape. In addition, $A$ may not be the smallest entry in the snakeshape.
The three snakeshapes below are valid KYF tableaux.
& 3 & 4 & & 7\
9 & 2 & 1 & 8 & 6 & 5
2 & 3 & & & 5\
2 & 1 & 7 & 6 & 4
4 & 5 & & & 2\
4 & 5 & 6 & 3 & 1
The snakeshapes below are not valid KYF tableaux. The first violates condition (iii), the second violates condition (ii), and the third. fourth, and fifth violate condition (iv).
& 3 & 4\
4 & 2 & 1
3 & 4 &\
2 & 1 & 4
4 & 6 & & & 2\
4 & 5 & 7 & 3 & 2
3 & 2\
1 & 2
1 & 3\
1 & 2
Let $\tau$ be a valid KYF tableau. We show how to insert positive integer $x$ into $\tau$ to obtain a new valid KYF tableau that may or may not be different than $\tau$. Notice that this insertion procedure is different than Hecke insertion and shifted Hecke insertion in that the algorithm does not proceed row-by-row or column-by-column. As before, we designate a specific box of the resulting tableau, $\tau'$, as being the box where the insertion terminated. We will later use this notion to define recording tableaux.
The rules for inserting any positive integer $x$ into $\tau$ are as follows:
1. If $x$ is equal to the smallest entry in $\tau$, then $\tau=\tau'$. In this case, we say the insertion terminated in the top cell of the first column of $\tau=\tau'$. If this is not the case, continue to step 2.
Inserting $1$ into the tableau on the left or $2$ into the tableau on the right will not change the tableaux.
& 2 &\
4 & 1 & 3
3 & &\
2 & 5 & 4
2. Attach a new box
x
just to the left of $\tau$ in the lower row.
3. Find all the entries of $\tau$ that are greater than or equal to $x$ and sort them: $$x\leq a_1 \leq a_2 \leq \ldots \leq a_k.$$ If in $\tau$ we have
A\
A
, then we consider the $A$ in the upper row to be the larger of the two.
4. If $a_i=a_{i+1}$, then replace $a_{i+1}$ with $*$.
5. Now put a box
just above
x
and move the $a_i$ chainwise according to the following rule: $a_1$ moves into the new box, $a_2$ moves to $a_1$’s original position, $a_3$ moves to $a_2$’s original position, etc. The box that was occupied by $a_k$ disappears. If it was located in the lower row, then the left and right parts of the snake are concatenated.
6. If there is a box with $*$ and no box directly above it, delete this box and concatenate the left and right parts of the snake. If this happens, we say the insertion terminated at the box in the upper row of the column directly to the right of this concatenation. Otherwise, a box was added in the insertion process, and we say the insertion terminated at this new box.
7. Replace any pair
A\
$*$
with
A\
A
.
Let’s insert 3 into $\tau=$
3 & 4 &\
2 & 4 & 1
. We first attach
3
as shown below.
& 3 & 4 &\
3 & 2 & 4 & 1
Next, we locate and order $$3\leq (a_1=3) \leq (a_2=4) \leq (a_3 =4),$$ and we replace $a_3$ with $*$. After shifting the boxes as in YF5, we have
3 & 4 & &\
3 & 2 & $*$ & 1
.
We then delete the box with $*$ to obtain the final KYF tableau below.
3 & 4 &\
3 & 2 & 1
.
This insertion terminates at the box at the top of the third column.
Notice also that inserting 1 into $\tau$ does not change the tableau, and this insertion terminates at the box at the top of the first column.
We insert 2 into KYF tableau
& & 2 &\
4 & 3 & 2 & 1
. We first attach a box containing 2 to the left of the original tableau. We then locate and order $$2 \leq (a_1=2) \leq (a_2=2)\leq (a_3=3) \leq (a_4=4).$$ We replace $a_2$ with $*$ and shift the boxes to obtain
2 & & 3 &\
2 & 4 & $*$ & 1
.
After performing the last step of the insertion procedure, we end with the tableau shown below.
2 & & 3 &\
2 & 4 & 3 & 1
This insertion terminated at the box at the top of the first column of the resulting tableau.
As usual, for a word $w=w_1w_2\ldots w_n$, we define $P_{YF}(w)$ by setting $$P_{YF}(w_1\ldots w_k)=P_{YF}(w_1\ldots w_{k-1}) \leftarrow w_k.$$
\[ex:1334241\] The sequence of KYF tableaux below shows the intermediate tableaux obtained in computing $P_{YF}(1334241)$, which is shown on the right.
\
1
&\
3 & 1
3 &\
3 & 1
& 3 &\
4 & 3 & 1
3 & 4 &\
2 & 4 & 1
4 & 3 &\
4 & 2 & 1
4 & 3 &\
4 & 2 & 1
We need the next definition to define the recording tableaux for KYF insertion.
\[def:KYFsetvalued\] A *standard set-valued KYF tableau* is a snakeshape whose boxes are filled with finite, nonempty subsets of positive integers that satisfy the following conditions, where $\bar{A},\bar{B},\bar{C}$ are subsets, $\bar{A}< \bar{B}$ when max$(\bar{A})<$min$(\bar{B})$, and the letters $[n]$ are used exactly once for some $n$:
- for any pair
|[B]{}\
|[A]{}
the inequality $\bar{A}< \bar{B}$ holds;
- to the right of any
|[B]{}\
|[A]{}
, there are no numbers from the interval $[\text{max}(\bar{A}),\text{min}(\bar{B})]$ in either the upper or lower rows;
- if the position above
|[A]{}
is not occupied yet, then to the right of
|[A]{}
there are no entries greater than min($\bar{A}$) in either the lower or upper rows;
A recording tableau for a word $w=w_1w_2\ldots w_n$ is a standard set-valued KYF tableau and is obtained as follows. Begin with $Q_{YF}(\varnothing)=\varnothing$. If the insertion of $w_k$ into $P_{YF}(w_1\cdots w_{k-1})$ resulted in adding a new box to $P_{YF}(w_1\cdots w_{k-1})$, add this same box with label $k$ to $Q_{YF}(w_1\ldots w_{k-1})$ to obtain $Q_{YF}(w_1\ldots w_k)$.
If the insertion of $w_k$ into $P_{YF}(w_1\cdots w_{k-1})$ did not change the shape of $P_{YF}(w_1\cdots w_{k-1})$, obtain $Q_{YF}(w_1\ldots w_k)$ from $Q_{YF}(w_1\ldots w_{k-1})$ by adding the label $k$ to the box where the insertion terminated.
In Example \[ex:1334241\], we computed $P_{YF}(1334241)$. We repeat this computation on the top row and show the corresponding steps of building $Q_{YF}(1334241)$ on the bottom row.
\
1
&\
3 & 1
3 &\
3 & 1
& 3 &\
4 & 3 & 1
3 & 4 &\
2 & 4 & 1
4 & 3 &\
4 & 2 & 1
4 & 3 &\
4 & 2 & 1
\
1
&\
2 & 1
3 &\
2 & 1
& 3 &\
4 & 2 & 1
5 & 3 &\
4 & 2 & 1
5 & 3 &\
4 & 2 & 16
57 & 3 &\
4 & 2 & 16
We next define a reverse insertion procedure so that given a pair $(P_{YF}(w),Q_{YF}(w))$, we can recover $w=w_1\ldots w_n$.
First locate the box containing the largest label of $Q_{YF}(w)$, call the label $n$ and its column $c$, and find the corresponding box in $P_{YF}(w)$. Let $x$ denote the label in the leftmost box of the bottom row of $P_{YF}(w)$.
- If the label $n$ of $Q$ was not the only label in its box and was located in the upper row of the first column, then $P_{YF}(w)=P_{YF}(w_1\ldots w_{n-1}) \leftarrow s$, where $s$ is the smallest entry in $P_{YF}(w_1\ldots w_{n-1})$. Finally, $Q_{YF}(w_1\ldots w_{n-1})$ is obtained from $Q_{YF}(w)$ by removing the label $n$.
In all remaining cases, $P_{YF}(w)=P_{YF}(w_1\ldots w_{n-1}) \leftarrow x$, and we describe how to constuct $P_{YF}(w_1\ldots w_{n-1})$. In each case, $Q_{YF}(w_1\ldots w_{n-1})$ is obtained from $Q_{YF}(w)$ by removing the label $n$. If $n$ is the only label in its box, the box is removed.
1. If the label $n$ of $Q$ was the only label in its box:
1. Delete the leftmost square in the bottom row.
2. Let $k$ denote the largest entry in $P_{YF}(w)$, and sort the entries of $P_{YF}(w)$ as shown below: $$x \leq b_1 \leq b_2\leq\ldots\leq b_t=k.$$ If we have
A\
A
, then we consider the $A$ in the upper row to be the larger of the two.
3. If $b_i=b_{i+1}$, replace $b_{i}$ with $*$.
4. Move the $b_i$ chainwise according the following rule: $b_1$ moves into $b_2$’s position, $b_2$ moves into $b_3$’s position, etc until $b_{t-1}$ moves into $b_t$’s position. The box that was occupied by $b_1$ disappears.
5. Place $b_t$ in the position determined by the shape of $Q_{S,n-1}$.
6. Replace any pair
$*$\
A
with
A\
A
.
2. If the label $n$ of $Q$ was not the only label in its box and the largest entry in $P_{YF}(w)$ is $k$:
1. Add the column
\
k
directly to the left of column $c$ to $P_{YF}(w)$.
2. Sort the entries of $P_{YF}(w)$ as shown below: $$k=b_1\geq b_2 \geq b_2 \ldots \geq x=b_t.$$ If we have
A\
A
, then we consider the $A$ in the upper row to be the larger of the two.
3. If $b_i=b_{i+1}$, replace $b_{i+1}$ with $*$.
4. Move the $b_i$ chainwise according to the following rule: $b_1$ moves into the new box
, $b_2$ moves to $b_1$’s original position, $b_3$ moves to $b_2$’s original position, etc. The box that was occupied by $b_t=x$ disappears.
5. Replace any pair
$*$\
A
with
A\
A
.
The steps above clearly reverse the KYF insertion steps, giving us the result below.
KYF insertion and reverse KYF insertion define mutually inverse bijections between the set of words on $\mathbb{N}$ and the set of pairs $(P_{YF},Q_{YF})$ consisting of a KYF tableau and a set-valued KYF tableau of the same shape.
Omitted for brevity.
KYF growth and decay {#sec:KYFgrowth}
--------------------
As before, given any word $w=w_1w_2\cdots w_k$, containing $n \leq k$ distinct numbers, we can create an $n \times k$ array with an $X$ in the $w_i^{th}$ square from the bottom of column $i$. Note that there can be multiple $X$’s in the same row but is at most one $X$ per column.
We will label the corners of each square with a snakeshape, label some of the horizontal edges of the squares with a specific inner corner by writing the column of the inner corner, and label some vertical edges corresponding to where a 2 was added to the bottom shape to obtain the top shape. For example, if the corner at the bottom of the veritcal edge is labeled with 221, the corner at the top is labeled 2221, and the vertical edge has label 3, this means that the third column of 2’s in 2221 is the column that was added when going from 221 to 2221. We begin by labeling all corners in the bottom row and left side of the diagram with the empty shape, $\varnothing$.
To complete the labeling of the corners, suppose the corners $\mu$, $\lambda$, and $\nu$ are labeled, where $\mu$, $\lambda$, and $\nu$ are as in the picture below. We label $\gamma$ according to the following rules.
\(A) at (-1,-1) [$\lambda$]{}; (B) at (1,-1) [$\nu$]{}; (C) at (-1,1) [$\mu$]{}; (D) at (1,1) [$\gamma$]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
**If the square contains an X:**
- If $\lambda=\nu=\varnothing$, then $\gamma=\mu=1$ and the top edge is labeled 1.
- If $\lambda=\mu=\nu$, then $\gamma=1\lambda$.
- If $\mu=2\lambda$ and the left edge is labeled $i$, then $\gamma=2\lambda$, the top edge is labeled $i+1$, and the right edge is labeled 1.
- If $\mu\neq\lambda$, $\mu\neq 2\lambda$, and $\mu\neq1$, then $\gamma=2\lambda$ and the right edge is labeled 1.
\(A) at (-1,-1) [$\varnothing$]{}; (B) at (1,-1) [$\varnothing$]{}; (C) at (-1,1) [1]{}; (D) at (1,1) [1]{}; (E) at (0,0) [$X$]{}; (F) at (0,1.25)[1]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
\(A) at (-1,-1) [$\lambda$]{}; (B) at (1,-1) [$\lambda$]{}; (C) at (-1,1) [$\lambda$]{}; (D) at (1,1) [1$\lambda$]{}; (E) at (0,0) [$X$]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
\(A) at (-1,-1) [$\lambda$]{}; (B) at (1,-1) [$\lambda$]{}; (C) at (-1,1) [$2\lambda$]{}; (D) at (1,1) [2$\lambda$]{}; (E) at (0,0) [$X$]{}; (F) at (0,1.25) [$i+1$]{}; (E) at (-1.25,0) [$i$]{}; (G) at (1.25,0) [$1$]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
\(A) at (-1,-1) [$\lambda$]{}; (B) at (1,-1) [$\lambda$]{}; (C) at (-1,1) [$\mu$]{}; (D) at (1,1) [2$\lambda$]{}; (E) at (0,0) [$X$]{}; (G) at (1.25,0) [$1$]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
**If the square does not contain an X and $\mu=\lambda$ or $\nu=\lambda$ with no edge label between $\lambda$ and $\nu$:**
- If $\mu=\lambda$, then set $\gamma = \nu$ and label the top edge with the same label as the bottom edge if one exists. If $\nu=\lambda$, then $\gamma = \mu$ with the right edge labeled with the same label as the left edge if such a label exists.
\(A) at (-1,-1) [$\lambda$]{}; (B) at (1,-1) [$\nu$]{}; (C) at (-1,1) [$\lambda$]{}; (D) at (1,1) [$\nu$]{}; (E) at (0,1.25) [i]{}; (F) at (0,-.75)[i]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
\(A) at (-1,-1) [$\lambda$]{}; (B) at (1,-1) [$\lambda$]{}; (C) at (-1,1) [$\mu$]{}; (D) at (1,1) [$\mu$]{}; (E) at (-1.25,0) [$i$]{}; (G) at (1.25,0) [$i$]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
**If $\nu=\lambda$ and the bottom edge is labeled 1:**
- If $\nu=\lambda$ and the bottom edge is labeled 1, then $\gamma=\mu$, the top edge is labeled 1, and the label on the left edge (if one exists) is the same as the label on the right edge.
\(A) at (-1,-1) [$\lambda$]{}; (B) at (1,-1) [$\lambda$]{}; (C) at (-1,1) [$\mu$]{}; (D) at (1,1) [$\mu$]{}; (E) at (0,-.75) [$1$]{}; (G) at (0,1.25) [$1$]{}; (H) at (-1.25,0) [$j$]{}; (I) at (1.25,0) [$j$]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
**If no previous cases apply, then set $\gamma=2\lambda$ and follow the rule below:**
- If the bottom edge is labled $i$, then label the right edge $i$. If the left edge is labled $j$, then label the top edge $j+1$.
\(A) at (-1,-1) [$\lambda$]{}; (B) at (1,-1) [$\nu$]{}; (C) at (-1,1) [$\mu$]{}; (D) at (1,1) [$2\lambda$]{}; (E) at (0,-.75) [$i$]{}; (G) at (1.25,0) [$i$]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
\(A) at (-1,-1) [$\lambda$]{}; (B) at (1,-1) [$\nu$]{}; (C) at (-1,1) [$\mu$]{}; (D) at (1,1) [$2\lambda$]{}; (E) at (-1.25,0) [$j$]{}; (G) at (0,1.25) [$j+1$]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
\(A) at (-1,-1) [$\lambda$]{}; (B) at (1,-1) [$\nu$]{}; (C) at (-1,1) [$\mu$]{}; (D) at (1,1) [$2\lambda$]{}; (F) at (0,-.75) [$i$]{}; (H) at (1.25,0) [$i$]{}; (E) at (-1.25,0) [$j$]{}; (G) at (0,1.25) [$j+1$]{}; (A) – (B) (C) – (D) (B) – (D) (A) – (C);
We call the resulting array the KYF growth diagram of $w$. For example, continuing with the word from Example \[ex:1334241\], $w=1334241$, we would have the diagram below.
Below is the KYF growth diagram for the word 1334241.
\(00) at (0,0) [$\varnothing$]{}; (10) at (1,0) [$\varnothing$]{}; (20) at (2,0) [$\varnothing$]{}; (30) at (3,0) [$\varnothing$]{}; (40) at (4,0) [$\varnothing$]{}; (50) at (5,0) [$\varnothing$]{}; (01) at (0,1) [$\varnothing$]{}; (.5.5) at (.5,.5) [$X$]{}; (11) at (1,1) [$1$]{}; (21) at (2,1) [$1$]{}; (31) at (3,1) [$1$]{}; (41) at (4,1) [$1$]{}; (51) at (5,1) [$1$]{}; (02) at (0,2) [$\varnothing$]{}; (12) at (1,2) [$1$]{}; (22) at (2,2) [$1$]{}; (32) at (3,2) [$1$]{}; (1.52.5) at (1.5,2.5) [$X$]{}; (42) at (4,2) [$1$]{}; (52) at (5,2) [$11$]{}; (03) at (0,3) [$\varnothing$]{}; (13) at (1,3) [$1$]{}; (23) at (2,3) [$11$]{}; (33) at (3,3) [$21$]{}; (43) at (4,3) [$21$]{}; (53) at (5,3) [$21$]{}; (2.52.5) at (2.5,2.5) [$X$]{}; (04) at (0,4) [$\varnothing$]{}; (14) at (1,4) [$1$]{}; (24) at (2,4) [$11$]{}; (34) at (3,4) [$21$]{}; (44) at (4,4) [$121$]{}; (54) at (5,4) [$221$]{}; (3.53.5) at (3.5,3.5) [$X$]{}; (4.51.5) at (4.5,1.5) [$X$]{}; (5,53.5) at (5.5,3.5) [$X$]{}; (60) at (6,0) [$\varnothing$]{}; (61) at (6,1) [$1$]{}; (62) at (6,2) [$11$]{}; (63) at (6,3) [$21$]{}; (64) at (6,4) [$221$]{}; (70) at (7,0) [$\varnothing$]{}; (71) at (7,1) [$1$]{}; (72) at (7,2) [$11$]{}; (73) at (7,3) [$21$]{}; (74) at (7,4) [$221$]{}; (6.5.5) at (6.5,.5) [$X$]{}; (6.51.15) at (6.5,1.15) [1]{}; (6.52.15) at (6.5,2.15) [1]{}; (6.53.15) at (6.5,3.15) [1]{}; (6.54.15) at (6.5,4.15) [1]{}; (6.93.5) at (6.9,3.5) [1]{}; (2.92.5) at (2.9,2.5) [1]{}; (3.92.5) at (3.9,2.5) [1]{}; (5.93.5) at (5.9,3.5) [1]{}; (4.93.5) at (4.9,3.5) [2]{}; (4.53.1) at (4.5,3.15) [2]{}; (5.54.1) at (5.5,4.15) [3]{};
(00)–(10)–(20)–(30)–(40)–(50)–(60)–(70) (01)–(11)–(21)–(31)–(41)–(51)–(61)–(71) (02)–(12)–(22)–(32)–(42)–(52)–(62)–(72) (03)–(13)–(23)–(33)–(43)–(53)–(63)–(73) (04)–(14)–(24)–(34)–(44)–(54)–(64)–(74) (00)–(01)–(02)–(03)–(04) (10)–(11)–(12)–(13)–(14) (20)–(21)–(22)–(23)–(24) (30)–(31)–(32)–(33)–(34) (40)–(41)–(42)–(43)–(44) (50)–(51)–(52)–(53)–(54) (60)–(61)–(62)–(63)–(64) (70)–(71)–(72)–(73)–(74);
Let $\mu_0 = \varnothing \subseteq \mu_1 \subseteq \ldots \mu_k$ be the seqence of snakeshapes across the top of the growth diagram, and let $\nu_0=\varnothing \subseteq \nu_1 \subseteq \ldots \nu_n$ be the sequence of snakeshapes on the right side of the KYF growth diagram. These sequences correspond to KYF tableaux $Q(w)$ and $P(w)$, respectively.
If the edge between $\nu_i$ and $\nu_{i+1}$ is labeled $j$, then $2\nu_i=\nu_{i+1}$, and a column of $i$’s is added in the $j^{th}$ column of $\nu_i$ to obtain $\nu_{i+1}$. If the edge between $\mu_i$ and $\mu_{i+1}$ is labeled $j$, then $\mu_i=\mu_{i+1}$ and the label $i+1$ of $Q(w_1\cdots w_{i+1})$ is placed in the box at the top of the the $j^{th}$ column of $Q(w_1\cdots w_i)$.
In the example above, we have
$P=$
4 & 3 &\
4 & 2 & 1
$Q=$
57 & 3 &\
4 & 2 & 16
,
which agrees with the insertion and recording tableau from Example \[ex:1334241\]. As for Hecke insertion and shifted Hecke insertion, this is not a coincidence.
For any word $w$, the KYF tableau and set-valued KYF tableau $P$ and $Q$ obtained from the sequence of snakeshapes across the right side of the KYF growth diagram for $w$ and across the top of the KYF growth diagram for $w$, respectively, are $P_{YF}(w)$ and $Q_{YF}(w)$, the KYF insertion and recording tableau for $w$.
Suppose the square under consideration is in row $t$ and column $s$. Assume also the $X$ in column $s$ is in row $r$. We may or may not have $r=t$. We will argue the result by induction on $t+s$. The base case is trivial. As in the previous growth diagram proofs, we have the following lemmas.
Oriented as in the square above, $|\nu/\lambda|\leq 1$.
Oriented as in the square above, to obtain $\mu$ from $\lambda$, we add either one box of a column of 2 boxes.
Rule 1 follows from the fact that inserting $t$ into the tableau
t
does not change the tableau. Rules 2 applies when inserting positive integer $t$ that is strictly larger than all other entries of the KYF tableau. Thus a box with $t$ will be added directly to the left of the tableau. In Rule 3, we are adding integer $t$ to a tableau that already has 2 boxes labeled $t$ in the $i^{th}$ column, and $t$ is the largest integer in the tableau. Following the insertion rules, we end with a column of $t$’s in the first column, hence the right edge is labeled 1. The special corner of this insertion is in the $(i+1)^{th}$ column, the column where the $t$ on the top row of the original KYF tableau was deleted in the insertion procedure.
In Rule 4, there is exactly one $t$ in insertion tableau $\mu$. Thus, inserting a $t$ will result in a tableau where the first column has two boxes filled with a $t$.
If $\lambda=\nu$ as in Rule 5, then $r>t$. Thus no integer is being inserted as we move from $\mu$ to $\gamma$, so $\mu=\gamma$ and the edge labeles are unchanged. Similarly, if if $\lambda=\mu$, then there is no $X$ to the left of the square we are considering, and so no boxes labeled $t$ in the KYF tableau $\gamma$. Thus there is no change in the insertion and recording tableaux.
Rule 6 follows directly from insertion rule YF1.
In considering Rule 7, note that $r<t$ because if not, we would be in the situation described in Rule 5. In Rule 7, $\gamma=2\lambda$ because no matter what $r$ is inserted, since $\mu\neq\lambda$, there is at least one entry of $\mu$ that is larger than $r$. This means that when we insert $r$ into insertion tableau $\mu$, a box with $r$ will be added to the first column, and after the shifting of YF5, there will be a box directly above this new box with $r$. An entry $t$ will shift into the position of any box that was “lost” when shifting during the insertion of $r$ into $\lambda$, and if there were two boxes labeled $t$ in $\mu$, the second will be deleted in YF6. It follows that $\gamma=2\lambda$.
If the bottom edge is labeled $i$, then insertion of $r$ into insertion tableau $\lambda$ included the situation described by YF6 in column $i$ of $\lambda$. When we insert $r$ into insertion tableau $\mu$, column $i$ now becomes
t\
$*$
and thus
t\
t
in $\gamma$. This means that the edge between $\nu$ and $\gamma$ should be labeled $i$, indicating where the $t$’s are in insertion tableau $\gamma$.
If the left edge is labeled $j$, then the difference between insertion tableaux $\lambda$ and $\mu$ is a column
t\
t
in column $j$ of $\mu$. The process of inserting $r$ into insertion tableau $\mu$ will match that of inserting $r$ into insertion tableau $\lambda$ until we get to the point where we are shifting the entries $t$ and $*$. Since $t$ is the largest entry in $\mu$, the insertion of $r$ into $\mu$ will end with deleting the unmatched $*$ from column $j+1$ of the insertion tableau $\gamma$, and therefore the top edge is labeled $j+1$.
We can also formulate the rules for the reverse insertion as follows. The proof is omitted for brevity.
**Reverse Rules**
1. If $\mu=\gamma=1$, $\nu=\varnothing$, at the top edge is labeled 1, then the square contains an $X$ and $\lambda=\varnothing$.
2. If $\mu=\nu$ and $\gamma=1\mu=1\nu$, then the square contains an $X$ and $\lambda=\mu=\nu$.
3. $\mu=\gamma=2\nu$, the top edge is labeled $i$ and the right edge is labeled 1, then there is an $X$ in the square, $\lambda=\nu$ and the left edge is labeled $i-1$.
4. If $\gamma=2\nu$ and the right edge is labeled 1, then the square contains an $X$ and $\lambda=\nu$.
5. If $\gamma=\nu$, then set $\lambda=\mu$ and label the bottom edge with the same label as the top edge if one exists. If $\gamma=\mu$, then $\lambda=\nu$ and the left edge has the same label as the right edge if such a label exists.
6. If $\gamma=\mu$ and the top edge is labeled 1, then $\lambda=\nu$, the bottom edge is labeled 1, and the left edge has the same label as the right edge.
7. If no previous cases apply, then $\lambda$ is obtained from $\gamma$ by removing the first 2. If the right edge is labeled $i$, then the bottom edge is labeled $i$. If the top edge is labeled $j$, then the left edge is labeled $j-1$.
Other examples {#sec:other}
==============
Binary tree deformations {#sec:bin}
------------------------
\[ex:BinWord\] The lifted binary tree is shown on the left, where vertices can be naturally labeled by bit strings: 0, 1, 10, 11, 100, 101,$\ldots$. The graph *BinWord* with the same set of vertices and rank function is shown on the right. In *BinWord*, an element $x$ covers $y$ if $y$ can be obtained from $x$ by deleting a single digit from $x$, and in addition, 1 covers 0. The lifted binary tree and *BinWord* form a dual graded graph, see [@F Example 2.4.1].
\(0) at (0,0) ; (1) at (0,1) ; (11) at (2,2) ; (10) at (-2,2) ; (100) at (-3,3) ; (101) at (-1,3) ; (110) at (1,3) ; (111) at (3,3) ; (1000) at (-3.5,4) ; (1001) at (-2.5,4) ; (1010) at (-1.5,4) ; (1011) at (-.5,4) ; (1100) at (.5,4) ; (1101) at (1.5,4) ; (1110) at (2.5,4) ; (1111) at (3.5,4) ; (0)–(1)–(11)–(111)–(1111) (1)–(10)–(100)–(1000) (100)–(1001) (10)–(101)–(1010) (101)–(1011) (11)–(110)–(1100) (110)–(1101) (111)–(1110);
\(0) at (0,0) ; (1) at (0,1) ; (11) at (2,2) ; (10) at (-2,2) ; (100) at (-3,3) ; (101) at (-1,3) ; (110) at (1,3) ; (111) at (3,3) ; (1000) at (-3.5,4) ; (1001) at (-2.5,4) ; (1010) at (-1.5,4) ; (1011) at (-.5,4) ; (1100) at (.5,4) ; (1101) at (1.5,4) ; (1110) at (2.5,4) ; (1111) at (3.5,4) ; (0)–(1)–(11)–(111)–(1111) (1)–(10)–(100)–(1000) (100)–(1001) (10)–(101)–(1010) (101)–(1011) (11)–(110)–(1100) (110)–(1101) (111)–(1110) (101)–(11) (110)–(10) (100)–(1010) (100)–(1100) (101)–(1001) (101)–(1101) (110)–(1110) (110)–(1010) (111)–(1101) (111)–(1011);
We form a Pieri deformation by interpreting the graph in the context of the ring of quasisymmetric functions, $\QSym$ (see [@EC2]). If we interpret “1” as $L_1$, “10” as $L_2$, “100” as $L_3$, 11001 as $L_{131}$, and so on, we see that $x$ covers $y$ in the graph on the right exactly when $x$ appears in the product $L_y\cdot L_1$, where $L_\alpha$ is the fundamental quasisymmetric function with the usual product. Let $D$ be the operator that subtracts 1 from the rightmost number in the subscript or deletes the rightmost number if it is 1. For example, $D(L_{14})=L_{13}$ and $D(L_{1421})=L_{142}$. Then $x$ covers $y$ in the graph on the left exactly when $y=D(x)$. To form the Pieri deformation, we can let $f=L_1+L_{11}+L_{111}+\ldots$. In other words, the multiplicity of the edge from $x$ down to $y$ is the coefficient of $x$ in $y\cdot (L_1+L_{11}+\ldots)$.
\(0) at (0,0) ; (1) at (0,1) ; (11) at (2,2) ; (10) at (-2,2) ; (100) at (-3,3) ; (101) at (-1,3) ; (110) at (1,3) ; (111) at (3,3) ;
(0)–(1)–(11)–(111) (1)–(10)–(100) (10)–(101) (101) (11)–(110);
\(0) at (0,0) ; (1) at (0,1) ; (11) at (2,2) ; (10) at (-2,2) ; (100) at (-3,3) ; (101) at (-1,3) ; (110) at (1,3) ; (111) at (3,3) ;
(0)–(1)–(11)–(111) (1)–(10)–(100) (10)–(101) (11)–(110) (101)–(11) (110)–(10) (101)–(1) (110)–(1) (111) to \[bend right=10\] (1) (0)–(11) (0) to \[bend right=10\] (111);
Using $A=\QSym$ and $D$ and $f$ as defined above, it is easy to see that $D(f)=L_\varnothing+f=id+f$. Using the Hopf algebra structure of $\QSym$, it is clear that $D$ is a derivation by Lemma \[lem:derivation\]. Thus the resulting graph is a dual filtered graph by Theorem \[thm:pieri\].
\[ex:L\] Consider the Hopf algebra of multi-quasisymmetric functions, $\mQSym$, defined in [@LP]. This Hopf algebra has a basis of multi-fundamental quasisymmetric functions indexed by compositions, $\{\tilde L_\alpha\}$. Notice that a graph with vertices indexed by $\{\tilde L_\alpha\}$ has the same vertices as in Example \[ex:BinWord\]. To define the product, we first define the concept of a multiword. Let $u=u_1u_2\cdots u_k$ be a word. We call $w=w_1w_2\cdots w_m$ a *multiword* of $u$ if there is a surjective and non-decreasing map $t:[m]\to[k]$ so that $w_j=u_{t(j)}$. For example, 11335662 and 133335562 are multiwords of 13562. Let $u=u_1u_2\cdots u_k$ and $v=v_1v_2\cdots v_l$ be two words, and assume that all letters $v_i$ and $u_j$ are distinct. A word $w=w_1w_2\cdots w_m$ is a *multishuffle* of $u$ and $v$ if
- for any $i\in [1,m-1]$, $w_i\neq w_{i+1}$, and
- when restricted to alphabets $v_i$ and $u_j$, $w$ becomes a multiword of $v$ and $u$, respectively.
For example, 1818373567627 is a multishuffle of 13562 and 87. If $u$ or $v$ contain repeated letters, obtain the multishuffle by first replacing $u$ and $v$ with words with distinct letters, taking the multishuffle product, and then replacing the letters in the result with the original letters. For example, if multishuffling 121 with 1, we can instead multishuffle 132 with 4 to get $1324+13242+41342+\ldots$, and then substitute the original letters to get $1214+12111+11211+\ldots$. Note that multiplicities may occur in this case.
To multiply $\tilde L_\alpha$ and $\tilde L_\beta$, first choose words $w_\alpha$ and $w_\beta$ such that the composition of the descent set of $u$ is $\alpha$ and of $v$ is $\beta$. For example, if $\alpha=(2,1)$ and $\beta$ is $(1)$, we can choose $w_\alpha=231$ and $w_\beta=1$. Say that $w_\alpha$ is on $[n]$ for some $n\in\mathbb{N}$, and let $w_\beta[n]$ be the word obtained by adding $n$ to each letter of $w_\beta$. In the previous example, $w_\beta[3]=4$. Now $$\tilde L_\alpha \tilde L_\beta=\displaystyle\sum _w \tilde L_{\mathcal{C}(w)},$$ where we sum over multishuffles of $w_\alpha$ and $w_\beta[n]$ and $\mathcal{C}(w)$ is the composition associated to the descent set of word $w$. For example, $\tilde L_{(2,1)}\tilde L_{(1)}=\tilde L_{(2,2)}+\tilde L_{(2,1,1)}+\tilde L_{(3,1)}+\tilde L_{(12121)}+\ldots$, where the terms shown correspond to multishuffles 2314, 2431, 2341, 4243141.
Define the coproduct using the *cuut coproduct* of a word $w$: $\blacktriangle(w_1w_2\cdots w_k)=\varnothing\otimes w_1\cdots w_k+
w_1\otimes w_1\cdots w_k+w_1\otimes w_2\cdots w_k+w_1w_2\otimes w_2\cdots w_k+\ldots+w_1\cdots w_k\otimes w_k+w_1\cdots w_k\otimes \varnothing.$ Then let $$\Delta(\tilde L_\alpha)=\displaystyle\sum_{u\otimes u' \text{ in } cuut(w_\alpha)}\tilde L_{\mathcal{C}(u)}\otimes \tilde L_{\mathcal{C}(u')}.$$ For example, $$\Delta(\tilde L_{(2,1)})=\varnothing\otimes \tilde L_{(2,1)}+\tilde L_{(1)}\otimes \tilde L_{(2,1)}+\tilde L_{(1)}\otimes \tilde L_{(1,1)}+\tilde L_{(2)}\otimes \tilde L_{(1,1)}+\tilde L_{(2)}\otimes\tilde L_{(1)}+\tilde L_{(2,1)}\otimes \tilde L_{(1)}+\tilde L_{(2,1)}\otimes \varnothing$$ since $$\blacktriangle(231)=
\varnothing\otimes 231+2\otimes 231+2\otimes 31+23\otimes 31 +23\otimes 1+231\otimes 1+231\otimes\varnothing.$$
Taking $A$ to be $\mQSym$ with the basis $\{\tilde L_\alpha \}$ indexed by compositions, we create a dual filtered graph by taking $f=\tilde L_{(1)}$ and $D=\xi\circ\Delta$ with $g=\tilde L_{(1)}$. The first five ranks are shown below.
\(0) at (0,0) ; (1) at (0,1) ; (11) at (2,2) ; (10) at (-2,2) ; (100) at (-3,3) ; (101) at (-1,3) ; (110) at (1,3) ; (111) at (3,3) ; (1000) at (-3.5,4) ; (1001) at (-2.5,4) ; (1010) at (-1.5,4) ; (1011) at (-.5,4) ; (1100) at (.5,4) ; (1101) at (1.5,4) ; (1110) at (2.5,4) ; (1111) at (3.5,4) ; (1) to \[out=60,in=100,distance=.5cm\] (1) (1) to \[out=60,in=100,distance=.5cm\] (1) (11) to \[out=60,in=100,distance=.5cm\] (11) (10) to \[out=60,in=100,distance=.5cm\] (10) (100) to \[out=-60,in=-100,distance=.5cm\] (100) (101) to \[out=-60,in=-100,distance=.5cm\] (101) (110) to \[out=-60,in=-100,distance=.5cm\] (110) (111) to \[out=-60,in=-100,distance=.5cm\] (111) (1000) to \[out=60,in=100,distance=.5cm\] (1000) (1001) to \[out=60,in=100,distance=.5cm\] (1001) (1010) to \[out=60,in=100,distance=.5cm\] (1010) (1011) to \[out=60,in=100,distance=.5cm\] (1011) (1100) to \[out=60,in=100,distance=.5cm\] (1100) (1101) to \[out=60,in=100,distance=.5cm\] (1101) (1110) to \[out=60,in=100,distance=.5cm\] (1110) (1111) to \[out=60,in=100,distance=.5cm\] (1111) (0)–(1)–(11)–(111)–(1111) (1)–(10)–(100)–(1000) (100)–(1001) (10)–(101)–(1010) (101)–(1011) (11)–(110)–(1100) (110)–(1101) (111)–(1110);
\(0) at (0,0) ; (1) at (0,1) ; (11) at (2,2) ; (10) at (-2,2) ; (100) at (-3,3) ; (101) at (-1,3) ; (110) at (1,3) ; (111) at (3,3) ; (1000) at (-3.5,4) ; (1001) at (-2.5,4) ; (1010) at (-1.5,4) ; (1011) at (-.5,4) ; (1100) at (.5,4) ; (1101) at (1.5,4) ; (1110) at (2.5,4) ; (1111) at (3.5,4) ; (0)–(1)–(11)–(111)–(1111) (1)–(10)–(100)–(1000) (100)–(1001) (10)–(101)–(1010) (101)–(1011) (11)–(110)–(1100) (110)–(1101) (111)–(1110) (101)–(11) (110)–(10) (100)–(1010) (100)–(1100) (101)–(1001) (101)–(1101) (110)–(1110) (110)–(1010) (111)–(1101) (111)–(1011) (101)–(1) (110)–(1) (1001)–(10) (1010) to \[bend right=5\] (10) (1010) to \[bend left=5\] (10) (1010) to \[bend left =15\] (1) (1010)–(11) (1011) to \[bend right =5\] (11) (1100) to \[bend left = 5\](10) (1101)–(10) (1101) to \[bend right =15\] (1) (1101) to \[bend left=5\] (11) (1101) to \[bend right =5\] (11) (1110)–(11);
This is a dual filtered graph by Theorem \[thm:pieri\].
The following proposition shows that we have another instance of Möbius via Pieri phenomenon, just like in the case of Young’s lattice.
The Pieri construction above is a Möbius deformation of the dual graded graph in Example \[ex:BinWord\].
We can transform the poset [*[BinWord]{}*]{} from Example \[ex:BinWord\] into the poset of subwords shown below by ignoring the first 1 in every word. In doing this, the element $\varnothing$ in *BinWord* disappears, $1$ becomes $\varnothing$ in the subword poset, $10$ becomes $0$, $11001$ becomes $1001$ and so on. With this change, $x<y$ in *BinWord* exactly when $x$ can be found as a subword of $y$, ignoring the inital 1’s in each.
\(1) at (0,1) ; (11) at (2,2) ; (10) at (-2,2) ; (100) at (-3,3) ; (101) at (-1,3) ; (110) at (1,3) ; (111) at (3,3) ; (1000) at (-3.5,4) ; (1001) at (-2.5,4) ; (1010) at (-1.5,4) ; (1011) at (-.5,4) ; (1100) at (.5,4) ; (1101) at (1.5,4) ; (1110) at (2.5,4) ; (1111) at (3.5,4) ; (1)–(11)–(111)–(1111) (1)–(10)–(100)–(1000) (100)–(1001) (10)–(101)–(1010) (101)–(1011) (11)–(110)–(1100) (110)–(1101) (111)–(1110) (101)–(11) (110)–(10) (100)–(1010) (100)–(1100) (101)–(1001) (101)–(1101) (110)–(1110) (110)–(1010) (111)–(1101) (111)–(1011);
Given a word $y=y_1y_2\cdots y_n$, define its *repetition set* $\mathcal{R}(y)=\{i:y_{i-1}=y_i\}.$ An embedding of $x$ in $y$ is a sequence $1\leq i_1<i_2<\ldots <i_k\leq n$ such that $x=y_{i_1}y_{i_2}\cdots y_{i_k}$. It is a *normal embedding* if $\mathcal{R}(y)\subseteq \{i_1,i_2,\ldots, i_k\}$. For two words, $x$ and $y$, let $\binom{y}{x}_n$ denote the number of normal embeddings of $x$ in $y$. For example, $\binom{10110}{10}_n=1$ and $\binom{1010}{10}_n=3$. Now, from Theorem 1 in [@Bj], $$\mu(x,y)=(-1)^{|y|-|x|}\binom{y}{x}_n.$$
We show that the coefficient of $\tilde L_\beta$ in $\tilde L_\alpha \tilde L_{(1)}$ is the number of normal embeddings of $\alpha$ into $\beta$ after viewing $\alpha$ and $\beta$ as sequences of 0’s and 1’s and deleting the first 1 in each by showing the unique way to insert strings of alternating 1’s and 0’s into $\alpha$ at a specific location to obtain $\beta$ in the multishuffle. So, for example, the coefficient of $\tilde L_{(1111)}$ in $\tilde L_{(11)}\tilde L_{(1)}$ is zero since $\binom{111}{1}_n=0$ and the coefficient of $\tilde L_{(122)}$ in $\tilde L_{(12)} \tilde L_{(1)}$ is $3$ since $\binom{1010}{10}_n=3$.
Choose word $w_\alpha$ on $[n]$ and multishuffle it with $n+1$. We note that adding two 0’s to $\alpha$ that are not separated by a 1 in $\alpha$ in the multishuffle means increasing the length of an increasing segment of $w_\alpha$ by two, which is impossible as adding $n+1$ to the end of an increasing segment is the only way to increase its length, and the letters of $w_\alpha$ must stay in the same relative order in the multishuffle. Similarly for adding two 1’s to $w_\alpha$ that are not separated by a 0 in $\alpha$. This agrees with the Möbius function since there are no normal embeddings in this case, e.g. $\binom{1010001}{101}_n=0$.
Suppose we want to insert a segment of alternating 1’s and 0’s in $\alpha$ in a place with a 0 (if anything) on the left and a 1 (if anything) on the right. Let $w_\alpha$ be $w_1w_2\cdots w_{k-1}w_k w_{k+1}\cdots w_n$, where the $w_k$ and $w_{k+1}$ correspond to the 1 and 0 of $\alpha$ mentioned above. The only way to add an alternating sequence beginning with 0 is to insert the string $(n+1)w_k (n+1) w_k (n+1)\ldots$ between $w_k$ and $w_{k+1}$. The resulting word, $$w_1\cdots w_{k-1}w_k(n+1)w_k(n+1)w_k\cdots w_{k+1}
\cdots w_n$$ is clearly a multishuffle of $w_\alpha$ and $n+1$. Similarly, to add an alternating sequence beginning with 1, “delete" the $w_k$ from $w$ and replace it with $(n+1)w_k(n+1)w_k(n+1)\cdots$. The resulting word, $$w_1\cdots w_{k-1}(n+1)w_k(n+1)w_k\cdots w_{k+1}\cdots w_n$$ is a multishuffle of $w_\alpha$ and $n+1$.
If instead we insert the alternating string in a place of $\alpha$ with a 1 on the left and a 0 on the right, we only need to consider strings beginning with 0 as the 1 at the beginning of the string can be added as in the previous case. To insert an alternating string beginning with 0, we must insert the segment $(n+1)w_k(n+1)w_k\cdots$ between $w_k$ and $w_{k+1}$ of $w_\alpha$.
If we insert the alternating string in place of $\alpha$ with 0’s to the left and right, we only need to consider strings that begin with 1. To do this, delete the $w_k$ from $w$ and insert the segment $(n+1)w_k(n+1)w_k\cdots$ between $w_{k-1}$ and $w_{k+1}$
If we insert the alternating string in place of $\alpha$ with 1’s to the left and right, we only need to consider strings that begin with 0. To do this, insert the segment $(n+1)w_k(n+1)w_k\cdots$ between $w_k$ and $w_{k+1}$.
Poirer-Reutenauer deformations {#sec:PR}
------------------------------
\[ex:SYTtree\] The *SYT-tree* has as vertices standard Young tableau, and standard Young tableau $T_1$ covers standard Young tableau $T_2$ if $T_2$ is obtained from $T_1$ by deleting the box with largest entry. It is shown below on the left. It is dual to the *Schensted graph*, which is shown on the right.
The *Schensted graph* is constructed using RSK insertion. A standard Young tableau $T_1$ covers $T_2$ if $T_1$ appears in the product of $T_2$ with the single-box standard tableau, where multiplication of standard Young tableaux is as follows. Suppose $T_1$ and $T_2$ are standard Young tableaux with row words $\r(T_1)$ on $[n]$ and $\r(T_2)$ on $[m]$, respectively. Then let $T_1\cdot T_2$ be the sum of tableaux obtained by first lifting $\r(T_1)$ to $[n+m]$ in all ways that preserve relative order and then RSK inserting the corresponding lifting of $\r(T_2)$. For example, in multiplying
1 & 2
by itself, we would get the sum of $(P(12)\leftarrow 3) \leftarrow 4$, $(P(13)\leftarrow 2 )\leftarrow 4$, $P((14)\leftarrow 2)\leftarrow 3$, $(P(23)\leftarrow 1)\leftarrow 4$, $(P(24)\leftarrow 1)\leftarrow 3$, and $(P(34)\leftarrow 1)\leftarrow 2$.
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,2) ; (21) at (2,4) ; (12) at (-2,4) ; (123) at (-3,6) ; (312) at (-1,6) ; (213) at (1,6) ; (321) at (3,6) ; (empty)–(1)–(12)–(123) (12)–(312) (1)–(21)–(321) (21)–(213);
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,2) ; (21) at (2,4) ; (12) at (-2,4) ; (123) at (-3,6) ; (312) at (-1,6) ; (213) at (1,6) ; (321) at (3,6) ; (empty)–(1)–(12)–(123) (12)–(312) (1)–(21)–(321) (21)–(213) (312)–(21) (213)–(12);
Let $f=T_1+T_2+T_3+\ldots$, where $T_i$ is the one-row tableau with row reading word $123\ldots i$, and create a Pieri deformation by adding edges to the *Schensted graph* so that $a_2(T_i,T_j)$ is the coefficient of $T_j$ in $T_i\cdot f$.
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,2) ; (21) at (2,4) ; (12) at (-2,4) ; (123) at (-3,6) ; (312) at (-1,6) ; (213) at (1,6) ; (321) at (3,6) ; (empty)–(1)–(12)–(123) (12)–(312) (1)–(21)–(321) (21)–(213);
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,2) ; (21) at (2,4) ; (12) at (-2,4) ; (123) at (-3,6) ; (312) at (-1,6) ; (213) at (1,6) ; (321) at (3,6) ; (empty)–(1)–(12)–(123) (12)–(312) (1)–(21)–(321) (21)–(213) (312)–(21) (213)–(12) (12)–(empty) (123) to \[bend right =25\] (empty) (123) to \[bend left=10\] (1) (312)–(1) (213)–(1);
To see that this gives a dual filtered graph, take $A$ to be the algebra of standard Young tableaux with multiplication as defined above. Note that $A$ is dual to the Poirier-Reutenauer Hopf algebra [@PR]. Take $D$ to be the operator that deletes the box with the largest entry, and notice that $T_1$ covers $T_2$ in *SYT-tree* exactly when $D(T_1)=T_2$. The operator $D$ is a derivation by Lemma \[lem:derivation\] using the fact that $\Delta(T)=\sum (u\otimes v)$, where we sum over words $u$ and $v$ such that $\r(T)$ is a shuffle of $u$ and $v$. Lastly, it is easy to see that $D(f)=\varnothing + f$. Thus by Theorem \[thm:pieri\], we have a dual filtered graph.
\[ex:KPR\] Take $A$ to be the $K$-theoretic Poierier-Reuntenauer bialgebra (KPR) defined in [@PP] with a basis of $K$-Knuth classes of initial words (i.e. words containing only $1,2,\ldots,n$ for some $n$) on $\mathbb{N}$. The $K$-Knuth relations are as follows:
$pp\equiv p$ for all $p$\
$pqp\equiv qpq$ for all $q,p$\
$pqs\equiv qps$ and $spq\equiv sqp$ for $p<s<q$
See [@BuchSam] for details on this relation.
Multiplication of two classes $[w_1]$ and $[w_2]$ is done by shuffling each pair of elements, one from $[w_1]$ and one from $[w_2[n]]$, where $w_1$ is a word on $n$ and $w_2[n]$ is obtained by adding $n$ to each letter in $w_2$, and writing the result as a sum of classes. For example, $$[[12]]\cdot[[312]]=[[53124]]+[[51234]]+[[35124]]+[[351234]]+[[53412]]+[[5351234]].$$ The coproduct is similarly defined by taking the sum of the coproduct, $\Delta(w_1\cdots w_k)=\varnothing\otimes w_1\cdots w_k+std(w_1)\otimes std(w_2\cdots w_k)+\ldots
+std(w_1\cdots w_{k-1}w_k)+ std(w_1\cdots w_k)\otimes \varnothing$, of each element in the class and writing the result as a sum of classes. Here, $std(w)$ sends $w$ to the unique word with the same relative order using all letters $\{1,2,\ldots,n\}$ for some $n$. For example, $std(13375)=12243$. Using this, we compute, $$\Delta([[12]])=[[\varnothing]]\otimes [[12]]+[[1]]\otimes[[1]]+[[1]]\otimes[[12]]+[[12]]\otimes[[1]]+[[12]]\otimes[[\varnothing]].$$ Letting $g=[1]$ and $f=[1]$, we create a Pieri construction using Theorem \[thm:pieri\]. The partial dual filtered graph is shown below. The first five ranks are shown completely, and there are a few additional elements shown in the graph on the right to illustrate all elements that cover $[212]$. To see that there is an upward edge from $[3412]$ to $[3124]$, we first notice that $34124 \in [3124]$ and that $3412\otimes 1$ appears in the coproduct of 34124.
In the figure below, $K$-Knuth equivalence classes of words are represented by an increasing tableau. Note that there may be more than one increasing tableau in any given class.
(empty) at (0,1) [$\varnothing$]{}; (1) at (0,2) [$\tiny\young(1)$]{}; (21) at (-2,4) [$\tiny\young(1,2)$]{}; (12) at (2,4) [$\tiny\young(12)$]{}; (321) at (-4,6) [$\tiny\young(1,2,3)$]{}; (213) at (-2,6) [$\tiny\young(13,2)$]{}; (212) at (0,6) [$\tiny\young(12,2)$]{}; (312) at (2,6) [$\tiny\young(12,3)$]{}; (123) at (4,6) [$\tiny\young(123)$]{}; (4321) at (-9,9) [$\tiny\young(1,2,3,4)$]{}; (3214) at (-8,9) [$\tiny\young(14,2,3)$]{}; (4213) at (-7,9)[$\tiny\young(13,2,4)$]{}; (4312) at (-6,9) [$\tiny\young(12,3,4)$]{}; (3213) at (-5,9) [$\tiny\young(13,2,3)$]{}; (3212) at (-4,9) [$\tiny\young(12,2,3)$]{}; (3412) at (-3,9)[$\tiny\young(12,34)$]{}; (2413) at (-2,9) [$\tiny\young(13,24)$]{}; (2312) at (-.5,9) [$\tiny\young(12,23)$]{}; (3123) at (1,9) [$\tiny\young(123,3)$]{}; (2134) at (2.5,9) [$\tiny\young(134,2)$]{}; (3124) at (4,9) [$\tiny\young(124,3)$]{}; (4123) at (5.5,9) [$\tiny\young(123,4)$]{}; (2123) at (7,9) [$\tiny\young(123,2)$]{}; (1234) at (8.5,9) [$\tiny\young(1234)$]{}; (empty)–(1)–(21)–(321)–(4321) (21)–(213) (21)–(212) (21) to \[bend right =10\] (312) (12)–(212) (12)–(312) (12)–(123) (12) to \[bend left =10\] (213) (1)–(12)–(123)–(1234) (321)–(3214) (321)–(4213) (321)–(4312) (321)–(3213) (321)–(3212) (213)–(3214) (213)–(3412) (213)–(2413) (213)–(2312) (213)–(2134) (312)–(4213) (312)–(4312) (312)–(3212) (312)–(3123) (312)–(3124) (312)–(4123) (123)–(3123) (123)–(2134) (123)–(3124) (123)–(4123) (123)–(2123) (123)–(1234) (212)–(3213) (212)–(2123) (1) to \[out=60,in=100,distance=.5cm\] (1) (12) to \[out=-60,in=-100,distance=.5cm\] (12) (21) to \[out=-60,in=-100,distance=.5cm\] (21) (123) to \[out=-60,in=-100,distance=.5cm\] (123) (321) to \[out=-60,in=-100,distance=.5cm\] (321) (213) to \[out=60,in=100,distance=.5cm\] (213) (213) to \[out=60,in=100,distance=.6cm\] (213) (212) to \[out=-60,in=-100,distance=.5cm\] (212) (212) to \[out=-60,in=-100,distance=.6cm\] (212) (312) to \[out=60,in=100,distance=.5cm\] (312) (3214) to \[out=60,in=100,distance=.5cm\] (3214) (3214) to \[out=60,in=100,distance=.6cm\] (3214) (4213) to \[out=60,in=100,distance=.5cm\] (4213) (4213) to \[out=60,in=100,distance=.6cm\] (4213) (3213) to \[out=60,in=100,distance=.5cm\] (3213) (3213) to \[out=60,in=100,distance=.6cm\] (3213) (3212) to \[out=60,in=100,distance=.5cm\] (3212) (3212) to \[out=60,in=100,distance=.6cm\] (3212) (3123) to \[out=60,in=100,distance=.5cm\] (3123) (3123) to \[out=60,in=100,distance=.6cm\] (3123) (2134) to \[out=60,in=100,distance=.5cm\] (2134) (2134) to \[out=60,in=100,distance=.6cm\] (2134) (3124) to \[out=60,in=100,distance=.5cm\] (3124) (3124) to \[out=60,in=100,distance=.6cm\] (3124) (2123) to \[out=60,in=100,distance=.5cm\] (2123) (2123) to \[out=60,in=100,distance=.6cm\] (2123) (4321) to \[out=60,in=100,distance=.5cm\] (4321) (4312) to \[out=60,in=100,distance=.5cm\] (4312) (3412) to \[out=60,in=100,distance=.5cm\] (3412) (2413) to \[out=60,in=100,distance=.5cm\] (2413) (2312) to \[out=60,in=100,distance=.5cm\] (2312) (4123) to \[out=60,in=100,distance=.5cm\] (4123) (1234) to \[out=60,in=100,distance=.5cm\] (1234); (212) edge node\[right\][$ $]{} (312); (3213) edge \[bend right=70\] (4213); (3412) edge \[bend left =40\] (3124);
(empty) at (0,1) [$\varnothing$]{}; (1) at (0,2) [$\tiny\young(1)$]{}; (21) at (-2,3) [$\tiny\young(1,2)$]{}; (12) at (2,3) [$\tiny\young(12)$]{}; (321) at (-4,5) [$\tiny\young(1,2,3)$]{}; (213) at (-2,5) [$\tiny\young(13,2)$]{}; (212) at (0,5) [$\tiny\young(12,2)$]{}; (312) at (2,5) [$\tiny\young(12,3)$]{}; (123) at (4,5) [$\tiny\young(123)$]{}; (4321) at (-9,7) [$\tiny\young(1,2,3,4)$]{}; (3214) at (-8,7) [$\tiny\young(14,2,3)$]{}; (4213) at (-7,7)[$\tiny\young(13,2,4)$]{}; (4312) at (-6,7) [$\tiny\young(12,3,4)$]{}; (3213) at (-5,7) [$\tiny\young(13,2,3)$]{}; (3212) at (-4,7) [$\tiny\young(12,2,3)$]{}; (3412) at (-3,7)[$\tiny\young(12,34)$]{}; (2413) at (-2,7) [$\tiny\young(13,24)$]{}; (2312) at (-.5,7) [$\tiny\young(12,23)$]{}; (3123) at (1,7) [$\tiny\young(123,3)$]{}; (2134) at (2.5,7) [$\tiny\young(134,2)$]{}; (3124) at (4,7) [$\tiny\young(124,3)$]{}; (4123) at (5.5,7) [$\tiny\young(123,4)$]{}; (2123) at (7,7) [$\tiny\young(123,2)$]{}; (1234) at (8.5,7) [$\tiny\young(1234)$]{}; (32312) at (-2,8) [$\tiny\young(12,23,3)$]{}; (23123) at (-.5,8) [$\tiny\young(123,23)$]{}; (32123) at (2,8) [$\tiny\young(123,2,3)$]{}; (323123) at (0,9) [$\tiny\young(123,23,3)$]{}; (empty)–(1)–(21)–(321)–(4321) (321)–(3214) (21) to \[bend left=40\] (3213) (21)–(213) (1)–(212)–(3212) (212)–(2123) (212)–(2312) (12)–(312) (12) to \[bend right=40\] (3123) (12)–(123) (1)–(12) (213)–(2134) (213)–(2413) (213)–(4213) (312)–(3124) (312)–(3412) (312)–(4312) (123)–(1234) (123)–(4123) (212)–(32312) (212) to \[bend left=40\] (23123) (212) to \[bend right =10\] (32123) (212) to \[bend right =15\] (323123);
Malvenuto-Reutenauer deformations {#sec:MR}
---------------------------------
\[ex:PermTrees\] Below we have dual graphs where vertices are permutations. On the left, a permutation $\sigma$ covers $\pi$ if $\pi$ is obtained from $\sigma$ by deleting the the rightmost number (in terms of position in the permutation). On the right, a permutation $\sigma$ covers $\pi$ if $\pi$ is obtained from $\sigma$ by deleting the largest number in the permutation. There are other similar constructions, which can be found in [@F Example 2.6.8].
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,1) [1]{}; (21) at (-2,3) [21]{}; (12) at (2,3) [12]{}; (321) at (-3.5,5) [321]{}; (213) at (-2,5) [213]{}; (312) at (-.5,5) [312]{}; (123) at (3.5,5) [123]{}; (132) at (2,5) [132]{}; (231) at (.5,5) [231]{}; (empty)–(1)–(21)–(321) (312)–(21)–(213) (1)–(12)–(123) (231)–(12)–(132);
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,1) [1]{}; (21) at (-2,3) [21]{}; (12) at (2,3) [12]{}; (321) at (-3.5,5) [321]{}; (213) at (-2,5) [213]{}; (312) at (-.5,5) [312]{}; (123) at (3.5,5) [123]{}; (132) at (2,5) [132]{}; (231) at (.5,5) [231]{}; (empty)–(1)–(21)–(321) (231)–(21)–(213) (1)–(12)–(123) (312)–(12)–(132);
Consider the Hopf algebra of permutations with the shuffle product and coproduct defined by $\Delta(w)=\sum std(u)\otimes std(v)$, where the concatenation of $u$ and $v$ is $w$ and $std(w)$ sends $w$ to the unique permutation with the same relative order. For example, $std(1375)=1243$. Take $D$ to be the operator that deletes the rightmost letter of the permutation. Then we see that $a_1(v,w)$ is the coefficient of $v$ in $D(w)$. We create a Pieri deformation by letting $f=1+12+123+\ldots$ and $a_2(v,w)$ be the coefficient of $w$ in $f\cdot v$. Clearly $D(f)=\varnothing+f$, and we use Lemma \[lem:derivation\] to see tha $D$ is a derivation. It then follows from Theorem \[thm:pieri\] that the resulting graph is a dual filtered graph.
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,1) [1]{}; (21) at (-2,3) [21]{}; (12) at (2,3) [12]{}; (321) at (-3.5,5) [321]{}; (213) at (-2,5) [213]{}; (312) at (-.5,5) [312]{}; (123) at (3.5,5) [123]{}; (132) at (2,5) [132]{}; (231) at (.5,5) [231]{}; (empty)–(1)–(21)–(321) (312)–(21)–(213) (1)–(12)–(123) (231)–(12)–(132);
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,1) [1]{}; (21) at (-2,3) [21]{}; (12) at (2,3) [12]{}; (321) at (-3.5,5) [321]{}; (213) at (-2,5) [213]{}; (312) at (-.5,5) [312]{}; (123) at (3.5,5) [123]{}; (132) at (2,5) [132]{}; (231) at (.5,5) [231]{}; (empty)–(1)–(21)–(321) (231)–(21)–(213) (1)–(12)–(123) (312)–(12)–(132) (12)–(empty) (213)–(1) (231)–(1) (123) to \[bend right=5\] (1) (123) to \[bend left=5\] (empty);
\[ex:mMR\] We next describe a $K$-theoretic analogue of the Malvenuto-Reutenauer Hopf algebra. For details, see [@LP]. A small multi-permutation or $\mathfrak{m}$-permutation of $[n]$ is a word $w$ in the alphabet $1,2,\ldots,n$ such that no two consecutive letters in $w$ are equal. Now let the small multi-Malvenuto-Reutenauer Hopf algebra, $\mathfrak{m}$MR be the free $\mathbb{Z}$-module of arbitrary $\mathbb{Z}$-linear combinations of multi-permutations. Recall the definition of the multishuffle product from Example \[ex:L\]. Given two $\mathfrak{m}$-permutations $w=w_1\cdots w_k$ and $u=u_1\cdots u_l$, define their product to be the multishuffle product of $w$ with $u[n]$, where $w$ contains exactly the numbers $1,2,\ldots,n$ and $u[n]=(u_1+n)(u_2+n)\ldots(u_l+n)$.
To define the coproduct, we must define the *cuut coproduct*, $\blacktriangle(w)$, for any word $w=w_1\cdots w_k$. We have $\blacktriangle(w)=\varnothing\otimes w_1\cdots w_k+
w_1\otimes w_1\cdots w_k+w_1\otimes w_2\cdots w_k+w_1w_2\otimes w_2\cdots w_k+\ldots+w_1\cdots w_k\otimes w_k+w_1\cdots w_k\otimes \varnothing$. Let $st(w)$ send a word $w$ to the unique $\mathfrak{m}$-permutation $u$ of the same length such that $w_i\leq w_j$ if and only if $u_i\leq u_j$ for each $1\leq i,j \leq l(w)$. Finally, define the coproduct in $\mathfrak{m}$MR by $\Delta(w)=st(\blacktriangle(w))$.
Using $g=1$, $D=\xi\circ\Delta$, and $f=1$, we form the dual filtered graph partially shown below. Notice that $D(w_1\ldots w_k)=w_1\cdots w_k + w_1\cdots w_{k-1}$, so $D(f)=\varnothing+f$.
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,1) [1]{}; (12) at (-2,2) [12]{}; (21) at (2,2) [21]{}; (121) at (-3.5,4) [121]{}; (123) at (-2.5,4) [123]{}; (132) at (-1.5,4) [132]{}; (312) at (-.5,4) [312]{}; (213) at (.5,4) [213]{}; (231) at (1.5,4) [231]{}; (321) at (2.5,4) [321]{}; (212) at (3.5,4) [212]{}; (empty)–(1)–(12)–(123) (12)–(132) (12)–(231) (12)–(121) (21)–(212) (21)–(312) (21)–(213) (1)–(21)–(321) (12)–(121) (21)–(212) (1) to \[out=60,in=100,distance=.5cm\] (1) (21) to \[out=-60,in=-100,distance=.5cm\] (21) (12) to \[out=-60,in=-100,distance=.5cm\] (12) (121) to \[out=60,in=100,distance=.5cm\] (121) (212) to \[out=60,in=100,distance=.5cm\] (212) (321) to \[out=60,in=100,distance=.5cm\] (321) (132) to \[out=60,in=100,distance=.5cm\] (132) (123) to \[out=60,in=100,distance=.5cm\] (123) (312) to \[out=60,in=100,distance=.5cm\] (312) (213) to \[out=60,in=100,distance=.5cm\] (213) (231) to \[out=60,in=100,distance=.5cm\] (231);
(empty) at (0,0) [$\varnothing$]{}; (1) at (0,1) [1]{}; (12) at (-2,2) [12]{}; (21) at (2,2) [21]{}; (121) at (-3.5,4) [121]{}; (123) at (-2.5,4) [123]{}; (132) at (-1.5,4) [132]{}; (312) at (-.5,4) [312]{}; (213) at (.5,4) [213]{}; (231) at (1.5,4) [231]{}; (321) at (2.5,4) [321]{}; (212) at (3.5,4) [212]{}; (empty)–(1)–(12)–(123) (12)–(132) (12)–(312) (1)–(21)–(321) (21)–(231) (21)–(213) (121) to \[bend right =40\] (1) (212) to \[bend left=40\] (1);
Stand-alone examples
--------------------
\[ex:FibonacciGraph\] There is another graph with the same set of vertices as the Young-Fibonacci lattice called the Fibonacci graph. The vertices of the Fibonacci graph are words on the alphabet $\{1,2\}$, or snakeshapes, and a word $w$ is covered by $w'$ if $w'$ is obtained from $w$ by adding a 1 at the end or by changing any 1 into a 2. In the graph that is dual to the Fibonacci graph, $w$ is covered by $w'$ if $w$ is obtained from $w'$ by deleting a 1, and the multiplicity of the edge between $w$ and $w'$ is the number of ways to delete a 1 from $w'$ to get $w$. The pair form a dual graded graph shown below, see [@F Example 2.3.7], where the numbers next to the edges denote multiplicity.
(empty) at (0,-1) [$\varnothing$]{}; (1) at (0,1) [1]{}; (11) at (0,3) [11]{}; (2) at (1.5,3) [2]{}; (111) at (0,5) [111]{}; (21) at (1.5,5) [21]{}; (12) at (3,5) [12]{}; (1111) at (0,7) [1111]{}; (211) at (1.5,7) [211]{}; (121) at (3,7) [121]{}; (112) at (4.5,7) [112]{}; (22) at (6,7) [22]{}; (empty)–(1)–(11)–(111)–(1111) (1)–(2) (11)–(12)–(22) (11)–(21)–(22) (111)–(211) (111)–(121) (111)–(112) (2)–(21)–(211) (12)–(121);
(empty) at (0,-1) [$\varnothing$]{}; (1) at (0,1) [1]{}; (11) at (0,3) [11]{}; (2) at (1.5,3) [2]{}; (111) at (0,5) [111]{}; (21) at (1.5,5) [21]{}; (12) at (3,5) [12]{}; (1111) at (0,7) [1111]{}; (211) at (1.5,7) [211]{}; (121) at (3,7) [121]{}; (112) at (4.5,7) [112]{}; (22) at (6,7) [22]{}; (label2) at (-.2,2) ; (label3) at (-.2,4) ; at (-.2,6) ; at (1.3,6) ; at (3.95,6) ; (empty)–(1)–(11)–(111)–(1111) (2)–(21)–(211) (2)–(12)–(112) (21)–(121)–(12);
We construct a Pieri deformation of this dual graded graph by adding downward-oriented edges so that in the resulting set of downward-oriented edges, there is an edge from $w'$ to $w$ for every way $w$ can be obtained from $w'$ by deleting at least one 1. For example, there are six edges from 1111 to 11 since there are six ways to delete two 1’s from 1111 to obtain 11.
(empty) at (0,-1) [$\varnothing$]{}; (1) at (0,1) [1]{}; (11) at (0,3) [11]{}; (2) at (1.5,3) [2]{}; (111) at (0,5) [111]{}; (21) at (1.5,5) [21]{}; (12) at (3,5) [12]{}; (1111) at (0,7) [1111]{}; (211) at (1.5,7) [211]{}; (121) at (3,7) [121]{}; (112) at (4.5,7) [112]{}; (22) at (6,7) [22]{}; (empty)–(1)–(11)–(111)–(1111) (1)–(2) (11)–(12)–(22) (11)–(21)–(22) (111)–(211) (111)–(121) (111)–(112) (2)–(21)–(211) (12)–(121);
(empty) at (0,-1) [$\varnothing$]{}; (1) at (0,1) [1]{}; (11) at (0,3) [11]{}; (2) at (1.5,3) [2]{}; (111) at (0,5) [111]{}; (21) at (1.5,5) [21]{}; (12) at (3,5) [12]{}; (1111) at (0,7) [1111]{}; (211) at (1.5,7) [211]{}; (121) at (3,7) [121]{}; (112) at (4.5,7) [112]{}; (22) at (6,7) [22]{}; (label2) at (-.2,2) ; (label3) at (-.2,4) ; at (-.2,6) ; at (1.3,6) ; at (3.5,6) ; at (-.75,5) ; at (-.75,3) ; at (.8,4.5) ; (empty)–(1)–(11)–(111)–(1111) (2)–(21)–(211) (2)–(12)–(112) (21)–(121)–(12) (121)–(2) (11) to \[bend right=30\] (empty) (111) to \[bend left=40\] (empty) (111) to \[bend right=30\] (1) (1111) to \[bend right=40\] (empty) (1111) to \[bend left=30\] (1) (1111) to \[bend right=40\] (11) (211) to \[bend left=20\] (2) (112) to \[bend left=20\] (2);
The resulting graph is a dual filtered graph.
Consider an algebra structure on words in alphabet $\{1,2\}$ with shuffle product, as in [@F]. Consider $D$ to be the operator that changes any $2$ into a $1$ or deletes the last digit if it is $1$. According to [@F Lemma 2.3.9], $D$ is a derivation such that the coefficient of $v$ in $D(w)$ is the multiplicity $a_1(v,w)$ for the Fibonacci graph. Take $f = 1 + 11 + 111 + \ldots$. It is easy to see that $D(f) = id + f$ and that the coefficient of $w$ in $fv$ is the multiplicity $a_2(v,w)$. Thus, Theorem \[thm:pieri\] applies, and the statement follows.
\[ex:polys\] The dual filtered graph on the polynomial ring shown below is that mentioned in Section \[sec:differential\]. We take $U$ to be multiplication by $x$ and $D=e^{\frac{\partial}{\partial x}}-1$. It is easy to check that $DU-UD=D+I$, so the result is a dual filtered graph.
\(1) at (0,0) [1]{}; (x) at (0,2) [$x$]{}; (x2) at (0,4) [$x^2$]{}; (x3) at (0,6) [$x^3$]{}; (x4) at (0,8) [$x^4$]{}; (x5) at (0,10) [$x^5$]{}; (1)–(x)–(x2)–(x3)–(x4)–(x5);
\(1) at (0,0) [1]{}; (x) at (0,2) [$x$]{}; (x2) at (0,4) [$x^2$]{}; (x3) at (0,6) [$x^3$]{}; (x4) at (0,8) [$x^4$]{}; (x5) at (0,10) [$x^5$]{}; at (-.1,9) ; at (-.1,7) ; at (-.1,5) ; at (-.1,3) ; at (.6,4) ; at (-.55,6) ; at (-1.1,5) ; at (.57,7) ; at (1.05,6) ; at (1.7,5) ; (1)–(x)–(x2)–(x3)–(x4)–(x5) (x2) to \[bend right=20\] (1) (x3) to \[bend left=30\] (1) (x3) to \[bend left=20\] (x) (x4) to \[bend right =20\] (x2) (x4) to \[bend right =30\] (x) (x4) to \[bend right =40\] (1) (x5) to \[bend left =20\] (x3) (x5) to \[bend left =30\] (x2) (x5) to \[bend left =40\] (x) (x5) to \[bend left =50\] (1);
Enumerative theorems via up-down calculus {#sec:enum}
=========================================
Let $\varnothing$ be the minimal element of a dual filtered graph satisfying $DU-UD=1+D$. Let $T(n,k)$ be the number of ways $n$ labeled objects can be distributed into $k$ nonempty parcels. We have $$T(n,k) = k! \cdot S(n,k),$$ where $S(n,k)$ is the Stirling number of the second kind.
For any dual filtered graph, the coefficient of $\varnothing$ in $D^k U^n (\varnothing)$ is $T(n,k)$.
We think of replacing the fragments $DU$ inside the word by either $UD$, $D$, or $1$. This way the initial word gets rewritten until there is no $DU$ in any of the terms: $$D^k U^n = D^{k-1} U D U^{n-1} + D^k U^{n-1} + D^{k-1} U^{n-1} = \ldots.$$ Only the terms of the form $U^t$ that appear at the end can contribute to the coefficient we are looking for, since $D (\varnothing) = 0$. Among those, only the terms $U^0=1$ can contribute. Thus, we are looking for all the terms where $D$’s and $U$’s eliminated each other.
It is easy to see that each $D$ eliminates at least one $U$, and each $U$ must be eliminated by some $D$. The number of ways to match $D$’s with $U$’s in this way is exactly $T(n,k)$.
Let $f^{\lambda}$ be the number of increasing tabeleaux of shape $\lambda$. Let $g^{\lambda}_n$ be the number of set-valued tableaux of shape $\lambda$ and content $1, \ldots, n$. Let $F(n)$ denote the [*[Fubini number]{}*]{}, or [*[ordered Bell number]{}*]{} - the number of ordered set partitions of $[n]$.
We have $$\sum_{|\lambda| \leq n} f^{\lambda} g^{\lambda}_n = F(n).$$
The left side is clearly the coefficient of $\varnothing$ in $(D + D^2 + \dotsc + D^n) U^n (\varnothing)$. It remains to note that $\sum_{k=1}^n T(n,k) = F(n)$.
This is the analogue of the famous Frobenius-Young identity. Of course, this also follows from bijectivity of Hecke insertion. The advantage of our proof is that a similar result exists for any dual filtered graph.
The following result is analogous to counting oscillating tableaux.
For any dual filtered graph the coefficient of $\varnothing$ in $(D+U)^n (\varnothing)$ is equal to the number of set partitions of $[n]$ with parts of size at least $2$.
As before, the desired coefficient is equal to the number of ways for all $D$’s to eliminate all $U$’s via the commutation relation. Each factor in the product $$(D+U)(D+U)\dotsc(D+U)$$ thus either eliminates one of the factors to the right of it, or is eliminated by a factor to the left. Grouping together such factors, we get a set partition with parts of size at least $2$. On the other hand, any such set partition corresponds to a choice of $D$ in the first factor of each part and of $U$’s in the rest. Thus, it corresponds to term $1$ after such $D$’s eliminate such $U$’s. The statement follows.
[99]{}
N. Bergeron, T. Lam and H. Li, Combinatorial Hopf algebras and towers of algebras—dimension, quantization and functorality, *Algebr. Represent. Theory* **15** (2012), no. 4, pp. 675–696.
J.-E.. Björk, Rings of differential operators, North-Holland Pub. Co. (1979).
A. Björner, The Möbius function of subword order. In Invariant theory and tableaux (Minneapolis, MN, 1988) IMA Vol. Math. Appl. **19** Springer, New York, 1990, pp. 118- 124
A. Björner and R. Stanley, An analogue of Young’s lattice for compositions, unpublished.
A. Buch, A Littlewood Richardson rule for the K-theory of Grassmannians, *Acta Mathematica* Vol. 189 (2002), pp. 37-78.
A. Buch, A. Kresch, M. Shimozono, H. Tamvakis, and A. Yong, Stable Grothendieck polynomials and K-theoretic factor sequences, *Math. Annalen* **2** , Vol. 340, (2008), pp. 359-382.
A. Buch and M. Samuel, $K$-theory of miniscule varieties, *Journal für die reine und angewandte Mathematik (Crelles Journal)*, (2014).
E. Clifford, H. Thomas, and A. Yong, $K$-theoretic Schubert calculus for OG$(n,2n+1)$ and jeu de taquin for shifted increasing tableaux, *J. Reine Angew. Math.* **690** (2014), pp. 51-63.
S. Fomin, Generalized Robinson-Schensted-Knuth correspondences (in Russian), *Zapiski Nauchn. Sem. LOMI* **155** (1986), 156-175.
S. Fomin, Duality of Graded Graphs, *Journal of Algebraic Combinatorics* **3** (1994), 357-404.
S. Fomin, Schensted algorithms for dual graded graphs, *Journal of Algebraic Combinatorics* **4** (1995), no. 1, pp 5-45.
Z. Hamaker, A. Keilthy, R. Patrias, L. Webster, Y. Zhang, and S. Zhou, Shifted Hecke insertion and $K$-theory of OG$(n,2n+1)$, preprint, arXiv:1510.08972.
D.E. Knuth, Permutations, matrices, and generalized Young tableaux, *Pacific J. Math.* **34** (1970), 709-727.
T. Lam, Quantized dual graded graphs, *Elec. J. Combin.* Vol. 17 (2010), no.1, paper 88.
T. Lam and P. Pylyavskyy, Combinatorial Hopf Algebras and K-Homology of Grassmannians, *Int. Math. Res. Not.***2007** (2007), rnm 125.
T. Lam and M. Shimozono, unpublished.
T. Lam and M. Shimozono, Dual graded graphs for Kac-Moody algebras, *Algebra and Number Theory* **1** (2007), pp. 451-488.
I.G., Macdonald, Symmetric Functions and Hall Polynomials, Second Edition, Clarendon Press, Oxford, (1995).
J. Nzeutchap, Dual graded graphs and Fomin’s $r$-correspondences associated to the Hopf algebras of planar binary trees, quasi-symmetric functions and noncommutative symmetric functions. In: FPSAC (2006).
O. Ore, Theory of non-commutative polynomials, *Ann. of Math.* **34** (1933), pp. 480-508.
R. Patrias and P. Pylyavskyy, $K$-theoretic Poirier-Reutenauer bialgebra, *Discrete Mathematics* **339** (2016), pp.1095-1115.
S. Poirier and C. Reutenauer, Hopf algebras of tableaux, *Ann. Sci. Math. Quebec* **19** (1995), pp. 79-90.
G. de B. Robinson, On the representations of $S_n$, *Amer. J. Math.* **60** (1938), pp. 745-760.
B. Sagan, Shifted tableaux, Schur $Q$-functions, and a conjecture of R.P. Stanley, *J. Combin. Theory Ser. A* **45** (1987), pp. 62-102.
C.E. Schensted, Longest increasing and decreasing subsequences, *Canad. J. Math.* **13** (1961), pp. 179-191.
R. Stanley, Differential posets, *Amer.Math.Soc.* **1** (1988), 919-961.
R. Stanley, Enumerative Combinatorics, Vol 1, Cambridge, (2011).
R. Stanley, Enumerative Combinatorics, Vol 2, Cambridge, (1999).
H. Thomas and A. Yong, A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus, *Algebra and Number Theory J.* **3** (2009), No. 2, pp. 121-148.
H. Thomas and A. Yong, Longest increasing subsequences, plancherel-type measure, and the Hecke insertion algorith, *Adv. Appl. Math.* **46** (2011), pp. 610-642.
H. Thomas and A. Yong, The direct sum map on Grassmannians and jeu de taquin for increasing tableaux, *Int. Math. Res. Not.* **12** (2011), pp. 2766-2793.
D.R. Worley, A theory of shifted Young tableaux, Ph.D. thesis, MIT, 1984
A. Young, Qualitative substitutional analysis (third paper). *Proc. Lond. Math. Soc. (2)* **28** (1927), pp. 255-292.
[^1]: R.P. was supported by NSF grant DMS-1148634.
[^2]: P.P. was supported by NSF grants DMS-1068169, DMS-1148634, DMS-1351590, and Sloan Fellowship.
[^3]:
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---
abstract: 'The [*Chandra*]{} Carina Complex contains 200 known O- and B type stars. The [*Chandra*]{} survey detected 68 of the 70 O stars and 61 of 127 known B0-B3 stars. We have assembled a publicly available optical/X-ray database to identify OB stars that depart from the canonical $L_{\rm X}/L_{\rm bol}$ relation, or whose average X-ray temperatures exceed $1$ keV. Among the single O stars with high $kT$ we identify two candidate magnetically confined wind shock sources: Tr16-22, O8.5 V, and LS 1865, O8.5 V((f)). The O4 III(fc) star HD 93250 exhibits strong, hard, variable X-rays, suggesting it may be a massive binary with a period of $>30$ days. The visual O2 If\* binary HD 93129A shows soft 0.6 keV and hard $1.9$ keV emission components, suggesting embedded wind shocks close to the O2 If\* Aa primary, and colliding wind shocks between Aa and Ab. Of the 11 known O-type spectroscopic binaries, the long orbital-period systems HD 93343, HD 93403 and QZ Car have higher shock temperatures than short-period systems such as HD 93205 and FO 15. Although the X-rays from most B stars may be produced in the coronae of unseen, low-mass pre–main sequence companions, a dozen B stars with high $L_{\rm X}$ cannot be explained by a distribution of unseen companions. One of these, SS73 24 in the Treasure Chest cluster, is a new candidate Herbig Be star.'
author:
- 'Marc Gagné, Garrett Fehon, and Michael R. Savoy'
- 'David H. Cohen'
- 'Leisa K. Townsley, Patrick S. Broos, and Matthew S. Povich'
- 'Michael F. Corcoran'
- 'Nolan R. Walborn'
- Nancy Remage Evans
- 'Anthony F.J. Moffat'
- Yaël Nazé
- 'Lida M. Oskinova'
title: 'Carina OB Stars: X-ray Signatures of Wind Shocks and Magnetic Fields'
---
Introduction
============
The [*Chandra*]{} Carina Complex Project (CCCP) [@townsley11a] survey area contains over 200 massive stars: the luminous blue variable (LBV) $\eta$ Car, the Wolf Rayet (WR) stars WR 22, WR 24 and WR 25, 70 known O stars, and 127 B0-B3 stars with determined spectral types and photometry consistent with a distance of 2.3 kpc to the Carina cluster Trumpler 16 (Tr 16) [@smith06]. In the next section we describe the database of OB stars in more detail. Briefly, the massive star population in Carina can be divided into groups based on spectral type, luminosity class and binarity: LBV/WR stars (4), early-O binaries (6), early-O single dwarfs and giants (10), late-O binaries (9), late-O dwarfs and giants (44), single O-type supergiants (1), B0-B3 stars (127), and B5-B7 stars (3).
In addition to the OB stars with measured spectral types considered here and by @naze11, this volume contains four other articles which discuss the massive-star population in Carina: @povich11a present a list of 94 candidate OB stars, selected on basis on their X-ray emission and infrared spectral energy distributions (SED); their spectral types have not been determined. @evans11 identify candidate late-B stars in Tr 16 based on their UBV photometry. @townsley11a analyze the CCCP spectra of the three WR stars and @parkin11 present a detailed X-ray spectral/temporal analysis of the double spectroscopic binary QZ Car, O9.7 I + O8 III. In this paper, we examine the X-ray, optical, and infrared characteristics of the 200 OB stars with determined spectral types and use X-ray spectra and light curves of this large sample of well-studied OB stars to better understand the physical mechanisms that produce X-rays in massive stellar systems.
The ubiquitous X-ray emission from O and early B stars is generally thought to arise in shocks embedded in the powerful radiation-driven winds of these stars. However, there are exceptions to this paradigm, with a subset of early-type stars showing harder and stronger X-ray emission than can be explained by embedded wind shocks (EWS) alone. We point to three recent X-ray surveys that have examined these questions in some detail: the [*XMM-Newton*]{} survey of NGC 6231 in the Sco OB1 association [@sana06a; @sana06b], the [*Chandra*]{} survey of Tr 16 [@evans03; @evans04], and the [*Chandra*]{} Orion Ultradeep Project [@stelzer05].
The Orion Nebula Cluster contains only two O stars, and both have unusual X-ray emission: $\theta^1$ Ori C, O7 Vp [@gagne05a], and $\theta^2$ Ori A, O9.5 SB [@feigelson02; @schulz06]. Of the early-B stars, @stelzer05 identify two groups: strong-wind sources (earlier than B4) that give rise to X-rays in wind shocks (relevant to this study), and weak-wind sources (later than B4) whose X-rays, when detected, may be produced by late-type pre-main-sequence (PMS) companions [see @evans11].
In their study of Tr 16 and part of Trumpler 14 (Tr 14), @evans03 [@evans04] analyzed the ACIS-I spectra of some of the prominent O stars in Carina, identifying HD 93250, O4 III(fc), and Tr16-244, O3/4 I, as highly unusual, and proposing a new colliding-wind binary system: Tr16-22, O8.5 V. Aside from these anomalously active stars, @evans03 find that the canonical $L_{\rm X} \approx 10^{-7} L_{\rm bol}$ relationship is obeyed.
In their comprehensive study of the OB stars in NGC 6231, @sana06b confirm the $L_{\rm X}/L_{\rm bol}$ relationship in the soft [*XMM-Newton*]{} bands below 2.5 keV, but show a breakdown in the correlation in the 2.5-10 keV hard band. They also confirm the prominent kink in $L_{\rm X}/L_{\rm bol}$ below $\log L_{\rm bol} = 38$ first noted by @berghoefer97 [see also @naze11].
In this paper, we focus mainly on three emission mechanisms, in addition to EWS, to understand the X-rays observed from O and B stars in the CCCP: colliding wind shocks (CWS), magnetically confined wind shocks (MCWS), and coronal emission from unseen pre-main–sequence (PMS) companions, though we discuss other mechanisms in §1.5. In particular, we focus on those stars that show high $L_{\rm X}/L_{\rm bol}$, hard X-ray spectra, or show notable time variability.
Embedded Wind Shocks
--------------------
The default mechanism for X-ray production on O and early B stars is embedded wind shocks (EWS), generally assumed to be associated with the Doppler deshadowing instability intrinsic to line driving [@lucy80; @lucy82; @owocki88; @feldmeier97]. The instability predicts that wind streams having different velocities interact, shock heating a modest fraction of the wind beyond a few tenths of a stellar radius. Several stellar radii above the photosphere, the shocked portion of the wind is traveling at speeds approaching the wind terminal velocity of thousands of km s$^{-1}$. [@dessart05a; @dessart05b].
But the relative velocities of the interacting wind streams are generally only a few hundred km s$^{-1}$, generating shock temperatures of just a few million K. This shock-heated plasma radiates a line-dominated soft X-ray spectrum. This soft X-ray emission generally shows little variability, indicating that there are numerous separate shock-heated regions of the wind at any given time. Wind clumping is a universal phenomenon in hot-star winds [@lepine99; @lepine00; @moffat08] and @feldmeier03 showed that the wind clumping affects the emergent X-ray line profiles.
The soft X-ray emission is attenuated by the colder, X-ray absorbing wind in which it is embedded. This wind attenuation affects the shapes of individual line profiles, as is evident in high-resolution X-ray spectroscopic studies of nearby O stars, which also confirm Doppler broadening commensurate with the wind terminal velocity, lending strong support to the EWS mechanism [@kahn01; @cassinelli01; @zhekov07; @walborn09; @cohen10].
The wind attenuation can harden the overall X-ray spectrum for early O stars with very high mass-loss rates. The attenuation of the X-rays from embedded wind shocks also affects the emergent X-ray luminosity. Empirically, a scaling of emergent X-ray luminosity with bolometric luminosity, $L_{\rm X} \approx 10^{-7} L_{\rm bol}$, has been long known [@pallavicini81], and is largely confirmed in the broad OB population in the CCCP [@naze11]. No strong theoretical expectation for this scaling has been found, although wind attenuation can explain it in principle [@owocki99].
For Carina O stars with X-ray emission dominated by the EWS mechanism, we should expect to see soft X-ray emission (characterized by $kT < 1$ keV) at a level corresponding to $L_{\rm X} \approx 10^{-7} L_{\rm bol}$, and with little variability. For B stars, significantly lower relative X-ray luminosities are seen, with values as low as $L_{\rm X} \approx 10^{-9}
L_{\rm bol}$ by spectral subtype B2 V [@cohen97].
While embedded wind shocks presumably exist in all early-type stars with strong radiation-driven winds, detailed X-ray studies of nearby O and early B stars show that a fraction of them have properties that are at odds with the EWS scenario. These usually involve higher X-ray luminosities and harder X-ray spectra than is predicted to arise in embedded wind shocks alone, and often involve X-ray time variability on orbital or rotational timescales. When high-resolution X-ray spectra are available, these often show narrower X-ray emission lines, as well. Corroborating evidence from other wavelengths generally shows evidence for either wind-wind collisions in binary systems or magnetic confinement of the wind. In the next four subsections, we summarize some of these theoretical and observational studies.
Colliding Wind Shocks
---------------------
In binary systems where both stars have strong winds, shock heating in the wind interaction zone between the two stars can lead to stronger X-ray emission than that seen in the EWS scenario [@stevens92; @pittard09]. The X-ray properties of CWS sources depend on the wind and orbital parameters of the two stars, and they vary with orbital phase in the case of systems with eccentric orbits. @antokhin04 predict that X-ray luminosity scales as the reciprocal of the binary separation. In many O+O, WR+O, and LBV+O-type systems in which both components have high mass-loss rates, CWS X-rays tend to dominate those from embedded wind shocks, with X-ray flux increasing for more equal-momentum winds and intermediate separations.
For example, the O+O binaries in the [*XMM-Newton*]{} survey of hot stars [@naze09] have, on average, $\sim3$ times higher $L_{\rm X}/L_{\rm bol}$ than single O stars. We note that most single and binary O stars in the [*XMM*]{} survey show relatively low-temperature shocks. In fact, the only O+O binaries in the [*XMM*]{} surveys with hard X-ray spectra are the long-period Carina binaries HD 93403, O5 III(fc) + O7 V [@naze09], and HD 93343, O8 V + O7-8.5 V [@antokhin08]. Long-period, WR+O and LBV+O binaries like WR 140 and $\eta$ Car also show very high-temperature shocks [@pollock05; @corcoran01].
@pittard09 model the colliding wind emission from pairs of O6 V + O6 V stars with varying orbital periods; their simulations predict peak temperatures of 3-4 MK and 10-20 MK for models [cwb1]{} (3-d period) and [cwb2]{} (10-d period), respectively. We will examine these issues in more detail in §6.3.
It can sometimes be difficult to confirm the binary nature of systems with X-ray properties indicative of CWS emission. Spectroscopic or photometric orbital variations are difficult to measure when the orbital inclination is low, although persistent observing [@hoffmeister08] and complementary approaches [@nelan04] reveal long-suspected companions in CWS systems. After many years of monitoring, @gamen08 derived a 207.7-d orbital period for the Carina WN6h colliding-wind binary WR 25. Observations of non-thermal radio emission often provide clues about the presence of wind-wind interactions in binaries, even when binary orbital parameters cannot be measured [@dougherty00; @debecker07]. Ideally, X-ray monitoring over multiple orbital periods is needed to show the orbital modulation expected from colliding wind systems, at least those with sufficiently high eccentricities.
Magnetically Confined Wind Shocks
---------------------------------
Where strong, large-scale magnetic fields exist on early-type stars, wind streams from opposite hemispheres are channeled toward the magnetic equator, where they collide head-on, leading to strong shocks and associated X-rays [@babel97]. MCWS X-ray emission can be differentiated from EWS X-rays by its temperature distributions [@wojdowski05], which can be dominated by plasma with temperatures of 30-50 million K. For oblique dipole configurations, the X-rays can be rotationally modulated due to differential occultation of the shock-heated plasma by the star [@gagne97].
The prototype MCWS source is $\theta^1$ Orionis C, O7 Vp, the illuminating star of the Orion Nebula, which has a 1.5 kG oblique dipole field, measured with Zeeman spectropolarimetry [@donati02; @wade06]. [*Chandra*]{} grating spectroscopy shows a high X-ray luminosity (corresponding to $L_{\rm X} > 10^{-6} L_{\rm bol}$) and peak X-ray emission measure $\sim 33$ MK. The X-rays vary (by roughly 20%) in a manner consistent with the rotationally modulated magnetic field configuration, and these X-ray properties are well explained by MHD simulations of the confined wind of this star [@gagne05a].
Early B stars of the chemically peculiar B2p class show very strong dipole magnetic fields, though they have much weaker winds than the O star $\theta^1$ Ori C. Some, like the prototype $\sigma$ Ori E, have modest, relatively hard X-ray emission [@sanz-forcada04; @skinner08], while other B2p stars have no X-ray emission or emission at levels below the detection threshold. Even the stronger X-ray sources among these Bp stars have X-ray luminosities of only $L_{\rm X} \approx 10^{30}$ ergs s$^{-1}$ cm$^{-2}$ [@drake94].
The ability to channel a wind depends both on the strength of the magnetic field and (inversely) on the density of the wind. These scalings are verified in detail by MHD simulations [@uddoula02; @gagne05a]. Some O stars with strong winds and weak fields, such as $\zeta$ Ori, have X-ray emission consistent with the EWS scenario [@cohen06].
On the other hand, the other well-studied magnetic O stars such as HD 108, O7 Ifpe, and HD 191612, 06.5 IIIf, have $L_{\rm X}/L_{\rm bol}\approx-6.2$ and $kT\approx0.3$ keV. They are more X-ray active than typical EWS sources, but substantially cooler than $\theta^1$ Ori C [@donati06a; @naze07; @naze10]. Also curious is the early B star, $\tau$ Scorpii, B0.2 V, which has a highly structured (non-dipole) field [@donati06b], strong and hard X-ray emission, but no evidence for wind confinement of the X-ray emitting plasma [@cohen03; @ignace10].
At this point, there are too few O and early B stars with well-characterized magnetic fields (aside from several dozen Bp stars), and their X-ray properties are too heterogeneous, to make definitive statements about what X-ray signatures will be seen from magnetic OB stars in Carina. However, we can say that it is possible that strong X-ray emission may be produced in Carina O stars via magnetically confined shocks if conditions are right. Overall, though, the general X-ray properties from the MCWS mechanism – elevated X-ray luminosities, hard X-ray spectra, and time variability – overlap with those from the CWS mechanism, though the variability in CWS systems is orbital, whereas the variability in MCWS is often rotationally modulated.
Unseen Pre–Main-Sequence Companions
-----------------------------------
The three X-ray emission mechanisms described above convert the kinetic energy of the supersonic wind into shocks and thermal X-ray emission. As the mass-loss rate and terminal wind speed diminish towards later spectral type, embedded wind shocks, colliding wind shocks, and magnetically confined wind shocks should be much weaker.
In star-forming regions, at spectral subtype $\sim$B2, $L_{\rm X}$ falls below a few $10^{30}$ ergs s$^{-1}$ [@caillault94; @daniel02], and the upper end of the low-mass PMS X-ray luminosity function begins to outshine the massive stars. It is now generally understood that hard, variable X-ray emission from main-sequence mid-B to late-A type stars is produced by unseen, lower-mass companions [@gagne95; @briggs03]. In most cases, the X-rays provide the only clue to the presence of cool companions. In a reverse application, @evans11 use X-ray detections in the CCCP and BV photometry to identify unknown B stars in Tr 16.
The situation for the PMS intermediate-mass stars, i.e., the Herbig Ae/Be stars, is quite different. Many are copious, often hard X-ray emitters, despite the fact that they have weak winds and radiative outer envelopes, so they are not expected to have wind shocks, or solar-type magnetic activity. @skinner04 consider low-mass PMS companions and an intrinsic, rotational shear dynamo to explain the X-rays from Herbig Ae/Be stars, but so far the mechanism is not known.
Other X-ray Emission Mechanisms
-------------------------------
In this paper, we assume that the soft X-ray emission from most O stars is produced in shock-heated, collisionally ionized, thermal plasmas. Specifically, plasmas whose densities are high enough to be modeled by equilibrium ionization codes like APEC.
However, other emission mechanisms have been proposed to explain the X-ray properties of massive stars. Most fundamentally, @pollock07 suggests that the X-rays from O supergiants like $\zeta$ Orionis, O9.7 Ib, are produced in the far wind where densities are low, and where the Coulumb collisional mean-free path is many stellar radii, so that the shocked wind is far from collisional or thermal equilibrium. In the @pollock07 scenario, [*ions*]{}, not electrons, ionize ions, and the shocks are collisionless; shock heating occurs via plasma processes and magnetic fields. The major difficulty with the @pollock07 paradigm is that the X-ray spectra of O stars appear to be produced by collisionally ionized, thermal plasmas with electron-ion bremsstrahlung continua [c.f., @raasen08].
The low-resolution ACIS CCD spectra from the CCCP cannot directly address the fundamental nature of the shock heating; we use the APEC model to estimate plasma temperatures and emission measures because the APEC model fits the CCD spectra well with only a few free parameters. When time variability or high-temperature components are observed, we examine secondary processes like colliding wind shocks, magnetically confined wind shocks, or coronal emission from unseen, lower-mass, pre–main-sequence companions, though other mechanisms have been proposed.
For example, the rigidly rotating magnetosphere model [@townsend05; @townsend08] and its successor, the rigid-field hydrodynamic model @townsend07, have been used to model the X-ray, UV and H$\alpha$ emission of the He-strong stars like $\sigma$ Ori E. This model may have applications to other B stars with strong magnetic fields like $\tau$ Sco, B0.2 V.
For the O9.5 V spectroscopic binary $\theta^2$ Ori A, @schulz06 proposed an interaction between the magnetospheres of the spectroscopic primary and secondary at periastron to explain its x-ray spectrum and time variability. Magnetic fields have yet to be seen on either component, though.
@cassinelli08 proposed a model for OB stars in which the thin fast wind forms bow shocks on dense, slowly moving clumps. The adiabatic shocks produce peak emission measure near $\log T = 6.5$, with some emission out to $\log T = 7.5$. It is unlikely that this model can operate in stars with strong magnetic fields like $\tau$ Sco or $\theta^1$ Ori C, but it might be viable in non-magnetic OB stars, though the effects of radiative cooling should be added to test the model further.
@waldron09 proposed a highly accelerated diamagnetic plasmoid model to explain the production of high-temperature plasma very close to the photospheres of some O stars. This is essentially the opposite mechanism proposed by @cassinelli08. In the diamagnetic plasmoid model, the blobs are ejected from the surface and plow into the cooler, slowly moving plasma at the base of the wind.
The origin of magnetic fields on massive stars is still not fully known. They could be generated by subsurface convection or via differential rotation, or they could be fossil fields. In the @spruit02 dynamo model, and its adaptation by @mullan05 [@mullan06], differential rotation and Tayler instabilities in the interior magnetic field lead to surface magnetic fields. Finally, we note that some of the known OB stars in Carina could harbor neutron star companions, producing strong, accretion-driven X-ray emission [e.g., @davidson73].
The discovery of magnetic fields on some massive stars, in combination with colliding winds from massive binaries, leads to a number of complicated but intriguing X-ray emission mechanisms. In this paper, we will focus on the X-ray diagnostics – variability, high $L_{\rm X}/L_{\rm col}$ or high $kT$ – that may signal the presence of magnetic fields or colliding wind shocks in the Carina massive stars.
Carina OB Star Optical Catalog
==============================
The goal of this study is to characterize the X-ray properties of the O and early-B stars with reliably determined spectral types in the $\sim 1$ square degree CCCP field, and to correlate those X-ray properties with effective temperature, bolometric luminosity, and binary properties. The optically selected sample is biased towards stars that are bright from 3900–4800 Å, and those that are close to the well-studied clusters Trumpler 14, 15 and 16, Bochum 10 and 11, and Collinder 228. This sample complements the X-ray selected massive-star candidates of @povich11a, though these are also biased towards luminous stars because of the correlation between $L_{\rm X}$ and $L_{\rm bol}$. The optically selected sample is biased to stars with $A_V < 3$ mag while the candidate massive stars of @povich11a are mostly obscured stars with $A_V > 3$ mag.
We began with the [@skiff09 and references therein] selecting stars with spectral type O2-B3. To these we added stars in @tapia03 [@smith06; @degioia01; @walborn95; @masseyandjohnson93], including two B5 stars, and a B7 star. For some individual stars, mostly spectroscopic binaries and Ofc stars, we used the updated spectral types of @morgan55 [@morrisonandconti80; @morrell88; @rauw00; @rauw01; @freyhammer01; @albacete02; @niemela06; @rauw09; @gosss10; @walborn10; @sota11].
The spatial distribution of the 200 OB stars is shown in Figure 1, where the clusters Trumpler 15, 14, and 16 are visible from N to S at $-59\arcdeg 24\arcmin$, $-59\arcdeg 36\arcmin$, and $-59\arcdeg 42\arcmin$, respectively. Many of the confirmed massive stars seen in Fig.1 are coincident with the low-mass populations in the rich clusters (Tr 14, Tr 15, Tr 16, and Coll 228), as expected. However, a significant fraction lie well outside the clusters, supporting the idea that star formation in Carina has been ongoing for a considerable length of time, allowing stars to drift or be ejected from their natal groupings. This is discussed further by @feigelson11 [@povich11a; @povich11b].
Johnson UBV photometry was gleaned primarily from the photoelectric catalog of and its update [@mermilliod94], and the CCD survey of Trumpler 14 and 16 by @masseyandjohnson93. For a few close visual binaries, we obtained UBV from @massey01 [@vazquez96; @forte81; @forte76].
We note that a number of other CCD and photoelectric photometry studies have been published on the Carina hot stars, particularly for those in Trumpler 14 and 16. For example, the compilation of @reed05, lists $V$, $U-B$, and $B-V$ from a number of reliable photometric surveys. Unfortunately, only 101 of the 200 OB stars in the CCCP are listed in the @reed05 catalog. Some surveys like @degioia01 and the [*Tycho*]{} catalog measured $B$ and $V$. After analyzing color-magnitude diagrams from a number of these studies, we chose the compilations of @mermilliod94 and @masseyandjohnson93 as our primary references because they provided consistent UBV measurements for all but one star: HD 92937, B2.5 II:.
For the 70 O stars, effective temperature was interpolated on the basis of spectral type, using the observational $T_{\rm eff}$ scale of @martins05. For consistency, we used the O-star synthetic colors of @martins06 to estimate $(B-V)_0$ and bolometric correction. We note that the Martins et al. work uses solar abundance, wind- and line-blanketed, non-LTE model atmospheres, that, for OB stars, lead to lower $T_{\rm eff}$ and $L_{\rm bol}$ for a given observed spectral type than did previous work.
For the 130 B stars, consistent sets of NLTE synthetic colors, effective temperatures, and bolometric corrections were not available, though NLTE effects are expected to be smaller in the lower-luminosity Carina B dwarfs. We used the intrinsic colors of @wegner94 to estimate $(B-V)_0$. For the B stars, we used the bolometric corrections and effective temperature scale of @bessell98.
Reddening and Bolometric Luminosity
-----------------------------------
This study and the study of @naze11 rely on two methods for determining $L_{\rm bol}$: (1) the spectral energy distribution (SED) fitting method of @povich11a, and (2) the more traditional method employing color excess and bolometric correction. We recall that in the color excess method, $M_{\rm bol} = V - A_V - {\rm DM} + {\rm BC}_V$, $R = A_V/E(B-V)$ and $E(B-V) = (B-V) - (B-V)_0$, where ${\rm BC}_V$ is the $V$-band bolometric correction and DM is the distance modulus.
Of the 200 OB stars in our spectroscopic sample, 182 have high-quality UBV, 2MASS JHK and [*Spitzer*]{} IRAC photometry in the VelaÐCarina Point-Source Archive [PI: S. Majewski, @povich11b]. @povich11a used the SED fitting method of @robitaille07, which applies an ISM absorption model to the revised ATLAS9 stellar atmosphere models of @castelli04. Though we recognize that the ATLAS9 LTE model atmospheres do not account for non-LTE wind and line blanketing effects, @povich11a show that LTE and non-LTE atmospheres produce similar broad-band optical/infrared SEDs, and hence very similar estimates of $A_V$ and $L_{\rm bol}$.
Recognizing that a number of stellar and ISM parameters can adequately fit a 10-point SED, @povich11a generate a family of solutions for each SED. The 182 OB stars in the validation sample have the advantage of having well-determined spectral types, and hence $T_{\rm eff}$, thereby significantly reducing the number of statistically acceptable solutions, and significantly constraining $A_V$ and $L_{\rm bol}$.
The SED fitting depends on the adopted reddening law, which effectively specifies the scaling between $A_V$ and the absorption in each photometric band. In the traditional color excess method, this comes down to choosing $R_V=A_V/E(B-V)$. As described in @povich11a, the estimates for $R_V$ in Carina have varied from an ISM value of 3.1 or 3.2 [@turner80a; @turner80b] to above 5.0 [@herbst76].
Table 1 of @walborn95 illustrates the interplay between $R$ and DM in the Carina clusters. Throughout the CCCP, we adopt ${\rm DM} = 11.81$, corresponding to $d=2.3$ kpc, the distance to $\eta$ Carinae derived from the Doppler velocity and proper motion of the expanding Homunculus nebula [@smith06]. @walborn95 find that for this distance modulus, $R=4$ provides the best fit to the O stars in Tr 16, though star-to-star, and cluster-to-cluster variations probably do exist. Similarly, @povich11a find better agreement between the $L_{\rm bol}$ estimates using $B-V$ and those derived from the optical-infrared SED method using $R_V=4$ than using $R_V=3.1$ or $R_V=5$. Hence, we and @povich11a adopt an average $R_V=4$ and $d=2.3$ kpc to all the stars in Carina. We do not attempt to account for star-to-star variations in the reddening law.
We note that, based on Walborn’s analysis of six O stars in Tr 14, $R_V=4$ implies ${\rm DM} = 12.33$ if the Tr 14 and Tr 16 are coeval (and thus the O stars in both clusters have the same average luminosity). A more likely solution, however, is that Tr 14 is younger than Tr 16 and the main-sequence O stars are sub-luminous relative to the average class V calibration.
For 181 of the 200 stars in the Carina OB database, we compute $L_{\rm bol}$ using both the SED (tag name [loglbol\_sed]{}) and color excess ([loglbol\_bv]{}) methods. We note that one star, MJ 501=Tr 16 74, B1 V, has unreliable $B-V$. For these 181, the offset in $A_V$ (SED - BV) is $\Delta A_V = 0.1$ with a standard deviation $\sigma=0.25$ mag. This is a lower bound on the systematic uncertainty in $A_V$. The offset in $\log L_{\rm bol}$ (SED - BV) is $\Delta \log L_{\rm bol} = 0.04$ with a standard deviation $\sigma=0.14$ mag.
We note though that CMFGEN model atmospheres [@hillier01] produce more EUV, NIR, and MIR emission for a given $T_{\rm eff}$ and $\log g$ than non-LTE model atmospheres, especially for O supergiants. Because we wish to exploit the full set of optical, near- and mid-infrared photometry, and to maintain consistency with the rest of the CCCP, we use the SED-derived $L_{\rm bol}$ values whenever possible, despite this deficiency. Future updates to the OB catalog will contain new $L_{\rm bol}$ estimates, based on updated SED-fitting codes.
The FITS table headers and the CDS header files list the name, unit, and fortran format code for all columns in the OB catalog. The first 38 columns are devoted to the optical/infrared database and include the following columns (or tags): IAU J2000 name, default designation (label), HD name, CPD name (from the 1919 list of Cannon, Pickering & Draper), LS or ALS name [@reed05], Feinstein Trumpler 14/15/16, Collinder 228, or Bochum 10/11 designation, and @masseyandjohnson93 designations. J2000 R.A. and Decl. are listed in decimal degrees and sexagesimal hours and degrees. Also listed are spectral type, spectral type reference, binary flag, notes, $V$, $U-B$, $B-V$, $Q$ [@masseyandjohnson93], and UBV reference. We list the derived quantities $(B-V)_0$, $E(B-V)$, BC, and $A_V$, $\log L_{\rm bol}$ (solar units), and $\log T_{\rm eff}$ as determined via the color excess method, the SED fitting method, and the adopted value. We note that the label tag contains the commonly used designation used in the text, tables and figures, but that the other designations in the electronic files are precisely those used by [*Simbad*]{}, and by association other databases, for rapid searching.
Table 1 shows a cross section of the optical database for the most luminous stars, sorted by $\log L_{\rm bol}$. Figure 2 shows the resulting H-R diagram. We overlay the [*average*]{} main sequence of @martins05b from their sample of 1-5 Myr-old stars. The B-star main sequence is from @dejager87 [see also @povich11a].
The zero-age main sequence (ZAMS) is clearly delineated in Fig. 2, as is the extent of the terminal-age main sequence (TAMS). The supergiants and bright giants are shown with white dots. The uncertainty in $L_{\rm bol}$ is dominated by random photometric errors (0.1 mag) and by systematic uncertainties, especially in the distance modulus and $R=A_V/E(B-V)$, on the order of 0.3 mag. As noted earlier, we use a global reddening law; we do not account for possible star-to-star variations in $R$. Despite these uncertainties, much of the spread above the ZAMS appears to indicate a real spread in age. We note that the most luminous OB star in Carina, QZ Car, is a quadruple system [@parkin11]. $T_{\rm eff}$ for the candidate Herbig Be star SS73 24 was estimated from its optical-infrared SED (see §6.4.1).
All but two of the 70 O stars were detected with at least 3 source counts in the CCCP: Tr14-27, O9 V, Tr15-18, O9 I/II:(e:). Tr15-18 also has the highest $A_V$ of all 200 OB stars; it may have avoided detection because of its high column density. While $A_V=5.66$ mag for Tr15-18 is high among the known OB stars, it is typical of the X-ray selected candidate OB stars identified by @povich11a. Finally, we note that Tr15-19, O9 V, and Tr15-21, B0 III, are well below the ZAMS; they may be a background stars, not members of Trumpler 15 (Tr 15).
To estimate the completeness of the optical OB catalog, we consider the populations of OB stars with measured spectral types, and the 94 X-ray-detected stars without measured spectral types discovered by @povich11a. Using an initial mass function (IMF) exponent $\Gamma = -1.3$ [@kroupa02], normalized to the 47 LBV, WR, and O2-O8.5 V stars with mass $M > 20 M_{\odot}$, we expect to find 24 stars in the mass range $15-20 M_{\odot}$, compared to 27 O9 and O9.5 stars, and 130 stars in the mass range $7-15 M_{\odot}$, compared to 123 main-sequence B0-B2.5 stars. Note, we are considering only the primary stars. This suggests that, if the number of early-mid O stars is complete, then the number of late-O and early-B stars is also nearly complete.
However, the 94 stars identified by @povich11a with $L_{\rm bol} > 10^4 L_{\odot}$ could have main-sequence spectral types as early as O4. I.e., the @povich11a result implies that a substantial fraction of early-O to early-B stars do not have measured spectral types. To estimate this fraction more precisely, we note that 140 stars in our optical catalog have $L_{\rm bol} > 10^4 L_{\odot}$, and that 76% are detected in the CCCP. Applying this detection fraction to the 94 stars in Table 3 of @povich11a suggests approximately 124 stars with $L_{\rm bol} > 10^4 L_{\odot}$. I.e., an additional $\sim 30$ luminous stars were not detected with [*Chandra*]{}. This suggests a total population of 140 + 94 + 30 = 264 luminous stars; our optical catalog is thus $\sim 53\%$ complete.
Thus if the @povich11a result is confirmed, then a significant fraction (up to half) of the OB stars in the CCCP region have yet to be spectroscopically identified. These stars have higher than average $A_V$ and generally reside outside the well-studied clusters in Carina. Similarly, @wright10 find that a large fraction of the B-star population in the Cyg OB2 star-forming region is not identified.
Carina OB Star X-ray catalog
============================
The 200 OB star optical positions were matched to the list of 14368 CCCP sources [see @broos11a for a detailed description of the catalog matching]. The astrometric systems in the two catalogs were well aligned ($\Delta\alpha = 0.005\arcsec$ and $\Delta\delta = 0.012\arcsec$), and the optical and X-ray positions reported for individual stars were in good agreement (the median offset was $r\approx0.19\arcsec$). This initial pass through the CCCP X-ray data produced 118 matches (out of 200), including 67 of 70 O stars, and 51 of 130 B stars.
In order to estimate count upper limits for the 82 OB stars not detected in the automated [*ACIS Extract*]{} procedure described by @broos11a, we determined optimal source and background regions for the undetected OB stars at their optical positions, accounting for the [*Chandra*]{} point spread function and the presence of nearby X-ray point sources.
The full [*ACIS Extract*]{} procedure was run again on the combined list of 14368 sources, 82 undetected OB stars, and a number of photometrically selected candidate mid-to-late B stars [@evans11]. The [*ACIS Extract*]{} procedure used in the CCCP produces source, background and net counts in three bands: hard (2-8 keV), soft (0.5-2 keV) and total (0.5-8 keV). The procedure also estimates the probability $P_{\rm B}$ in these three bands that the source counts were produced by fluctuations in the observed local background.
Although they were not identified in the automated source detection procedure, an additional 11 stars met the detection criteria used in the CCCP catalog: at least 3 extracted counts and $P_{\rm B} \leq 0.01$ in a least one of the three bands [@broos11a]. They are: Coll228-66, O9.5 V, LS 1745, B2 III, HD 305515, B1.5 Vsn:, Tr16-13, B1 V, HD 305534, B0.5 V: + B1 V:, Tr16-29, B2 V, LS 1866, B2 V, Tr16-33, B2 V, HD 93342, B1 Iab-Ib, Coll228-81, B0.5 V, HD 93723, B3 III. Thus, the only undetected O stars are Tr14-27, O9 V, Tr15-18, O9 I/II:(e:). Thus the OB catalog contains 129 X-ray matches (out of 200), including 68 of 70 O stars, and 61 of 127 B0-B3 stars. We note that the three stars with spectral types later than B3 were not detected in X-rays.
For the undetected stars, the 90% count rate upper limit, [netcounts\_hi\_t]{} is used to derive a photon flux upper limit corrected for the mean effective area in the 0.5-8 keV band, the effective exposure time, and the PSF fraction of the selected source extraction region [using eqn. 1 in @broos11a].
The 200 OB stars have been divided into three groups: (1) 63 undetected stars with $P_{\rm B} > 0.01$ for which we calculate a photon flux upper limit, (2) an additional 51 X-ray detected OB stars with 3-50 counts, for which we also determine median energy, $E_{\rm med} =~$[medianenergy\_t]{} and the 0.5-8 keV absorbed (uncorrected) energy flux, $f_{\rm X} =~$[energyflux\_t]{} [see eqns. 1 and 2, @broos11a], and (3) 78 stars with at least 50 net counts for which @naze11 also derive $kT$ and $N_{\rm H}$ by fitting the ACIS spectra in XSPEC with a one- or two-temperature VAPEC emission model, and a two-component [tbabs]{} absorption model.
The results of the XSPEC fitting procedure are described fully by @naze11, but the absorption modeling deserves mention here. For all 200 OB stars, we have estimated $A_V$ and converted that absorption into an ISM column density using $N_{\rm H}/A_V = 1.6\times10^{21}~{\rm cm}^2~{\rm mag}^{-1}$ [@vuong03; @getman05][^1].
In the XSPEC fits, @naze11 fix the first ISM column density parameter, and allow a second column density parameter to vary. Many OB-star CCD spectra appear to require this extra absorption, presumably caused by absorption of shocked emission in the overlying wind.
The absorbed X-ray fluxes, used in this paper were computed in XSPEC with the first absorption parameter set to $N^{\rm ISM}_{\rm H}$, and the second absorption parameter at its best-fit value. The [*unabsorbed*]{} fluxes were computed using the best-fit emission model, the ISM absorption parameter set to zero, and the second absorption parameter at its best-fit value. This way the unabsorbed flux represents the flux emerging from the far-wind of the star, corrected for ISM absorption. $L_{\rm X}$ is the corresponding 0.5-8 keV unabsorbed X-ray luminosity assuming $d=2.3$ kpc, [fitluminosity\_tc]{}.
We note that @naze11 derive the absorption-corrected 0.5-10 keV X-ray luminosity, related to tag name [fcto]{}, assuming the same distance. We note that 0.5-8.0 keV is the total band of the CCCP, and 0.5-10 keV is the default energy range of previous [*XMM-Newton*]{} surveys [e.g., @sana06b]. The two X-ray luminosities are very similar, with an average offset of 0.008 in the log, and RMS deviation of 0.033 in the log.
We note that the four X-ray brightest stars were piled up in the CCCP ACIS-I data: HD 93129A, O2 If${\ast}$, HD 93205, O3 V + O8 V, HD 93250, O4 III(fc), and QZ Car, O9.7 I + O8 III. A more extensive set of QZ Car data have been fully analyzed by @parkin11. In the X-ray catalog, we use the average QZ Car parameters in Table 4 of @parkin11. For HD 93129A and HD 93205 @naze11 performed their two-temperature XSPEC analysis on the pileup-corrected spectrum [for more details see @broos11a]. For HD 93250, we used a series of [*Chandra*]{} ACIS-S observations; these data and the XSPEC spectral analysis are described in Appendix A.
Throughout this paper, we use $kT_{\rm avg}$, the emission-measure weighted mean $kT$ for the 78 OB stars with XSPEC fit parameters. For the 53 stars with one-temperature XSPEC fits, $kT_{\rm avg} = kT_1$. For the 24 stars with two-temperature XSPEC fits, and for QZ Car’s three-temperature fit, $kT_{\rm avg} = \Sigma kT_i n_i / \Sigma n_i$, where $n_i$ is the normalization parameter of the $i$th temperature component. The volume emission measure is $4\pi d^2\Sigma n_i$.
Thus, in addition to the many parameters produced by the extended [*ACIS Extract*]{} procedures described by @broos11a, our OB X-ray database contains an additional 30 parameters to describe the results of the two-temperature XSPEC fits of @naze11. Merging the X-ray parameters with the optical parameters for each star produces a set of tables with 200 rows (one row for each OB star). We used the following column names (i.e., IDL structure tags) extensively to create the figures and tables in this paper: [label]{}, [sptype]{}, [ktmean]{}, [loglbol]{}, [logt]{}, [probks\_single]{}, [probks\_merge]{}, [probnosrc\_t]{}, [netcounts\_t]{}, [netcounts\_hi\_t]{}, [fitluminosity\_tc]{}, [medianenergy\_t]{}, [energyflux\_t]{}, where [\_t]{} refers to the total 0.5-8 keV band, and [\_tc]{} refers to total band, absorption-corrected.
The full set of optical/X-ray tables are available electronically as a machine-readable tables from the CDS, a FITS binary table file, and an IDL save file[^2]. The catalog can also be queried from the Vizier database at the CDS.
X-ray Spectral Results
======================
Tables 2 and 3 summarize the X-ray properties for the single O stars and the known O+O binaries, respectively, sorted by $\log L_{\rm X}/L_{\rm bol}$. Table 4 summarizes the X-ray spectral and timing analysis results for the 28 X-ray brightest B stars, sorted by $\log f_{\rm X}$. From Tables 2, 3, and 4 we find:
1. The great majority of single and binary O stars have $-6.8 < \log L_{\rm X}/L_{\rm bol} < -8.0$ and have soft X-ray spectra characterized by $kT_{\rm avg} < 0.8$ keV.
2. A small number of single O stars have hard X-ray spectra characterized by $kT_{\rm avg} \gtrsim 1$ keV: HD 93250, O4 III(fc), MJ 496, O8.5 V, and MJ 449 = LS 1865, O8.5 V((f)).
3. The spectroscopic O+O binaries with $P_{\rm orb} > 10$ days like QZ Car, HD 93403 and HD 93343 show hard X-ray spectra characterized by $kT_{\rm avg} > 1$ keV.
4. The O+O binaries with $P_{\rm orb} < 10$ days like HD 93205, HD 93161A, FO 15 and five other known short-period O+O binaries in Tr16 have softer X-ray spectra characterized by $kT_{\rm avg} < 1$ keV.
5. The shortest-period binary in Carina, FO15, O5.5 Vz + O9.5 V, has low $\log L_{\rm X}/L_{\rm bol} \approx -7.65$.
6. All but one of the B stars with $\log L_{\rm X} \gtrsim 31$ have $kT_{\rm avg} \gtrsim 1$ keV
7. All but one of the B stars with $\log L_{\rm X} < 31$ or $\log f_{\rm X} < -14$ have relatively soft X-ray spectra, with $kT_{\rm avg} < 0.6$ keV.
The $L_{\rm X}/L_{\rm bol}$ patterns have been discussed in detail by @naze11. We recall that the ISM-corrected $L_{\rm X}$ is computed for the 78 stars with more than 50 counts in XSPEC, whereas $f_{\rm X}$, which is not corrected for ISM absorption, was computed for all the X-ray detected stars in [*ACIS Extract*]{} (except HD 93250, HD 93129A, HD 93403 and QZ Car). Noting that $f_{\rm bol} = {L_{\rm bol}} / {4\pi d^2}$, in Figure 3 we show $\log f_{\rm X}$ versus $\log f_{\rm bol}$ using the same symbol definitions as Figs. 1 and 2 to highlight the location of early and late, single and binary O stars, B stars, and hard and soft X-ray spectra.
In Figs. 3 and 4, the 51 X-ray detected stars with 3-50 counts are shown as filled diamonds; 49 are B stars (pink). We categorize the OB stars as [*hard*]{} (with $kT_{\rm avg} > 1.8$ keV, shown as triangles in scatter plots), [*medium-energy*]{} (with $kT_{\rm avg}$ in the range 1-1.8 keV, squares), and [*soft*]{} (with $kT_{\rm avg} < 1$ keV, circles). In Figs. 3 and 4a, the upper dotted lines represent $\log f_{\rm X}/f_{\rm bol} = {-7}$ and the lower dashed lines represent $\log L_{\rm X}/L_{\rm bol} = {-7.23}$, the mean of $\log L_{\rm X}/L_{\rm bol}$ with $kT_{\rm avg} < 1$ keV.
The upper and lower panels of Figure 5 show the number distributions of $\log L_{\rm X}$ and $\log f_{\rm X}$, respectively, for O and B stars: early-O binaries are black, early-O single stars are blue, late-O binaries are dark green, late-O single stars are light green, and B stars are pink. Fig. 3 shows two populations of O stars (green and blue/black symbols): (i) a population of relatively cool X-ray sources clustered around $\log f_{\rm X}/f_{\rm bol} \approx Ð7.55$, though there are some hotter sources in this group, and (ii) a group of sometimes hotter X-ray sources with higher $\log f_{\rm X}/f_{\rm bol}$. This trend is emphasized in Figure 4, which plots $\log L_{\rm X}/L_{\rm bol}$ versus $kT_{\rm avg}$ and $\log f_{\rm X}/f_{\rm bol}$ versus median energy $E_{\rm med}$ for the same samples as Fig. 3. The dotted and dashed vertical lines coincide with the dividing lines between soft, medium and hard symbol definitions used in Figs. 1-4. The soft, low $L_{\rm X}/L_{\rm bol}$ stars are clumped in the lower left of both panels of Fig. 4. The horizontal dotted and dashed lines correspond to $\log L_{\rm X}/L_{\rm bol} = -7$ and $-7.23$. This latter value is the mean of $\log L_{\rm X}/L_{\rm bol}$ for the stars with $kT_{\rm avg} < 1$ keV [@naze11].
The upper and lower panels of Figure 5 show the distributions of $\log L_{\rm X}$ and $\log f_{\rm X}$, respectively, for the samples shown in Figs. 3 and 4. The upper and lower panels of Figure 6 show the distributions of $\log L_{\rm X}/L_{\rm bol}$ and $\log f_{\rm X}/f_{\rm bol}$, respectively, for the samples shown in Figs. 3-5. The histograms suggest two distributions, as do Figs. 3 and 4: a group of stars centered around $\log L_{\rm X}/L_{\rm bol} < -6.8$ and $kT_{\rm avg} < 1$ keV, and a group with somewhat higher $\log L_{\rm X}/L_{\rm bol}$ and $kT_{\rm avg} \geq 1$ keV. We note a similar pattern for the B stars.
The upper and lower panels of Figure 7 show the distributions of $kT_{\rm avg}$ and median energy $E_{\rm med}$, respectively, for the samples in Figs. 3-6. The upper panel of Fig. 7 reflects the trend seen in the top panel of Fig. 4: a large distribution of OB stars with $kT\approx0.5$ keV, and a small, broad distribution of stars with $kT > 1$ keV. The lower panel of Fig. 7 shows no clear segregation; $E_{\rm med}$ is a less useful measure of intrinsic X-ray temperature, because higher absorption also produces higher median energy.
Figs. 3, 4, 6 and 7a suggest two populations of OB stars. The first is a low activity group with $\log L_{\rm X}/L_{\rm bol} < -6.8$ and $kT < 1$ keV. We suggest that for most of these stars, conventional embedded wind shocks produce most of the X-ray emission. We suggest that the stars with $kT_{\rm avg} > 1$ keV require an additional X-ray emission mechanism to produce the enhanced activity and hotter X-ray shocks. We will explore these results in more detail for single O stars, astrometric binaries, O+O binaries, and B stars in §6 and §7 below.
Table 4 lists the names, spectral types, and selected X-ray properties of the 28 X-ray brightest early-B stars, sorted by $f_{\rm X}$. We note that 14 of the 15 X-ray brightest early-B stars in Table 4 have $\log L_{\rm X}/L_{\rm bol} > -7$, that 14 of those 15 have $kT_{\rm avg} \gtrsim 1$ keV, and that 7 of the top 12 are probably variable on long time scales, though none show the characteristic rapid rise and slower decay of strong PMS flares (see Figs. 9 and 10). For the B stars (in pink), Fig. 4b shows a wide range of median energy and $f_{\rm X}/f_{\rm bol}$, with only a slight positive correlation.
We address the origin of the X-ray emission among the 130 early B stars by comparing the X-ray flux distributions of the early B stars and the X-ray detected low-mass stars. Figure 8 shows histograms of log photon flux for 61 X-ray detected early B stars (in pink), 69 B-star upper limits (in brown), and the 14250 CCCP sources not associated with massive stars (in gray hatching). The vast majority of the general CCCP source population is associated with low-mass PMS stars [@feigelson11]. The PMS histogram has been normalized to the 130 B stars for comparison. We note that $f_{\rm X}$ cannot be computed for the undetected sources, so we plot the 0.5-8 keV photon flux (or the 90% confidence upper limit), corrected for the time-averaged PSF fraction and effective area at the location of each star.
At first glance, Fig. 8 suggests that, overall, the X-ray detected B stars have higher flux than the X-ray detected low-mass PMS stars. For example, the 61 X-ray detected B stars have mean X-ray flux $f_{\rm X} = 4.6\times10^{-15}$ ergs cm$^{-2}$ s$^{-1}$, with a standard deviation $\sigma=2.7\times10^{-15}$ ergs cm$^{-2}$ s$^{-1}$. The upper limits span one order of magnitude in photon flux, illustrating the non-spatial uniformity of the CCCP survey [c.f. @townsley11a]. Fig. 8 suggests that the CCCP is only complete to a photon flux of $\sim 10^{-3}$ counts ks$^{-1}$ cm$^{-2}$.
Adjusting for this bias, two results emerge from Fig. 8: (i) the B stars with log photon flux below $-2.3$, corresponding to $\log f_{\rm X} < -14$ ergs cm$^{-2}$ s$^{-1}$, appear to have the same flux distribution as the low-mass PMS stars, with a companion fraction $\sim 50\%$, and (ii) the distribution of B stars with $\log f_{\rm X} > -14$ ergs cm$^{-2}$ s$^{-1}$ cannot be explained by a distribution of typical, low-mass PMS companions.
Result (i) is consistent with the results of @evans11: most X-ray detected B stars probably harbor coronal PMS companions. Result (ii) suggests either (1) the brightest dozen or so B stars formed with hyperactive companions, or (2) that their X-rays are produced by some intrinsic shock mechanism.
These stars are: SS73 24, Be pec, Tr16-64, B1.5 Vb, Tr16-10, B0 V, Tr16-5, B1 V, Tr14-28, B2 V, HD 93501, B1.5 III:, Coll228-68, B1 Vn, Tr14-124, B1 V, HD 93190, B0 IV:ep, Tr14-18, B1.5 V, LS 1813, B2 V, Tr16-11, B1.5 V, Tr14-19, B1 V, and Tr14-29, B1.5 V.
X-ray Timing Analysis
=====================
We have looked for time variability in the CCCP event data of the 129 X-ray detected OB stars in three ways: (1) we visually examined the sequenced and stacked corrected photon flux light curves, (2) we used the merged Kolmogorov-Smirnov test probability, $P_{\rm KS} =~$ [probks\_merge]{} in the CCCP data products [@broos10], and (3) we used a maximum likelihood procedure to divide the event data into blocks of approximately constant count rate. The latter, an IDL procedure [mlb\_acis]{} developed by E. Flaccomio, was used by @wolk05 to identify flares in the [*Chandra*]{} Orion Ultradeep Project (COUP). An X-ray source is variable if an event list produces more than one maximum likelihood block. In [mlb\_acis]{}, the significance thresholds are calibrated as a function of counts using a large set of simulated constant count rate sources. For each star, the number of maximum likelihood blocks, assuming at least 5 counts per block, and 95% confidence, are listed in the last column of Tables 2, 3, and 4.
We note that the maximum likelihood procedure can find two maximum likelihood blocks for a constant source, if that source is observed in two OBSIDs at significantly different off-axis angles. This was not an issue in the study of @wolk05 because the COUP observations were all centered on the Trapezium cluster – stars appeared at the same off-axis angle. We note that the photon flux curves produced by [*ACIS Extract*]{} are corrected for effective area and do not suffer this bias.
The sequenced photon flux light curves of the visually identified sources are shown in Figures 9 and 10. Some very bright sources like HD 93403 in the lower left panel of Fig. 9, show significant long- and short-term variability, and others like FO 15 in the upper left panel of Fig. 9 show clear long-term variability. None of the OB stars showed the characteristic rapid rise and slower decay of strong coronal flares [@wolk05].
Notes on Individual Massive Star Systems
========================================
In this section we examine the X-ray, optical, infrared, and radio properties of the most X-ray luminous OB stars, dividing them into four categories based on spectral type and binary separation: astrometric O binaries, spectroscopic O+O binaries, apparently single O stars, including weak-wind stars, and B stars.
Single O-type Stars
-------------------
Table 2 and Fig. 4 suggest that most single O stars have low $L_{\rm X}/L_{\rm bol}$ and low $kT$. Below we consider the three single O stars with $\log L_{\rm X}/L_{\rm bol} > -7$ and $kT > 1$ keV: HD 93250, MJ 496 = Tr16-22, and MJ 449 = LS 1865.
### HD 93250, O4 III(fc)
The most X-ray luminous O star in Carina, and possibly the most enigmatic, is the Ofc star HD 93250, located east of Tr 14, and north of Tr 16. Though long classified as O3.5 V((f+)) [@walborn02; @skiff09], @walborn10 have reclassified it as O4 III(fc). They write, “The Ofc category consists of normal spectra with $\lambda\lambda$4647-4650-4652 emission lines of comparable intensity to those of the Of defining lines $\lambda\lambda$4634-4640-4642.”
HD 93250 was observed far off-axis in a series of HETG/ACIS-S observations of $\eta$ Car (see Appendix A). The merged ACIS-S spectrum in Figure 11 shows strong and emission lines and strong Fe K$\alpha$ emission at 6.7 keV, mostly from , confirming the presence of extremely hot plasma ranging in temperature from 6-40 MK. Its mean 0.5-8 keV X-ray luminosity is $L_{\rm X} \approx 1.5\times10^{33}$ ergs s$^{-1}$, similar to the value found by @sanchawala07a.
Figure 12 shows the $L_{\rm X}$ light curve of HD 93250 based on the XSPEC fits to the individual ACIS-I and ACIS-S datasets. In all but one OBSID, its X-ray luminosity is marginally consistent with a single value, $L_{\rm X} \approx 1.45\times10^{33}$ ergs s$^{-1}$. During the ACIS-I OBSID 4495 when the source was marginally piled up, $L_{\rm X} = 1.7\pm0.1\times10^{33}$ ergs s$^{-1}$, suggesting $\sim20\%$ time variability on time scales of months (between observations), not unlike the variability seen in a series of 21 [*XMM-Newton*]{} spectra of HD 93250 [@rauw09].
Based on the spectral fits presented in Appendix A, we use the wind mass column to derive a mass-loss rate $\dot{M}=1.4\pm0.5\times10^{-6} M_{\odot}$ yr$^{-1}$. This is a factor of 3-4 smaller than the value estimated by @puls96, who did not allow for wind clumping, yet somewhat larger than the recent estimate of @martins05b, $\dot{M}=6\pm2\times10^{-7} M_{\odot}$ yr$^{-1}$.
@oskinova07 have shown that the conventional treatment of wind clumping commonly used in stellar atmosphere codes [e.g., @martins05b] leads to an underestimate of empirical mass-loss rates by a factor of few. The mass-loss rate for HD 93250 derived from the CCCP spectral fits appears to be generally consistent with the findings of @oskinova07.
Along with HD 93129A, HD 93250 is one of only two non-thermal radio O stars in Carina, and @debecker07 list it as a “suspected binary” on the basis of its X-ray and non-thermal radio emission. @rauw09 note however that the only radio detection of HD 93250 is at 8.6 GHz [@leitherer95]. The radio flux is assumed to be non-thermal because, if it were thermal, the 8.6 GHz flux would suggest a mass-loss rate $\dot{M} > 10^{-5} M_{\odot}$ yr$^{-1}$.
Though HD 93250 shows all the signatures of a colliding wind binary, an optical companion has yet to be detected. @nelan04 [and Nelan & Walborn 2010, private communication] found no evidence of a luminous astrometric companion in their [*HST*]{} FGS data, and @rauw09 detected no statistically significant radial velocity variations ($\sigma = 1.3$ km s$^{-1}$ using seven absorption lines over six epochs).
As @rauw09 point out, we are left with two plausible scenarios: (i) a wide, massive binary that has evaded detection, or (ii) magnetically confined wind shocks around a single magnetic O4 star.
Magnetically confined wind shocks are difficult to reconcile with strong non-thermal radio emission. Though non-thermal radio emission would be produced in magnetized wind shocks, most of that emission would occur within a few stellar radii of the visible photosphere, well inside the radio photosphere. At 8.6 GHz, the wind of the O4 giant is optically thick out to hundreds of stellar radii [e.g., @debecker04].
Given the similarity between the primary of HD 93403, O5 III(fc), and HD 93250, O4 III(fc), in both its optical and X-ray spectral properties, and given the observed correlation between shock temperature and binary period (see Fig. 13 and §6.3), we speculate that HD 93250 is a wide binary with an orbital period of 30-60 days. If the primary also possesses a strong magnetic field, high-S/N X-ray grating spectra will be needed to untangle the origin of the strong X-ray shocks.
Detections at two or more radio frequencies are needed to measure HD 93250’s spectral index, and to confirm its non-thermal radio emission. In addition, spectro-polarimetric monitoring is needed to look for surface magnetic fields and measure the primary’s rotation period. Ground-based optical interferometry [e.g., @patience08] is needed to look for a luminous companion inside the $\sim20$ AU limit imposed by the [*HST*]{} FGS.
### Tr16-22 = MJ 496, O8.5 V
The single O8.5 V dwarf MJ 496=Tr 16 22 was detected with [*Chandra*]{} by @albacete-colombo08 and @sanchawala07a and with [*XMM-Newton*]{} by @naze09 and @antokhin08. @morrell01 use Tr16-22 as a spectroscopic template for the O8 V secondary in the colliding wind binary HD 93205. Compared to the results in Table 3, @albacete-colombo08 derive similar, though slightly lower, $kT_1$, $kT_2$, $L_{\rm X}$, and $L_{\rm bol}$, and list Tr16-22 as a probable binary. Similarly, @evans03 single out Tr16-22 as a possible colliding wind binary. We note however that Tr16-22’s $L_{\rm X}$ and $\log L_{\rm X}/L_{\rm bol}$ are higher than those of any late-O + late-O binary in Carina. As such, Tr16-22 is a good candidate for magnetically confined wind shocks. High-S/N spectro-polarimetric monitoring are needed to detect and measure its magnetic geometry.
### MJ 449 = CPD-59 2610 = LS 1865, O8.5 V((f))
The last of the apparently single O stars with very hot X-ray shocks is MJ 449 = LS 1865, O8.5 V((f)). We find $\log L_{\rm X} = 31.42\pm0.20$, $\log L_{\rm X}/L_{\rm bol} = -6.90$, $kT_1= 0.59_{-0.3}^{+0.1}$, and $kT_2 > 3.1$ keV, with comparable emission measure in the two thermal components [@naze11]. Though MJ 449 is not shown in Fig. 10, it is most likely time variable, with $P_{\rm KS} = 16\%$, and three maximum likelihood blocks. Like Tr16-22, MJ 449 is a candidate for magnetically confined wind shocks, though its lower $\log L_{\rm X}/L_{\rm bol}$ suggests that other emission mechanisms may be at work, including coronal emission from a PMS companion.
Astrometric Binaries
--------------------
We first examine the orbital properties and radio emission of the four astrometric binaries detected with [*HST*]{}. @nelan04 used a series of [*HST*]{} Fine Guidance Sensor observations (in high angular resolution mode 1r) to look for luminous companions around O and early B stars in Tr 14 and 16. The observations were typically sensitive to binary separations as low as $\sim10$ mas, or $\sim23$ AU at $d=2.3$ kpc. We consider the four O stars in Table 1 of @nelan10. We note that Tr16-31=MJ 484, B0.5 V is a wide binary with a separation of $352\pm2$ mas ($\sim810$ AU); it is not detected in the CCCP.
### HD 93129A, O2 If$\ast$
HD 93129A is the earliest O star in Carina, and one of the earliest known in the Galaxy. The O3.5 V((f)) star HD 93129B is located $2.74\arcsec$ to the southeast ($6300$ AU at the distance of Carina) – HD 93129A and B are clearly resolved in the [*Chandra*]{} data. @nelan04 [@nelan10] discovered a luminous companion, HD 93129Ab, at a separation of $0.053\arcsec\pm0.003\arcsec$ ($\approx122$ AU at 2.3 kpc), and $\Delta V = 0.9\pm0.05$ mag. @nelan04 estimate the primary’s mass to be $M_{\rm Aa} \approx 110 M_{\odot}$, and the secondary’s mass to be $M_{\rm Ab}\approx 70 M_{\odot}$, implying an approximate spectral type o3.5 v for Ab, very similar to HD 93129B. An unpublished analysis by @maiz-apellaniz10 indicates that the Aa/Ab system may be approaching periastron in $\sim2020$.
HD 93129A is a strong non-thermal radio source [@benaglia06]. @benaglia10 resolve the emission at 2.37 GHz into what appears to be a bow shock structure, consistent with a wind collision zone at the interface of Aa and Ab. Wide, massive binaries like HD 93129A are sometimes strong non-thermal radio sources because the wind collision zone is outside the radio photosphere of both stars [c.f., @debecker07].
Although hard X-ray emission and non-thermal radio emission are usually associated with colliding wind shocks in WR+O binaries, the [*Chandra*]{} ACIS-I data do not allow us to establish the location of the X-ray shocks within the HD 93129A system. We note that the system’s $\log L_{\rm X}/L_{\rm bol} = -6.85$ is higher than typical single, early-O stars in Table 2, and the spectrum requires a 1.9 keV emission component with about 12% of the total emission measure.
A [*Chandra*]{} high-energy transmission grating spectrum of HD 93129A shows broad, asymmetric line profiles, and moderately weak He-like forbidden lines of and that suggest that most of the X-rays are produced in wind shocks a few $R_{\star}$ from the photosphere [@cohen11]. The most likely explanation for HD 93129A’s X-ray and radio properties is that both mechanisms are operating: embedded wind shocks close to Aa and Ab, and colliding wind shocks between Aa and Ab.
### HD 303308, O4.5 V((fc))
HD 303308 is the fifth most X-ray luminous O binary in Carina with a soft spectrum ($kT\approx 0.25$ keV) and $\log L_{\rm X}/L_{\rm bol} \approx -6.89$. @nelan10 find a binary separation of $15\pm2$ mas, corresponding to $\sim 35$ AU at $d=2.3$ kpc, the closest of their five astrometric binaries. The apparent V magnitude difference was $\Delta V=1.0\pm0.3$. @sota11 find a spectral type of O4.5 V((fc)).
From the standpoint of its X-ray emission, HD 303308 is quite typical of other early-mid type O stars, though we cannot rule out some colliding wind emission in the zone between the two stars. We note that HD 303308 showed some evidence of time variability, with four maximum likelihood blocks, and $P_{\rm KS}=20\%$.
### Tr16-9 = MJ 481, O9.5 V
@nelan10 find a binary separation of $16\pm2$ mas for Tr16-9 = MJ 481, corresponding to $\sim 35$ AU at $d=2.3$ kpc, the second closest of their five astrometric binaries. MJ 481 shows low-level time variability, $\log L_{\rm X}/L_{\rm bol} = -7.17$, and a higher-than-average $kT_1 = 0.70_{-0.1}^{+0.2}$ keV [@naze11]. Like HD 303308, we cannot rule out some colliding wind emission.
O+O Spectroscopic Binaries
--------------------------
The O+O binaries listed in Table 3 are among the best studied in Carina and among them HD 93403, HD 93205, HD 93343, Tr16-112, and QZ Car have been proposed as colliding wind binaries [@rauw09; @morrell01; @parkin11].
We note however the wide variety of post-shock temperatures in these systems in Table 3. We find that the longer-period O+O spectroscopic binaries like HD 93403 (15.09 d), QZ Car (20.72 d), and HD 93343 (44.15 d) have high $kT$ shocks, and the shorter-period systems like HD 93205 (6.08 d), Tr16-112 (4.02 d), and FO 15 (1.41 d), show only soft X-ray shocks. Figure 13 plots $kT_{\rm avg}$ versus orbital period for the 11 spectroscopic binaries in Table 3. The orbital periods are from the compilations of @rauw09 [@nelan10].
For multiple systems like QZ Car, we use the orbital period of the primary in Fig. 13, because the primary pair accounts for most of the X-ray luminosity. For example, in QZ Car the A1/A2 pair with the O9.7 I primary accounts for $\sim 74\%$ of the system’s X-ray flux [see Table 5, @parkin11]. For Tr16-110, O7 V + O8 V+ O9 V, a $2+1$ spectroscopic binary, the primary and secondary orbital periods are 3.63 and 5.03 days, respectively [Table 7, @rauw09]. We use $P = 3.63$ in Fig. 13, though both periods are short; either period yields the same correlation. Similarly, for Tr16-104, an eclipsing O7 V + O9.5 V binary bound to a B0.2 IV star, we use the 2.15-d primary orbital period. The orbital period of the B0.2 IV star is either 285 d or 1341 d [@rauw01]. We ignore the possible contribution of wind shocks with the distant B0.2 IV star.
Fig. 13 shows a correlation between orbital period and $kT$, with a Pearson’s linear correlation coefficient of 0.95. The correlation hinges on the three long-period systems with $kT_{\rm avg} > 1$ keV: HD 93403, QZ Car, and HD 93343. A larger sample of O+O binaries is needed to confirm a correlation between orbital period or binary separation and average shock temperature. We note that the wide binaries $\eta$ Car with $P_{\rm orb}\approx 5.54$ yr [@gull09], and WR 25 (WN6h + O) with $P_{\rm orb}\approx 207.7$ d [@gamen08], are consistent with this correlation.
If such a correlation is confirmed, then the presence of a close, luminous companion (separation $\lesssim 5R_{\star}$) may significantly reduce the observed post-shock temperatures, and thus the wind speed at the shock, whether those shocks are produced throughout the wind (EWS) or in the wind collision zone (CWS). Put another way, there may not be enough room to allow the winds to ramp up to terminal speed. In systems like FO 15 with low $L_{\rm X}/L_{\rm bol}$ the emission measure and, hence, the density in the X-ray shocks may also be reduced.
Including the astrometric binaries in our analysis, it appears that, as the binary separation increases beyond a few tens of AU, the colliding wind shocks become weaker, as expected from adiabatic expansion [e.g., @antokhin04], and the X-rays are dominated by embedded wind shock emission produced within a few $R_{\star}$. We emphasize that for most O+O binaries, both mechanisms – embedded wind shocks and colliding wind shocks – may be at work.
### HD 93403, O5 III(fc) + O7 V
The 15.093-d O5 III(fc) + O7 V eccentric ($e=0.234$) binary HD 93403 is the most X-ray active O+O binary in Carina, with $\log L_{\rm X} \approx33.1$ ergs s$^{-1}$ and $\log L_{\rm X}/L_{\rm bol} \approx -6.4$. In the CCCP data, HD 93403 was observed in OBSIDs 9484, 9486 and 9891 [for the observation log see Table 1 of @townsley11a]. @naze11 fit the combined ACIS-I spectrum with $\log N_{\rm H}^{\rm ISM} = 21.49$ cm$^{-2}$, and find $N_{\rm H}^{\star} = 21.5_{-0.07}^{+0.06}$, $kT_1 = 0.62_{-0.01}^{+0.02}$ and $kT_2 = 1.6_{-0.05}^{+0.06}$, with approximately equal emission measure in each temperature component.
We note that @walborn10 recently reclassified the primary of HD 93403 as O5 III(fc) on the basis of its emission lines. Using the common ephemeris of @rauw00, $P=15.093$ d and $T_0 = 51355.14$ MJD, OBSIDs 9486 and 9491 occurred from phase $0.15$ to $0.28$, and OBSID 9484 occurred from phase $0.44$ to $0.48$, near apastron, with similar exposure times in each phase range. We note that throughout this study we use the [*Chandra*]{} convention ${\rm MJD}={\rm HJD} - 2400000.5$, rather than the more common convention ${\rm HJD} - 2450000.0$ to express $T_0$. The lower left panel of Fig. 9 shows the sequenced light curve of HD 93403.
In the most comprehensive examination of HD 93403’s X-ray light curve to date, @rauw02 use a series of [*ROSAT*]{} HRI and [*XMM-Newton*]{} MOS1 photometry and PN spectra to show a clear orbital modulation of its X-ray luminosity with orbital phase, with X-ray maximum occurring at $\phi = 0.0$ (periastron). The lower mean count rate at the beginning of the HD 93403 light curve, corresponding to $0.44 < \phi < 0.48$, suggests an X-ray minimum near apastron.
The short-term variability seen throughout the three [*Chandra*]{} observations, though small, is statistically significant. The rapid variability amplitude varies from $5-15\%$, and is comparable to the 20% long-term variability amplitude reported by @rauw02.
@rauw02 used a greater distance to Carina, $d = 3.2$ kpc, and a higher ISM column density, $\log N_{\rm H}^{\rm ISM} = 21.56$ cm$^{-2}$. They found a 0.5-2.5 keV $\log L_{\rm X} \approx 33.57$ ergs s$^{-1}$. Applying a distance $d = 2.3$ kpc yields $\log L_{\rm X} \approx 33.28$ ergs s$^{-1}$ in the 0.5-2.5 keV band for the mean of the four [*XMM*]{} observations.
Using @rauw02’s higher ISM column density, we find a CCCP 0.5-2.5 keV $\log L_{\rm X} \approx 33.19$ ergs s$^{-1}$ during the CCCP observations in 2008 August, only 19% lower than the [*XMM-Newton*]{} observations in 2000 December and 2001 February. Overall the CCCP spectral and timing analysis of HD 93403 confirms the results of @rauw02.
### HD 93205, O3.5 V((f)) + O8 V
@morrell01 present the definitive study on HD 93205 from an extensive set of ground-based spectroscopic, [*IUE*]{} far ultraviolet, and [*ROSAT*]{} X-ray observations. They find a $6.0803\pm0.0004$-d orbital period, primary and secondary masses of $52-60 M_{\odot}$ and $22-25 M_{\odot}$, an orbital inclination of about $55\arcdeg$, an orbital eccentricity $e=0.37$, an apsidal period of $185\pm16$ yr, and phase-locked X-ray variability, with X-ray maximum near periastron.
HD 93205’s CCCP light curve in Fig. 9 shows a slow steady decline over the 1 day observation, which began and ended at MJD 53977.794 and 53978.817. From the combined ephemeris in Table 4 of @morrell01 in which $T_0 = 50498.589$ MJD, OBSID 4495 began at orbital phase $\phi=0.21\pm0.04$ and ended at phase $\phi=0.38\pm0.04$, during which the average X-ray parameters were $\log L_{\rm X} \approx 32.55$ ergs s$^{-1}$, $\log L_{\rm X}/L_{\rm bol} = -6.82$, $kT_1 = 0.23\pm0.01$, and $kT_2 =0.74\pm0.03$, with most of the emission measure in the cooler component [@naze11].
This is the end of the declining phase in the [*ROSAT*]{} HRI light curve in Fig. 8 of @morrell01. Our results thus support the X-ray periodicity of @morrell01. Its high $L_{\rm X}/L_{\rm bol}$ is similar to other binaries with early-O primaries. As with the other short-period binaries, the colliding wind shock temperatures do not exceed 1 keV.
### Tr16-112 = LS 1874 = CPD-59 2641 = MJ 535, O5.5-O6 V(n)((fc)) + B2 V
@rauw09 presented a detailed multiplicity study of three X-ray luminous Carina O stars as part of the X-Mega project[^3]: Tr16-112, HD 93343 and HD 93250. Based on a long series of medium- and high-resolution spectra of Tr16-112, they obtained spectral types of O5.5-6 V((f)) and B2 V, for the primary and secondary. @sota11 revised the primary spectral type to O5.5-O6 V(n)((fc)).
@rauw09 find an orbital period $P = 4.0157$ d, eccentricity $e=0.15 \pm 0.01$, inclination $i = 54\arcdeg^{+4}_{-3}$, primary and secondary masses $M_1 = 41.0\pm2.5 M_{\odot}$ and $M_2 = 11.7\pm0.6 M_{\odot}$, and periastron passage time $T_0 = 54559.810\pm0.060$ MJD. They estimate that the primary fills $\frac{3}{4}$ of its Roche lobe.
In the CCCP data, Tr16-112 was observed in OBSID 6402 in 2006 August during which time it showed no signs of variability, with $P_{\rm KS} = 53\%$ and only one maximum likelihood block (95% confidence). OBSID 6402 began and ended at MJD 53977.78 and 53978.83, corresponding to orbital phases 0.06 and 0.32 in Tr16-112. That is close to periastron passage, though we note that the orbit is not highly eccentric. This short-period binary has $kT_{\rm avg}\approx0.35$ keV.
### HD 93343, O8 V + O7-8.5 V
@rauw09 present the first comprehensive analysis of the double-lined spectroscopic binary HD 93343. They find a set of sharp lines associated with the less massive O8 V secondary moving in opposite phase to the higher-mass O8 V primary. They note that the spectral type of the secondary could be in the range O7-O8.5 V. They find a period of $\approx 44.15$ days, though they were not able to establish an absolute ephemeris.
For HD 93343 we note $\log L_{\rm X} = 31.66\pm0.24$, $\log L_{\rm X}/L_{\rm bol} \approx -6.98$, $kT_1 = 0.70_{-0.09}^{+0.13}$, and $kT_2 > 6.5$ keV, with comparable emission measure in each temperature component. HD 93343 showed no evidence of time variability in OBSID 6402.
@antokhin08 found $kT_1 = 0.29^{+0.04}_{-0.05}$, $kT_2 = 2.8^{+1.5}_{-1.4}$, and $\log L_{\rm X}\approx32.2$, a factor of 3 more luminous in the 0.4-10 keV band with [*XMM*]{} than was found in the same band with [*Chandra*]{} [@naze11], with most of the added luminosity in the [*XMM*]{} fit in the softer component.
Further optical spectroscopic monitoring is needed to establish a firm ephemeris for HD 93343. Further [*Chandra*]{} observations would provide a better estimate of the shock temperatures, and a third estimate of $L_{\rm X}$ to confirm the variability. An X-ray maximum near periastron would signal a colliding-wind binary. Spectropolarimetric monitoring could reveal the presence of large-scale magnetic fields on one or the other star.
For now, we consider HD 93343 a CWS X-ray source, so we include it in Fig. 13. We note though that the hard shocks seen on HD 93343 with [*Chandra*]{} are not predicted by @pittard09 in their colliding wind binary model [cwb3]{}, an O6 V + O8 V binary with a 10.7-d period, the closest model to HD 93343. The predicted temperature distributions are somewhat consistent with the @antokhin08 result, however.
### The Short-Period Eclipsing Binary FO 15, O5.5 Vz + O9.5 V
The double-lined eclipsing binary FO 15 was first typed by @forte81 and studied in detail by @niemela06. Its circular period is $1.413560 \pm 0.000003$ d with $T_0 = 53159.0\pm0.2$ MJD, $i\approx80\arcdeg$, and primary and secondary masses $M_1 = 30.4\pm1.0 M_{\odot}$ and $M_2 = 15.8\pm1.0 M_{\odot}$. The stellar radii are $R_1\approx7.5 R_{\odot}$ and $R_2\approx 5.3 R_{\odot}$ and the Roche lobe radii are just $8.3 R_{\odot}$ and $6.15 R_{\odot}$. The semi-major axis is $a = 19.1 R_{\odot}$, implying the photospheres are separated by $\sim3 R_1$.
The “z" spectral type indicates strong $\lambda 4686$ absorption suggesting the primary is on the ZAMS [c.f., @walborn97]. Located in the south pillars [see @povich11b], FO 15 is a ZAMS-age O+O binary, with a separation of only $3 R_{\star}$, in a region of active star formation. Moreover, @niemela06 note the absence of wind features in both stars at most phases. They suggest that both O dwarfs may be weak-wind stars [@martins05b].
The top left panel of Fig. 9 shows the sequenced light curve of FO 15, showing a steady decline from the beginning of the first observation through the end of the second. OBSID 6402 began and ended at MJD 53977.78 and 53978.83, corresponding to orbital phases $\phi=0.231\pm0.001$ and $0.978\pm0.001$. OBSID 9483 began and ended at MJD 54706.97 and 54707.68, corresponding to orbital phases $\phi=0.083\pm0.002$ and $0.591\pm0.002$. Note that OBSIDs 6402 and 9483 were separated by nearly 729 days, i.e., 515 orbital cycles.
FO 15 is noteworthy because it has the lowest $L_{\rm X}/L_{\rm bol}$ of any O binary in the CCCP: $\log L_{\rm X}/L_{\rm bol} = -7.65$. It is relatively soft with $kT=0.5\pm0.3$ keV, yet it is clearly time variable. It is very young, it has the shortest orbital period in our sample, and it is one of only four weak-wind systems in Carina [see also @naze11].
The soft X-ray spectrum of FO 15 (indicating low-velocity shocks) and low $L_{\rm X}/L_{\rm bol}$ (indicting low shock densities) suggest that in very close, short-period binaries the winds may not have reached $v_{\infty}$ when they collide, or alternatively, the luminous companion may inhibit wind acceleration and mass-loss via radiative braking [@gayley97] as a result of non-radial radiative forces (from the distorted shape of the star).
B Stars
-------
B stars provide a transition in the X-ray properties of massive stars. Early B stars are often detected in X-rays, though their detection fraction is typically less than 50%. Like most late-O stars, B stars are often soft, low $L_{\rm X}/L_{\rm bol}$ sources [@cohen97; @evans03], but in some cases they are particularly hard and strong X-ray sources (e.g. $\tau$ Sco, B0.2 V), which has a complex magnetic geometry [@donati06b]. The Carina Nebula Complex objects provide an excellent sample to explore this interesting spectral region further.
@evans11 examine a group of photometrically selected, candidate late-B stars in Tr 16. For this group, @evans11 find no stars with $\log L_{\rm X} > 31$ and suggest that the X-ray detected late-B stars in Tr 16 probably harbor unseen, lower-mass, late-type pre–main sequence companions.
With regard to the B stars, the main results of this study are that (i) many B stars with moderate X-ray fluxes ($\log f_{\rm X} < -14$ ergs cm$^{-2}$ s$^{-1}$, corresponding approximately to $\log L_{\rm X} < 31$ ergs s$^{-1}$) probably harbor unseen, coronal PMS companions, and (ii) that most of the B stars with very high X-ray fluxes ($\log f_{\rm X} > -14$ ergs cm$^{-2}$ s$^{-1}$, corresponding approximately to $\log L_{\rm X} > 31$ ergs s$^{-1}$, [*cannot*]{} be accounted for by unseen, coronal companions.
Because their winds are too weak to produce much embedded wind shock emission, these active B stars may be similar to the magnetic B stars like $\tau$ Scorpii, B0.2 V [@cohen03; @donati06b; @petit11] or the He-strong B2 Vp stars like $\sigma$ Ori E [@shore90; @sanz-forcada04; @townsend05].
Thus, the X-ray active B stars at the top of Table 4 merit detailed follow-up ground-based spectroscopic, photometric, and spectro-polarimetric monitoring. Below we discuss the two most remarkable B stars in the CCCP survey in more detail.
### The Candidate Herbig Be Star SS73 24
The most X-ray active B star in Carina is the emission-line star Hen 3-485 = Tr 16 MJ 640 = SS73 24 = WRA 642, with $\log L_{\rm X} = 31.70$ and $\log L_{\rm X}/L_{\rm bol} = -5.65$. SS73 24 was classified by Sanduleak and Stephenson (1973) as Be! pec, remarking in the footnote to their Table 1, “Our plates show numerous weak emission lines some of which appear to be or \[\]." SS73 24 is located close to the Treasure Chest Cluster [@smith05] in the southern portion of the Carina Nebula, and is a strong infrared emission source (see Fig. 14).
Based on its emission lines, spectral type, and strong infrared excess, Miroshnichenko (2007) classified Hen 3-485 = SS73 24 as a dust-forming B\[e\] star, and listed it as a candidate FS CMa star. He proposed, “that these objects are binary systems that are currently undergoing or have recently undergone a phase of rapid mass exchange, associated with a strong mass loss and dust formation. A new name, FS CMa stars, and classification criteria are proposed for the unclassified B\[e\] stars." Based on its X-ray and infrared emission, and its association with the star-forming Treasure Chest cluster, we show that SS73 24 is most likely a young, accreting Herbig Be star, not an evolved B\[e\] binary.
In Fig. 14a we show the UBV/JHK/IRAC/MIPS $0.3-24~\mu$m SED of SS73 24, though we note that the Spitzer-Vela Survey IRAC $3.5-8~\mu$m photometry is mildly saturated. Moreover, its $U-B=-0.71$ and $B-V=0.70$ colors indicate relatively strong emission-line veiling, which we do not model.
The UBV/JHK/MIPS photometry were fit using the @robitaille06 YSO models. The SED fitting method described by @povich11b generates a set of best-fit models from a large grid of star/disk/envelope models. The set of most likely models yields the following parameters: $L_{\rm bol} = 6\,000 \pm 2\,000 L_{\odot}$, $T_{\rm eff} = 25\,000\pm 1\,400~{\rm K}$, and $A_V = 2.0\pm0.2~{\rm mag}$, consistent with a 9-11 $M_{\odot}$ B2 dwarf, a warm accretion disk, and a cooler circumstellar envelope (blue, green, and red curves, respectively, in Fig. 14b).
SS73 24 was observed for 69 ks during OBSID 6578 and 9483, separated by $\sim 400$ days. It showed no strong short-term variability during either observation, and no long-term variability between observations. Its grouped ACIS spectrum was fit with a single-temperature, solar-abundance APEC emission model, and two [tbabs]{} absorption components: one fixed at $N_{\rm H}=3.2\times10^{21}~{\rm cm}^{-2}$ representing ISM absorption with $A_V = 2.0~{\rm mag}$, the other column density (a free parameter) to represent circumstellar absorption from the disk/envelope. We note that SS73 24 is the most X-ray luminous B star in Carina with $\log L_{\rm X}\approx 31.7$ ergs s$^{-1}$, and one of the hardest with $kT\sim 3~{\rm keV}$, comparable to only a few other known Herbig Be stars [@hamaguchi00; @zinnecker94; @stelzer06].
It should be noted that the origin of hard X-rays on Herbig Be stars is not known. @vink05 show that the flaring X-ray Herbig Be star MCW 297 has an astrometric binary companion (a lower-mass PMS star) that may the source of the hard X-ray flare. That said, SS73 24 is not flaring in the CCCP data and its $L_{\rm X}$ is higher than any other non-flaring PMS star in the CCCP.
### HD 93501, B1.5 III:
The B1.5 III: star HD 93501=Coll 228 96 was detected by @levato90 as SB2, though no orbital parameters were published. The star is noteworthy because of its extremely hard X-ray spectrum, that we model as very hot thermal plasma ($kT > 6$ keV, 90% confidence lower bound), with little excess column density. It’s spectrum can also be fit a power-law with index $\Gamma = 1.3\pm 0.2$. Assuming HD 93501 is a member of the Collinder 228 cluster at $d = 2.3$ kpc, its $\log L_{\rm X}\approx 31.1$ ergs s$^{-1}$. It was observed in three [*Chandra*]{} OBSIDs with no indication of long- or short-term variability. HD 93501’s very hard spectrum, high $L_{\rm X}$, and lack of variability is not typical of coronal PMS stars. HD 93501’s X-ray spectrum is unique among the 78 OB stars with more than 50 counts.
Summary and Discussion
======================
We have assembled an optical/X-ray catalog of the 200 known O and early-B stars in the [*Chandra*]{} Carina Complex Project to study the X-ray emission mechanisms on young, massive stars. Like @naze11, we find that most O stars in Carina approximately follow the well-established relation between $L_{\rm X}$ and $L_{\rm bol}$, and generally have soft X-ray spectra, with characteristic temperatures $kT < 0.8$ keV.
There are numerous examples of O and B stars, though, that do not follow this trend. We examine their X-ray properties with an eye to classifying their primary underlying X-ray emission mechanism as (1) embedded wind shocks, (2) colliding wind shocks, (3) magnetically confined wind shocks, (4) low-mass, coronal PMS companions, or (5) some other mechanism, possibly related to magnetic fields.
Fig. 4 provides a good overview of the X-ray/optical data: $\approx\frac{2}{3}$ of the O and early-B stars lie on the cool side of the $L_{\rm X}/L_{\rm bol}$ versus $kT$ diagram. Most of these stars have X-ray properties consistent with embedded wind shock sources, with a characteristic average $kT\approx0.5$ keV and $\log L_{\rm X}/L_{\rm bol} \approx -7.2$.
Some of the well-known spectroscopic binaries in Table 3 also lie on the cool side of Fig. 4a, suggesting perhaps a universal X-ray emission mechanism for all O stars. However, Fig. 13 suggests a different trend. It shows a correlation between orbital period and $kT$, with a Pearson’s linear correlation coefficient of 0.95. The correlation, which hinges on the longest-period binaries, requires confirmation using a larger sample of binaries.
If the trend is confirmed, we suggest that in short-period binaries like HD 93205 and FO 15, the winds have not reached $v_{\infty}$ when they collide or, alternatively, the luminous companion may inhibit wind acceleration via radiative braking or some other mechanism [@gayley97]. In some very wide systems like HD 93129A, O2 If\*, it appears that embedded wind shocks (close to the O2 photosphere of Aa) and colliding wind shocks in the far wind collision zone between Aa and Ab are both at work.
Then there is the curious case of HD 93250, the O4 III(fc) star. We argue in §6.2.1 [see also @rauw09] that HD 93250’s strong X-ray and radio emission are evidence for colliding wind shocks. Given the $kT$ versus $P_{\rm orb}$ trend seen in Fig. 13, we suggest that it could be an O+O binary with a period $>30$ days, and/or a magnetic O star, and that further observations are needed: multi-frequency radio interferometry to measure its spectral index, ground-based optical interferometry to look for a luminous companion, and optical spectro-polarimetry to look for surface magnetic fields.
We are left with two additional candidate magnetic O stars with high $L_{\rm X}/L_{\rm bol}$ and $kT$: Tr16-22 = MJ 496, O8.5 V, and CPD-59 2610 = MJ 449 = LS 1865, O8.5 V((f)). So far, magnetic fields have not been reported on any Carina OB stars.
Among the B stars, Fig. 8, shows a large population of B stars, some X-ray detected, some not, whose photon fluxes and upper limits are consistent with a distribution of unseen, low-mass, coronal PMS stars. Superposed on this distribution is a group of B stars whose high-$L_{\rm X}$ cannot be explained by a distribution of ordinary coronal, PMS companions. The situation here is reminiscent of the X-ray active Herbig Be stars: either their companions are hyperactive compared to other low-mass PMS stars, or there is some intrinsic magnetic mechanism in these B stars. The dozen or so B stars near the top of Table 4 could, for example, be related to the magnetic B stars like $\tau$ Sco, or the He-strong Bp stars.
The B star with the highest $L_{\rm X}$ and the OB star with the highest $L_{\rm X}/L_{\rm bol}$, SS73 24, is located adjacent to the pillar in the Treasure Chest Cluster (see Fig. 15), and it shows a prominent mid-infrared excess (Fig. 14). We model its $0.3-24~\mu$m SED as an accretion disk + envelope + B2 photosphere. SS73 24 appears to be the first Herbig Be star in Carina.
This research has made extensive use of the SIMBAD and [*Vizier*]{} databases, operated at CDS, Strasbourg, France. This work was partially supported by SAO/[*Chandra*]{} X-ray Observatory grants GO8-9014X and G09-0019A (PI: Gagné). The CCCP is supported by [*Chandra*]{} X-ray Observatory grant GO8-9131X (PI: Townsley) and by the ACIS Instrument Team contract SV4-74018 (PI: Garmire), issued by the [*Chandra*]{} X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of NASA under contract NAS8-03060. AFJM is grateful to NSERC (Canada) and FQRNT (Québec) for financial assistance. NRW acknowledges the support of the Space Telescope Science Institute, operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. MSP is supported by an NSF Astronomy and Astrophysics Postdoctoral Fellowship under award AST-0901646. The authors wish to thank the anonymous referee for many helpful suggestions.
[*Facilities:*]{} .
Appendix: [*Chandra*]{} Spectral Analysis of HD 93250
=====================================================
In addition to the CCCP ACIS-I observation of Tr14 (OBSID 4495), HD 93250 was observed far off-axis in seven additional grating observations of $\eta$ Car. Even though HD 93250 is located $\sim 7.5\arcmin$ NNW of $\eta$ Car, the satellite roll angle placed HD 93250 on one of the ACIS-S chips during many of the $\eta$ Car monitoring observations. Even though the PSF is quite large at $7.5\arcmin$ off-axis, HD 93250 is the only bright source there. Pileup was mitigated by the effects of comatic aberration and vignetting at those off-axis angles, and the reduced effective area in zeroth order. The [*Chandra*]{} observation log is shown in Table 5.
The data were reduced in the usual way [see @broos11a] and spectra were extracted using the [*CIAO*]{} tool [psextract]{} using a $12\arcsec$ source circle, and a nearby source-free background region, away from the dispersed HEG and MEG spectra of $\eta$ Car. The spectra were grouped using the [*ACIS Extract*]{} tool [ae\_group\_spectra]{}. The ACIS-S spectra were merged with the [*CIAO*]{} tool [acisspec]{} for a combined exposure time of 238 ks (see Table 5). We note that @evans04 performed a similar wind-absorption analysis of the OBSID 6402 ACIS-I spectrum of HD 93250, and found two temperature components at 0.55 and 2.33 keV.
To exploit the high S/N of the merged ACIS-S spectrum, we fit a rather complex XSPEC spectral model. As before, we modeled the cold, neutral ISM with [tbabs]{}, freezing $\log N_{\rm H}^{\rm ISM} = 21.48$. Assuming that some of the emission arises from embedded wind shocks close to HD 93250, and that the harder emission forms in colliding wind shocks where the wind is less dense, we used a [windtabs]{} absorption component [@leutenegger10] for the cool emission component. The main advantage of [windtabs]{} over [tbabs]{} is that (i) it is not a simple slab model – the emission and absorption is distributed radially throughout the wind via a specified onset shock radius and wind velocity law, and (ii) the absorption model accounts for the ionization of the wind. The only free parameter in [windtabs]{} is the characteristic mass column in g cm$^{-2}$, $$\Sigma_{\star} = \frac{\dot{M}}{4 \pi R_{\star} v_{\infty}}.$$
To estimate the distribution of absorbing material in the wind of HD 93250 we adopt wind parameters $R_0 = 1.5 R_{\star}$ (the onset radius for X-ray shocks) and $\beta=0.7$ (the wind velocity law) that are consistent with recent X-ray line-profile measurements of O supergiants [@cohen10; @cohen06]. We note that a direct finite-disk corrected CAK [@castor75] model with fixed $\bar{Q}$ and $\alpha$ also produces $\beta\approx0.75$.
For the APEC emission model and the [windtabs]{} and [tbabs]{} absorption models we adopt the solar abundance values of Asplund et al. (2009). Though we expect differing levels of CNO processed material in the atmospheres and winds of the Carina O stars, we note that the CNO lines and absorption edges occur longward of 20 Å where the [*Chandra*]{} ACIS-I spectra are extremely weak. The ACIS spectra are truncated below 0.5 keV (above 25 Å).
The best-fit [tbabs(windtabs(apec)+apec+apec)]{} model is shown in Fig. 11 in red. The ACIS-I and ACIS-S spectra were then fit individually to measure $L_{\rm X}$ during each OBSID. In Fig. 12, we merged the last three spectra, OBSID 7341, 7342 and 7189 from 2006 June.
The best-fit mass column $\Sigma_{\star}=0.03\pm0.01$ g cm$^{-2}$ yields a time-averaged ISM-corrected X-ray luminosity $\log L_{\rm X} = 33.18$. We note that this is very close to the value found by @naze11 fitting only the ACIS-I data without [windtabs]{}, $\log L_{\rm X} = 33.19$. The benefit of using [windtabs]{} is that it provides a physically meaningful estimate of the wind mass column.
Using a terminal wind speed $v_{\infty}=3\,000$ km s$^{-1}$ [@martins05b] and a radius $R=17.7 R_{\odot}$ implied by $\log L_{\rm bol}/L_{\odot} = 5.95$ and $\log T_{\rm eff} = 4.62$, the best-fit mass column yields a mass-loss rate $\dot{M}=1.4\pm0.5\times10^{-6} M_{\odot}$ yr$^{-1}$.
Albacete Colombo, J. F., Morrell, N. I., Rauw, G., Corcoran, M. F., Niemela, V. S., & Sana, H. 2002, , 336, 1099
Albacete-Colombo, J. F., Damiani, F., Micela, G., Sciortino, S., & Harnden, F. R., Jr. 2008, , 490, 1055
Antokhin, I. I., Owocki, S. P., & Brown, J. C. 2004, , 611, 434
Antokhin, I. I., Rauw, G., Vreux, J.-M., van der Hucht, K. A., & Brown, J. C. 2008, , 477, 593
Babel, J., & Montmerle, T. 1997, , 485, L29
Benaglia, P., Koribalski, B., & Albacete Colombo, J. F. 2006, , 23, 50
Benaglia, P., Dougherty, S. M., Phillips, C., Koribalski, B., & Tzioumis, T. 2010, Revista Mexicana de Astronomia y Astrofisica Conference Series, 38, 41
Berghoefer, T. W., Schmitt, J. H. M. M., Danner, R., & Cassinelli, J. P. 1997, , 322, 167
Bessell, M. S., Castelli, F., & Plez, B. 1998, , 333, 231
Briggs, K. R., & Pye, J. P. 2003, , 345, 714
Broos, P. S., Townsley, L. K., Feigelson, E. D., Getman, K. V., Bauer, F. E., & Garmire, G. P. 2010, , 714, 1582
Broos, P. S., et al. 2011, , submitted (CCCP Catalog Paper)
Caillault, J.-P., Gagné, M., & Stauffer, J. R. 1994, , 432, 386
Cassinelli, J. P., Miller, N. A., Waldron, W. L., MacFarlane, J. J., & Cohen, D. H. 2001, , 554, L55
Cassinelli, J. P., Ignace, R., Waldron, W. L., Cho, J., Murphy, N. A., & Lazarian, A. 2008, , 683, 1052
Castelli, F., & Kurucz, R. L. 2004, arXiv:astro-ph/0405087
Castor, J. I., Abbott, D. C., & Klein, R. I. 1975, , 195, 157
Cohen, D. H., Cassinelli, J. P., & Macfarlane, J. J. 1997, , 487, 867
Cohen, D. H., de Messi[è]{}res, G. E., MacFarlane, J. J., Miller, N. A., Cassinelli, J. P., Owocki, S. P., & Liedahl, D. A. 2003, , 586, 495
Cohen, D. H., Leutenegger, M. A., Grizzard, K. T., Reed, C. L., Kramer, R. H., & Owocki, S. P. 2006, , 368, 1905
Cohen, D. H., Leutenegger, M. A., Wollman, E. E., Zsarg[ó]{}, J., Hillier, D. J., Townsend, R. H. D., & Owocki, S. P. 2010, , 405, 2391
Cohen, D. H., Wollman, E. E., & Leutenegger, M. A. 2011, arXiv:1009.5669, Proceedings of IAU Symposium 272: Active OB Stars
Corcoran, M. F., et al. 2001, , 562, 1031
Daniel, K. J., Linsky, J. L., & Gagn[é]{}, M. 2002, , 578, 486
Davidson, K., & Ostriker, J. P. 1973, , 179, 585
De Becker, M., Rauw, G., Pittard, J. M., Antokhin, I. I., Stevens, I. R., Gosset, E., & Owocki, S. P. 2004, , 416, 221
De Becker, M. 2007, , 14, 171
DeGioia-Eastwood, K., Throop, H., Walker, G., & Cudworth, K. M. 2001,, 549, 578
de Jager, C., & Nieuwenhuijzen, H. 1987, , 177, 217
Dessart, L., & Owocki, S. P. 2005, , 437, 657
Dessart, L., & Owocki, S. P. 2005, , 432, 281
Donati, J.-F., Babel, J., Harries, T. J., Howarth, I. D., Petit, P., & Semel, M. 2002, , 333, 55
Donati, J.-F., Howarth, I. D., Bouret, J.-C., Petit, P., Catala, C., & Landstreet, J. 2006, , 365, L6
Donati, J.-F., et al. 2006, , 370, 629
Donati, J.-F., & Landstreet, J. D. 2009, , 47, 333
Dougherty, S. M., & Williams, P. M. 2000, , 319, 1005
Drake, S. A., Linsky, J. L., Schmitt, J. H. M. M., & Rosso, C. 1994, , 420, 387
Evans, N. R., Seward, F. D., Krauss, M. I., Isobe, T., Nichols, J., Schlegel, E. M., & Wolk, S. J. 2003, , 589, 509
Evans, N. R., Schlegel, E. M., Waldron, W. L., Seward, F. D., Krauss, M. I., Nichols, J., & Wolk, S. J. 2004, , 612, 1065
Evans, N. R., et al. 2011, , submitted (CCCP Tr16 B Stars Paper)
Feigelson, E. D., Broos, P., Gaffney, J. A., III, Garmire, G., Hillenbrand, L. A., Pravdo, S. H., Townsley, L., & Tsuboi, Y. 2002, , 574, 258
Feigelson, E. D., et al. 2011, , submitted (CCCP Clustering Paper)
Feldmeier, A., Puls, J., & Pauldrach, A. W. A. 1997, , 322, 878
Feldmeier, A., Oskinova, L., & Hamann, W.-R. 2003, , 403, 217
Forte, J. C. 1976, , 25, 271
Forte, J. C., & Orsatti, A. M. 1981, , 86, 209
Freyhammer, L., Clausen, J., Arentoft, T., & Sterken, C. 2001, Eta Carinae and Other Mysterious Stars: The Hidden Opportunities of Emission Spectroscopy, 242, 195
Gagné, M., Caillault, J.-P., & Stauffer, J. R. 1995, , 450, 217
Gagne, M., Caillault, J.-P., Stauffer, J. R., & Linsky, J. L. 1997, , 478, L87
Gagn[é]{}, M., Oksala, M. E., Cohen, D. H., Tonnesen, S. K., ud-Doula, A., Owocki, S. P., Townsend, R. H. D., & MacFarlane, J. J. 2005, , 628, 986
Gamen, R., et al. 2008, Revista Mexicana de Astronomia y Astrofisica Conference Series, 33, 91
Gayley, K. G., Owocki, S. P., & Cranmer, S. R. 1997, , 475, 786
Getman, K. V., Feigelson, E. D., Grosso, N., McCaughrean, M. J., Micela, G., Broos, P., Garmire, G., & Townsley, L. 2005, , 160, 353
Gosset, E., Naz[é]{}, Y., Sana, H., Rauw, G., & Vreux, J.-M. 2009, , 508, 805
Gull, T. R., et al. 2009, , 396, 1308
Hamaguchi, K., Terada, H., Bamba, A., & Koyama, K. 2000, , 532, 1111
Herbst, W. 1976, , 208, 923
Hillier, D. J., & Lanz, T. 2001, Spectroscopic Challenges of Photoionized Plasmas, 247, 343
Hoffmeister, V. H., Chini, R., Scheyda, C. M., Schulze, D., Watermann, R., N[ü]{}rnberger, D., & Vogt, N. 2008, , 686, 310
Ignace, R., Oskinova, L. M., Jardine, M., Cassinelli, J. P., Cohen, D. H., Donati, J.-F., Townsend, R. H. D., & ud-Doula, A. 2010, , 721, 1412
Kahn, S. M., Leutenegger, M. A., Cottam, J., Rauw, G., Vreux, J.-M., den Boggende, A. J. F., Mewe, R., Güdel, M. 2001, , 365, L312
Kroupa, P. 2002, Science, 295, 82
Leitherer, C., Chapman, J. M., & Koribalski, B. 1995, , 450, 289
L[é]{}pine, S., & Moffat, A. F. J. 1999, , 514, 909
L[é]{}pine, S., et al. 2000, , 120, 3201
Leutenegger, M. A., Cohen, D. H., Zsarg[ó]{}, J., Martell, E. M., MacArthur, J. P., Owocki, S. P., Gagn[é]{}, M., & Hillier, D. J. 2010, , 719, 1767
Levato, H., Malaroda, S., Garcia, B., Morrell, N., & Solivella, G. 1990, , 72, 323
Lucy, L. B., & White, R. L. 1980, , 241, 300
Lucy, L. B. 1982, , 255, 286
Marchenko, S. V., Moffat, A. F. J., St-Louis, N., & Fullerton, A. W. 2006, , 639, L75
Ma[í]{}z Apell[á]{}niz, J., et al. 2010, arXiv:1010.5680
Maiz Apellaniz, J. 2007, HST Proposal, 11294
Martins, F., Schaerer, D., & Hillier, D. J. 2005, , 436, 1049
Martins, F., Schaerer, D., Hillier, D. J., Meynadier, F., Heydari-Malayeri, M., & Walborn, N. R. 2005, , 441, 735
Martins, F., & Plez, B. 2006, , 457, 637
Massey, P., & Johnson, J. 1993, , 105, 980
Massey, P., DeGioia-Eastwood, K., & Waterhouse, E. 2001, , 121, 1050
Mermilliod, J.-C. 1994, Bulletin d’Information du Centre de Donnees Stellaires, 45, 3
Moffat, A. F. J. 2008, Clumping in Hot-Star Winds, 17
Morgan, W. W., Code, A. D., & Whitford, A. E. 1955, , 2, 41
Morrell, N., Garcia, B., & Levato, H. 1988, , 100, 1431
Morrell, N. I., et al. 2001, , 326, 85
Morrison, N. D., & Conti, P. S. 1980, , 239, 212
Mullan, D. J., & MacDonald, J. 2005, , 356, 1139
Mullan, D. J., & Waldron, W. L. 2006, , 637, 506
Naz[é]{}, Y., Rauw, G., Pollock, A. M. T., Walborn, N. R., & Howarth, I. D. 2007, , 375, 145
Naz[é]{}, Y. 2009, , 506, 1055
Naz[é]{}, Y., Ud-Doula, A., Spano, M., Rauw, G., De Becker, M., & Walborn, N. R. 2010, , 520, A59
Naz[é]{}, Y., et al. 2011, , submitted (CCCP Massive Star $L_{\rm X}/L_{\rm bol}$ Paper)
Nelan, E. P., Walborn, N. R., Wallace, D. J., Moffat, A. F. J., Makidon, R. B., Gies, D. R., & Panagia, N. 2004, , 128, 323
Nelan, E. P., Walborn, N. R., Wallace, D. J., Moffat, A. F. J., Makidon, R. B., Gies, D. R., & Panagia, N. 2010, , 139, 2714
Niemela, V. S., Morrell, N. I., Fern[á]{}ndez Laj[ú]{}s, E., Barb[á]{}, R., Albacete Colombo, J. F., & Orellana, M. 2006, , 367, 1450
Oskinova, L. M., Hamann, W.-R., Feldmeier, A. 2007, , 476, 1331
Owocki, S. P., Castor, J. I., & Rybicki, G. B. 1988, , 335, 914
Owocki, S. P., & Cohen, D. H. 1999, , 520, 833
Pallavicini, R., Golub, L., Rosner, R., Vaiana, G. S., Ayres, T., & Linsky, J. L. 1981, , 248, 279
Parkin, E. R., et al. 2011, , submitted (CCCP QZ Car Paper)
Patience, J., Zavala, R. T., Prato, L., Franz, O., Wasserman, L., Tycner, C., Hutter, D. J., & Hummel, C. A. 2008, , 674, L97
Pereira, C. B., Franco, C. S., & de Ara[ú]{}jo, F. X. 2003, , 397, 927
Petit, V., Wade, G. A., Drissen, L., Montmerle, T., & Alecian, E. 2008, , 387, L23
, V., [Massa]{}, D. L., [Marcolinot]{}, W. L. F., [Wade]{}, G. A.. & [Ignace]{}, R. 2011, , in press
Pittard, J. M. 2009, , 396, 1743
Pollock, A. M. T., Corcoran, M. F., Stevens, I. R., & Williams, P. M. 2005, , 629, 482
Pollock, A. M. T. 2007, , 463, 1111
Povich, M. S., et al. 2011, , submitted (CCCP Massive Star Candidates Paper)
Povich, M. S., et al. 2011, , submitted (CCCP IR YSOs Paper)
Preibisch, T., et al. 2011, , submitted (CCCP HAWK-I Paper)
Puls, J., et al. 1996, , 305, 171
Raassen, A. J. J., van der Hucht, K. A., Miller, N. A., & Cassinelli, J. P. 2008, , 478, 513
Rauw, G., Sana, H., Gosset, E., Vreux, J.-M., Jehin, E., & Parmentier, G. 2000, , 360, 1003
Rauw, G., Naz[é]{}, Y., Carrier, F., Burki, G., Gosset, E., & Vreux, J.-M. 2001, , 368, 212
Rauw, G., Vreux, J.-M., Stevens, I. R., Gosset, E., Sana, H., Jamar, C., & Mason, K. O. 2002, , 388, 552
Rauw, G., Naz[é]{}, Y., Fern[á]{}ndez Laj[ú]{}s, E., Lanotte, A. A., Solivella, G. R., Sana, H., & Gosset, E. 2009, , 398, 1582
Reed, C. 2005, VizieR Online Data Catalog, 5125, 0
Robitaille, T. P., Whitney, B. A., Indebetouw, R., Wood, K., & Denzmore, P. 2006, , 167, 256
Robitaille, T. P., Whitney, B. A., Indebetouw, R., & Wood, K. 2007, , 169, 328
Sana, H., Rauw, G., Naz[é]{}, Y., Gosset, E., & Vreux, J.-M. 2006, , 372, 661
Sana, H., Gosset, E., Rauw, G., Sung, H., & Vreux, J.-M. 2006, , 454, 1047
Sanchawala, K., Chen, W.-P., Lee, H.-T., Chu, Y.-H., Nakajima, Y., Tamura, M., Baba, D., & Sato, S. 2007, , 656, 462
Sanz-Forcada, J., Franciosini, E., & Pallavicini, R. 2004, , 421, 715
Schulz, N. S., Testa, P., Huenemoerder, D. P., Ishibashi, K., & Canizares, C. R. 2006, , 653, 636
Shore, S. N., & Brown, D. N. 1990, , 365, 665
Smith, N. 2006, , 644, 1151
Skiff, B. A. 2009, VizieR Online Data Catalog, 1, 2023
Skinner, S. L., G[ü]{}del, M., Audard, M., & Smith, K. 2004, , 614, 221
Skinner, S. L., Sokal, K. R., Cohen, D. H., Gagn[é]{}, M., Owocki, S. P., & Townsend, R. D. 2008, , 683, 796
Skinner, S. L., Zhekov, S. A., G[ü]{}del, M., Schmutz, W., & Sokal, K. R. 2010, , 139, 825
Smith, N., Stassun, K. G., & Bally, J. 2005, , 129, 888
Smith, N. 2006, , 367, 763
Spruit, H. C. 2002, , 381, 923
Sota, A., Maíz Apellániz, J., Walborn, N.R., Alfaro, E.J., Barbá, R.H., Morrell, N.I., Gamen, R.C., & Arias, J.I. 2011, , submitted
Stelzer, B., Flaccomio, E., Montmerle, T., Micela, G., Sciortino, S., Favata, F., Preibisch, T., & Feigelson, E. D. 2005, , 160, 557
Stelzer, B., Micela, G., Hamaguchi, K., & Schmitt, J. H. M. M. 2006, , 457, 223
Stevens, I. R., Blondin, J. M., & Pollock, A. M. T. 1992, , 386, 265
Tapia, M., Roth, M., V[á]{}zquez, R. A., & Feinstein, A. 2003, , 339, 44
Townsend, R. H. D., & Owocki, S. P. 2005, , 357, 251
Townsend, R. H. D., Owocki, S. P., & Ud-Doula, A. 2007, , 382, 139
Townsend, R. H. D. 2008, , 389, 559
Townsley, L. K., et al. 2011, , submitted (CCCP Intro Paper)
Turner, D. G., Grieve, G. R., Herbst, W., & Harris, W. E. 1980, , 85, 1193
Turner, D. G., & Moffat, A. F. J. 1980, , 192, 283
ud-Doula, A., & Owocki, S. P. 2002, , 576, 413
Vazquez, R. A., Baume, G., Feinstein, A., & Pado, P. 1996, Revista Mexicana de Astronomia y Astrofisica Conference Series, 4, 131
Vink, J. S., O’Neill, P. M., Els, S. G., & Drew, J. E. 2005, , 438, L21
Vuong, M. H., Montmerle, T., Grosso, N., Feigelson, E. D., Verstraete, L., & Ozawa, H. 2003, , 408, 581
Wade, G. A., Fullerton, A. W., Donati, J.-F., Landstreet, J. D., Petit, P., & Strasser, S. 2006, , 451, 195
Walborn, N. R. 1995, Revista Mexicana de Astronomia y Astrofisica Conference Series, 2, 51
Walborn, N. R., & Blades, J. C. 1997, , 112, 457
Walborn, N. R., et al. 2002, , 123, 2754
Walborn, N. R., Nichols, J. S., & Waldron, W. L. 2009, , 703, 633
Walborn, N. R., Sota, A., Ma[í]{}z Apell[á]{}niz, J., Alfaro, E. J., Morrell, N. I., Barb[á]{}, R. H., Arias, J. I., & Gamen, R. C. 2010, , 711, L143
Waldron, W. L., & Cassinelli, J. P. 2009, , 692, L76
Wegner, W. 1994, , 270, 229
Wojdowski, P. S., & Schulz, N. S. 2005, , 627, 953
Wolk, S. J., Harnden, F. R., Jr., Flaccomio, E., Micela, G., Favata, F., Shang, H., & Feigelson, E. D. 2005, , 160, 423
Wright, N. J., Drake, J. J., Drew, J. E., & Vink, J. S. 2010, , 713, 871 Zhekov, S. A., & Palla, F. 2007, , 382, 1124
Zinnecker, H., & Preibisch, T. 1994, , 292, 152
![ Spatial distribution of the 200 OB stars with determined spectral types in Carina. Stars detected in the CCCP are shown as filled symbols. Early-O binaries are black, early-O single stars are blue, late-O binaries are dark green, late-O single stars are light green, and B stars are pink. For the 78 OB stars with more than 50 counts hard, medium, and soft X-ray stars are shown as triangles, squares, and circles, respectively. The 51 X-ray detected stars with fewer than 50 counts are shown as filled diamonds. The open diamonds represent undetected stars (mostly B stars, in pink). The grey outline shows the boundaries of the CCCP and the contours show boundaries of the large-scale clustering of low-mass stars, reproduced from Figure 1 of @feigelson11. See §4 for symbol definitions. []{data-label="f1"}](f1.pdf)
![ HR diagram of the 200 OB stars in the CCCP field of view. See Fig. 1 and §4 for symbol definitions. The most luminous stars are labeled. []{data-label="f2"}](f2.pdf)
![ 0.5-8.0 keV X-ray flux at Earth versus bolometric flux for the full sample of X-ray detected OB stars. See Fig. 1 for symbol definitions. Note: the ACIS-I CCCP survey data of HD 93250, HD 93129A, QZ Car, and HD 93205 were piled up; $f_{\rm X}$ was not computed in [*ACIS Extract*]{} for these four bright stars. The upper dotted and lower dashed lines represent $\log f_{\rm X}/f_{\rm bol} = -7$ and $-7.23$, respectively.[]{data-label="f3"}](f3.pdf)
![ [*Top panel:*]{} $\log L_{\rm X}/L_{\rm bol}$ versus $kT_{\rm avg}$ for the 78 OB stars with XSPEC parameters and more than 50 net ACIS counts. See Fig. 1 for symbol definitions. The upper dotted and lower dashed lines represent $\log L_{\rm X}/L_{\rm bol} = -7$ and $-7.23$, respectively. For stars with $kT_{\rm avg} < 1$ keV, the mean of $\log L_{\rm X}/L_{\rm bol}$ is $-7.23$. [*Bottom planel:*]{} $\log f_{\rm X}/f_{\rm bol}$ versus $E_{\rm med}$ for the full sample of X-ray detected OB stars. See Fig. 1 for symbol definitions. []{data-label="f4"}](f4a.pdf "fig:") ![ [*Top panel:*]{} $\log L_{\rm X}/L_{\rm bol}$ versus $kT_{\rm avg}$ for the 78 OB stars with XSPEC parameters and more than 50 net ACIS counts. See Fig. 1 for symbol definitions. The upper dotted and lower dashed lines represent $\log L_{\rm X}/L_{\rm bol} = -7$ and $-7.23$, respectively. For stars with $kT_{\rm avg} < 1$ keV, the mean of $\log L_{\rm X}/L_{\rm bol}$ is $-7.23$. [*Bottom planel:*]{} $\log f_{\rm X}/f_{\rm bol}$ versus $E_{\rm med}$ for the full sample of X-ray detected OB stars. See Fig. 1 for symbol definitions. []{data-label="f4"}](f4b.pdf "fig:")
![ [*Top panel:*]{} $\log L_{\rm X}$ histogram for the 78 OB stars with XSPEC parameters and more than 50 net ACIS counts. [*Bottom panel:*]{} $\log f_{\rm X}$ histogram for the full sample of X-ray detected OB stars. As in Figs. 1-4, early-O binaries are shown in black, early-O single stars in blue, late-O binaries in dark green, late-O single stars in light green, and B stars in pink. []{data-label="f5"}](f5a.pdf "fig:") ![ [*Top panel:*]{} $\log L_{\rm X}$ histogram for the 78 OB stars with XSPEC parameters and more than 50 net ACIS counts. [*Bottom panel:*]{} $\log f_{\rm X}$ histogram for the full sample of X-ray detected OB stars. As in Figs. 1-4, early-O binaries are shown in black, early-O single stars in blue, late-O binaries in dark green, late-O single stars in light green, and B stars in pink. []{data-label="f5"}](f5b.pdf "fig:")
![ [*Top panel:*]{} $\log L_{\rm X}/L_{\rm bol}$ histogram for the 78 OB stars with XSPEC parameters and more than 50 net ACIS counts. [*Bottom panel:*]{} $\log f_{\rm X}/f_{\rm bol}$ histogram for the full sample of X-ray detected OB stars. []{data-label="f6"}](f6a.pdf "fig:") ![ [*Top panel:*]{} $\log L_{\rm X}/L_{\rm bol}$ histogram for the 78 OB stars with XSPEC parameters and more than 50 net ACIS counts. [*Bottom panel:*]{} $\log f_{\rm X}/f_{\rm bol}$ histogram for the full sample of X-ray detected OB stars. []{data-label="f6"}](f6b.pdf "fig:")
![ [*Top panel:*]{} $kT_{\rm avg}$ histogram for the 78 OB stars with XSPEC parameters and more than 50 net ACIS counts. [*Bottom planel:*]{} $E_{\rm med}$ histogram for the full sample of X-ray detected OB stars. For the 51 OB stars with fewer than 50 counts and no XSPEC fits, [*hard*]{} corresponds approximately to $E_{\rm med} > 1.5$ keV. []{data-label="f7"}](f7a.pdf "fig:") ![ [*Top panel:*]{} $kT_{\rm avg}$ histogram for the 78 OB stars with XSPEC parameters and more than 50 net ACIS counts. [*Bottom planel:*]{} $E_{\rm med}$ histogram for the full sample of X-ray detected OB stars. For the 51 OB stars with fewer than 50 counts and no XSPEC fits, [*hard*]{} corresponds approximately to $E_{\rm med} > 1.5$ keV. []{data-label="f7"}](f7b.pdf "fig:")
![ Photon flux histogram of the 61 X-ray sources with B-star primaries (in pink), the 69 B-star upper limits (in brown), and the relative frequency of the 14250 CCCP sources (in gray) with measured photon flux, and not associated with a known massive star (WR or OB star). Most of the 14250 are associated with lower-mass, coronal, pre–main-sequence stars. This figure and Table 4 suggest two B star populations: a group with $\log$ photon flux below $-2.3$ (corresponding to $\log f_{\rm X} < -14$) whose X-ray properties are similar to those of the general coronal PMS population, and an active group with $\log$ photon flux above $-2.3$ (corresponding to $\log f_{\rm X} > -14$), the majority of whose X-rays are probably [*not*]{} produced by coronal PMS stars. []{data-label="f8"}](f8.pdf)
![ Sequenced photon flux light curves of FO 15 (O5.5 Vz + O9.5 V), HD 92607 (O8 V), HD 93190 (B0 IV:ep), HD 93205 (O3 V + O8 V), HD 93403 (O5.5 I + O7 V), and LS 1813 (B2 V). Listed are $\log L_{\rm X}$, $\log L_{\rm X}/L_{\rm bol}$, $kT_{\rm avg}$, and probability of constancy, $P_{\rm KS}$. The blue curves without error bars are the running count rates, corrected for effective area (in photons ks$^{-1}$ cm$^{-2}$). The corrected count rates (in black) and median energy (in red) are binned to show variability. The vertical dotted lines indicated large time gaps between adjacent observations. []{data-label="f9"}](f9.pdf)
![ Sequenced light curves of Tr16-11 (B1.5 V), Tr16-10 (B0 V), LS 1809 (O7 V), LS 1864 (B1 V). Listed are $\log L_{\rm X}$, $\log L_{\rm X}/L_{\rm bol}$, $kT_{\rm avg}$, and probability of constancy, $P_{\rm KS}$. The blue curves without error bars are the running count rates, corrected for effective area (in photons ks$^{-1}$ cm$^{-2}$). The corrected count rates (in black) and median energy (in red) are binned to show variability. The vertical dotted lines indicated large time gaps between adjacent observations. []{data-label="f10"}](f10.pdf)
![ Merged off-axis zeroth-order ACIS-S spectrum of HD 93250, O4 III(fc) and best-fit [tbabs\*(windtabs(apec)+apec+apec)]{} parameters. In this model all three emission components are absorbed by cold, neutral ISM gas, and the 0.5 keV embedded wind shocks are absorbed by warm, partially ionized, solar-abundance plasma distributed throughout the wind. The best-fit mass column is $\Sigma_{\star}=0.03\pm0.01$ g cm$^{-2}$. []{data-label="f11"}](f11.pdf)
![ $L_{\rm X}$ light curve of HD 93250, O4 III(fc), assuming the same model as in Fig. 11, indicating significant long-term variability. []{data-label="f12"}](f12.pdf)
![ $kT_{\rm avg}$ with 90% confidence limits versus primary orbital period for the 11 spectroscopic binaries in Table 3. []{data-label="f13"}](f13.pdf)
![ [*Upper panel:*]{} Optical-infrared SED of SS73 24 and the best-fit (black) Robitaille et al. (2006) YSO model (star + disk + envelope). [*Lower panel:*]{} SS73 24’s SED suggests a $6\pm2\times10^3 L_{\odot}$ photosphere with $T_{\rm eff}=25\pm1.4$ kK (dashed, blue line), a large disk (solid, green line), and cooler envelope (solid, red line). []{data-label="f14"}](f14a.pdf "fig:") ![ [*Upper panel:*]{} Optical-infrared SED of SS73 24 and the best-fit (black) Robitaille et al. (2006) YSO model (star + disk + envelope). [*Lower panel:*]{} SS73 24’s SED suggests a $6\pm2\times10^3 L_{\odot}$ photosphere with $T_{\rm eff}=25\pm1.4$ kK (dashed, blue line), a large disk (solid, green line), and cooler envelope (solid, red line). []{data-label="f14"}](f14b.pdf "fig:")
![ [*Spitzer*]{} IRAC + MIPS 24 $\mu$m image of SS73 24 and the nearby Treasure Chest cluster. SS73 24’s proximity to the most active star-forming portion of the Treasure Chest suggests it is a very young Herbig Be star. []{data-label="f15"}](f15.pdf)
[llllcclccccc]{} QZ Car & Coll 228 33 & & HD 93206 & 10 44 22.92 & -59 59 35.9 & O9.7 I + O8 III & 6.30 & -0.81 & 0.14 & 2.21 & 6.23\
HD 93129A & Tr 14 1 & MJ 177 & CPD-58 2618A & 10 43 57.46 & -59 32 51.4 & O2 If\* & 7.26 & -0.81 & 0.25 & 2.32 & 6.18\
HD 93250 & Tr 16 180 & & CPD-58 2661 & 10 44 45.02 & -59 33 54.7 & O4 III(fc) & 7.41 & -0.85 & 0.17 & 1.87 & 5.95\
HD 93403 & & & CPD-58 2680 & 10 45 44.11 & -59 24 28.2 & O5 III(fc) + O7 V & 7.27 & -0.76 & 0.22 & 1.95 & 5.95\
HD 92964 & Bo 10 44 & & CPD-58 2581 & 10 42 40.58 & -59 12 56.6 & B2.5 Ia & 5.38 & -0.66 & 0.26 & 1.61 & 5.87\
HD 93632 & Bo 11 1 & & CPD-59 2696 & 10 47 12.64 & -60 05 50.9 & O5 I-IIIf & 8.36 & -0.73 & 0.30 & 2.72 & 5.87\
HD 93205 & Tr 16 179 & MJ 342 & CPD-59 2587 & 10 44 33.75 & -59 44 15.4 & O3.5 V((f)) + O8 V & 7.76 & -0.94 & 0.08 & 1.60 & 5.79\
HD 93160 & Coll 232 & MJ 229 & CPD-58 2631C & 10 44 07.26 & -59 34 30.5 & O6 III & 7.88 & -0.68 & 0.07 & 2.01 & 5.70\
HD 93130 & Coll 228 1 & MJ 208 & CPD-59 2556 & 10 44 00.38 & -59 52 27.5 & O6 III & 8.11 & -0.75 & 0.19 & 2.20 & 5.68\
HD 93843 & & & CPD-59 2732 & 10 48 37.77 & -60 13 25.6 & O5 III(fc) & 7.32 & -0.94 & -0.04& 1.17 & 5.64\
HD 93128 & Tr 14 2 & MJ 157 & CPD-58 2617 & 10 43 54.41 & -59 32 57.4 & O3.5 V((fc)) & 8.81 & -0.79 & 0.19 & 2.21 & 5.63\
Tr15-18 & Tr 15 18 & & & 10 44 36.36 & -59 24 20.3 & O9 I/II:(e:) & 11.28& -0.09 & 1.11 & 5.66 & 5.61\
HD 93873 & & & CPD-58 2747 & 10 48 55.21 & -59 26 48.3 & B1 Ia & 7.74 & -0.50 & 0.45 & 2.60 & 5.60\
HD 305525 & Coll 228 98 & MJ 661 & CPD-59 2665 & 10 46 05.70 & -59 50 49.4 & O4 V & 10.03& -0.22 & 0.57 & 3.64 & 5.59\
HD 303308 & Tr 16 7 & MJ 480 & CPD-59 2623 & 10 45 05.92 & -59 40 06.1 & O4.5 V((fc)) & 8.19 & -0.82 & 0.14 & 1.72 & 5.57\
HD 93129B & & & CPD-58 2618B & 10 43 57.65 & -59 32 53.8 & O3.5 V((fc)) & 8.84 & -0.79 & 0.23 & 2.16 & 5.53\
Tr16-112 & Tr 16 112 & MJ 535 & CPD-59 2641 & 10 45 16.51 & -59 43 37.0 & O5.5-O6 V(n)((fc)) + B2 V-III & 9.28 & -0.69 & 0.29 & 2.68 & 5.51\
Coll228-12 & Coll 228 12 & MJ 366 & CPD-59 2592 & 10 44 36.76 & -59 54 24.9 & B1 Ib & 9.52 & -0.21 & 0.68 & 4.03 & 5.50\
Tr16-100 & Tr 16 100 & MJ 380 & CPD-59 2600 & 10 44 41.77 & -59 46 56.3 & O6 V & 8.65 & -0.77 & 0.19 & 2.18 & 5.50\
HD 93222 & Coll 228 6 & & CPD-59 2590 & 10 44 36.25 & -60 05 28.9 & O7 V((f)) & 8.10 & -0.89 & 0.05 & 1.74 & 5.46\
HD 93342 & & & CPD-58 2674 & 10 45 17.57 & -59 23 37.5 & B1 Iab-Ib & 9.09 & -0.40 & 0.60 & 3.52 & 5.45\
HD 93161B & & & & 10 44 09.09 & -59 34 35.4 & O6.5 V(f) & 8.60 & -0.77 & 0.23 & 2.10 & 5.43\
Tr16-244 & Tr 16 244 & MJ 257 & & 10 44 13.20 & -59 43 10.3 & O3/4 If & 10.78& -0.32 & 0.68 & 4.00 & 5.42\
Tr16-110 & Tr 16 110 & MJ 517 & CPD-59 2636 & 10 45 12.88 & -59 44 19.2 & O7 V + O8 V + O9 V & 9.31 & -0.64 & 0.29 & 2.74 & 5.40\
HD 93204 & Tr 16 178 & MJ 340 & CPD-59 2584 & 10 44 32.34 & -59 44 31.0 & O5.5 V((fc)) & 8.48 & -0.88 & 0.09 & 1.70 & 5.39\
HD 93190 & & & CPD-58 2637 & 10 44 19.61 & -59 16 59.0 & B0 IV:ep & 8.57 & -0.80 & 0.32 & 2.38 & 5.39\
FO 15 & & MJ 596 & & 10 45 36.32 & -59 48 23.4 & O5.5 Vz + O9.5 V & 12.05& -0.20 & 0.86 & 4.96 & 5.31\
HD 305619 & & & CPD-59 2727 & 10 48 15.54 & -60 15 56.9 & O9.7 Ib & 9.43 & -0.55 & 0.44 & 3.05 & 5.30\
Coll228-97 & Coll 228 97 & MJ 691 & CPD-59 2673 & 10 46 22.46 & -59 53 20.5 & O5 V(n)((f)) & 10.42& -0.48 & 0.41 & 3.28 & 5.29\
HD 93161A & & & & 10 44 08.82 & -59 34 34.5 & O8 V + O9 V & 8.56 & -0.75 & 0.20 & 1.94 & 5.28\
HD 303304 & & MJ 709 & CPD-58 2697 & 10 46 35.69 & -59 37 00.6 & O7 V & 9.71 & -0.62 & 0.33 & 2.85 & 5.28\
HD 93146 & Coll 228 65 & & CPD-59 2555 & 10 44 00.15 & -60 05 09.9 & O7 V((f)) & 8.44 & -0.91 & 0.02 & 1.46 & 5.26\
HD 305523 & Coll 228 32 & MJ 336 & CPD-59 2580 & 10 44 29.48 & -59 57 18.2 & O9 II & 8.50 & -0.83 & 0.18 & 1.97 & 5.24\
HD 92607 & & & CPD-59 2404 & 10 40 12.42 & -59 48 10.0 & O8 V & 8.23& -0.87 & 0.00 & 1.34 & 5.21\
Tr16-34 & Tr 16 34 & MJ 516 & CPD-59 2635 & 10 45 12.71 & -59 44 46.0 & O8 V + O9.5 V & 9.27 & -0.73 & 0.23 & 2.37 & 5.21\
Tr14-20 & Tr 14 20 & MJ 115 & CPD-58 2611 & 10 43 46.70 & -59 32 54.8 & O6 V & 9.65 & -0.63 & 0.20 & 2.35 & 5.17\
HD 305439A & & & CPD-59 2479A & 10 42 10.34 & -59 58 00.8 & B0 Ia & 9.56 & -0.52 & 0.51 & 2.98 & 5.17\
HD 303311 & Tr 16 98 & MJ 351 & CPD-58 2652 & 10 44 37.45 & -59 32 55.2 & O5 V & 9.03 & -0.86 & 0.13 & 1.59 & 5.15\
HD 305532 & Coll 228 38 & MJ 593 & CPD-59 2650 & 10 45 34.04 & -59 57 26.8 & O6 V & 10.19 & -0.70 & 0.32 & 2.80 & 5.15\
HD 303316 & & MJ 22 & CPD-59 2518 & 10 43 11.19 & -59 44 21.1 & O6 V & 9.64 & -0.76 & 0.26 & 2.30 & 5.14\
HD 305524 & Coll 228 7 & MJ 404 & CPD-59 2602 & 10 44 45.23 & -59 54 41.6 & O7 V((f)) & 9.32 & -0.75 & 0.24 & 2.12 & 5.12\
HD 93249 & Tr 15 1 & & CPD-58 2659 & 10 44 43.88 & -59 21 25.0 & O9 III & 8.36 & -0.75 & 0.14 & 1.53 & 5.11\
Tr14-8 & Tr 14 8 & MJ 192 & CPD-58 2620 & 10 43 59.93 & -59 32 25.4 & O6.5 V & 9.45 & -0.78 & 0.05 & 2.00 & 5.10\
Tr16-104 & Tr 16 104 & MJ 408 & CPD-59 2603 & 10 44 47.30 & -59 43 53.2 & O7 V((f)) + O9.5 + B0.2 IV & 8.82 & -0.79 & 0.14 & 1.55 & 5.09\
Tr16-23 & Tr 16 23 & MJ 484 & CPD-59 2626 & 10 45 05.79 & -59 45 19.6 & O9 III & 10.00 & -0.61 & 0.37 & 2.91 & 5.06\
HD 93343 & Tr 16 182 & MJ 512 & CPD-59 2633 & 10 45 12.21 & -59 45 00.4 & O8 V + O7-8.5 V & 9.60 & -0.75 & 0.24 & 2.33 & 5.06\
LS 1809 & & & CPD-58 2608 & 10 43 41.24 & -59 35 48.2 & O7 V & 10.44 & -0.40 & 0.50 & 3.08 & 5.03\
HD 305520 & Coll 228 4 & & CPD-59 2560 & 10 44 05.86 & -59 59 41.5 & B1 Ib & 8.70 & -0.71 & 0.18 & 2.00 & 5.01\
Tr14-9 & Tr 14 9 & MJ 165 & & 10 43 55.41 & -59 32 49.3 & O8 V & 9.73 & -0.68 & 0.32 & 2.42 & 5.01\
[lclccccrc]{} Tr16-22 & 104508.23-594607.0 & O8.5 V & 32.25 & -6.30 & 1.70 & -6.39 & 14 & 1\
HD 93250 & 104445.04-593354.6 & O4 III(fc) & 33.12 & -6.41 & 2.30 & & 20 & 1\
LS 1865 & 104454.70-595601.8 & O8.5 V((f)) & 31.42 & -6.90 & 3.09 & -7.12 & 6 & 3\
HD 303311 & 104437.47-593255.3 & O5 V & 31.77 & -6.96 & 0.42 & -7.47 & 33 & 1\
LS 1821 & 104357.46-600528.3 & O8.5 V & 31.25 & -7.01 & 0.30 & -7.55 & 74 & 1\
Tr16-100 & 104441.80-594656.4 & O6 V & 32.06 & -7.02 & 0.35 & -7.44 & 88 & 1\
Tr14-20 & 104346.69-593254.7 & O6 V & 31.71 & -7.03 & 0.34 & -7.77 & 98 & 1\
HD 92607 & 104012.42-594810.1 & O8 V & 31.75 & -7.04 & 0.19 & -7.42 & 1 & 2\
Tr14-5 & 104353.63-593328.4 & O9 V & 30.98 & -7.05 & 0.67 & -7.18 & 33 & 1\
HD 93146 & 104400.16-600509.8 & O7 V((f)) & 31.77 & -7.07 & 0.24 & -7.52 & 50 & 1\
HD 93129B & 104357.65-593253.7 & O3.5 V((fc)) & 32.03 & -7.08 & 0.51 & -7.56 & 84 & 1\
CPD-59 2661 & 104553.71-595703.9 & O9.5 V & 31.08 & -7.09 & 2.40 & -7.22 & 19 & 1\
Tr14-21 & 104348.70-593324.2 & O9 V & 31.13 & -7.10 & 0.53 & -7.52 & 48 & 1\
HD 93128 & 104354.40-593257.4 & O3.5 V((fc)) & 32.11 & -7.10 & 0.39 & -7.52 & 26 & 3\
HD 93222 & 104436.23-600529.0 & O7 V((f)) & 31.94 & -7.11 & 0.27 & -7.59 & 11 & 3\
HD 93843 & 104837.74-601325.7 & O5 III(fc) & 32.08 & -7.14 & 0.40 & -7.43 & 26 & 2\
Tr14-9 & 104355.36-593248.8 & O8 V & 31.44 & -7.14 & 0.19 & -7.81 & 39 & 1\
HD 305438 & 104243.71-595416.6 & O8 V((f)) & 31.17 & -7.16 & 0.38 & -7.62 & 60 & 3\
HD 305536 & 104411.04-600321.8 & O9.5 V & 31.24 & -7.16 & 1.01 & -7.40 & 57 & 1\
HD 93160 & 104407.26-593430.5 & O6 III & 32.10 & -7.18 & 0.25 & -7.68 & 2 & 2\
Tr16-3 & 104506.70-594156.6 & O8.5 V & 31.04 & -7.18 & 0.36 & -7.74 & 49 & 1\
HD 303316 & 104311.17-594420.8 & O6 V & 31.53 & -7.19 & 0.30 & -7.77 & 89 & 1\
Tr16-115 & 104520.57-594251.1 & O9.5 V & 31.01 & -7.20 & 0.32 & -7.65 & 18 & 1\
Tr14-8 & 104359.92-593225.4 & O6.5 V & 31.48 & -7.20 & 0.23 & -7.83 & 89 & 1\
Tr16-244 & 104413.19-594310.1 & O3/4 If & 31.80 & -7.21 & 0.62 & -7.33 & 34 & 1\
HD 93028 & 104315.33-601204.3 & O9 IV & 31.23 & -7.21 & 0.14 & -7.74 & 10 & 1\
HD 93161B & 104409.08-593435.3 & O6.5 V(f) & 31.75 & -7.26 & 0.54 & -7.62 & 11 & 1\
HD 305524 & 104445.27-595441.5 & O7 V((f)) & 31.43 & -7.28 & 0.61 & -7.60 & 10 & 1\
HD 305518 & 104343.99-594817.9 & O9.5 V & 31.27 & -7.29 & 0.38 & -7.96 & 61 & 1\
HD 93204 & 104432.34-594431.0 & O5.5 V((fc)) & 31.67 & -7.29 & 0.30 & -7.70 & 9 & 1\
HD 93027 & 104317.92-600803.1 & O9.5 IV & 31.08 & -7.30 & 0.27 & -7.81 & 34 & 1\
HD 305539 & 104633.07-600412.9 & O7 & 31.16 & -7.32 & 0.48 & -7.81 & 25 & 4\
HD 305532 & 104534.04-595726.7 & O6 V & 31.39 & -7.34 & 0.23 & -7.88 & 83 & 1\
LS 1892 & 104622.48-595320.4 & O5 V(n)((fc)) & 31.53 & -7.35 & 0.55 & -7.64 & 47 & 2\
HD 93249 & 104443.88-592125.1 & O9 III & 31.33 & -7.36 & 0.46 & -7.69 & 45 & 1\
HD 305525 & 104605.70-595049.5 & O4 V & 31.76 & -7.41 & 0.33 & -7.92 & 41 & 1\
HD 305523 & 104429.47-595718.1 & O9 II & 31.41 & -7.41 & 0.30 & -8.06 & 9 & 1\
Tr16-21 & 104436.73-594729.5 & O8 V & 30.97 & -7.47 & 0.15 & -7.91 & 14 & 1\
HD 93576 & 104653.84-600441.9 & O9 IV & 30.91 & -7.50 & 0.34 & -7.86 & 86 & 1\
LS 1809 & 104341.24-593548.1 & O7 V & 31.03 & -7.58 & 0.44 & -7.93 & 1 & 2\
HD 93130 & 104400.38-595227.5 & O6 III & 31.65 & -7.61 & 0.63 & -8.00 & 33 & 1\
HD 303304 & 104635.70-593700.7 & O7 V & 31.23 & -7.64 & 0.74 & -7.87 & 7 & 1\
HD 305612 & 104716.41-600539.9 & O9 V & 30.80 & -7.64 & 0.54 & -7.93 & 94 & 1\
HD 93632 & 104712.63-600550.8 & O5 I-IIIf & 31.77 & -7.67 & 0.56 & -7.92 & 94 & 1\
HD 305619 & 104815.50-601556.9 & O9.7 Ib & 31.20 & -7.68 & 0.54 & -8.18 & 68 & 1\
Tr15-19 & 104435.91-592335.7 & O9 V & & & & -6.85 & 10 &\
CPD-58 2627 & 104402.44-592936.3 & O9 III & & & & -7.67 & 1 &\
Coll228-67 & 104400.43-600559.8 & O9 V & & & & -7.80 & 29 &\
Bo11-5 & 104715.29-600538.8 & O9 V: & & & & -8.04 & 11 &\
Tr14-127 & 104400.94-593545.7 & O9 V & & & & -8.07 & 33 &\
Coll228-66 & 104359.45-600513.3 & O9.5 V & & & & -8.10 &&\
Tr15-2 & 104443.77-592117.2 & O9.5 III: & & & & -8.32 & 19 &\
Tr14-27 & 104343.89-593346.1 & O9 V & & & & -8.98 &&\
Tr15-18 & 104436.35-592420.3 & O9 I/II:(e:) & & & & &&\
Tr15-20 & 104435.12-592328.1 & O9 V: & & & & &&\
[lclcccccc]{} HD 93403 & 104544.13-592428.1 & O5 III(fc) + O7 V & 15.093 & 33.11 & -6.41 & 1.00 &9 & 3\
HD 93205 & 104433.74-594415.4 & O3 V + O8 V & 6.0803 & 32.55 & -6.82 & 0.30 &3 & 1\
HD 93129A & 104357.47-593251.3 & O2 If$\ast$ & & 32.91 & -6.85 & 0.74 & 46 & 1\
HD 303308 & 104505.90-594006.0 & O4.5 V((fc)) & & 32.25 & -6.89 & 0.25 & 20 & 4\
HD 93161A & 104408.84-593434.4 & O8 V + O9 V & 8.566 & 31.94 & -6.92 & 0.53 & 81 & 1\
HD 93343 & 104512.23-594500.5 & O8 V + O7-8.5 V & 44.15 & 31.66 & -6.98 & 3.17 & 30 & 1\
Tr16-9 & 104505.84-594307.7 & O9.5 V & & 31.27 & -7.17 & 0.70 &9 & 2\
Tr16-34 & 104512.72-594446.2 & O8 V + O9.5 V & 2.30000 & 31.56 & -7.23 & 0.60 & 26 & 5\
Tr16-110 & 104512.88-594419.3 & O7 V + O8 V + O9 V & 3.62864, 5.034 & 31.74 & -7.24 & 0.63 & 19 & 4\
QZ Car & 104422.91-595935.9 & O9.7 I + O8 III & 20.72, 5.999 & 32.55 & -7.26 & 1.03 &5 & 1\
Tr16-23 & 104505.79-594519.7 & O9 III & & 31.38 & -7.26 & 0.32 & 83 & 1\
Tr16-104 & 104447.31-594353.3 & O7 V((f)) + O9.5 + B0.2 IV & 2.1529 & 31.38 & -7.29 & 0.53 & 49 & 1\
Tr16-1 & 104508.21-594049.6 & O9.5 V + B0.3 V & 1.4693 & 30.87 & -7.30 & 0.28 & 38 & 1\
Tr16-112 & 104516.52-594337.1 & O5.5-O6 V(n)((fc)) + B2 V-III & 4.0157 & 31.78 & -7.31 & 0.35 & 53 & 1\
FO 15 & 104536.33-594823.5 & O5.5 Vz + O9.5 V & 1.1414 & 31.24 & -7.65 & 0.50 & 41 & 5\
[lclccccrc]{} SS73 24 & 104557.13-595643.1 & Be pec & 31.70 & -5.65 & 3.13 & -13.07 & 19 & 2\
Tr16-64 & 104504.75-594053.7 & B1.5 Vb & 31.37 & -6.05 & 2.73 & -13.48 & 30 & 2\
Tr16-10 & 104430.34-593726.8 & B0 V & 31.44 & -6.99 & 1.68 & -13.56 & 32 & 2\
Tr16-5 & 104454.06-594129.4 & B1 V & 31.18 & -6.43 & 2.94 & -13.64 & 0 & 4\
Tr14-28 & 104343.55-593403.4 & B2 V & 31.24 & -5.98 & 2.74 & -13.66 & 0 & 3\
HD 93501 & 104622.02-600118.8 & B1.5 III: & 31.13 & -6.94 & $>6.$ & -13.69 & 30 & 1\
Coll228-68 & 104400.17-600607.7 & B1 Vn & 31.06 & -6.46 & 2.54 & -13.76 & 2 & 3\
Tr14-124 & 104405.84-593511.6 & B1 V & 31.13 & -6.59 & 1.96 & -13.77 & 12 & 2\
HD 93190 & 104419.63-591658.6 & B0 IV:ep & 31.12 & -7.84 & 2.35 & -13.77 & 1 & 2\
Tr14-18 & 104357.96-593353.4 & B1.5 V & 31.13 & -6.25 & 2.13 & -13.77 & 40 & 1\
LS 1813 & 104345.04-595325.0 & B2 V & 31.04 & -6.34 & 1.96 & -13.81 & 0 & 2\
Tr16-11 & 104422.51-593925.4 & B1.5 V & 31.00 & -6.35 & 0.98 & -13.88 & 0 & 4\
Tr14-19 & 104358.45-593301.5 & B1 V & 31.07 & -6.54 & 2.38 & -13.92 & 92 & 1\
HD 92644 & 104031.71-594643.9 & B1.5 III & 31.12 & -6.82 & 0.24 & -13.98 & 52 & 1\
Tr14-29 & 104405.09-593341.4 & B1.5 V & 30.91 & -6.62 & 2.49 & -14.02 & 16 & 2\
HD 305533 & 104513.44-595753.1 & B0.5 Vnn:(shell) &&&& -14.16 & 20 & 1\
HD 305599 & 104924.95-594944.0 & B0 Ib & 31.28 & -6.94 & 0.23 & -14.19 & 17 & 2\
HD 305439A & 104210.35-595800.9 & B0 Ia & 31.11 & -7.63 & 0.56 & -14.20 & 7 & 2\
Tr16-126 & 104359.86-593524.1 & B1 V &&&& -14.24 & 66 & 2\
HD 305538 & 104546.52-600513.5 & B0 V &&&& -14.27 & 30 & 3\
HD 305556 & 104343.42-602027.7 & B0 Ib & 31.17 & -7.36 & 0.28 & -14.27 & 9 & 1\
Tr14-22 & 104348.82-593335.2 & B2 V &&&& -14.33 & 64 & 1\
HD 92741 & 104112.33-595825.0 & B1.5 II: &&&& -14.64 & 2 & 2\
LS 1864 & 104450.39-595545.0 & B1 V &&&& -14.70 & 7 & 1\
HD 93026 & 104316.35-591027.2 & B1.5 V &&&& -14.82 & 6 & 1\
HD 303225 & 104135.44-593945.6 & B1.5 V &&&& -14.90 & 9 & 1\
HD 305606 & 104925.87-600137.3 & B2 V &&&& -14.92 & 1 & 1\
[lcccc]{} 2000-11-19 & 51867.12 & HETG & 632 & 89545.8\
2003-05-02 & 52761.50 & HETG & 3745 & 94533.0\
2004-09-21 & 53269.73 & NONE & 4495 & 56634.4\
2005-09-05 & 53618.62 & HETG & 5400 & 33727.4\
2005-10-22 & 53665.13 & HETG & 5399 & 39670.4\
2006-06-19 & 53905.75 & HETG & 7341 & 53265.7\
2006-06-22 & 53908.43 & HETG & 7342 & 49313.2\
2006-06-23 & 53909.74 & HETG & 7189 & 17721.5\
[^1]: We note that the $N_{\rm H}/A_V$ ratio is robust to very high $A_V$, but that the scatter in the $N_{\rm H}$ versus $A_V$ diagram is quite large, especially at high $A_V$.
[^2]: The FITS and IDL save files are available from the authors upon request.
[^3]: http://lheawww.gsfc.nasa.gov/users/corcoran/xmega/xmega.html
|
---
abstract: 'We constrain the proper motions of five OB stars associated with candidate stellar wind bow shocks in the Carina Nebula using *HST* ACS imaging over 9–10 year baselines. These proper motions allow us to directly compare each star’s motion to the orientation of its [[candidate]{}]{} bow shock. Although these stars are saturated in our imaging, we assess their motion by the shifts required to minimize residuals in their Airy rings. The results limit the direction of each star’s motion to sectors less than 90$\degr$ wide. None of the five stars are moving away from the Carina Nebula’s central clusters as runaway stars would be, confirming that [[a candidate]{}]{} bow shock is not necessarily indicative of a runaway star. Two of the five stars are moving tangentially relative to the orientation of their [[candidate]{}]{} bow shocks, both of which point at the OB cluster Trumpler 14. In these cases, the large-scale flow of the interstellar medium, powered by feedback from the cluster, appears to dominate over the motion of the star in producing the observed [[candidate]{}]{} bow shock. The remaining three stars all have some component of motion toward the central clusters, meaning that we cannot distinguish whether their [[candidate]{}]{} bow shocks are indicators of stellar motion, of the flow of ambient gas, or of density gradients in their surroundings. In addition, these stars’ lack of outward motion hints that the distributed massive-star population in Carina’s South Pillars region formed in place, rather than migrating out from the association’s central clusters.'
author:
- |
Megan M. Kiminki,$^{1}$[^1] Nathan Smith,$^{1}$ Megan Reiter$^{2}$ and John Bally$^{3}$\
$^{1}$Steward Observatory, University of Arizona, 933 N. Cherry Avenue, Tucson, AZ 85721, USA\
$^{2}$Department of Astronomy, University of Michigan, 311 West Hall, 1085 S. University Avenue, Ann Arbor, MI 48109, USA\
$^{3}$Department of Astrophysical and Planetary Sciences, University of Colorado, UCB 389, Boulder, CO 80309, USA
bibliography:
- 'ms.bib'
date: 'Accepted 2017 March 9. Received 2017 March 7; in original form 2016 November 21.'
title: Proper motions of five OB stars with candidate dusty bow shocks in the Carina Nebula
---
\[firstpage\]
regions – open clusters and associations: individual: Carina Nebula – proper motions – stars: early-type – stars: kinematics and dynamics
Introduction {#sec:intro}
============
Feedback from massive stars impacts their surroundings on scales ranging from the shaping of their immediate circumstellar environment to the reionization of the universe. Stellar wind bow shocks, falling on the former end of that scale, provide important information about a star’s history and environment. Bow shocks are produced when the relative velocity between a star and the surrounding interstellar medium (ISM) is supersonic [@baranov1971; @vanburenmccray1988]. They typically appear as arc-shaped features in optical line emission [e.g., @kaper1997; @bally2000; @brownbomans2005; @brownsbergerromani2014] and/or thermal infrared continuum emission from dust [e.g., @vanburen1995; @noriegacrespo1997; @comeronpasquali2007; @france2007; @gaspar2008; @peri2012; @winston2012; @kobulnicky2016]. These features mark the sweeping-up of ambient material between the stellar wind termination shock and a second shock from the supersonic motion. The orientation of the bow shock arc depends on the direction of the relative motion, although it can be skewed by density gradients in the environment [@wilkin2000]. The arc’s standoff distance from the star depends on the pressure balance between the stellar wind and the ISM and hence on the magnitude of their relative motion and the density of the ISM. Bow-shock-like structures may also be produced where dust in a photoevaporative flow is stalled by radiation pressure rather than the stellar wind [@ochsendorf2014a; @ochsendorf2014b; @ochsendorftielens2015; @ochsendorf2015]. The asymmetric stellar wind bubbles of slower-moving stars may also have a similar appearance in the mid-infrared [@mackey2015; @mackey2016].
Arc-shaped structures around massive stars have commonly been considered a marker of high stellar velocities, under the assumption that the relative motion between star and ISM is dominated by the absolute motion of the star [@vanburen1995; @kaper1997; @gvaramadzebomans2008; @gvaramadze2010; @kobulnicky2010; @gvaramadze2011a; @gvaramadze2011b]. The typical velocity of an O-type star relative to its surroundings is $\sim10$ km s$^{-1}$ [@blaauw1961; @cruzgonzalez1974; @giesbolton1986; @tetzlaff2011], comparable to the speed of sound in an region, but 20–30% of O-type stars are “runaways” with velocities $\gtrsim40$ km s$^{-1}$ [@blaauw1961; @cruzgonzalez1974; @stone1991; @tetzlaff2011]. The high speeds of runaway stars are imparted through dynamical interactions in a cluster [@poveda1967; @giesbolton1986; @fujiiportegieszwart2011], [[through the disruption of a binary system when the companion star explodes as a supernova]{}]{} [@blaauw1961], or through a two-step scenario involving both processes [@pflammaltenburgkroupa2010]. Runaways make up 50–100% of field O-type stars, the O-type population found outside clusters and associations [@dewit2005; @schilbachroser2008; @gvaramadze2012a].
The question of whether all field O-type stars are runaways, or whether a small fraction formed in isolation, is of key importance to our understanding of massive star formation. The monolithic collapse model [@mckeetan2003; @krumholz2005b; @krumholz2009] permits truly isolated massive star formation, albeit rarely, while the competitive accretion model [@zinnecker1982; @bonnell2001a; @bonnell2001b; @bonnell2004] requires that massive stars form exclusively in clusters. In observational studies, the presence of a bow shock [[or candidate bow shock]{}]{} is sometimes taken as a clue that a given massive field star did not form in situ. For example, HD 48229 and HD 165319 were part of the $4\pm2$% of all O-type stars identified by @dewit2004 [@dewit2005] as likely candidates for isolated massive star formation. Bow shocks were later discovered around both sources [@gvaramadzebomans2008; @gvaramadze2012a], calling into question their origins in the field.
While 70% of bow shocks [[and bow-shock-like structures]{}]{} are located in relatively isolated environments consistent with runaway stars [@kobulnicky2016], the rest are found around OB stars in clusters and associations. These stars have sometimes been interpreted as runaway interlopers from other regions [@gvaramadze2011a]. However, the assumption that the motion of the ambient ISM is negligible relative to that of the star may not always be valid, particularly in and around giant regions. In many cases, bow shock orientations suggest that feedback-driven ISM flows are relevant. @povich2008 observed that bow shocks in the massive star-forming regions M17 and RCW 49 are oriented toward those regions’ central clusters, suggesting that global expansion of the regions is the dominant component of the relative star–ISM velocity. Similarly, several bow shocks in Cygnus OB2 point toward the association’s interior [@kobulnicky2010], as do more than half of the candidate bow shocks in the Carina Nebula [@smith2010b; @sexton2015]. The Galactic Plane survey of @kobulnicky2016 found that roughly 15% of infrared bow shocks are pointed at regions, while another 8% face bright-rimmed clouds; they also noted that bow shock orientations are correlated on small scales, indicative of the influence of external forces. @povich2008 refer to such feedback-facing bow shocks as “interstellar weather vanes,” tracing [[photoevaporative flows off local dense gas and/or large-scale gas motions driven by cluster feedback.]{}]{} @kobulnicky2016 call them “in-situ bow shocks,” reflecting their origin around presumably non-runaway OB stars.
[[When the motion of the star dominates over the motion of the surrounding ISM, as it does for runaway stars, the bow shock is expected to point in the direction of the star’s motion. @vanburen1995 surveyed bow shocks around known runaway stars and found that the bow shocks were preferentially aligned with their host stars’ proper-motion vectors. However, they used proper motions measured in an absolute reference frame, not corrected for Galactic rotation and solar peculiar motion and thus not necessarily representative of a star’s motion relative to the surrounding ISM. More recent surveys by @peri2012 [@peri2015], again of bow shocks around known runaways stars, did correct proper motions [[for Galactic rotation]{}]{} and noted a similar, albeit qualitative, tendency for alignment. Individual runaway stars are also often observed to be moving in the direction of their bow shocks [e.g., @moffat1998; @moffat1999; @comeronpasquali2007]. But what about bow shock around stars that have not already been identified as runaways? @kobulnicky2016 compiled a sample of bow shocks without any selection on their host stars’ kinematics. They found that more than 50 measured proper motions had velocity–bow shock misalignments of more than 45, although again, they were working with absolute proper motions rather than local. The relationship between stellar motion and bow shock orientation for stars in clusters and associations remains largely unexplored. ]{}]{}
[[To further investigate this relationship,]{}]{} we measure *local* proper motions for five [[massive stars in the Carina Nebula (listed in Table \[tab:obstab\]), each of which is associated with a candidate bow shock from @sexton2015 [which includes objects first identified by @smith2010b]. @smith2010b and @sexton2015 identified a total of 39 “extended red objects” (EROs) in the Carina Nebula.]{}]{} These EROs exhibit extended, often arc-shaped, morphology [[in *Spitzer* Infrared Array Camera (IRAC) 8.0 $\mu$m images. Nine of the @sexton2015 EROs are clearly resolved arcs and are classified as morphological bow shock candidates; one of our stars (ALS 15206) is associated with one of these sources (ERO 2). Another eight of the @sexton2015 EROs lack resolved morphologies at 8.0 $\mu$m but have infrared colors that rule out emission from young stellar objects (YSOs) and polycyclic aromatic hydrocarbons (PAHs). The remaining four of our stars are associated with sources in this category, known as color bow shock candidates.]{}]{}
-------------------- ----------------- ------------ ----------- ------------------------ ----------------------------- ------- ------------- ---------------- ------------- ----------------
ERO Star ID R.A. Dec. Spectral [[V]{}]{} ACS Date 1 Exp. time 1 Date 2 Exp. time 2
No.$^{\mathrm{a}}$ (J2000) (J2000) type [[(mag)]{}]{} field (s) (s)
2 ALS 15206 10:44:00.9 -59:35:46 O9.2 V$^{\mathrm{b}}$ [[10.7$^{\mathrm{d}}$]{}]{} TR14 2005 Jul 17 2 $\times$ 500 2015 Jun 28 2 $\times$ 520
23 TYC 8626-2506-1 10:44:30.2 -59:26:13 O9 V$^{\mathrm{b}}$ [[10.9$^{\mathrm{e}}$]{}]{} TR14 2005 Jul 17 2 $\times$ 500 2015 Jun 28 2 $\times$ 520
24 CPD-59 2605 10:44:50.4 -59:55:45 B1 V$^{\mathrm{c}}$ [[11.1$^{\mathrm{f}}$]{}]{} POS27 2006 Mar 18 2 $\times$ 500 2015 Mar 12 2 $\times$ 560
25 HDE 305533 10:45:13.4 -59:57:54 B1 V$^{\mathrm{a}}$ [[10.6$^{\mathrm{f}}$]{}]{} POS26 2006 Mar 16 2 $\times$ 500 2015 Mar 12 2 $\times$ 560
31 HD 93576 10:46:53.8 -60:04:42 O9.5 IV$^{\mathrm{b}}$ [[9.6$^{\mathrm{d}}$]{}]{} POS20 2006 Mar 15 2 $\times$ 500 2015 Mar 11 2 $\times$ 455
-------------------- ----------------- ------------ ----------- ------------------------ ----------------------------- ------- ------------- ---------------- ------------- ----------------
\
---------------------------------------------- -------------------------------------------------
$^{\mathrm{a}}$From @sexton2015. [[$^{\mathrm{d}}$From @reed2003.]{}]{}
$^{\mathrm{b}}$From @sota2014. [[$^{\mathrm{e}}$From @hog2000.]{}]{}
$^{\mathrm{c}}$From @vijapurkardrilling1993. [[$^{\mathrm{f}}$From @masseyjohnson1993.]{}]{}
---------------------------------------------- -------------------------------------------------
[[Our five target stars reside in the Carina Nebula: their visual magnitudes [[(see Table \[tab:obstab\])]{}]{}, spectral types, and extinctions [@povich2011b] confirm that they are unlikely to be foreground or background objects.]{}]{} The Carina Nebula is home to nearly 70 O-type and evolved massive stars [@smith2006a], including some of the earliest known O-type stars [@walborn2002a] and the luminous blue variable $\eta$ Carinae [@davidsonhumphreys1997]. At 2.3 kpc [@smith2006b], it is one of the closest and least-extincted massive star-forming regions. Its two central clusters, Trumpler (Tr) 14 and Tr 16, contain about half of its massive-star population. The rest is spread across $\sim$30 pc, mostly in a region of ongoing star formation known as the South Pillars [@smith2000]. Emission-line profiles show that feedback from the central clusters is driving the expansion of multiple shells of ionized gas [@damiani2016], resulting in a global expansion of the region at $\pm$15–20 km s$^{-1}$ [@walbornhesser1975; @walborn2002b; @walborn2007]. It is easy to envision that the inward-facing orientations of many of Carina’s [[candidate]{}]{} bow shocks are the result of this supersonic, feedback-driven ISM expansion [@sexton2015] or that they are shaped by interaction with dense photoevaporative flows. Here, we explore whether those interpretations are valid and to what degree these bow shocks are shaped by the motion and structure of the ISM versus the motion of their driving stars.
The organization of this paper is as follows: In Section \[sec:obs\], we describe our multiepoch *Hubble Space Telescope* (*HST*) observations, our image alignment procedure, and our method for measuring proper motions. We present our results and compare the stellar motions to the orientations of their associated [[bow shock candidates]{}]{} in Section \[sec:results\]. Section \[sec:disc\] discusses the implications and limitations of our results, and Section \[sec:conc\] summarizes our conclusions.
Observations and Analysis {#sec:obs}
=========================
*HST* ACS Imaging {#subsec:hst}
-----------------
We have conducted a large-scale multiepoch survey of the Carina Nebula using the Wide Field Camera (WFC) of *HST*’s Advanced Camera for Surveys (ACS). All observations were made with the F658N filter, which captures emission from H$\alpha$ and \[\] $\lambda$6584. Our imaging coverage is shown in Figure \[fig:overview\], where each small rectangle is one orbit made up of three overlapping pairs of [CR-SPLIT]{} exposures. Orbital pointings were designed to target features of particular interest in star formation (pillars, [[Herbig-Haro objects]{}]{}, etc.) as well as the central Tr 14 and Tr 16 clusters. [[The pointings in Figure \[fig:overview\] are labelled according to their designations in the *HST* data archive.]{}]{}
The first epoch of our ACS observations was taken in 2005–2006 (GO-10241 and GO-10475, PI: N. Smith; see @smith2010a). The same set of observations was repeated in 2014–2015 (GO-13390 and GO-13791, PI: N. Smith). This second epoch was designed to replicate the first as closely as possible in pointing and position angle in order to minimize position-dependent systemic errors when measuring proper motions. Owing to changes in the *HST* Guide Star Catalog between epochs, we were unable to duplicate the orientation angle of the central segment of the Tr 14 mosaic (marked in orange in Figure \[fig:overview\]) and of Positions 25 and 30. Those observations were rotated by $\sim$180.
Also marked in Figure \[fig:overview\] are the locations of EROs from @sexton2015. Our survey serendipitously imaged the stars associated with seven EROs. However, as discussed below, we were unable to constrain the proper motions of the two ERO-associated stars in the central part of the Tr 14 mosaic (the part for which the orientation angle changed beween epochs), leaving us with a sample of five. Full details of the observations of each of these five stars are given in Table \[tab:obstab\].
Image Alignment and Stacking {#subsec:align}
----------------------------
Our image alignment procedure, which adapts the methods of @anderson2008a [@anderson2008b], @andersonvandermarel2010, and @sohn2012, is described in detail in @reiter2015a [@reiter2015b] and @kiminki2016. In summary, we find the positions of uncrowded, unsaturated stars in individual exposures and use those positions to relate each image to a master, distortion-free reference frame. We use the program [ img2xym\_WFC.09x10]{} [@andersonking2006], which uses an array of effective point spread functions (PSFs) and has the option to fit a spatially constant perturbation PSF to account for telescope breathing and other focus changes. The measured stellar positions were then corrected for geometric distortion [@anderson2006].
A master reference frame with a pixel scale of 50 mas was constructed for each orbital pointing, aligned with north in the $+y$ direction. The six overlapping images from each epoch of a given pointing were stacked into two reference-frame master images (one per epoch) using the stacking algorithm of @anderson2008a. Object positions in the master images are directly comparable between epochs to an alignment accuracy of approximately 1 mas ($\sim$ 1 km s$^{-1}$ over a 9–10 year baseline at the distance of the Carina Nebula). We found that including a perturbation PSF in fitting stellar positions did not improve the alignment precision, but we address other possible effects of *HST* focus changes in Section \[subsec:pms\].
In all cases, the master reference frames are not tied to an absolute proper-motion zero point. Instead, the zero point is based on the average motion of several hundred well-measured stars in the image. In other words, the bulk motion of the Carina Nebula is removed, as are smaller differences in the large-scale motion of Carina’s clusters and subclusters. Features that are locally stationary, like bow shocks, are expected to be stationary in our reference frames, allowing direct measurement of the motion of stars relative to their surroundings.
Measuring Local Proper Motions of Saturated Stars {#subsec:pms}
-------------------------------------------------
With the images from two epochs on the same reference frame, measuring local proper motions for unsaturated stars in our stacked images is as simple as comparing their PSF-derived positions between epochs. However, most of the OB stars observed, including those associated with [[candidate]{}]{} bow shocks, are saturated in our ACS images (which were all $\sim$500 s long). We were unable to reconstruct the PSF core to perform traditional astrometry. Instead, we used the positions of the extended Airy rings, which are clearly visible for these stars in these deep, high-resolution images. The left column of Figure \[fig:resid\] shows the first-epoch image of all five [[ERO-associated]{}]{} stars for which we measured proper motions.
As marked in Figure \[fig:resid\], we identified four regions in the outer PSF of each star, avoiding diffraction spikes, saturation bleeding, and close companions. We then found the pixel offset in $x$ (west-east) and $y$ (north-south) by which the first epoch needed to be shifted in order to minimize the sum over those four regions of the absolute value of the flux difference between epochs. The best-fit offset was computed using the [AMOEBA]{} algorithm, the IDL implementation of the downhill simplex function minimization method [@neldermead1965; @press1992]. [AMOEBA]{} requires an input estimate. As recommended by @press1992, we run the algorithm twice, giving it a random starting estimate on the first run and then starting the second run at the best-fit parameters of the first. The resulting best-fit difference images (unshifted second epoch minus best-fit shifted first epoch) are shown in the middle column of Figure \[fig:resid\]. Although the Airy rings do not disappear completely in the difference images, there are no systematic differences in residual flux between quadrants. The best-fit offset for each star, in pixels over the 9–10 year baseline, is plotted in the right column of Figure \[fig:resid\].
This method does not produce accurate results for saturated stars in fields that experienced significant rotation between epochs. The asymmetry of the ACS WFC PSF [@andersonking2006; @mahmudanderson2008] causes the flux distribution in the outer PSF to be orientation-dependent. When the fields are rotated into alignment, the asymmetric flux distribution introduces an apparent shift of up to several pixels. Consequently, as mentioned in Section \[subsec:hst\], we were unable to measure the true shifts of the two ERO-associated stars in the central part of our Tr 14 mosaic (see Figure \[fig:overview\]). These stars were removed from further analysis and are not shown in Figure \[fig:resid\]; our final sample consists of the five ERO-associated stars listed in Table \[tab:obstab\].
To characterize the uncertainties in our fits, we ran several different tests. First, we adjusted the size and placement of the boxes used to calculate the residuals, then refit. These adjustments proved to have a negligible effect on the resulting best-fit offset. Second, we repeated the full fitting process 100 times and measured the standard deviation among the results. These ranged from 0.03 to 0.4 pixels depending on the star. Finally, we applied 100 random \[$x$,$y$\] offsets to the first-epoch images and repeated the full fitting process again for each, to determine how well we could recover the expected (artificial $+$ true) offsets. The standard deviation in the offsets recovered via this approach ranged from 0.04 to 0.23 pixels. For each star, we adopt the greater of the two uncertainty values as the formal uncertainty.
Changes in *HST* focus from both short-term thermal breathing and long-term non-thermal effects [e.g., @coxlallo2012] could induce an apparent offset between epochs by redistributing flux in the PSF. To evaluate the magnitude of this effect, we downloaded a Tiny Tim model PSF [@krist2011] for each star, at its observed chip position, for the appropriate focus value taken from the *HST* focus model [@dinino2008; @niemilallo2010; @coxniemi2011]. The true shift between “epochs” of Tiny Tim models is zero, so any measured shift would be a false positive. We ran the pair of Tiny Tim models for each star through our fitting procedure and measured apparent offsets of 0.035 to 0.22 pixels, which are illustrated by the shaded gray regions in the third column of Figure \[fig:resid\]. In most cases, the focus-induced shifts are small and/or distributed roughly evenly about the origin. For HDE 305533 (ERO 24), however, the focus changes induced a systematic $-x, +y$ offset. Removing this shift would reduce the magnitude of the observed proper motion of HDE 305533 by roughly half, but would have only a small effect on its direction of motion.
Results {#sec:results}
=======
As the plots in the right column of Figure \[fig:resid\] demonstrate, each of the five [[stars associated with candidate bow shocks]{}]{} traveled no more than $\sim$0.5 pixels (25 mas) in any direction over their 9–10 year baselines. The measured pixel offsets are given in Table \[tab:pmtab\] along with the corresponding proper motion components, the total transverse velocity, and the position angle of the proper motion vector. The best-fit local transverse velocities range from 16 to 35 km s$^{-1}$; the red arrows in Figure \[fig:overview\] show the expected travel distances over 10$^5$ yr. However, the uncertainties on most of the measured velocities are relatively large: most of the stars have motion consistent with zero within 1–2 $\sigma$. Only HD 93576 has motion significant at the 3$\sigma$ level, in the $x$ direction, although it has negligible $y$ (north–south) motion. We argue in Section \[subsec:up\] below that the true proper moions are likely on the smaller side of the allowed ranges. Even so, the results for all five stars constrain their directions of motion to sectors less than 90 wide.
--------- ----------------- -------------- -------------- --------------------------- ----------------- --------------------------------- ----------------
ERO No. Star ID $\delta x$ $\delta y$ $\mu_{\alpha}\cos\delta $ $\mu_{\delta}$ v$_{\textrm{T}}$$^{\mathrm{a}}$ Position angle
(pixels) (pixels) (mas yr$^{-1}$) (mas yr$^{-1}$) (km s$^{-1}$) (deg E of N)
2 ALS 15206 -0.39 (0.16) 0.35 (0.28) 2.0 (0.8) 1.8 (1.4) 29 (17) 48 (25)
23 TYC 8626-2506-1 -0.27 (0.29) -0.45 (0.39) 1.4 (1.4) -2.3 (2.0) 29 (27) 149 (35)
24 CPD-59 2605 0.36 (0.37) 0.46 (0.20) -2.0 (2.1) 2.5 (1.1) 35 (25) 322 (31)
25 HDE 305533 -0.39 (0.18) 0.40 (0.19) 2.2 (1.0) 2.2 (1.1) 34 (16) 45 (19)
31 HD 93576 0.26 (0.04) 0.04 (0.07) -1.4 (0.2) 0.2 (0.4) 16 ( 5) 279 (15)
--------- ----------------- -------------- -------------- --------------------------- ----------------- --------------------------------- ----------------
\
---------------------------------------------------------------------------
Uncertainties for each quantity are listed in parentheses.
$^{\mathrm{a}}$Total transverse velocity, assuming a distance of 2.3 kpc.
---------------------------------------------------------------------------
In Figure \[fig:spitzer\], we compare the local proper motions of the stars to the orientations of their associated [[candidate]{}]{} bow shocks. The latter were determined by @sexton2015 based on the peaks of the 8.0 $\micron$ flux. (ERO 25, associated with HDE 305533, does not have a measured orientation.) In these three-color *Spitzer* IRAC images, the [[candidate]{}]{} stellar wind bow shocks appear as extended red (8.0 $\micron$) features, while nearby stars are prominent in blue (3.6 $\micron$) and green (4.5 $\micron$). We indicate the stars’ motions with white arrows (lengths arbitrarily scaled for visiblity) and show the range of possible directions with dotted white lines. The [[orientations of the candidate bow shocks]{}]{}, where known, are denoted by cyan arrows, and the outer yellow arrows show the directions to the various OB clusters.
{width="0.85\linewidth"}
The uppermost panels in Figure \[fig:spitzer\] show ALS 15206 (with ERO 2 from @sexton2015) and TYC 8626-2506-1 (ERO 23). Both of these stars are closer to Tr 14 than to Tr 16 (see Figure \[fig:overview\])[[,]{}]{} both [[are associated with candidate]{}]{} bow shocks pointing at Tr 14, [[and both have proper motions directed]{}]{} tangentially to the orientation of their [[candidate]{}]{} bow shocks. [[The radial velocity of ALS 15206 is poorly constrained, as it is a probable spectroscopic binary (Kiminki et al. in preparation), but is consistent with being drawn from the radial velocity distribution of Tr 14 (@penny1993 [@garcia1998]; Kiminki et al. in preparation). No radial velocity data exist for TYC 8626-2506-1.]{}]{} [[Thus based on the proper motions of their associated stars, the]{}]{} relative motion shaping [[EROs 2 and 23]{}]{} appears to be dominated by the motion of the surrounding ISM, expanding outward from Tr 14. Unseen density gradients may also play a role, but the motion of the stars themselves do not look to be influencing the directions of [[these candidate bow shocks]{}]{}. [[They]{}]{} may truly be acting as “weather vanes,” tracing the large-scale flows of the ISM.
The middle row of Figure \[fig:spitzer\] shows CPD-59 2605 (ERO 24) and HDE 305533 (ERO 25). These stars’ [[candidate]{}]{} bow shocks are not arc-shaped at IRAC resolutions; @sexton2015 were able to measure an orientation for ERO 24 but not for ERO 25. Both stars are in Carina’s South Pillars region, and are roughly 7.5 pc northeast of the nominal center [@wu2009] of the sparse open cluster Cr 228. ERO 24 points north toward Tr 16, suggesting that it is influenced by feedback-driven outflows. However, its associated star (CPD-59 2605) has a local proper motion to the northwest, consistent with the [[orientation of the candidate bow shock]{}]{} within the uncertainties. It is thus not possible to distinguish between the effects of ISM flows and stellar motion in the case of ERO 24, as both may be relevant to shaping that [[feature]{}]{}. [[No radial velocity data exist for CPD-59 2605.]{}]{}
Relative to its surroundings, HDE 305533 (ERO 25) is moving to the northeast, away from the WNH star HD 93131 and the small group of late O-type stars that make up the center of Cr 228. Its path hints at an ejection from Cr 228, although at its observed speed it would have covered the 7.5 pc from Cr 228 in just 220,000 yr (but see discussion in Section \[subsec:up\] below on the likelihood that our measured proper motions are upper limits). The age and extent of Cr 228 are also poorly constrained, as it has often been considered an extension of Tr 16 [@walborn1995; @smithbrooks2008], while X-ray data show it to be a discrete collection of groups and subclusters without a clear center [@feigelson2011]. The origin of HDE 305533 is therefore not clearly evident. [[Its radial velocity [-18 km s$^{-1}$; @levato1990] is typical for the massive stars in Cr 228 and the South Pillars region (@levato1990; Kiminki et al. in preparation) and comparable to the radial velocity of the surrounding gas pillars [@rebolledo2016].]{}]{}
Finally, the bottom panel of Figure \[fig:spitzer\] shows HD 93576, the binary system [@levato1990] associated with ERO 31. HD 93576 lies on the outskirts of the small open cluster Bochum 11 (Bo 11), located in the southeastern part of the South Pillars. Bo 11 is home to an estimated 1000 stars [@dias2002], including the O5 supergiant HD 93632 [@sota2014] and three additional O-type stars besides HD 93576. Photometric analysis indicates that the cluster is 3–5 Myr old [@fitzgeraldmehta1987; @patatcarraro2001; @preibisch2011c]; the presence of an O5I star suggests that 3 Myr is more likely. As Figure \[fig:spitzer\] shows, the proper motion vector of HD 93576 is closely aligned with the orientation of its [[candidate]{}]{} bow shock, which in turn points nearly directly away from the center of Bo 11. This configuration suggests that HD 93576 was ejected from Bo 11 and that its subsequent supersonic motion produced the observed [[candidate]{}]{} bow shock. [[Its systemic radial velocity (-8 km s$^{-1}$; Kiminki et al. in preparation) is commensurate with the radial velocities of the other massive members of Bo 11 (@levato1990; Kiminki et al. in preparation) and the nearby dense gas [@rebolledo2016].]{}]{} But its observed proper motion (15 km s$^{-1}$) and current position (1.9 pc from the center of Bo 11) indicate an ejection date just 130,000 yr ago. Perhaps HD 93576 was ejected 2–3 Myr after the formation of Bo 11 [possible; see @ohkroupa2016]. Alternately, it could have originated outside the Carina Nebula [[and have a coincidental agreement in radial velocities]{}]{}, or the magnitude of its proper motion could be smaller than measured (see discussion in Section \[subsec:up\]). In addition, its [[candidate]{}]{} bow shock is also generally directed toward the interior of the Carina Nebula, so a contribution from ISM flows driven by cluster feedback cannot be ruled out regardless of the origin of the star.
Discussion {#sec:disc}
==========
Proper Motions as Upper Limits {#subsec:up}
------------------------------
For four of the five [[candidate]{}]{} bow shock host stars in our sample, we measure local proper motions of $\sim$30 km s$^{-1}$, with associated uncertainties of $>15$ km s$^{-1}$. (This total includes HDE 305533, whose observed motion may include a contribution from focus changes as described in Section \[subsec:pms\].) Several lines of reasoning support the interpretation of these measured velocities as upper limits, with the true proper motions lying on the small side of the allowed range. First, the typical velocity of an O-type star relative to its surrounding is $\sim$ 10 km s$^{-1}$ [@blaauw1961; @cruzgonzalez1974; @giesbolton1986; @tetzlaff2011]. Of course, the stars in our sample are arguably not typical, given their association with [[candidate]{}]{} dusty bow shocks. Space velocities of 30 km s$^{-1}$ may qualify them as runaway stars, depending on the choice of runaway classification criteria. None of the five stars measured here are moving with trajectories that could have originated in Tr 14 or Tr 16, although HDE 305533 (ERO 25) and HD 93576 (ERO 31) may have come from the smaller open clusters Cr 228 and Bo 11, respectively. An object moving at 30 km s$^{-1}$ would cover 60 pc in 2 Myr [the average estimated age of Tr 14/16; @walborn1995; @smith2006a; @preibisch2011c], and these stars are all significantly closer than that to any possible clusters of origin in the Carina Nebula. It is possible that all four of the stars with measured proper motions of $\sim$30 km s$^{-1}$ were ejected more recently, but that scenario would still not explain their directions of motion. Similarly, it is possible that all four are interlopers in the Carina Nebula, originating from another cluster, but the chance of encountering four such stars in our small sample is low. [[And as described in Section \[sec:results\], the radial velocities of our sample stars, where available, agree with the radial motions of the surrounding stars and gas, consistent with more local origins.]{}]{}
In addition, speeds of 30 km s$^{-1}$ are inconsistent with the relative star–ISM velocities computed for Carina’s [[candidate]{}]{} bow shocks by @sexton2015. The pressure balance governing a standard bow shock makes it possible to estimate the relative star–ISM velocity as a function of measured standoff distance by making reasonable assumptions about stellar wind velocity, mass-loss rate, and ISM density. @sexton2015 measured the standoff distances of nine EROs in the Carina Nebula and found an average star–ISM velocity of 17 km s$^{-1}$. For ERO 2 (associated with ALS 15206), the relative star–ISM velocity was a barely-supersonic 7 km s$^{-1}$. Similar relative velocities for bow shocks in the massive star-forming region RCW 38 were reported by @winston2012. These numbers have substantial uncertainties due to the assumptions that go into their calculation, but they still suggest somewhat lower stellar velocities. Consider ERO 2 (ALS 15206): The orientation of the [[candidate]{}]{} bow shock indicates that the direction of the highest relative star–ISM velocity is to the northwest. We have measured that ALS 15206 is moving to the northeast, tangential to its [[candidate]{}]{} bow shock. If the relative star–ISM velocity in the direction of the [[candidate]{}]{} bow shock is on the order of 7 km s$^{-1}$, the relative star–ISM velocity in a different direction cannot be substantially higher than that, although the picture may be complicated if there are density gradients in the ISM.
For these reasons, it is unlikely that the measured [[stars associated with candidate bow shocks]{}]{} are moving as fast as 30 km s$^{-1}$ relative to their surroundings. The local proper motions given here should thus be treated as upper limits. HD 93576 may be an exception, as its westward motion is measured at 3$\sigma$ significance (but this raises questions about its possible origin in Bo 11, as discussed in Section \[sec:results\]).
Comparison to Absolute Proper Motions {#subsec:abs}
-------------------------------------
[[All five of the stars in our sample have proper motions listed in the USNO CCD Astrograph Catalog [UCAC4; @zacharias2013], and ALS 15206, TYC 8626-2506-1, and HD 93576 also have proper motions in the Tycho-2 Catalogue [@hog2000] and *Gaia* Data Release 1 [DR1; @gaia2016a; @gaiabrown2016; @lindegren2016]. (Note that Tycho-2 and *Gaia* DR 1 are not wholly independent measurements, as the latter incorporates positional information from the former.) The UCAC4, Tycho-2, and *Gaia* DR1 proper motions are measured in an absolute reference frame and are therefore not directly comparable to the local proper motions measured here. We would expect to see a roughly constant offset between these absolute proper motions and our local ones, with that offset representing the bulk motion of the Carina Nebula relative to the Sun. We plot the available absolute proper motions for each star, along with our measured local proper motions, in Figure \[fig:comp\]. Contrary to expectations, Figure \[fig:comp\] does not show a consistent offset between local and absolute proper motions. The UCAC4 proper motions in particular do not follow any apparent trend relative to the local proper motions or the Tycho-2 and *Gaia* DR1 data. The differences between catalogues suggest that there may be systematic effects in the literature measurements that are not taken into account in the published uncertainties.]{}]{}
For further comparison, we correct the *Gaia* proper motions, where available, to the rest frame of the Carina Nebula in two ways. First, we formally correct for Galactic rotation and solar peculiar motion, as in @moffat1998 [@moffat1999] and @comeronpasquali2007. We adopt Oort’s constants $A=15\pm1$ km s$^{-1}$ kpc$^{-1}$ and $B=-12\pm1$ km s$^{-1}$ kpc$^{-1}$ [@feastwhitelock1997; @elias2006; @bovy2017] and components of the solar peculiar velocity $(U_{\sun},V_{\sun},W_{\sun})=(10\pm1,12\pm1,7\pm1)$ km s$^{-1}$ [@feastwhitelock1997; @elias2006; @schonrich2010; @tetzlaff2011]. The corrected proper motions are plotted in Figure \[fig:gaia\]. For all three stars, the corrected *Gaia* proper motions are $\le1.3$ mas yr$^{-1}$ ($\le14$ km s$^{-1}$), supporting our interpretation that these stars are not runaways. The corrected *Gaia* proper motion of ALS 15206 (ERO 2) is, like our measured motion, directed to the northeast, tangential to the orientation of its candidate bow shock. The corrected *Gaia* motion of TYC 8626-2506-1 (ERO 23) is to the southwest, into its candidate bow shock, although its 1$\sigma$ uncertainties overlap with those of our measured motion to the southeast. The corrected *Gaia* motion of HD 93576 (ERO 31) is also consistent with our data, although the *Gaia* results indicate a smaller velocity to the west, suggesting a longer time since ejection from Bo 11.
We also perform an empirical correction to the local reference frame: we compute the weighted mean proper motion of the 38 O-type stars in the Carina Nebula in *Gaia* DR1 [which is roughly half the total O-type population of the region; e.g., @smith2006a; @gagne2011; @alexander2016] and subtract that from the absolute *Gaia* proper motions of the three sample stars. The results are consistently $\sim1.1$ mas yr$^{-1}$ ($\sim12$ km s$^{-1}$) offset from the results of formally correcting for Galactic rotation and solar peculiar motion. For ALS 15206 (ERO 2) and TYC 8626-2560-1 (ERO 23), the empirical correction brings the corrected *Gaia* proper motions into better agreement with our results. For HD 93576 (ERO 31), the empirical correction produces worse agreement with our results and suggests that the star is moving to the east, away from its candidate bow shock and toward Bo 11. The different parts of the Carina Nebula may have different large-scale motions not properly accounted for in these corrections. Future *Gaia* data releases, extending into Carina’s intermediate-mass population, will allow more precise and locally-specific corrections to the local reference frame.
Interpreting Bow Shocks in Giant Regions {#subsec:interp}
----------------------------------------
In our subsample of bow shock candidates in the Carina Nebula, EROs 2 and 23 face the OB cluster Tr 14, while EROs 24 and 31 point more generally toward Tr 14 and 16. The majority of bow shock candidates in the full @sexton2015 sample also point in toward the clusters rather than out as would be expected for runaway stars. @sexton2015 hypothesized that these [[candidate]{}]{} bow shocks are markers of large-scale ISM flows driven by cluster feedback. The ionized gas in the Carina Nebula is known to be globally expanding at 15–20 km s$^{-1}$ [@walbornhesser1975; @walborn2002b; @walborn2007], with multiple local centers of expansion including Tr 14 [@damiani2016]. Feedback-driven outflows are also thought to explain the orientations of bow shocks in other massive star-forming regions [@povich2008; @winston2012] and the correlation of bow shock orientations on small angular scales [@kobulnicky2016]. Our results are broadly compatible with this interpretation, but they indicate that the factors influencing a bow shock [[or bow-shock-like structure]{}]{} cannot be deduced solely from its orientation.
The associated stars of EROs 2 and 23 are not moving in the direction of their [[infrared]{}]{} arcs, which are thus likely shaped by feedback from Tr 14. These two objects confirm that, at least in this environment, [[apparent]{}]{} bow shock orientation does not always follow stellar motion. In contrast, the associated stars of EROs 24, 25, and 31 are moving roughly toward Tr 14 and 16. Both stellar motion and ISM flows could be relevant in setting the orientation of these three [[candidate]{}]{} bow shocks, demonstrating that cluster-facing [[bow-shock-like structures]{}]{} are not necessarily clear markers of the motion of the ISM.
Feedback may also be affecting bow shocks in giant regions in other ways besides large-scale outflows. For instance, the pile-up of dust in bow shock arcs depends on the presence of dust in the surrounding region, as the hot winds of OB stars do not make dust effectively. This dust may originate in photoevaporative flows off nearby molecular pillars, driven globally by ionizing radiation from the central clusters and locally by individual OB stars. @kobulnicky2016 found that eight percent of bow shocks across the Galactic Plane face bright-rimmed clouds, suggesting they are shaped by local photoevaporative flows. The arc-shaped dust waves around $\sigma$ and $\lambda$ Ori are also thought to be driven by photoevaporative flows off the edge of ionized bubbles [@ochsendorf2014a; @ochsendorf2014b; @ochsendorftielens2015; @ochsendorf2015]. Density gradients in the ISM can also affect bow shock symmetries [@wilkin2000] or create infrared arcs via uneven heating.
We inspected *Spitzer* images of the Carina Nebula to assess the relationship between Carina’s EROs, its molecular gas, and the distribution of warm dust. There is a possible tendency for EROs to be closer to dense pillars than expected from a random distribution, but there is no correlation between ERO orientation and the direction to the nearest pillar. ERO 31, for example, lies just 45 ($\sim$0.5 pc) from the edge of a prominent pillar, but points almost directly away from it. Multiband Imaging Photometer (MIPS) images at 24 $\micron$ reveal complex warm dust structures throughout the nebula, including around EROs 24 and 31. However, the origin and impact of these structures with respect to the EROs is unclear. Higher-resolution mid-infrared imaging is required to tease out the effects of density gradients and photoevaporative flows in shaping Carina’s EROs.
In any case, our main result is unaffected: in a giant region, [[the orientation of bow-shock-like structures]{}]{} may be determined by the ISM, by stellar motion, or by some combination of factors. It is worth reiterating that none of the five [[stars]{}]{} in our study are runaways from Tr 14 or 16. While this result is unsurprising given the orientation of their [[candidate]{}]{} bow shocks, it confirms that stars with [[bow-shock-like structures]{}]{} are not automatically runaways. The statistical preference for alignment between stellar motion and bow shock orientation, particularly among known runaway stars [@vanburen1995; @kobulnicky2016] suggests that stellar motion does dominate over ISM flows for bow shocks far from feedback-generating clusters. But within associations, assumptions about the implications of bow shocks [[and bow-shock-like structures]{}]{} [e.g., @kobulnicky2010; @gvaramadze2011a] should be made with caution.
Implications for the Origins of OB Associations {#subsec:assoc}
-----------------------------------------------
The local proper motions of CPD-59 2605 (ERO 24), HD 305533 (ERO 25), and HD 93576 (ERO 31) can also shed light on the origins of the distributed massive-star population in the Carina Nebula. Nearly half of Carina’s massive stars, including the WNH star HD 93131, are spread across roughly 20 pc in the South Pillars [@smith2006a]. Some of these massive stars are associated with small open clusters (Bo 11, Cr 228) and other groups and subclusters of young stars [@smith2010b; @feigelson2011]. However, *Herschel* imaging detected no massive protostars in the region, suggesting that the ongoing star formation in the South Pillars is limited to low- and intermediate-mass stars [@gaczkowski2013].
In the classic picture of clustered star formation [e.g., @ladalada2003], massive stars rarely form in a distributed mode as seen in the South Pillars. Instead, massive stars are born in clusters that may subsequently become unbound after gas dispersal and expand into OB associations [[[@tutukov1978; @hills1980; @ladalada1991; @ladalada2003]]{}]{}. In this picture, one would expect the Carina Nebula’s distributed massive stars to have formed in the central Trumpler clusters and drifted out to their current locations over several Myr.
Our proper motion results are inconsistent with this expectation, as all three of the massive South Pillars stars measured here are moving toward the Trumpler clusters, not away. These stars’ kinematics suggest that they were born in the South Pillars, possibly in one of the smaller open clusters, and support a model of star formation in which OB associations form directly as loose aggregates [e.g., @efremovelmegreen1998b; @clark2005]. A similar result has been observed for the Cyg OB2 association based on its substructure and lack of global expansion [@wright2014; @wright2016]. Further investigation of stellar kinematics in the South Pillars is needed to confirm this interpretation of the Carina Nebula’s distributed population.
Conclusions {#sec:conc}
===========
Using *HST* ACS imaging with 9–10 year baselines, we have measured the local proper motions (i.e., relative to the surrounding stars) of five OB stars associated with candidate bow shocks in the Carina Nebula. Because these stars are highly saturated in our data, we use precisely-aligned images to measure the shift in each star’s Airy rings between epochs. The results are largely upper limits, but we are able to constrain the direction of each star’s motion for comparison to the orientation of its [[candidate]{}]{} bow shock.
Stellar wind bow shocks are formed when the relative velocity between star and ISM is supersonic, but the bow shock alone does not indicate which component of the relative velocity dominates. Are bow shocks indicators of fast-moving runaway stars or do they mark the large-scale flow of the ISM? In our sample of five, we find two cases where the latter is likely the case, as the stars are moving at a tangent to the arc of their [[candidate]{}]{} bow shocks. In the other three cases, we conclude that the possible influences of ISM flows, ISM structure, and stellar motion cannot be separated, and that multiple factors could be relevant for each object. We consequently caution against overinterpreting the orientation of bow shocks [[and bow-shock-like structures]{}]{} in giant regions like the Carina Nebula.
In addition, none of the five stars measured here are runaways from the central OB clusters of the Carina Nebula, although two may have been ejected from smaller open clusters in the South Pillars. This finding emphasizes that bow shocks [[and bow-shock-like structures]{}]{} in giant regions are not definite markers of runaway stars. It also suggests that the distributed massive-star population in the Carina’s South Pillars formed along with the distributed low- and intermediate-mass population; the resulting OB association is not the expanding remnant of an embedded cluster but a loose collection of many small groups and clusters.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank Jay Anderson for providing us with his suite of PSF-fitting and image alignment software, and for his valuable instruction, guidance, and technical support. [[We also thank the anonymous referee for a constructive review.]{}]{} Support for this work was provided by NASA grants GO-13390 and GO-13791 from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. This work is based on observations made with the NASA/ESA *Hubble Space Telescope*, obtained from the Data Archive at the Space Telescope Science Institute. This work has made use of data from the European Space Agency (ESA) mission [*Gaia*]{} (<http://www.cosmos.esa.int/gaia>), processed by the [*Gaia*]{} Data Processing and Analysis Consortium (DPAC, <http://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the [*Gaia*]{} Multilateral Agreement.
\[lastpage\]
[^1]: E-mail: <[email protected]>
|
---
abstract: 'In continuation of our previous study (Phys.Rev.D 99 (2019) 4, 044012) we investigate the motion of charged particles in the $\gamma$-metric. We provide some examples of curled trajectories in the equatorial plane and escape trajectories outside the equatorial plane. Finally, we consider harmonic oscillations due to small displacements from stable circular orbits on the equatorial plane and compute the epicyclic frequencies for different values of deformation parameter and magnetic field.'
author:
- 'Carlos A. Benavides-Gallego'
- Ahmadjon Abdujabbarov
- Daniele Malafarina
- Cosimo Bambi
title: 'Quasi-harmonic oscillations of charged particles in static axially symmetric space-times immersed in a uniform magnetic field'
---
Introduction
============
Black holes in the general theory of relativity are entirely described by three parameters, namely mass, angular momentum and charge [@Israel:1967wq; @Carter:1971zc; @Gurlebeck:2015xpa]. We know that classical black hole solutions contain curvature singularities which are expected to be resolved by a theory of quantum-gravity [@Penrose:1964wq; @Hawking:1969sw]. This naturally leads to the question of whether such quantum-gravity effects remain confined within the horizon or may affect the exterior geometry, thus separating (astro)physical black holes from mathematical black holes. A hint towards the answer to this question comes from the study of non-singular gravitational collapse (see for example [@Bambi:2013caa; @Bambi:2016uda; @Chakrabarty:2019omm; @Malafarina:2017csn; @Carballo-Rubio:2019fnb; @Carballo-Rubio:2019nel; @Giddings:2019ujs; @Giddings:2017jts; @Barcelo:2015noa]), which suggests that the resolution of the singularity must affect the trapped surfaces in the space-time. Another hint comes from the investigation of exact solutions of Einstein’s equations that do not describe black holes. For example, it is well known that static axially symmetric vacuum solutions generically posses naked curvature singularities when multipole moments of higher order are present [@Weyl; @WeylI; @Quevedo1989; @Quevedo1990; @Bonnor1992; @Pastora1994] . This fact can be interpreted in two ways:
- A collapsing body must shed away all higher multipole moments before crossing the horizon threshold.
- The space-time resulting from the collapse of a non-spherical body requires quantum-gravity modification already at the horizon scale to account for the non-vanishing multipole moments.
The issue will likely be resolved once we will be able to precisely probe experimentally the geometry around astrophysical black hole candidates.
As of today, the nature of the geometry around extreme astrophysical compact objects is still unknown. However, recent results such as X-ray reflection spectroscopy [@Bambi:2016sac; @Cao:2017kdq; @Tripathi:2018lhx] or the image of the ‘shadow’ of the super-massive black hole candidate at the center of the galaxy M87 [@Akiyama:2019bqs] suggest that experimental tests of the so-called Kerr hypothesis [@Bambi:2015kza; @Berti:2015itd], namely the hypothesis that all astrophysical black hole candidates are described by the Kerr metric, may be possible in the near future.
In this work, we focus our attention on the Zipoy-Voorhees space-time, also known as $\gamma$-metric, which is a static, axially symmetric line element describing the gravitational field outside a prolate or oblate object [@Voorhees:1971wh; @Zipoy]. The space-time is continuously linked to the Schwarzschild metric through the value of one parameter $\gamma$, which is related to the non-vanishing multipole moments and in the limit of $\gamma=1$ it reduces to Schwarzschild. As mentioned before, for $\gamma\neq 1$ the line-element presents a curvature singularity at the surface $r=2m$, making the manifold geodesically incomplete [@Kodama:2003ch; @Herrera:1998rj; @Herrera:1998eq] . The properties of the $\gamma$-metric, its geodesics, interior solutions and geometrical aspects, have been studied by many authors [@Herrera:2000tz; @LukesGerakopoulos:2012pq; @Boshkayev:2015jaa; @Virbhadra:1996cz; @Papadopoulos:1981wr; @Stewart1982; @Herrera:2004ra]. In a series of previous articles, some of us investigated the properties of this space-time in connection with the possibility of observing departures from the Schwarzschild or Kerr line-elements in astrophysical observations [@Abdikamalov:2019ztb; @Toshmatov:2019qih; @Narzilloev:2020qdc].
In practice, to probe experimentally the geometry of extreme compact objects one can rely on different methods. Shadow and orbits are limited to only two super-massive black hole candidates and therefore to obtain a larger sample of observations one must look at X-ray binaries.
The Rossi X-ray Timing Explorer (RXTE) mission has provided thousands of observations of black holes transients [@Belloni:2012sv] and the study of X-ray binaries has been considered with great interest because it opens the possibility to probe fundamental physics [@Rezzolla:2003zx]. The X-rays emitted by binary systems are produced by matter falling from the donor (usually a normal star) to the accretor: a neutron star or a black hole. Thanks to the RXTE mission, complex variability patterns were discovered, including the detection of Quasi-Periodic Oscillation (QPO) at frequencies higher than 40Hz [@Belloni:2012sv]. According to Belloni et al., the QPOs “*open a new window onto fast phenomena in the innermost regions of an accretion disk.*” [@Belloni:2012sv]. The high-frequency QPOs give us information about the masses and radii of neutron stars [@Kluzniak:1997sh; @Miller:1997if], the masses and spin of the central objects [@Wagoner:2001uj; @Abramowicz:2001bi]. Furthermore, since the high-frequency QPOs are observed close to the orbital frequencies of the marginally stable circular orbits (MSO), the effects from strong gravity must be essential to explain their behavior [@Abramowicz:2004je; @Stella:1999sj; @Stuchlik:2008fy; @Kotrlova:2008xs]. Finally, from the analysis of the frequencies, it is also possible to obtain information about the electromagnetic field in the vicinity of black hole candidates.
Many of the observed black hole candidates have accretion discs formed by plasma whose dynamics can generate a regular magnetic field [@Kolos:2015iva]. In this sense, it is essential to consider the role of the electromagnetic field in the processes taking place in the surroundings of a black hole. Recent observations have suggested that the center of the Milky Way galaxy has a strong magnetic field that is not related to the accretion disc of the black hole [@Eatough:2013nva]. Hence, black holes can also be immersed in an external, large-scale electromagnetic field that can have a complicated structure in the vicinity of field source, but asymptotically (at large distances) its character can be close to being homogeneous [@Stuchlik:2015nlt]. In this sense, the idea of a black hole immersed in a uniform magnetic field has been used in Ref. [@Kolos:2015iva] as a model to explain the frequencies of the 3:2 high-frequency QPOs observed in the three Galactic microquasars GRS 1915+105, XTE 1550–564 and GRO1655–40, which cannot be explained by a model based on the frequencies of the geodesic epicyclic motion, if the accepted limits on the mass and spin of the black holes are taken into account [@Torok:2011qy; @Bambi:2012pa].
In the present work, we shall focus on the motion of charged test particles in the equatorial plane of the Zipoy-Voorhees space-time immersed in an external magnetic field to characterize the effects of the non-vanishing quadrupole moment on the QPOs (the case of QPOs for neutral particles was considered in [@Toshmatov:2019qih]). The motion of test particles in the magnetized Schwarzschild space-time was first considered in Ref. [@Wald:1974np]. Since then, solutions of Einstein’s equations in external magnetic fields have been widely studied (see for example [@Petterson:1975sg; @Kolos:2015iva; @Frolov:2010mi; @Narzilloev:2019hep]).
The article is organized as follows. In section \[III\], we review the $\gamma$-metric and its main properties together with the basics ideas of the description of a space-time immersed in an external magnetic field [@Wald:1974np]. In section \[IV\], following the Hamiltonian formalism presented in Ref. [@Kolos:2015iva], we discuss the charged particle motion in the magnetized Zipoy-Voorhees space-time and obtain the effective potential for charged test particles. Finally section \[V\] is devoted to the study of harmonic oscillations about the circular orbits of charged particles. Throughout the manuscript, we use the signature $(-,+,+,+)$, and use geometrized units thus setting $G=c=1$. Greeks indices run from $0$ to $3$.
Zipoy-Voorhees space-time immersed in a uniform magnetic field. \[III\]
=======================================================================
The Zipoy-Vorhees space-time, also known as $\gamma$-metric [@Voorhees:1971wh; @Zipoy] is a well known asymptotically flat vacuum solution of Einstein’s equations that belongs to the Weyl class of static, axially symmetric space-times [@Weyl]. In Erez-Rosen coordinates the line element is given by [@Herrera2000] $$\label{II.1}
ds^2=-Fdt^2+F^{-1}\left[Gdr^2+Hd\theta^2+(r^2-2mr)\sin^2\theta d\phi^2\right],$$ where $$\label{II.2}
\begin{aligned}
F(r)&=\left(1-\frac{2m}{r}\right)^\gamma,\\
G(r,\theta)&=\left(\frac{r^2-2mr}{r^2-2mr+m^2\sin^2\theta}\right)^{\gamma^2-1},\\
H(r,\theta)&=\frac{(r^2-2mr)^{\gamma^2}}{\left(r^2-2mr+m^2\sin^2\theta\right)^{\gamma^2-1}}.\\
\end{aligned}$$ When $\gamma=1$ the line element reduces to Schwarzschild in Schwarzschild coordinates. From the expansion of the gravitational potential it is easy to evaluate the total mass $M$ of the source as [@Herrera2000] $$\label{II.3}
M=\gamma m,$$ while the quadrupole moment $Q$ is given by $$\label{II.4}
Q=\frac{\gamma}{3}\left(1-\gamma^2\right)M^3.\\$$ From Eq. (\[II.4\]) we see that $\gamma>1$ correspond to oblate spheroids, while $\gamma<1$ correspond to prolate spheroids.
Symmetries in the space-time can be studied by means of Killing vectors. In the case of vacuum, stationary and axial symmetric space-times, we have two Killing vectors: one related to time translations and the other to spatial rotations about the symmetry axis. Such vectors are related to conserved quantities, energy and angular momentum, and satisfy the Killing equation [@Wald:1984rg; @Wald:1974np] $$\label{III.1}
\mathsterling_\xi g_{\mu\nu}=\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu=0,$$ where $\mathsterling_\xi g_{\mu\nu}$ is the Lie derivative of the metric tensor and $\nabla$ is the covariant derivative. One interesting property that follows form Eq.(\[III.1\]) is the relation $$\label{III.9}
\nabla^\nu\nabla_\nu\xi_\mu=-R^{\;\;\;\lambda}_{\mu}\xi_\lambda=-R^\lambda_{\;\;\;\mu}\xi_\lambda.$$
On the other hand, in curved space-times, the Maxwell’s equations for the vector potential $A_\mu$ in the Lorentz gauge ($\nabla_\mu A^\mu=0$) are given by [@Wald:1984rg; @MTW] $$\label{III.10}
\nabla^\nu\nabla_\nu A_\mu-R^\lambda_{\;\;\mu}A_\lambda=-4\pi J_\mu.$$ In this sense, if we consider $J_\mu=0$, Eq. (\[III.10\]) reduces to $$\label{III.11}
\nabla^\nu\nabla_\nu A_\mu=R^\lambda_{\;\;\mu}A_\lambda.$$ Note that there is a sign difference between Eqs. (\[III.9\]) and (\[III.11\]), so that the two equations coincide in the case of vacuum space-times. This means that, when $R^\lambda_{\;\;\;\mu}=0$, the Killing vector $\xi_\mu$ must satisfy the source-free Maxwell’s equations for a vector potential in the Lorentz gauge. In this sense, the Killing vector $\xi_\mu$ in vacuum is endowed with the property of being proportional to some vector potential $A_\mu$ and can be used to derive a solution for the electromagnetic field occurring when a stationary, axisymmetric space-time is placed in an external magnetic field aligned along the axis of symmetry [@Azreg-Ainou:2016tkt]. Hence, the Faraday tensor of the electromagnetic field $\mathrm{F}_{\mu\nu}$ can be expressed as [@Wald:1974np] $$\label{III.2}
\mathrm{F}_{\mu\nu}=\nabla_\mu\xi_\nu-\nabla_\nu\xi_\mu=-2\nabla_\nu\xi_\mu,$$ and the vector potential $A_{\mu}$ can be written as
$$\label{III.12}
A_\mu=C_1\xi^{(t)}_\mu+C_2\xi^{(\phi)}_\mu,$$
where $\xi^{(t)}_\mu$ and $\xi^{(\phi)}_\mu$ are the Killing vector fields associated with time translations and rotations about the symmetry axis. In this work, we consider the case of a magnetic field that is uniform, with magnitude $B$ at spatial infinity. If the field is oriented perpendicularly to the equatorial plane which is orthogonal to the symmetry axis, then the four-vector potential takes the form $$\label{III.13}
A_\mu=\frac{B}{2}\xi^{(\phi)}_\mu.$$ Consequently, the only nonzero component of the potential of the electromagnetic field is [@Wald:1984rg] $$\label{III.14}
A_\phi=\frac{B}{2}g_{\phi\phi}.$$ When applied to the $\gamma$-metric, given in Eq. (\[II.1\]), the vector potential is given by [@Benavides-Gallego:2018htf] $$\label{III.15}
A_\phi=\frac{B}{2}\left(1-\frac{2m}{r}\right)^{1-\gamma}r^2\sin^2\theta,$$ and it automatically reduces to the well-known vector potential in the Schwarzschild space-time for $\gamma$=1. We shall now move to consider the motion of test particles in the Zipoy-Voorhees geometry immersed in the magnetic field discussed above.
     
Charged particle dynamics \[IV\]
================================
To study the dynamics of charged particles we follow Ref. [@Kolos:2015iva]. The Hamiltonian for the charged particle can be written as $$\label{IV.1}
\mathcal{H}=\frac{1}{2}g^{\alpha\beta}(\pi_\alpha-qA_\alpha)(\pi_\beta-qA_\beta)+\frac{1}{2}m^2_0,$$ where $\pi_\alpha$ is the canonical four-momentum which is related to the kinematical four-momentum $p^\mu=m_0 u^\mu$ by the relation $$\label{IV.2}
\pi^\mu=p^\mu+qA^\mu,$$ and satisfies Hamilton’s equations $$\label{IV.3}
\begin{aligned}
\frac{dx^\mu}{d\zeta}&\equiv p^\mu=\frac{\partial \mathcal{H}}{\partial \pi_\mu},\\
\frac{d\pi_\mu}{d\zeta}&=-\frac{\partial \mathcal{H}}{\partial x^\mu}.\\
\end{aligned}$$ The affine parameter $\zeta$ is related to the proper time of the particle by the relation $\zeta=\tau/m_0$. For the line element in Eq. (\[II.1\]), using Eqs. (\[IV.1\]) and (\[IV.3\]), we obtain two constants of motion: the energy per unit mass $\mathcal{E}$ and the angular momentum per unit mass $\mathcal{L}$, which are given by $$\label{IV.4}
\begin{aligned}
\mathcal{E}&=\frac{E}{m_0}=\left(1-\frac{2m}{r}\right)^{\gamma}\frac{dt}{d\tau}, \\
\mathcal{L}&=\frac{L}{m_0}=r^2\sin^2\theta\left(1-\frac{2m}{r}\right)^{1-\gamma}\left[\frac{d\phi}{d\tau}+\mathcal{B}\right],
\end{aligned}$$ where we have introduced also $\mathcal{B}=qB/2m_0$. As expected, Eqs. (\[IV.4\]) reduce to the Schwarzschild case when $\gamma=1$.
  
By expressing the Hamiltonian in Eq. (\[IV.1\]) as
$$\label{IV.6}
\mathcal{H}=\frac{1}{2}\frac{F(r)}{G(r)} p^2_r+\frac{1}{2}\frac{F(r)}{H(r)}p^2_\theta+\frac{m^2_0}{2F(r)}\left(V_{\text{eff}}(r,\theta;\mathcal{B},\mathcal{L})-\mathcal{E}^2\right).$$
we can derive the effective potential $V_{\text{eff}}(r,\theta;\mathcal{B},\mathcal{L})$ as [@Benavides-Gallego:2018htf]
$$\label{IV.7}
V_{\text{eff}}(r,\theta;\mathcal{B},\mathcal{L})=\left(1-\frac{2m}{r}\right)^\gamma\left[1+\left(1-\frac{2m}{r}\right)^{\gamma-1}\left(\frac{\mathcal{L}}{r\sin\theta}-\mathcal{B}r\sin\theta \left(1-\frac{2m}{r}\right)^{1-\gamma}\right)^2\right].$$
The term in parentheses of Eq. (\[IV.7\]) is the central force potential for the specific angular momentum $\mathcal{L}$ and the electromagnetic potential energy given by the magnetic parameter $\mathcal{B}$. Once again, note that Eq. (\[IV.7\]) reduces to the Schwarzschild case when $\gamma=1$. As usual, for a given value of $\mathcal{E}$ the particle’s motion is restricted by the condition that $$\label{IV.7a}
\mathcal{E}^2=V_{\text{eff}}(r,\theta;\mathcal{L},\mathcal{B}).$$
Due to the static nature of the space-time, the effective potential of the $\gamma$-metric immersed in a uniform magnetic field is symmetric with respect to the change in sign for $(\mathcal{L},\mathcal{B})$, namely replacing $(\mathcal{L},\mathcal{B})$ with $(-\mathcal{L},-\mathcal{B})$ does not change the effective potential [@Kolos:2015iva; @Frolov:2010mi]. This symmetry enables us to distinguish the following two situations:
1. Minus configuration (**MC**): the magnetic field and angular momentum parameters have opposite signs and the Lorentz force is attracting the charged particle towards the axis of symmetry. This holds for $\mathcal{L}>0$, $\mathcal{B}<0$ or, equivalently, for $\mathcal{L}<0$, $\mathcal{B}>0$
2. Plus configuration (**PC**): the magnetic field and angular momentum parameters have the same sign and the Lorentz force is repulsive, pushing the particle away from the source. In this configuration we have $\mathcal{L}>0$, $\mathcal{B}>0$ or equivalently $\mathcal{L}<0$, $\mathcal{B}<0$).
Having fixed the direction of the symmetry axis $z$, a positive value of the angular momentum $\mathcal{L}>0$ means that the particle moves counter-clockwise. On the other hand, taking the particle’s charge $q>0$, in the **MC**, $\mathcal{B}<0$ corresponds to the magnetic field pointing in the negative direction (downwards), while in the **PC**, $\mathcal{B}>0$ corresponds to the magnetic field pointing upwards the $z$-axis.
In Fig. \[figure0\], we show the equipotential slices for the effective potential at the equatorial plane for different values of $\gamma$. From the figure, it is possible to see the regions where $V_{\text{eff}}$ has a local minimum. On the other hand, in Fig. \[figureI\], we plot sections ($y=0$) of the effective potential $V_{\text{eff}}$ as function of the Cartesian direction $x$ for different values of $\gamma$ at a fixed value of $\mathcal{B}$. We consider two case: $z=0$ and $z\rightarrow \infty$. To do so we use spherical coordinates given by $$\begin{array}{ccc}
x=r\sin\theta\cos\phi,&y=r\sin\theta\sin\phi,&z=r\cos\theta,
\end{array}$$ with $r=\sqrt{x^2+y^2+z^2}$. For $y=0$ (i.e. $\phi=0$), the last equation reduces to $x=r\sin\theta$ and $z=r\cos\theta$. Therefore, the effective potential takes the form,
$$V_{\text{eff}}(x,0,z;\mathcal{L},\mathcal{B})=\left(1-\frac{2 m}{\sqrt{x^2+z^2}}\right)^{\gamma } \left[\left(1-\frac{2 m}{\sqrt{x^2+z^2}}\right)^{\gamma -1} \left(\frac{\mathcal{L}}{x}-x \mathcal{B} \left(1-\frac{2 m}{\sqrt{x^2+z^2}}\right)^{1-\gamma }\right)^2+1\right].$$
At $z\rightarrow \infty$, the effective potential reduces to $$V_{\text{eff}}(x,0,z\rightarrow \infty;\mathcal{L},\mathcal{B})=1+\left(\frac{\mathcal{L}}{x}-\mathcal{B}x\right)^2,$$ which is the same expression as Schwarzschild [@Kolos:2015iva].
From Fig. \[figureI\] it can be seen that in the equatorial plane ($z=0$) notable differences in $V_{\text{eff}}$ for different values of $\gamma$ appear in the region where the role of the magnetic field is suppressed by gravity (i.e. small values of $x$). In this case, it is possible to distinguish between different values of $\gamma$. At large values of $x$, the role of the magnetic field dominates over the gravitational part of the effective potential, and differences between different values of $\gamma$ become smaller. Fig. \[figureI\] also shows that at $z\rightarrow\infty$, the behaviour of $V_{\text{eff}}$ does not depend on $\gamma$ and is the same as in the Schwarzschild case [@Kolos:2015iva] (see the solid grey lines in the figure). The reason for such a behavior lies in those terms which contain the dependence on $\gamma$ in Eq. (\[IV.7\]). Since these terms tend to $1$ when $z\rightarrow \infty$, the effective potential reduces to the Schwarzschild case as $z\rightarrow\infty$ as shown also for example in Ref. [@Kolos:2015iva]. This behavior is expected since both line elements ($\gamma$-metric and Schwarzschild) are asymptotically flat.
According to Eq. (\[IV.7\]), the effective potential is a function of two variables only: $r$ and $\theta$. This is due to the axial symmetry of the system which is in turn due to the symmetry of the background magnetic field and the symmetry of the geometry. Therefore, the extrema of the effective potential, can be obtained from $$\label{IV.8}
\begin{array}{ccc}
\partial_r V_{\text{eff}}(r,\theta;\mathcal{L},\mathcal{B})=0&\text{and}&\partial_\theta V_{\text{eff}}(r,\theta;\mathcal{L},\mathcal{B})=0.
\end{array}$$ From the second condition in Eq.(\[IV.8\]), we see that all extrema of $V_{\text{eff}}(r,\theta;\mathcal{L}, \mathcal{B})$ occur in the equatorial plane. Therefore, similarly to the Schwarzschild case, there are no off-equatorial circular orbits for charged particles.
On the other hand, from the first condition in Eq. (\[IV.8\]) restricted to the equatorial plane, we obtain the following polynomial equation $$\label{IV.9}
d\mathcal{L}^2+e\mathcal{L}+f=0\;,$$ with $$\label{IV.9a}
\begin{aligned}
d&=[r-m(1+2\gamma)](r-2m)^{2\gamma-2},\\\\
e&=2m\gamma\mathcal{B}r^{1+\gamma}(r-2m)^{\gamma-1},\\\\
f&=-(r-m)r^{2\gamma+2}\mathcal{B}^2-m\gamma r^{1+\gamma}(r-2m)^{\gamma}.
\end{aligned}$$
Equation (\[IV.9\]) is quadratic for the specific angular momentum. Therefore, if we solve for $\mathcal{L}$, we find that circular orbits are given by
$$\label{IV.10}
\mathcal{L}_{E\pm}=(r-2m)^{1-\gamma}\left[\frac{-m\mathcal{B}\gamma r^{\gamma+1}\pm(r-2m)^{1-\gamma}r^{\frac{1+\gamma}{2}}\mathcal{F}(r;\mathcal{B})}{r-m(1+2\gamma)}\right]\;,$$
with $$\mathcal{F}(r;\mathcal{B})=\sqrt{(r-2m)^{2\gamma-3}\left[m(r-2m)^\gamma(r-m(1+2\gamma))\gamma+\mathcal{B}^2(r-2m)r^{\gamma+1}(m(1+\gamma)-r)^2\right]}.$$
Equation (\[IV.9\]) can be used to obtain the innermost stable circular orbit (ISCO). There are two ways to compute the ISCO: either by solving the equation $\partial^2_r V_{\text{eff}}=0$, or by means of the local extrema $\mathcal{L}_{E(\text{ex})}$ of Eq. (\[IV.10\]). Hence, after using the implicit function theorem on Eq. (\[IV.9\]), we find the local extremum of $\mathcal{L}_{E\pm}(r;\mathcal{B})$ is given by [^1].
$$\label{IV.11}
\mathcal{L}_{E(\text{ex})}=\frac{-2m\gamma\mathcal{B}r^\gamma(r-2m)^{1-\gamma}[(r-m)\gamma-m] +(r-2m)^{1-\gamma}\mathcal{G}(r;\mathcal{B})}{(r-2m\gamma)(2\gamma-1)},$$
where the function $\mathcal{G}(r;\mathcal{B})$ is given by $$\label{IV.12}
\mathcal{G}(r;\mathcal{B})=\sqrt{4m^2\mathcal{B}^2r^{2\gamma}\gamma^2[m+\gamma(m-r)]^2-r^\gamma(r-2m)(r-2m\gamma)(2\gamma-1)\delta}.$$ with $$\label{IV.11a}
\delta=-\mathcal{B}^2r^{1+\gamma}(3r-2m+2(r-m)\gamma)+2m(r-2m)^{\gamma-2}\gamma(m+\gamma(m-r)).$$
\[tableII\]
$\gamma$ $\mathcal{B}=-0.2$ $\mathcal{B}=-0.1$ $\mathcal{B}=0$ $\mathcal{B}=0.1$ $\mathcal{B}=0.2$
---------- -------------------- -------------------- ------------------- ------------------- -------------------
$r_{\text{ISCO}}$ $r_{\text{ISCO}}$ $r_{\text{ISCO}}$ $r_{\text{ISCO}}$ $r_{\text{ISCO}}$
0.6 3.028 3.235 3.6944 2.987 2.618
0.7 3.411 3.609 4.3045 3.284 2.817
0.8 3.759 3.922 4.8832 3.549 3.033
0.9 4.066 4.291 5.4464 3.802 3.227
1 4.4139 4.6319 6.0000 3.9827 3.3552
1.1 4.763 4.981 6.54722 4.205 3.487
: \[tabI\] Numerical values of the innermost stable circular orbits ($r_{\text{ISCO}}$) as a function of $\gamma$ and $\mathcal{B}$. We set $m=1$, which is equivalent to using the rescaling $r\rightarrow r/m$ in the line element.
![Location of $r_{\text{ISCO}}$ as a function of $\gamma$ for different values of $\mathcal{B}$. Again, we set $m=1$. \[figureIA\]](ISCO.pdf)
The ISCO is located at the point of intersection of the functions $\mathcal{L}_{\text{E(ex)}}$ and $\mathcal{L}_{\text{E+}}$. In Table \[tabI\] some ISCO radii are evaluated for different values of $\gamma$ and $\mathcal{B}$. In Fig. \[figureIA\], we show the behaviour of $r_{\text{ISCO}}$ as a function of $\gamma$ for different values of $\mathcal{B}$. The case without an external magnetic field (i.e. the dashed line in Fig. \[figureIA\]) was studied in Ref. [@Chowdhury:2011aa]. There it was shown that the dependence of the ISCO radius upon the $\gamma$ parameter for neutral particles could be divided into three different regions: (i) For $\gamma < 1/\sqrt{5}$ there is no ISCO, (ii) for $ 1/\sqrt{5}<\gamma <1/2$, there are two disjoint regions for stable circular orbits (i.e. two separate allowed values of the ISCO), (iii) for $\gamma>1/2$, there is only one ISCO, similarly to the spherically symmetric case. A similar situation was described in Ref. [@Benavides-Gallego:2018htf].
  \
  \
  
In Fig. \[figureIA\], we show the value of $r_\text{ISCO}$ as function of $\gamma$ for $\mathcal{B}=\pm0.1$ and compare it with the case without magnetic field. In both cases, the effect of the magnetic field on the value of the ISCO is to reduce it with respect to the case $\mathcal{B}=0$. In addition, the ISCO for $\mathcal{B}=-0.1$ (i.e. the [**MC**]{} configuration) is larger than the ISCO radius for $\mathcal{B}=0.1$ (i.e. the [**PC**]{} configuration). Hence, the presence of a uniform magnetic field allows particles to move in stable circular orbits closer to the infinitely redshifted surface $r=2m$. This behaviour can be observed in Fig. \[figureII\], where we plot the behaviour of $\mathcal{L}_{\text{E}_+}$ and $\mathcal{L}_{\text{E(ex)}}$.
Similarly to the case of Schwarzschild immersed in a uniform magnetic field, the motion of charged particles in the $\gamma$-space-time is also bounded in the radial direction near the equatorial plane due to the term proportional to $\mathcal{B}$ in the effective potential. Nevertheless, this term vanishes for $\theta=0$ and so particles can escape along the polar direction, i.e. the $z$ direction, towards infinity.
In order to find the condition on the particle’s energy for the particle to escape, we consider the effective potential at $z\rightarrow\infty$ along a given direction $x$. Then $V_{\text{eff}}(x;z\rightarrow\infty)$ is given by (see gray solid line in Fig. \[figureI\]) $$\label{IV.15}
V_{\text{eff}}(x;z\rightarrow\infty)=1+\frac{\mathcal{L}^2}{x^2}-2\mathcal{L}\mathcal{B}+\mathcal{B}^2x^2.$$ The minimum of Eq. (\[IV.15\]) is located at $x=|\mathcal{L}/\mathcal{B}|$ and at this value of $x$ the energy is $$\label{IV.16}
\mathcal{E}_{\text{escape}}=\sqrt{1+2|\mathcal{L}\mathcal{B}|-2\mathcal{L}\mathcal{B}}.$$ Therefore, the condition for particles to escape is given by $$\label{IV.17}
\mathcal{E}\geq\mathcal{E}_{\text{scape}}=
\begin{cases}
\begin{array}{ccc}
1 \hspace{1cm}&&\text{if}\hspace{0.5cm}\mathcal{B}\geq 0\\\\
\sqrt{1-4\mathcal{BL}}&& \text{if}\hspace{0.5cm}\mathcal{B}<0\\
\end{array}
\end{cases}$$ The exact same condition was found by M. Kološ et. al in [@Kolos:2015iva]. This is due to the fact that Schwarzschild and $\gamma$-metric have the same behavior at $z\rightarrow\infty$ (see Eq. (\[IV.15\]) and Fig. \[figureI\]).
On the other hand, for charged particles on bound orbits the energy condition in Eq. (\[IV.7a\]) describes what is called a “*lake-like*” region where the particles are trapped between two radii. The condition for such a region is given by $$\label{IV.18}
\mathcal{E}<\mathcal{E}_{\text{scape}}.$$ Since the specific energy $\mathcal{E}$ is related by the specific angular momentum $\mathcal{L}$, the trapped region can be expressed by the condition
  \
  \
  
  \
  \
  
$$\label{IV.19}
\mathcal{L}_{\text{L}_-}<\mathcal{L}<\mathcal{L}_{\text{L}_+}.$$
Where the values of $\mathcal{L}_{\text{L}_-}$ and $\mathcal{L}_{\text{L}_+}$ are given by the condition $\mathcal{E}=\mathcal{E}_{\text{escape}}$. Therefore, for $\mathcal{B}\geq 0$ we have
$$\label{IV.20}
\mathcal{L}_{\text{L}\pm}=\frac{(r-2m) \mathcal{B} r^{\gamma +1}\pm r^{\gamma }\sqrt{(r-2m) r^{1-\gamma} \left(r^{\gamma }-(r-2m)^{\gamma }\right)} }{(r-2m)^{\gamma }}\;,$$
and for $\mathcal{B}<0$, $\mathcal{L}_{\text{L}_-}$ and $\mathcal{L}_{\text{L}_+}$ become $$\label{IV.21}
\mathcal{L}_{\text{L}_\pm}=\frac{-\mathcal{B} r^{\gamma +1} \left(2 r^{\gamma } (r-2m)^{1-\gamma }-(r-2m)\right)\pm r^{2 \gamma -1} \mathcal{K}(r,\mathcal{B})}{(r-2m)^{\gamma }}\;,$$ with $$\label{IV.21A}
\mathcal{K}(r,\mathcal{B})=\sqrt{r^{3 (1-\gamma )} (r-2m)^{1-2 \gamma } \left(r^{\gamma }-(r-2m)^{\gamma }\right) \left(4 (r-2m) \mathcal{B}^2 r^{2 \gamma +1}+(r-2m)^{2 \gamma }\right)}\;.$$
  
  
In Fig. \[figureII\], we show the behaviour of $\mathcal{L}_{\text{E}\pm}$, $\mathcal{L}_{\text{E(ex)}}$, and the behaviour of the so-called “[*lake-like*]{}” angular momentum functions $\mathcal{L}_{\text{L}_\pm}$ as functions of $r$ for different values of $\gamma$ and $\mathcal{B}$. In the figure, we see that the extremum function $\mathcal{L}_{\text{E}\pm}$, which determines the circular orbits in the equatorial plane, diverges when the charged particles move close to the photon sphere. In the figure, we can also see the value of $\mathcal{L}_{\text{ISCO}}$ from the intersection of $\mathcal{L}_{\text{E}\pm}$ with $\mathcal{L}_{\text{E(ex)}}$. Note that the position of $r_\text{ISCO}$ is affected by the presence of the magnetic field. Moreover, stable and unstable circular orbits are located at $r>r_{\text{ISCO}}$ and $r<r_{\text{ISCO}}$, respectively. On the other hand, regarding the lake angular momentum functions, we see that the region between $\mathcal{L}_{\text{L-}}$ and $\mathcal{L}_{\text{L+}}$ is smaller for $\mathcal{B}>0$ than the region in the case $\mathcal{B}<0$ (for a given value of $\gamma$). Finally, for fixed values of $\mathcal{B}\neq0$, the region between $\mathcal{L}_{\text{L-}}$ and $\mathcal{L}_{\text{L+}}$ becomes broader as $\gamma$ increases.
Charged particle trajectories
-----------------------------
The trajectories of charged particles immersed in a uniform magnetic field can be obtained by solving the equations of motion. Typically this can be done following two possible procedures, namely i) from the Lorentz equation or ii) from the Hamiltonian formalism. In the following, we use the former method. Then, a particle with (rest) mass $m_0$ and charge $q$ immersed in an electromagnetic field $\mathrm{F}_{\rho\sigma}$ satisfies the Lorentz force law [@MTW; @Wald:1984rg] $$\label{IV.22}
\frac{d^2x^\mu}{d\tau^2}+\Gamma^{\mu}_{\;\;\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau}=\frac{q}{m_0}g^{\mu\rho}\mathrm{F}_{\rho\sigma}\frac{dx^\sigma}{d\tau}.$$ where $\tau$ is the affine parameter of the particle’s trajectory and $\mathrm{F}_{\rho\sigma}$ is the electromagnetic Faraday tensor. In general, the tensor $\mathrm{F}_{\rho\sigma}$ is defined, in terms of the vector potential $A_\alpha$, as $$\label{IV.23}
\mathrm{F}_{\rho\sigma}=\partial_\rho A_\sigma-\partial_\sigma A_\rho,$$ and in the case of the $\gamma$-metric immersed in a uniform magnetic field the only non vanishing component of vector potential are given by Eq. so that the non-vanishing components of the electromagnetic tensor, expressed in $(t,r,\theta,\phi)$ coordinates, take the form $$\label{IV.25}
\begin{aligned}
\tilde{\mathrm{F}}_{r\phi}&=-\tilde{\mathrm{F}}_{\phi r}=2\mathcal{B} \sin ^2\theta(r-2m)^{-\gamma } r^{\gamma }[r-(\gamma+1)m],\\\\
\tilde{\mathrm{F}}_{\theta\phi}&=-\tilde{\mathrm{F}}_{\phi\theta}=2\mathcal{B} \sin\theta \cos\theta (r-2m)^{1-\gamma } r^{\gamma +1},
\end{aligned}$$ where we have defined $\tilde{\mathrm{F}}_{\rho\sigma}=q\mathrm{F}_{\rho\sigma}/m_0$. Note again that for $\gamma=1$, Eq. (\[IV.25\]) reduce to the case of Schwarzschild immersed in a uniform electromagnetic field [@Kolos:2015iva].
Using the conservation of angular momentum from Eq. (\[IV.4\]), we can obtain the equation of motion for the axial coordinate $\phi$. This equation has the form $$\label{IVC1}
\frac{d\phi}{d\tau}=\frac{\mathcal{L}}{r^2}\left(1-\frac{2m}{r}\right)^{\gamma-1}-\mathcal{B}\;,$$ while the equation of motion for the radial component takes the form
$$\left(\frac{dr}{d\tau}\right)^2=\left(1- \frac{m}{r}\right)^{\gamma ^2-1} \left[\mathcal{E}^2-\left(1-\frac{2 m}{r}\right)^{\gamma }-\frac{\left(r \mathcal{B} (2 m-r)+\mathcal{L} \left(1-\frac{2 m}{r}\right)^{\gamma }\right)^2}{r (2 m+r)}\right]\;.$$
Given the difficulty of solving analytically the equations in general one needs to resort to numerical analysis. In Figs. \[figure3\] and \[figure3a\], we show some examples of trajectories for charged particles on the equatorial plane. We considered both **PC** and **MC** configurations, and different values of $\gamma$. From the figures, it is possible to identify the infalling trajectories, and the circular and bounded orbits. In the case of bound orbits, the radial coordinate is constrained in the interval $r_a\leq r\leq r_p$, where $r_a$ and $r_p$ are the apoapsis and periapsis, respectively [@Kolos:2015iva]. From the figures, we can see two types of bounded trajectories: curled and non-curled. The former is obtained when the coordinate $\phi$ decreases during the particle motion, in contrast to non-curled trajectories, where $\phi$ always increases (see also Fig. \[figure3b\]). Furthermore, it is important to point out that these trajectories are specific to the charged particle motion in the asymptotically uniform magnetic fields, and can not occur in the case of uncharged particles, and more precisely they are possible only in the **PC** configuration. Similar curly trajectories were found in the Schwarzschild space-time immersed in a magnetic field [@Kolos:2015iva; @Frolov:2010mi; @Narzilloev:2019hep].
It is known that the motion of an electrically neutral test particle is always restricted to the equatorial plane and that a freely moving particle can escape to infinity only in this plane, even in the presence of an external magnetic field. Nevertheless, when we consider charged particles in the presence of an asymptotically uniform magnetic field, it is possible to have escape trajectories, which end at spatial infinity as $z \rightarrow \infty$ evolving along the magnetic field lines, as can be seen in the left and central panels of Fig. \[figure3c\].
Out of the equatorial plane, particle motion tends to be chaotic [@LukesGerakopoulos:2012pq; @Kopacek:2014moa; @Kopacek:2010yr; @Sota:1995ms]. The chaotic motion is related to the variations in the coordinate $\theta$. Nevertheless, as we will discuss in the next section, harmonic oscillations may occur when bounded orbits are located close to the equatorial plane. In Fig. \[figure3c\] left panel we show one example.
Harmonic oscillations of charged test particles {#V}
===============================================
It is known that if a particle is displaced slightly from the radius of a stable circular orbit, the particle will start to oscillate around its equilibrium value. For sufficiently small displacement, it will execute simple harmonic motion[@Wald:1984rg]. This oscillating motion is governed by the epicyclic frequencies.
Epicyclic frequencies
---------------------
To obtain the epicyclic frequencies for the $\gamma$ space-time immersed in a uniform magnetic field, we follow the analysis in Ref. [@Toshmatov:2019qih]. In this sense, we also restrict our analysis to the linear regime and consider radial and vertical oscillations separately.
The Hamiltonian given in Eq. (\[IV.1\]) can be expressed in the following form $$\label{V1}
\mathcal{H}=\frac{1}{2}g^{rr}p^2_r+\frac{1}{2}g^{\theta\theta}p^2_\theta+\frac{m^2_0}{2}
\left(g^{tt}\mathcal{E}^2+g^{\phi\phi}(\mathcal{L}-\mathcal{B}g_{\phi\phi})^2+1\right).$$ The normalization condition $u_\beta u^\beta=-1$ (where $u^\beta$ is the $4-$velocity of the charged particle), leads to $\mathcal{H}=0$. Hence, Eq. (\[V1\]) reduces to $$\label{V2}
g^{rr}p^2_r+g^{\theta\theta}p^2_\theta=-m^2_0
\left(g^{tt}\mathcal{E}^2+g^{\phi\phi}(\mathcal{L}-\mathcal{B}g_{\phi\phi})^2+1\right),$$ from which we can express the left-hand side as $$\label{V3}
g_{rr}\left(\frac{dr}{d\zeta}\right)^2+g_{\theta\theta}\left(\frac{d\theta}{d\zeta}\right)^2=
-\Pi(r,\theta),$$ where we have used the relation $p^\mu=m_0u^\mu$ and defined the function $\Pi(r,\theta)$ as $$\label{V4}
\Pi(r,\theta)=g^{tt}\mathcal{E}^2+g^{\phi\phi}(\mathcal{L}-\mathcal{B}g_{\phi\phi})^2+1.$$ In the case of charged particles moving on a plane with $\theta_0= \text{const}$ (as is the case for motion in the equatorial plane), Eq. (\[V3\]) takes the form, $$\label{V5}
g_{rr}\left(\frac{dr}{d\zeta}\right)^2=\mathcal{R}(r)=-\Pi(r,\theta_0).$$ On the other hand, if we consider charged particles moving at a fixed radial distance $r=r_0$ with $\theta\neq0$, then Eq. (\[V3\]) reduces to $$\label{V6}
g_{\theta\theta}\left(\frac{d\theta}{d\zeta}\right)^2=\Theta(\theta)=-\Pi(r_0,\theta).$$ Therefore, test charged particles will move on a circular orbit $r=r_0$ in the equatorial $\theta=\theta_0=\pi/2$ if [@Toshmatov:2019qih] $$\label{V7a}
\begin{array}{cc}
\mathcal{R}(r_0)=0,&\partial_r\mathcal{R}(r)|_{r_0}=0,
\end{array}$$ and $$\label{V7b}
\begin{array}{cc}
\Theta(\theta_0)=0,&\partial_\theta\Theta(\theta)|_{\theta_0}=0.
\end{array}$$
  \
  \
  \
  \
  
  \
  \
  \
  \
  
In the case of small perturbation along the radial direction, i.e. $r= r_0+\delta r$ with $\delta\theta=0$ and in the case of small perturbation in the vertical direction, i.e. $\theta=\theta_0+\delta\theta$ with $\delta r=0$, we can express the functions $\mathcal{R}(r)$ and $\Theta(\theta)$ in powers of $\delta r$ and $\delta \theta$, by expanding near the equilibrium positions $r_0$ and $\theta_0$. In this way, we obtain the equations governing the motion of test charged particles that slightly depart from a circular orbit. The Taylor expansions up to second order in $\delta r$ and $\delta \theta$ are given by [@Toshmatov:2019qih] $$\begin{aligned}
\label{V8}
\mathcal{R}(r)&=&\mathcal{R}(r_0)+\partial_r\mathcal{R}(r)|_{r_0}\delta r+\frac{1}{2}\partial^2_r\mathcal{R}(r)|_{r_0}\delta r^2+...\\
\Theta(\theta)&=&\Theta(\theta_0)+\partial_\theta \Theta(\theta)|_{\theta_0}\delta \theta+\frac{1}{2}\partial^2_\theta \Theta(\theta)|_{\theta_0}\delta \theta^2+...
\end{aligned}$$ which, after using Eqs. (\[V7a\]) and (\[V7b\]), allows to rewrite Eqs. (\[V5\]) and (\[V6\]) as $$\begin{aligned}
\label{V9}
g_{rr}\delta \dot{r}^2&=&\frac{1}{2}\partial^2_r\mathcal{R}(r)|_{r_0}\delta r^2,\\
g_{\theta\theta}\delta \dot{\theta}^2&=&\frac{1}{2}\partial^2_\theta \Theta(\theta)|_{\theta_0}\delta \theta^2.
\end{aligned}$$ Taking into account that the total energy of the orbit is conserved, one obtains [@Toshmatov:2019qih]. $$\begin{aligned}
\label{V10a}
\delta \dot{r}\left[g_{rr}\delta\ddot{r}-\frac{1}{2}\partial^2_r\mathcal{R}(r)|_{r_0}\delta r\right]&=&0,\\ \label{V10b}
\delta \dot{\theta}\left[g_{\theta\theta}\delta\ddot{\theta}-\frac{1}{2}\partial^2_\theta \Theta(\theta)|_{\theta_0}\delta \theta\right]&=&0.
\end{aligned}$$ It is clear that the trivial solutions $\delta\dot{r}=0$, and $\delta\dot{\theta}=0$ correspond to circular orbits. On the other hand, the other solution, given by the quantities inside square brackets of Eqs. (\[V10a\]) and (\[V10b\]), can be expressed in the form of harmonic oscillators as [@Abramowicz:2004tm] $$\begin{aligned}
\label{V11}
\delta \ddot{r}+\omega^2_r\delta r&=&0,\\
\delta \ddot{\theta}+\omega^2_\theta\delta \theta&=&0,
\end{aligned}$$ where $\omega^2_r$ and $\omega^2_\theta$ are the radial and vertical (latitudinal) frequencies, which are defined by $$\begin{aligned}
\label{V12}
\omega^2_r&=&-\frac{\partial^2_r\mathcal{R}(r)|_{r_0}}{2g_{rr}}=\frac{\partial^2_r\Pi(r,\theta_0)|_{r_0}}{2g_{rr}},\\
\omega^2_\theta&=&-\frac{\partial^2_\theta \Theta(\theta)|_{\theta_0}}{2g_{\theta\theta}}=\frac{\partial^2_\theta \Pi(r_0,\theta)|_{\theta_0}}{2g_{\theta\theta}}.
\end{aligned}$$ For the $\gamma$-metric immersed in uniform magnetic field, the function $\Pi(r,\theta)$ is given by
$$\label{V13}
\begin{aligned}
\Pi(r,\theta)&=1+\left(1-\frac{2m}{r}\right)^{\gamma-1}\left[\frac{\mathcal{L}}{r\sin\theta}-\mathcal{B}r\sin\theta\left(1-\frac{2m}{r}\right)^{1-\gamma}\right]^2-\left(1-\frac{2m}{r}\right)^{-\gamma}\mathcal{E}^2.
\end{aligned}$$
Therefore, the epicyclic frequencies are given by $$\label{V14}
\begin{aligned}
\omega^2_r&=\frac{[r(r-m)^{-2}(r-2m)]^{-\gamma^2}}{r^2(r^2-3mr+2m^2)^2}\big[\mathit{a}(r,\gamma)\mathcal{E}^2+\mathit{b}(r,\gamma)\mathcal{B}^2+\mathit{c}(r,\gamma)\mathcal{L}^2\big]\;,\\
\omega^2_\theta&=(r-m)^{2(\gamma ^2-1)} [(r-2m) r]^{-\gamma ^2-1} \left[\mathcal{L}^2 \left(r-2m\right)^{2 \gamma }r^{-2\gamma}-(r-2m)^2 r^2 \mathcal{B}^2\right]\;,
\end{aligned}$$
where we have defined $$\label{V15}
\begin{aligned}
\mathit{a}(r,\gamma)&=2 \gamma m r (2 m-r) ((\gamma -1) m+r)\;,\\
\mathit{b}(r,\gamma)&=r^2 (r-2 m)^2 \left(2 \gamma (\gamma +1) m^2-2 (\gamma +1) m r+r^2\right)\;,\\
\mathit{c}(r,\gamma)&=(r-2m)^{2\gamma}r^{-2\gamma}\big[2 (\gamma +1) (\gamma +2) m^2-6 (\gamma +1) m r+3 r^2\big],
\end{aligned}$$
and the values for $\mathcal{E}^2=V_{\text{eff}}(r,\pi/2)$ and $\mathcal{L}=\mathcal{L}_{\text{E}+}$ are given by Eqs. (\[IV.7\]) and (\[IV.10\]), respectively. There is a third fundamental frequency given by oscillations about the azimuthal angle $\phi$ which can be obtained from Eq. (\[IVC1\]). This is given by $$\label{V15a}
\omega^2_\phi=\left[\mathcal{L}r^{-1-\gamma}\left(r-2m\right)^{\gamma-1}-\mathcal{B}\right]^2.$$
In the case of vanishing magnetic field, we obtain again the same results as in Ref. [@Toshmatov:2019qih]. On the other hand, and differently from the neutral case, when $\mathcal{B}\neq 0$ we also have the so-called Larmor angular frequency $\omega_{L}$, which is associated with the uniform magnetic field itself, and is given by the relation [@Kolos:2015iva] $$\label{V16}
\omega_L=\frac{qB}{m_0}=2|\mathcal{B}|.$$
It is important to point out that the Larmor frequency $\omega_L$ does not dependent on the radial coordinate and therefore it becomes important at large distances, where the magnetic field dominates over the gravitational field.
In Fig. \[figure4\], we show the behaviour of $\omega_r$, $\omega_\theta$ and $\omega_\phi$ as functions of $r$ for different values of $\gamma$ and $\mathcal{B}$. For completeness we also show the corresponding value of the Larmor frequency $\omega_L$. When the $\gamma$-space-time is not immersed in a uniform magnetic field (i.e. when $\mathcal{B}=0$), $\omega_\phi$ and $\omega_\theta$ only coincide when $\gamma=1$ (i.e. Schwarzschild). Moreover, as expected, they asymptotically tend to zero as $r$ increases, similar to the Schwarzschild case [@Kolos:2015iva] (see the third row in Fig. \[figure4\]). In the same figure, it is possible to see that the radial frequency $\omega_r$ has a maximum value, which decreases as $\gamma$ increases. Moreover, we can see that $\omega_r$ vanishes at the ISCO. On the other hand, when $\mathcal{B}\neq0$, Fig. \[figure4\] shows different behaviors. It is worth noting that for $\mathcal{B}<0$ both $\omega_r$ and $\omega_\phi$ tend asymptotically to the Larmor frequency $\omega_L$, while $\omega_\theta$ tends to 0. Nevertheless, when $\mathcal{B}>0$, only the radial frequency $\omega_r$ tends to $\omega_L$ as $r$ increases.
Frequencies measured by distant observers
-----------------------------------------
The epicyclic frequencies derived in the previous section in Eqs. (\[V14\]) and (\[V15a\]) are measured with respect to the proper time of a comoving observer. Therefore, to obtain the frequencies measured by an observer at infinity, it is necessary to take into account the redshift factor $d\tau/dt$. Hence, we have that $$\label{VB1}
\Omega_i=\omega_i \frac{d\tau}{dt},$$ where $i=r$, $\theta$ or $\phi$. For the $\gamma$-metric, the redshift factor can be obtained from Eq. (\[IV.4\]) and is given by $$\label{VB2}
\frac{d\tau}{dt}=\frac{r^{-\gamma}(r-2m)^\gamma}{\mathcal{E}}.$$ Therefore, the frequencies in Eq. (\[VB1\]), are given by $$\label{VB3}
\Omega_i=\omega_i\frac{r^{-\gamma}(r-2m)^\gamma}{\mathcal{E}}.$$ It is important to point out that $\mathcal{E}=\mathcal{E}(r)$ in Eq. (\[VB3\]) is the specific energy at the circular orbit given by $\mathcal{E}=V_{\text{eff}}(r,\pi/2,\mathcal{L}_{E+})$. Therefore, we obtain
$$\label{VB4}
\begin{aligned}
\Omega^2_r&=\frac{r^{-\gamma(\gamma+2)}(r-m)^{2\gamma^2}(r-2m)^{\gamma(2-\gamma)}}{r^2(r^2-3mr+2m^2)}\left[\mathit{a}(r,\gamma)+\frac{\mathit{b}(r,\gamma)\mathcal{B}^2+\mathit{c}(r,\gamma)\mathcal{L}^2}{\mathcal{E}^2}\right]\;,\\
\Omega^2_\theta&=\frac{(r-m)^{2(\gamma^2-1)}(r-2m)^{-(\gamma-1)^2}r^{-(\gamma+1)^2}}{\mathcal{E}^2} \left[\mathcal{L}^2 \left(r-2m\right)^{2 \gamma }r^{-2\gamma}-(r-2m)^2 r^2 \mathcal{B}^2\right]\;,\\
\Omega^2_\phi&=\left[\frac{\mathcal{L}r^{-(1+2\gamma)}(r-2m)^{2\gamma-1}-\mathcal{B}r^{-\gamma}(r-2m)^\gamma}{\mathcal{E}}\right]^2.
\end{aligned}$$
In Fig. \[figure5\] we show the behaviour of $\Omega^2_r$, $\Omega^2_\theta$ and $\Omega^2_\phi$ as functions of $r$ for different values of $\gamma$ and $\mathcal{B}$. From the figure, we see that the latitudinal $\Omega_\theta$ and azimuthal $\Omega_\phi$ frequencies measured by a distant observer vanish as $r$ increases for all the values of $\gamma$ and $\mathcal{B}$. In particular, when $\mathcal{B}=0$ and $\gamma=1$, these frequencies coincide ($\Omega_\theta=\Omega_\phi$). Furthermore, for positive (negative) values of $\mathcal{B}$, $\Omega_\phi$ is always smaller (greater) than $\Omega_\theta$.
Conclusions
===========
Observations of stellar mass and super-massive black hole candidates rely on the measurement of the spectrum of light emitted by their accretion disks. In particular X-ray reflection spectroscopy appears to be a promising tool to probe the nature of the geometry in the vicinity of black hole candidates [@Bambi:2016sac; @Cao:2017kdq; @Tripathi:2018lhx]. The recent observation of the ‘shadow’ of the super-massive black hole candidate at the core of the galaxy M87 [@Akiyama:2019cqa; @Akiyama:2019brx; @Akiyama:2019sww] also suggests that the possibility to test experimentally the geometry in the vicinity of such astrophysical compact objects may soon be at hand.
In both cases, it is of paramount importance to know the behavior of the motion of test particles in the gas of the accretion disk, since the light emitted from the accretion disk is the only medium through which the measurements are made. The behaviors of test particles and, as a consequence, the properties of accretion disks, are influenced by several factors, besides the geometry, of which the most relevant are the presence of additional matter fields [@Joshi:2013dva; @Boshkayev:2018sbj; @Ilyas:2016nio], magnetic fields [@Wald:1974np; @Petterson:1975sg] and the particle’s spin [@Abramowicz:1979; @Semerak:1999qc; @Toshmatov:2019bda]. Therefore it is very important to know how such factors may alter the observational features of the accretion disks, in order to exclude the possibility of degeneracies such as, for example, the measurement of a black hole spin which could be also due to a non-spinning source with quadrupole moment[@Bambi:2013hza].
The behavior of test particles in the Kerr and Schwarzschild geometries has been studied in detail [@Thorne:1974ve; @Page:1974he; @Abramowicz:1988sp; @Abramowicz:1996ap; @Johannsen:2013asa; @Gimeno-Soler:2018pjd]. On the other hand, the same behavior in other exact solutions of Einstein’s vacuum field equations has not been thoroughly studied so far. One reasonable hypothesis is that if departures from the black hole geometry do exist they would manifest in the presence of quadrupole moment in the source of the gravitational field. In this respect, the $\gamma$-metric represents the ideal test-bed to investigate departures from the behavior expected by black holes.
Following a program initiated by some of us, we attempted here to characterize the behavior of charged test particles on orbits slightly departing from the ISCO of the $\gamma$-metric immersed in an external magnetic field. We have shown that the presence of a non-vanishing deformation parameter affects the motion of charged particles and changes the epicyclic frequencies.
As more and more precise measurements of the properties of super-massive black hole candidates become available, these models will allow testing the hypothesis that the geometry outside such objects is well described by the Kerr metric.
C.A.B.G wishes to dedicate this work to Prof. Alirio Calderón-Molina as a tribute to his memory. The authors want to thank M. Kološ for helpful discussion. The work of C.A.B.G. and C.B. was supported by the Innovation Program of the Shanghai Municipal Education Commission, Grant No. 2019-01-07-00-07-E00035, and the National Natural Science Foundation of China (NSFC), Grant No. 11973019. C.A.B.G. also acknowledges support from the China Scholarship Council (CSC), grant No. 2017GXZ019022 and the Nazarbayev University for hospitality. This research is supported by Grants No. VA-FA-F-2-008, No. MRB-AN-2019-29 and No. YFA-Ftech-2018-8 of the Uzbekistan Ministry for Innovative Development. This research is partially supported by an Erasmus+ exchange grant between SU and NUUz. A.A. is supported by a postdoc fund through PIFI of the Chinese Academy of Sciences. D.M. acknowledges support by Nazarbayev University Faculty Development Competitive Research Grant No. 090118FD5348 and by the Ministry of Education and Science of the Republic of Kazakhstan’s target program IRN: BR05236454.
[99]{} W. Israel, Phys. Rev. **164**, 1776-1779 (1967) doi:10.1103/PhysRev.164.1776 B. Carter, Phys. Rev. Lett. **26**, 331-333 (1971) doi:10.1103/PhysRevLett.26.331 N. Gürlebeck, Phys. Rev. Lett. **114**, no.15, 151102 (2015) doi:10.1103/PhysRevLett.114.151102 \[arXiv:1503.03240 \[gr-qc\]\]. R. Penrose, Phys. Rev. Lett. **14**, 57-59 (1965) doi:10.1103/PhysRevLett.14.57 S. Hawking and R. Penrose, Proc. Roy. Soc. Lond. A **A314**, 529-548 (1970) doi:10.1098/rspa.1970.0021 C. Bambi, D. Malafarina and L. Modesto, Phys. Rev. D **88**, 044009 (2013) doi:10.1103/PhysRevD.88.044009 \[arXiv:1305.4790 \[gr-qc\]\]. C. Bambi, D. Malafarina and L. Modesto, JHEP **04**, 147 (2016) doi:10.1007/JHEP04(2016)147 \[arXiv:1603.09592 \[gr-qc\]\]. H. Chakrabarty, A. Abdujabbarov, D. Malafarina and C. Bambi, Eur. Phys. J. C **80**, no.5, 373 (2020) doi:10.1140/epjc/s10052-020-7964-0 \[arXiv:1909.07129 \[gr-qc\]\]. D. Malafarina, Universe **3**, no.2, 48 (2017) doi:10.3390/universe3020048 \[arXiv:1703.04138 \[gr-qc\]\]. R. Carballo-Rubio, F. Di Filippo, S. Liberati and M. Visser, Phys. Rev. D **101**, 084047 (2020) doi:10.1103/PhysRevD.101.084047 \[arXiv:1911.11200 \[gr-qc\]\].
R. Carballo-Rubio, F. Di Filippo, S. Liberati and M. Visser, \[arXiv:1908.03261 \[gr-qc\]\]. S. B. Giddings, S. Koren and G. Treviño, Phys. Rev. D **100**, no.4, 044005 (2019) doi:10.1103/PhysRevD.100.044005 \[arXiv:1904.04258 \[gr-qc\]\]. S. B. Giddings, Nat. Astron. **1**, 0067 (2017) doi:10.1038/s41550-017-0067 \[arXiv:1703.03387 \[gr-qc\]\]. C. Barceló, R. Carballo-Rubio and L. J. Garay, Universe **2**, no.2, 7 (2016) doi:10.3390/universe2020007 \[arXiv:1510.04957 \[gr-qc\]\]. H. Weyl, Ann. Physik [**54**]{}, 117 (1917).
H. Weyl, Ann. Phys. (Leipzing) [**59**]{}, 185 (1919).
H. Quevedo, Phys. Rev. D, [**29**]{}, 2904 (1989).
H. Quevedo, Fortschr. Phys. [**58**]{} (1990) 10, 733-840
W.B. Bonnor, Gen Relat Gravit [**24**]{}, 551–574 (1992). https://doi.org/10.1007/BF00760137
J. L. Hernandez-Pastora and J. Martin, Gen Relat Gravit [**26**]{}, No.9, 877 (1994).
C. Bambi, A. Cardenas-Avendano, T. Dauser, J. A. Garcia and S. Nampalliwar, Astrophys. J. **842**, no.2, 76 (2017) doi:10.3847/1538-4357/aa74c0 \[arXiv:1607.00596 \[gr-qc\]\]. Z. Cao, S. Nampalliwar, C. Bambi, T. Dauser and J. A. Garcia, Phys. Rev. Lett. **120**, no.5, 051101 (2018) doi:10.1103/PhysRevLett.120.051101 \[arXiv:1709.00219 \[gr-qc\]\]. A. Tripathi, S. Nampalliwar, A. B. Abdikamalov, D. Ayzenberg, C. Bambi, T. Dauser, J. A. Garcia and A. Marinucci, Astrophys. J. **875**, no.1, 56 (2019) doi:10.3847/1538-4357/ab0e7e \[arXiv:1811.08148 \[gr-qc\]\].
K. Akiyama *et al.* \[Event Horizon Telescope\], Astrophys. J. Lett. **875**, no.1, L4 (2019) doi:10.3847/2041-8213/ab0e85 \[arXiv:1906.11241 \[astro-ph.GA\]\]. C. Bambi, Rev. Mod. Phys. **89**, no.2, 025001 (2017) doi:10.1103/RevModPhys.89.025001 \[arXiv:1509.03884 \[gr-qc\]\]. E. Berti *et al.*, Class. Quant. Grav. **32**, 243001 (2015) doi:10.1088/0264-9381/32/24/243001 \[arXiv:1501.07274 \[gr-qc\]\]. B. H. Voorhees, “Static axially symmetric gravitational fields,” Phys. Rev. D [**2**]{} (1970) 2119. doi:10.1103/PhysRevD.2.2119
D. M. Zipoy “Static axially symmetric gravitational fields,” Journal of Mathematical Physics 22, 1137 (1970). https://doi.org/10.1063/1.1705005
H. Kodama and W. Hikida, Class. Quant. Grav. **20**, 5121-5140 (2003) doi:10.1088/0264-9381/20/23/011 \[arXiv:gr-qc/0304064 \[gr-qc\]\]. L. Herrera, F. M. Paiva and N. Santos, Int. J. Mod. Phys. D **9**, 649-660 (2000) doi:10.1142/S021827180000061X \[arXiv:gr-qc/9812023 \[gr-qc\]\]. L. Herrera, F. M. Paiva and N. Santos, J. Math. Phys. **40**, 4064-4071 (1999) doi:10.1063/1.532943 \[arXiv:gr-qc/9810079 \[gr-qc\]\]. L. Herrera and J. Hernandez Pastora, J. Math. Phys. **41**, 7544-7555 (2000) doi:10.1063/1.1319517 \[arXiv:gr-qc/0010003 \[gr-qc\]\]. G. Lukes-Gerakopoulos, Phys. Rev. D **86**, 044013 (2012) doi:10.1103/PhysRevD.86.044013 \[arXiv:1206.0660 \[gr-qc\]\]. K. Boshkayev, E. Gasperin, A. Gutierrez-Pineres, H. Quevedo and S. Toktarbay, Phys. Rev. D **93**, no.2, 024024 (2016) doi:10.1103/PhysRevD.93.024024 \[arXiv:1509.03827 \[gr-qc\]\]. K. S. Virbhadra, gr-qc/9606004. D. Papadopoulos, B. Stewart and L. Witten, Phys. Rev. D **24**, 320-326 (1981) doi:10.1103/PhysRevD.24.320
B. W. Stewart , D. Papadopoulos, L. Witten *et al.*, Gen Relat Gravit [**14**]{}, 97–103 (1982). https://doi.org/10.1007/BF00756201
L. Herrera, G. Magli and D. Malafarina, Gen. Rel. Grav. **37**, 1371-1383 (2005) doi:10.1007/s10714-005-0120-1 \[arXiv:gr-qc/0407037 \[gr-qc\]\]. A. B. Abdikamalov, A. A. Abdujabbarov, D. Ayzenberg, D. Malafarina, C. Bambi and B. Ahmedov, Phys. Rev. D **100**, no.2, 024014 (2019) doi:10.1103/PhysRevD.100.024014 \[arXiv:1904.06207 \[gr-qc\]\]. B. Toshmatov, D. Malafarina and N. Dadhich, Phys. Rev. D **100**, no.4, 044001 (2019) doi:10.1103/PhysRevD.100.044001 \[arXiv:1905.01088 \[gr-qc\]\]. B. Narzilloev, D. Malafarina, A. Abdujabbarov and C. Bambi, \[arXiv:2003.11828 \[gr-qc\]\]. T. M. Belloni, A. Sanna and M. Mendez, Mon. Not. Roy. Astron. Soc. **426**, 1701 (2012) doi:10.1111/j.1365-2966.2012.21634.x \[arXiv:1207.2311 \[astro-ph.HE\]\]. L. Rezzolla, S. Yoshida, T. J. Maccarone and O. Zanotti, Mon. Not. Roy. Astron. Soc. **344**, L37 (2003) doi:10.1046/j.1365-8711.2003.07018.x \[arXiv:astro-ph/0307487 \[astro-ph\]\].
W. Kluzniak, Astrophys. J. [**509**]{}, L37 (1998) doi:10.1086/311748 \[astro-ph/9712243\]. M. Miller and F. K. Lamb, Astrophys. J. **499**, L37 (1998) doi:10.1086/311335 \[arXiv:astro-ph/9711325 \[astro-ph\]\]. R. V. Wagoner, A. S. Silbergleit and M. Ortega-Rodriguez, Astrophys. J. **559**, L25-L28 (2001) doi:10.1086/323655 \[arXiv:astro-ph/0107168 \[astro-ph\]\]. M. A. Abramowicz and W. Kluzniak, Astron. Astrophys. **374**, L19 (2001) doi:10.1051/0004-6361:20010791 \[arXiv:astro-ph/0105077 \[astro-ph\]\]. M. A. Abramowicz, W. Kluzniak, Z. Stuchlik and G. Torok, \[arXiv:astro-ph/0401464 \[astro-ph\]\]. L. Stella, M. Vietri and S. Morsink, Astrophys. J. **524**, L63-L66 (1999) doi:10.1086/312291 \[arXiv:astro-ph/9907346 \[astro-ph\]\]. Z. Stuchlík and A. Kotrlová, Gen. Rel. Grav. **41**, 1305-1343 (2009) doi:10.1007/s10714-008-0709-2 \[arXiv:0812.5066 \[astro-ph\]\]. A. Kotrlova, Z. Stuchlik and G. Torok, Class. Quant. Grav. **25**, 225016 (2008) doi:10.1088/0264-9381/25/22/225016 \[arXiv:0812.0720 \[astro-ph\]\]. M. Kološ, Z. Stuchlík and A. Tursunov, Class. Quant. Grav. [**32**]{}, no. 16, 165009 (2015) doi:10.1088/0264-9381/32/16/165009 \[arXiv:1506.06799 \[gr-qc\]\]. R. P. Eatough [*et al.*]{}, Nature [**501**]{}, 391 (2013) doi:10.1038/nature12499 \[arXiv:1308.3147 \[astro-ph.GA\]\]. Z. Stuchlík and M. Kološ, Eur. Phys. J. C [**76**]{}, no. 1, 32 (2016) doi:10.1140/epjc/s10052-015-3862-2 \[arXiv:1511.02936 \[gr-qc\]\]. G. Torok, A. Kotrlova, E. Sramkova and Z. Stuchlik, Astron. Astrophys. **531**, A59 (2011) doi:10.1051/0004-6361/201015549 \[arXiv:1103.2438 \[astro-ph.HE\]\]. C. Bambi, JCAP **09**, 014 (2012) doi:10.1088/1475-7516/2012/09/014 \[arXiv:1205.6348 \[gr-qc\]\]. R. M. Wald, Phys. Rev. D [**10**]{}, 1680 (1974). doi:10.1103/PhysRevD.10.1680 J. A. Petterson, Phys. Rev. D **12**, 2218-2225 (1975) doi:10.1103/PhysRevD.12.2218 V. P. Frolov and A. A. Shoom, Phys. Rev. D [**82**]{}, 084034 (2010) doi:10.1103/PhysRevD.82.084034 \[arXiv:1008.2985 \[gr-qc\]\]. B. Narzilloev, A. Abdujabbarov, C. Bambi and B. Ahmedov, Phys. Rev. D **99**, no.10, 104009 (2019) doi:10.1103/PhysRevD.99.104009 \[arXiv:1902.03414 \[gr-qc\]\]. L. Herrera, F. M. Paiva, and N. O. Santos, J. Math. Phys. 40, 4064 (1999), gr-qc/9810079.
R. M. Wald, “General Relativity,” doi:10.7208/chicago/9780226870373.001.0001 C.W. Misner, K.S. Thorne, J.A. Wheeler. [*Gravitation*]{} (Princeton University Press. New Jersey, 2017), ISBN:978-0-691-17779-3
M. Azreg-Aïnou, Eur. Phys. J. C [**76**]{}, no. 7, 414 (2016) doi:10.1140/epjc/s10052-016-4259-6 \[arXiv:1603.07894 \[gr-qc\]\]. C. A. Benavides-Gallego, A. Abdujabbarov, D. Malafarina, B. Ahmedov and C. Bambi, Phys. Rev. D [**99**]{}, no. 4, 044012 (2019) doi:10.1103/PhysRevD.99.044012 \[arXiv:1812.04846 \[gr-qc\]\]. A. N. Chowdhury, M. Patil, D. Malafarina and P. S. Joshi, Phys. Rev. D [**85**]{}, 104031 (2012) doi:10.1103/PhysRevD.85.104031 \[arXiv:1112.2522 \[gr-qc\]\]. O. Kopáček and V. Karas, Astrophys. J. [**787**]{}, 117 (2014) doi:10.1088/0004-637X/787/2/117 \[arXiv:1404.5495 \[astro-ph.HE\]\]. O. Kopacek, V. Karas, J. Kovar and Z. Stuchlik, Astrophys. J. [**722**]{}, 1240 (2010) doi:10.1088/0004-637X/722/2/1240 \[arXiv:1008.4650 \[astro-ph.HE\]\]. Y. Sota, S. Suzuki and K. i. Maeda, Class. Quant. Grav. [**13**]{}, 1241 (1996) doi:10.1088/0264-9381/13/5/034 \[gr-qc/9505036\]. M. A. Abramowicz and W. Kluzniak, Astrophysics [**300**]{}, 127 (2005) doi:10.1007/s10509-005-1173-z \[astro-ph/0411709\].
K. Akiyama [*et al.*]{} \[Event Horizon Telescope Collaboration\], Astrophys. J. [**875**]{}, no. 1, L1 (2019) doi:10.3847/2041-8213/ab0ec7 \[arXiv:1906.11238 \[astro-ph.GA\]\]. K. Akiyama [*et al.*]{} \[Event Horizon Telescope Collaboration\], Astrophys. J. [**875**]{}, no. 1, L2 (2019) doi:10.3847/2041-8213/ab0c96 \[arXiv:1906.11239 \[astro-ph.IM\]\]. K. Akiyama [*et al.*]{} \[Event Horizon Telescope Collaboration\], Astrophys. J. [**875**]{}, no. 1, L3 (2019) doi:10.3847/2041-8213/ab0c57 \[arXiv:1906.11240 \[astro-ph.GA\]\].
P. S. Joshi, D. Malafarina and R. Narayan, Class. Quant. Grav. **31**, 015002 (2014) doi:10.1088/0264-9381/31/1/015002 \[arXiv:1304.7331 \[gr-qc\]\]. K. Boshkayev and D. Malafarina, Mon. Not. Roy. Astron. Soc. **484**, no.3, 3325-3333 (2019) doi:10.1093/mnras/stz219 \[arXiv:1811.04061 \[gr-qc\]\]. B. Ilyas, J. Yang, D. Malafarina and C. Bambi, Eur. Phys. J. C **77**, no.7, 461 (2017) doi:10.1140/epjc/s10052-017-5014-3 \[arXiv:1611.03972 \[gr-qc\]\]. M. A. Abramowicz , M. Calvani, MNRAS, 189, 621 (1979)
O. Semerak, Mon. Not. Roy. Astron. Soc. **308**, 863-875 (1999) doi:10.1046/j.1365-8711.1999.02754.x B. Toshmatov and D. Malafarina, Phys. Rev. D **100**, no.10, 104052 (2019) doi:10.1103/PhysRevD.100.104052 \[arXiv:1910.11565 \[gr-qc\]\]. C. Bambi and D. Malafarina, Phys. Rev. D [**88**]{}, 064022 (2013) doi:10.1103/PhysRevD.88.064022 \[arXiv:1307.2106 \[gr-qc\]\]. K. S. Thorne, Astrophys. J. **191**, 507-520 (1974) doi:10.1086/152991 D. N. Page and K. S. Thorne, Astrophys. J. **191**, 499-506 (1974) doi:10.1086/152990 M. Abramowicz, B. Czerny, J. Lasota and E. Szuszkiewicz, Astrophys. J. **332**, 646 (1988) doi:10.1086/166683 M. Abramowicz, A. Lanza and M. Percival, \[arXiv:astro-ph/9611101 \[astro-ph\]\]. T. Johannsen, Phys. Rev. D [**87**]{}, no. 12, 124010 (2013) doi:10.1103/PhysRevD.87.124010 \[arXiv:1304.8106 \[gr-qc\]\]. S. Gimeno-Soler, J. A. Font, C. Herdeiro and E. Radu, Phys. Rev. D [**99**]{}, no. 4, 043002 (2019) doi:10.1103/PhysRevD.99.043002 \[arXiv:1811.11492 \[gr-qc\]\].
[^1]: We considered the function $$\begin{aligned}
\mathcal{W}&=\mathcal{L}^2 [r-m(2 \gamma +1)] (r-2m)^{2 \gamma -2}+2m\gamma \mathcal{B} r^{\gamma +1} (r-2m)^{\gamma -1}\mathcal{L}\\
& -m\gamma r^{\gamma +1} (r-2m)^{\gamma -1} -(r-m) \mathcal{B}^2 r^{2 \gamma +2},
\end{aligned}$$ from which $d\mathcal{L}/dr=-(\partial \mathcal{W}/\partial r)/(\partial \mathcal{W}/\partial \mathcal{L})$, where $\partial\mathcal{W}/\partial \mathcal{L}\neq 0$. Thus, using the condition $d\mathcal{L}/dr=0$ and solving for $\mathcal{L}$ we obtain Eq. (\[IV.11\]).
|
---
abstract: 'We study an origin of fermion mass hierarchy based on an extension of the standard model with vector-like fermions, using a bottom-up approach. It is shown that a magnitude of elements of Yukawa coupling matrices can become $O(1)$ and a Yukawa coupling unification can be realized in a theory beyond the extended model, if vector-like fermions mix with three families. In this case, small Yukawa couplings in the standard model can be highly sensitive to a small variation of matrix elements, and it suggests that the mass hierarchy occurs as a result of a fine tuning.'
author:
- |
Yoshiharu <span style="font-variant:small-caps;">Kawamura</span>[^1]\
[*Department of Physics, Shinshu University,* ]{}\
[*Matsumoto 390-8621, Japan*]{}\
date: 'September 9, 2019'
title: 'Study on fermion mass hierarchy due to vector-like fermions from the bottom up'
---
Introduction
============
One of the most fascinating riddles in particle physics is the origin of the fermion mass hierarchy and flavor mixing in the standard model (SM). Various intriguing attempts have been performed to solve it. Most of them are based on the top-down approach [@CEG; @F; @HHW; @F2; @GJ; @FN]. Starting from high-energy theories (HETs) such as grand unified theories and superstring theories or extensions of the SM with some flavor symmetries, Yukawa coupling matrices are constructed or ansatzes called texture zeros are proposed, to explain the flavor structure in the SM. In spite of endless efforts, we have not arrived at satisfactory answers, because a theory beyond the SM has not yet been confirmed and there is no powerful guiding principle to determine it. Although flavor symmetries are possible candidates[^2], any solid evidence has not yet been discovered.
In the exploration of the flavor physics, the bottom-up approach has also been carried out [@Ku; @AN; @St; @DR; @YK; @YK2]. Observed values of fermion masses and mixing angles become a springboard for study on the origin of flavor structure. In some cases, the analyses are made based on specific models. In other cases, the form of Yukawa coupling matrices can be specified to some extent by adopting a guiding principle and/or taking reasonable assumptions, independent of models in HETs, in the framework of the SM or its extension. This approach has also a limitation, because global U(3) symmetries exist in the fermion kinetic terms, Yukawa coupling matrices contain unphysical parameters, and the structure of HETs cannot be completely identified from experimental data alone. However, it can offer useful information on HETs, depending on how it is used. In concrete, we make plausible conjectures in the extension with extra particles and/or under additional assumptions, and then we can give some statements as theorems, by examining whether they are correct or not.
In this paper, based on the bottom-up approach, we make conjectures on Yukawa couplings and pursue whether they are realized or not in an extension of the SM including heavy vector-like fermions, without specifying HETs beyond the extension. We consider two conjectures. One is that [*a magnitude of Yukawa couplings can become $O(1)$ in a HET*]{}. It comes from Dirac’s naturalness. Here, Dirac’s naturalness means that the magnitude of dimensionless parameters on terms allowed by symmetries should be $O(1)$ in a fundamental theory. If it is true, the hierarchical structure of Yukawa couplings occurs after a transition from a HET to the extension of the SM or through a mechanism in some lower-energy physics. The other is that [*Yukawa couplings can be unified in a HET*]{}. It stems from a symmetry principle. It is deeply related to a grand unification based on SO(10) [@SO10] and E$_6$ [@E6]. One of our goals is to present our strategy, its availability, and its limitation, and hence we focus on the quark sector, in the following.
The outline of this paper is as follows. In the next section, we introduce our strategy based on the SM and explain our setup on an extension of the SM. In Sect. 3, we examine whether above-mentioned conjectures hold or not. In the last section, we give conclusions and discussions.
An extension of the standard model
==================================
Strategy
--------
Before we explain our setup, we introduce our strategy using the SM. Let us start with the quark sector in the SM, described by the Lagrangian density: $$\begin{aligned}
{\mathscr{L}}_{\rm SM}^{\rm quark} = \overline{q}_{{\rm L}i} i \SlashD q_{{\rm L}i}
+ \overline{u}_{{\rm R}i} i \SlashD u_{{\rm R}i}
+ \overline{d}_{{\rm R}i} i \SlashD d_{{\rm R}i}
- y_{ij}^{(u)} \overline{q}_{{\rm L}i} \tilde{\phi} u_{{\rm R}j}
- y_{ij}^{(d)} \overline{q}_{{\rm L}i} \phi d_{{\rm R}j} + {\rm h.c.},
\label{L-ky-quark}\end{aligned}$$ where $q_{{\rm L}i}$ are left-handed quark doublets, $u_{{\rm R}i}$ and $d_{{\rm R}i}$ are right-handed up- and down-type quark singlets, $i, j (=1, 2, 3)$ are family labels, summation over repeated indices is understood with few exceptions throughout this paper, $y_{ij}^{(u)}$ and $y_{ij}^{(d)}$ are Yukawa coupling matrices, $\phi$ is the Higgs doublet, $\tilde{\phi} = i \tau_2 \phi^*$ and h.c. stands for hermitian conjugation of former terms.
Here, we assume that there exists a set of privileged field variables ($q'_{\rm L}$, $u'_{\rm R}$, $d'_{\rm R}$) relating to the flavor structure. A candidate is a unitary basis of flavor symmetries [@YK; @YK2]. By the change of field variables as $$\begin{aligned}
&~& q_{\rm L} = N_q q'_{\rm L},~~u_{\rm R} = N_u u'_{\rm R},~~d_{\rm R} = N_d d'_{\rm R},
\label{unitary-SM}\end{aligned}$$ the above Lagrangian density is rewritten by $$\begin{aligned}
&~& {\mathscr{L}'}_{\rm SM}^{\rm quark}
= k_{ij}^{(q)} \overline{q}'_{{\rm L}i} i \SlashD q'_{{\rm L}j}
+ k_{ij}^{(u)} \overline{u}'_{{\rm R}i} i \SlashD u'_{{\rm R}j}
+ k_{ij}^{(d)} \overline{d}'_{{\rm R}i} i \SlashD d'_{{\rm R}j}
\nonumber \\
&~& ~~~~~~~~~~~~~~~~~~~~
- \left(y_1\right)_{ij} \overline{q}'_{{\rm L}i} \tilde{\phi} u'_{{\rm R}j}
- \left(y_2\right)_{ij} \overline{q}'_{{\rm L}i} \phi d'_{{\rm R}j} + {\rm h.c.},
\label{L-SM-quark-prime}\end{aligned}$$ where $N_q$, $N_u$, and $N_d$ are $3 \times 3$ complex matrices, $k_{ij}^{(q)}$, $k_{ij}^{(u)}$, and $k_{ij}^{(d)}$ are quark kinetic coefficients, and $\left(y_1\right)_{ij}$ and $\left(y_2\right)_{ij}$ are Yukawa coupling matrices for privileged fields. They yield the relations: $$\begin{aligned}
\hspace{-0.7cm}
&~& k^{(q)}_{ij} = \left(N_{q}^{\dagger}N_{q}\right)_{ij},~~
k^{(u)}_{ij} = \left(N_{u}^{\dagger}N_{u}\right)_{ij},~~
k^{(d)}_{ij} = \left(N_{d}^{\dagger}N_{d}\right)_{ij},~~
\label{k-I}\\
\hspace{-0.7cm}
&~& \left(y_1\right)_{ij} = \left(N_{q}^{\dagger} y^{(u)} N_{u}\right)_{ij}
= \left(N_{q}^{\dagger} {V_{\rm L}^{(u)}}^{\dagger}
y_{\rm diag}^{(u)} V_{\rm R}^{(u)} N_{u}\right)_{ij},~~
\label{y1SM}\\
\hspace{-0.7cm}
&~& \left(y_2\right)_{ij} = \left(N_{q}^{\dagger} y^{(d)} N_{d}\right)_{ij}
= \left(N_{q}^{\dagger} {V_{\rm L}^{(d)}}^{\dagger}
y_{\rm diag}^{(d)} V_{\rm R}^{(d)} N_{d}\right)_{ij}
= \left(N_{q}^{\dagger} {V_{\rm L}^{(u)}}^{\dagger} V_{\rm KM}
y_{\rm diag}^{(d)} V_{\rm R}^{(d)} N_{d}\right)_{ij},
\label{y2SM}\end{aligned}$$ where $V_{\rm L}^{(u)}$, $V_{\rm L}^{(d)}$, $V_{\rm R}^{(u)}$, and $V_{\rm R}^{(d)}$ are unitary matrices and, using them, the Yukawa coupling matrices are diagonalized as $$\begin{aligned}
V_{\rm L}^{(u)} y^{(u)} {V_{\rm R}^{(u)}}^{\dagger}
= y_{\rm diag}^{(u)} = {\rm diag}\left(y_u, y_c, y_t\right),~~
V_{\rm L}^{(d)} y^{(d)} {V_{\rm R}^{(d)}}^{\dagger}
= y_{\rm diag}^{(d)} = {\rm diag}\left(y_d, y_s, y_b\right).
\label{y-diag}\end{aligned}$$ $V_{\rm KM}$ is the Kobayashi-Maskawa matrix defined by [@KM] $$\begin{aligned}
V_{\rm KM} \equiv V_{\rm L}^{(u)} {V_{\rm L}^{(d)}}^{\dagger}.
\label{VKM}\end{aligned}$$ Using experimental values of quark masses, $y_{\rm diag}^{(u)}$, $y_{\rm diag}^{(d)}$, and $V_{\rm KM}$ are roughly estimated at the weak scale as[@PDG] $$\begin{aligned}
&~& y_{\rm diag}^{(u)} =
{\rm diag}\left(1.3 \times 10^{-5},~ 7.3 \times 10^{-3},~ 1.0\right)
= {\rm diag}\left(\lambda^7, \lambda^4, 1\right),
\label{yu-diag-value}\\
&~& y_{\rm diag}^{(d)}
= {\rm diag}\left(2.7 \times 10^{-5},~
5.5 \times 10^{-4},~ 2.4 \times 10^{-2}\right)
= {\rm diag}\left(\lambda^7, \lambda^5, \lambda^3\right),
\label{yd-diag-value}\\
&~& V_{\rm KM} = \left(
\begin{array}{ccc}
1 & \lambda & \lambda^3 \\
\lambda & 1 & \lambda^2 \\
\lambda^3 & \lambda^2 & 1
\end{array}
\right).
\label{VKM-lambda}\end{aligned}$$ In the final expressions, $\lambda^n$ means $\displaystyle{O\left(\lambda^n\right)}$ with $\lambda = \sin\theta_{\rm C} \cong 0.225$ ($\theta_{\rm C}$ is the Cabibbo angle [@C]).[^3] Physical parameters, in general, receive radiative corrections, and the above values should be evaluated by considering renormalization effects to match with their counterparts of HET at a high-energy scale.
${\mathscr{L}'}_{\rm SM}^{\rm quark}$ is supposed to be obtained as a result of a transition from a theory beyond the SM and can possess useful information on what is behind the flavor structure. Note that non-canonical matter kinetic terms appear in ${\mathscr{L}'}_{\rm SM}^{\rm quark}$.[^4]
Because the kinetic coefficients are hermitian and positive definite, they are written by $$\begin{aligned}
k_{ij}^{(q)} = \left(U_q^{\dagger} \left(J_{q}\right)^2 U_q\right)_{ij},~~
k_{ij}^{(u)} = \left(U_u^{\dagger} \left(J_{u}\right)^2 U_u\right)_{ij},~~
k_{ij}^{(d)} = \left(U_d^{\dagger} \left(J_{d}\right)^2 U_d\right)_{ij},
\label{k-UJ}\end{aligned}$$ where $U_q$, $U_u$, and $U_d$ are $3 \times 3$ unitary matrices, and $J_q$, $J_u$, and $J_d$ are real $3 \times 3$ diagonal matrices. Then, $N_q$, $N_u$, and $N_d$ are expressed by $$\begin{aligned}
N_q = V_q J_q U_q,~~ N_u = V_u J_u U_u,~~ N_d = V_d J_d U_d,
\label{NqNuNd}\end{aligned}$$ where $V_q$, $V_u$, and $V_d$ are $3 \times 3$ unitary matrices. As a magnitude of elements in unitary matrices is not beyond $1$, we obtain the inequalities: $$\begin{aligned}
\left(J_q\right)_{ii} \le O(1),~~ \left(J_u\right)_{ii} \le O(1),~~
\left(J_d\right)_{ii} \le O(1),~~ ({\rm no~~\!summation~~\!on}~~i)
\label{JqJuJd}\end{aligned}$$ from the requirement that the magnitude of matter kinetic coefficients should be at most $O(1)$ according to Dirac’s naturalness and due to the appearance of suppression factors in terms containing higher-dimensional operators. Then, from Eqs. (\[y2SM\]) and (\[yd-diag-value\]), we have the following no-go theorem.\
$[$Theorem$]$ [*If $k_{ij}^{(q)} \le O(1)$ and $k_{ij}^{(d)} \le O(1)$, the magnitude of elements of down-type Yukawa coupling matrices cannot go beyond $\displaystyle{O\left(10^{-2}\right)}$, i.e., $\displaystyle{\left(y_2\right)_{ij} \le O\left(\lambda^3\right)}$, in the framework of SM*]{}.
For reference, in a case with a large mixing such as $\displaystyle{\left(N_q\right)_{ij} = O(1)}$ and $\displaystyle{\left(N_d\right)_{ij} = O(1)}$, the magnitude of several components in $\displaystyle{\left(y_2\right)_{ij}}$ can be $\displaystyle{O\left(\lambda^3\right)}$. In contrast, the magnitude of the top-quark Yukawa coupling is sizable as $y_t = 1.0$, and that of various components in $\displaystyle{\left(y_1\right)_{ij}}$ can be $O(1)$ through a large mixing such as $\displaystyle{\left(N_q\right)_{ij} = O(1)}$ and $\displaystyle{\left(N_u\right)_{ij} = O(1)}$. Here and hereafter, we regard $O(1)$ as a number greater than $\lambda(\cong 0.225)$ and less than $\lambda^{-1}(\cong 4.44)$, i.e., $O(1)$ contains $1/\sqrt{2}$ and $1/2$.
Next, we examine whether the Yukawa couplings can be unified or not. Using ${\mathscr{L}'}_{\rm SM}^{\rm quark}$, we obtain the following no-go theorem.\
$[$Theorem$]$ [*Under the kinetic unification such as $k_{ij}^{(q)} = k_{ij}^{(u)} = k_{ij}^{(d)}$, the exact Yukawa coupling unification such as $\displaystyle{\left(y_1\right)_{ij} = \left(y_2\right)_{ij}}$ does not occur in the framework of SM*]{}.
It is understood from the observation that we obtain unrealistic features such that Yukawa coupling matrices have same eigenvalues, degenerate masses are derived between up- and down-type quarks, and the Kobayashi-Maskawa matrix becomes a unit matrix, if the kinetic coefficients are common and the up-type Yukawa coupling matrix is identical to the down-type one.
In the case that the unifications are required from a symmetry, symmetry breaking terms appear and ruin some unification relations in the broken phase. Here, we consider the case that the kinetic unification is destroyed. In this case, from Eqs. (\[y1SM\]) and (\[y2SM\]), the following relation should hold $$\begin{aligned}
\left(N^{\dagger}_q {V_{\rm L}^{(u)}}^{\dagger}
y_{\rm diag}^{(u)} V_{\rm R}^{(u)} N_{u}\right)_{ij}
= \left(N^{\dagger}_q {V_{\rm L}^{(u)}}^{\dagger} V_{\rm KM}
y_{\rm diag}^{(d)} V_{\rm R}^{(d)} N_{d}\right)_{ij},
\label{y1=y2SM}\end{aligned}$$ to realize $\displaystyle{\left(y_1\right)_{ij} = \left(y_2\right)_{ij}}$. Using Eqs. (\[yu-diag-value\]), (\[yd-diag-value\]), (\[VKM-lambda\]), and (\[y1=y2SM\]), we obtain the relation: $$\begin{aligned}
V_{\rm R}^{(u)} N_{u}
= \left(y_{\rm diag}^{(u)}\right)^{-1} V_{\rm KM} y_{\rm diag}^{(d)}
V_{\rm R}^{(d)} N_{d}
= \left(
\begin{array}{ccc}
\lambda^0 & \lambda^{-1} & \lambda^{-1} \\
\lambda^4 & \lambda & \lambda \\
\lambda^{10} & \lambda^{7} & \lambda^3
\end{array}
\right)V_{\rm R}^{(d)} N_{d}.
\label{Nu-Nd}\end{aligned}$$ If a fine tuning is absent in Eq. (\[Nu-Nd\]), we have $\displaystyle{\left(V_{\rm R}^{(d)} N_{d}\right)_{2j} \le O(\lambda)}$ and $\displaystyle{\left(V_{\rm R}^{(d)} N_{d}\right)_{3j} \le O(\lambda)}$, and then every element becomes small, i.e., $\displaystyle{\left(y_1\right)_{ij} = \left(y_2\right)_{ij}
\le O\left(\lambda^3\right)}$.
In this way, we have obtained no-go theorems on the Yukawa couplings using experimental data in the framework of SM, without specifying HETs. Through such a down-to-earth approach, knowledge and information on the flavor structure are expected to be accumulated, and some clues to the origin of flavor and hints on HETs are provided.
In the following, we apply this strategy in an extension of the SM.
Setup
-----
We adopt several assumptions.\
(a) A theory beyond the SM, which is referred to as HET, has higher gauge symmetries. It owns a seed of the flavor structure, and the form of Yukawa coupling matrices is determined by HET. Fermion kinetic terms do not necessarily take a canonical form, where the origin of flavor structure is unveiled [@YK; @YK2].\
(b) At a high-energy scale, the theory turns out to be an extension of the SM, i.e., a model with the SM gauge group G$_{\rm SM}(=$ SU(3)$_{\rm C} \times$ SU(2)$_{\rm L} \times$ U(1)$_{\rm Y}$) and extra particles. We refer to it as “SM + $\alpha$”. One should be careful not to confuse SM + $\alpha$ with HET.[^5]\
(c) The extra particles have large masses, compared with the weak boson mass, and the SM particles survive after the decoupling of heavy ones. There are 4th generation fermions and their mirror particles, as extra particles. Here, mirror particles are particles with opposite quantum numbers under G$_{\rm SM}$. A fermion and its mirror one obey a vector representation in pairs, and hence they are often referred to as a vector-like fermion.
We consider the Lagrangian density of quarks in SM + $\alpha$, described by $$\begin{aligned}
&~& {\mathscr{L}'}_{\rm SM+\alpha}^{\rm quark}
= k_{IJ}^{(q)} \overline{q'_{{\rm L}I}} i \SlashD q'_{{\rm L}J}
+ k_{IJ}^{(u)} \overline{u'_{{\rm R}I}} i \SlashD u'_{{\rm R}J}
+ k_{IJ}^{(d)} \overline{d'_{{\rm R}I}} i \SlashD d'_{{\rm R}J}
\nonumber \\
&~& ~~~~~~~~~~~~~~~~~~
+ k^{(q_{\rm m})} \overline{\left(q'_{{\rm L(m)}}\right)^{c}}
i \SlashD \left(q'_{{\rm L(m)}}\right)^c
+ k^{(u_{\rm m})} \overline{\left(u'_{{\rm R(m)}}\right)^c}
i \SlashD \left(u'_{{\rm R(m)}}\right)^c
\nonumber \\
&~& ~~~~~~~~~~~~~~~~~~
+ k^{(d_{\rm m})} \overline{\left(d'_{{\rm R(m)}}\right)^c}
i \SlashD \left(d'_{{\rm R(m)}}\right)^c
\nonumber \\
&~& ~~~~~~~~~~~~~~~~~~
- y^{(U)}_{IJ} \overline{q'_{{\rm L}I}} \tilde{\phi} u'_{{\rm R}J}
- y^{(D)}_{IJ} \overline{q'_{{\rm L}I}} \phi d'_{{\rm R}J} + {\rm h.c.}
\nonumber \\
&~& ~~~~~~~~~~~~~~~~~~
- y^{(u_{\rm m})} \overline{\left(u'_{{\rm R(m)}}\right)^{c}}
\left(q'_{{\rm L(m)}}\right)^c \tilde{\phi}^*
- y^{(d_{\rm m})} \overline{\left(d'_{{\rm R(m)}}\right)^{c}}
\left(q'_{{\rm L(m)}}\right)^c {\phi}^* + {\rm h.c.}
\nonumber \\
&~& ~~~~~~~~~~~~~~~~~~
- m_I^{(q_{\rm m})} \overline{q'_{{\rm L}I}} \left(q'_{{\rm L(m)}}\right)^c
- m_J^{(u_{\rm m})} \overline{\left(u'_{{\rm R(m)}}\right)^{c}} u'_{{\rm R}J}
- m_J^{(d_{\rm m})} \overline{\left(d'_{{\rm R(m)}}\right)^{c}} d'_{{\rm R}J}
+ {\rm h.c.},
\label{L-SM+alpha-quark}\end{aligned}$$ where $q'_{{\rm L}I}$ are counterparts of left-handed quark doublets, $u'_{{\rm R}I}$ and $d'_{{\rm R}I}$ are those of right-handed up- and down-type quark singlets, and $I$ and $J$ run from 1 to 4. $\displaystyle{\left(q'_{{\rm L(m)}}\right)^c}$, $\displaystyle{\left(u'_{{\rm R(m)}}\right)^c}$, and $\displaystyle{\left(d'_{{\rm R(m)}}\right)^c}$ are charge conjugations of mirror quarks $q'_{{\rm L(m)}}$, $u'_{{\rm R(m)}}$, and $d'_{{\rm R(m)}}$. We refer to $q'_{{\rm L}I}$, $u'_{{\rm R}I}$, $d'_{{\rm R}I}$, $\displaystyle{\left(q'_{{\rm L(m)}}\right)^c}$, $\displaystyle{\left(u'_{{\rm R(m)}}\right)^c}$, and $\displaystyle{\left(d'_{{\rm R(m)}}\right)^c}$ as quarks, collectively. Fields with prime represents privileged fields concerning the flavor structure. $k_{IJ}^{(q)}$, $k_{IJ}^{(u)}$, $k_{IJ}^{(d)}$, $k^{(q_{\rm m})}$, $k^{(u_{\rm m})}$, and $k^{(d_{\rm m})}$ are quark kinetic coefficients, $y^{(U)}_{IJ}$, $y^{(D)}_{IJ}$, $y^{(u_{\rm m})}$, and $y^{(d_{\rm m})}$ are Yukawa coupling matrices, and $m_I^{(q_{\rm m})}$, $m_J^{(u_{\rm m})}$, and $m_J^{(d_{\rm m})}$ are mass parameters.
The gauge quantum numbers and the chirality $\gamma_5$ of quarks are listed in Table \[T-SM+alpha\]. For reference, we list the gauge quantum numbers and the chirality of mirror quarks in Table \[T-mirrors\]. In Tables \[T-SM+alpha\] and \[T-mirrors\], Y is the weak hypercharge.
-----------------------------------------------------------------------------------------------------------
quarks in SM + $\alpha$ SU(3)$_{\rm C}$ SU(2)$_{\rm L}$ Y $\gamma_5$
----------------------------------------- ----------------- ----------------- ---------------- ------------
$q'_{{\rm L}I} = \left( $\bm{3}$ $\bm{2}$ $\frac{1}{6}$ $-1$
\begin{array}{c}
u'_{{\rm L}I} \\
d'_{{\rm L}I}
\end{array}
\right)$
$\left(q'_{{\rm L(m)}}\right)^c= \left( $\bm{3}$ $\bm{2}$ $\frac{1}{6}$ $1$
\begin{array}{c}
\left(u'_{{\rm L(m)}}\right)^c \\
\left(d'_{{\rm L(m)}}\right)^c
\end{array}
\right)$
$u'_{{\rm R}I}$ $\bm{3}$ $\bm{1}$ $\frac{2}{3}$ $1$
$\left(u'_{{\rm R(m)}}\right)^c$ $\bm{3}$ $\bm{1}$ $\frac{2}{3}$ $-1$
$d'_{{\rm R}I}$ $\bm{3}$ $\bm{1}$ $-\frac{1}{3}$ $1$
$\left(d'_{{\rm R(m)}}\right)^c$ $\bm{3}$ $\bm{1}$ $-\frac{1}{3}$ $-1$
-----------------------------------------------------------------------------------------------------------
: The gauge quantum numbers and the chirality of quarks in SM + $\alpha$.[]{data-label="T-SM+alpha"}
------------------------------------------------------------------------------------------------
mirror quarks SU(3)$_{\rm C}$ SU(2)$_{\rm L}$ Y $\gamma_5$
-------------------------- --------------------- ----------------- ---------------- ------------
$q'_{{\rm L(m)}}= \left( $\overline{\bm{3}}$ $\bm{2}$ $-\frac{1}{6}$ $-1$
\begin{array}{c}
u'_{{\rm L(m)}} \\
d'_{{\rm L(m)}}
\end{array}
\right)$
$u'_{{\rm R(m)}}$ $\overline{\bm{3}}$ $\bm{1}$ $-\frac{2}{3}$ $1$
$d'_{{\rm R(m)}}$ $\overline{\bm{3}}$ $\bm{1}$ $\frac{1}{3}$ $1$
------------------------------------------------------------------------------------------------
: The gauge quantum numbers and the chirality of mirror quarks in SM + $\alpha$.[]{data-label="T-mirrors"}
The Yukawa interactions and mass terms in ${\mathscr{L}'}_{\rm SM+\alpha}^{\rm quark}$ are compactly written by $$\begin{aligned}
{\mathscr{L}'}_{\rm SM+\alpha (Y, m)}^{\rm quark}
= - \overline{U'_{{\rm L}A}} M^{(U)}_{AB} U'_{{\rm R}B}
- \overline{D'_{{\rm L}A}} M^{(D)}_{AB} D'_{{\rm R}B}
+ {\rm h.c.} + \cdots,
\label{L-Y+M}\end{aligned}$$ where $A$ and $B$ run from 1 to 5, and $U'_{{\rm L}A}$, $U'_{{\rm R}A}$, $D'_{{\rm L}A}$, and $D'_{{\rm R}A}$ consist of 5 components such that $$\begin{aligned}
U'_{{\rm L}} =
\left(
\begin{array}{c}
u'_{{\rm L}I} \\
\left(u'_{{\rm R(m)}}\right)^c
\end{array}
\right),~~
U'_{{\rm R}} =
\left(
\begin{array}{c}
u'_{{\rm R}I} \\
\left(u'_{{\rm L(m)}}\right)^c
\end{array}
\right),~~
D'_{{\rm L}} =
\left(
\begin{array}{c}
d'_{{\rm L}I} \\
\left(d'_{{\rm R(m)}}\right)^c
\end{array}
\right),~~
D'_{{\rm R}} =
\left(
\begin{array}{c}
d'_{{\rm R}I} \\
\left(d'_{{\rm L(m)}}\right)^c
\end{array}
\right),
\label{U+D}\end{aligned}$$ and $M^{(U)}_{AB}$ and $M^{(D)}_{AB}$ are $5 \times 5$ complex matrices given by $$\begin{aligned}
M^{(U)} =
\left(
\begin{array}{cc}
y_{IJ}^{(U)} \phi^{0*} & m_I^{(q_{\rm m})} \\
m_J^{(u_{\rm m})} & y^{(u_{\rm m})} \phi^{0}
\end{array}
\right),~~
M^{(D)} =
\left(
\begin{array}{cc}
y_{IJ}^{(D)} \phi^{0} & m_I^{(q_{\rm m})} \\
m_J^{(d_{\rm m})} & y^{(d_{\rm m})} \phi^{0*}
\end{array}
\right),
\label{M(U)}\end{aligned}$$ respectively. The ellipsis in Eq. (\[L-Y+M\]) stands for terms containing a charged component of the Higgs doublet.
To apply the bottom-up approach to our model, we need to know the relationship between the privileged fields in ${\mathscr{L}'}_{\rm SM+\alpha}^{\rm quark}$ and mass eigenstates in the SM. In the following, we will see that the kinetic terms change into the canonical form and mass terms including Yukawa interactions are diagonalized by redefining field variables.
First, we pay attention to the fact that the quark kinetic coefficients are hermitian and are expressed by $$\begin{aligned}
&~& K^{(U_{\rm L})} =
\left(
\begin{array}{cc}
k_{IJ}^{(q)} & 0 \\
0 & k^{(u_{\rm m})}
\end{array}
\right) = {N_{\rm L}^{u}}^{\dagger}N_{\rm L}^{u},~~
K^{(D_{\rm L})} =
\left(
\begin{array}{cc}
k_{IJ}^{(q)} & 0 \\
0 & k^{(d_{\rm m})}
\end{array}
\right) = {N_{\rm L}^{d}}^{\dagger}N_{\rm L}^{d},~~
\nonumber \\
&~& K^{(U_{\rm R})} =
\left(
\begin{array}{cc}
k_{IJ}^{(u)} & 0 \\
0 & k^{(q_{\rm m})}
\end{array}
\right) = {N_{\rm R}^{u}}^{\dagger}N_{\rm R}^{u},~~
K^{(D_{\rm R})} =
\left(
\begin{array}{cc}
k_{IJ}^{(d)} & 0 \\
0 & k^{(q_{\rm m})}
\end{array}
\right) = {N_{\rm R}^{d}}^{\dagger}N_{\rm R}^{d},
\label{K(UL)}\end{aligned}$$ using $5 \times 5$ complex matrices $N_{\rm L}^{u}$, $N_{\rm L}^{d}$, $N_{\rm R}^{u}$, and $N_{\rm R}^{d}$ with $\displaystyle{\left(N_{\rm L}^{u}\right)_{IJ}= \left(N_{\rm L}^{d}\right)_{IJ}}$, $\displaystyle{\left(N_{\rm R}^{u}\right)_{55}= \left(N_{\rm R}^{d}\right)_{55}}$, $\displaystyle{\left(N_{\rm L}^{u}\right)_{I5}= 0}$, $\displaystyle{\left(N_{\rm L}^{u}\right)_{5J}= 0}$ and so on. Then, after the change of variables: $$\begin{aligned}
&~& U_{\rm L} =
\left(
\begin{array}{c}
u_{{\rm L}I} \\
\left(u_{{\rm R(m)}}\right)^c
\end{array}
\right) = N_{\rm L}^{u} U'_{\rm L},~~
D_{\rm L} =
\left(
\begin{array}{c}
d_{{\rm L}I} \\
\left(d_{{\rm R(m)}}\right)^c
\end{array}
\right) = N_{\rm L}^{d} D'_{\rm L},~~
\label{UL}\\
&~& U_{\rm R}=
\left(
\begin{array}{c}
u_{{\rm R}I} \\
\left(u_{{\rm L(m)}}\right)^c
\end{array}
\right) = N_{\rm R}^{u} U'_{\rm R},~~
D_{\rm R} =
\left(
\begin{array}{c}
d_{{\rm R}I} \\
\left(d_{{\rm L(m)}}\right)^c
\end{array}
\right) = N_{\rm R}^{d} D'_{\rm R},
\label{DL}\end{aligned}$$ we obtain the canonical type of quark kinetic terms: $$\begin{aligned}
&~& {\mathscr{L}}_{\rm SM+\alpha (k)}^{\rm quark}
= \overline{q_{{\rm L}I}} i \SlashD q_{{\rm L}I}
+ \overline{u_{{\rm R}I}} i \SlashD u_{{\rm R}I}
+ \overline{d_{{\rm R}I}} i \SlashD d_{{\rm R}I}
\nonumber \\
&~& ~~~~~~~~~~~~~~~~~~~~~
+ \overline{\left(q_{{\rm L(m)}}\right)^{c}}
i \SlashD \left(q_{{\rm L(m)}}\right)^c
+ \overline{\left(u_{{\rm R(m)}}\right)^c}
i \SlashD \left(u_{{\rm R(m)}}\right)^c
+ \overline{\left(d_{{\rm R(m)}}\right)^c}
i \SlashD \left(d_{{\rm R(m)}}\right)^c,
\label{L-kinetic}\end{aligned}$$ where $q_{{\rm L}I}$ and $\displaystyle{\left(q_{{\rm L(m)}}\right)^c}$ are SU(2)$_{\rm L}$ doublets given by $$\begin{aligned}
q_{{\rm L}I} =
\left(
\begin{array}{c}
u_{{\rm L}I} \\
d_{{\rm L}I}
\end{array}
\right),~~
\left(q_{{\rm L(m)}}\right)^c =
\left(
\begin{array}{c}
\left(u_{{\rm L(m)}}\right)^c \\
\left(d_{{\rm L(m)}}\right)^c
\end{array}
\right),
\label{qqc}\end{aligned}$$ respectively.
Here, we give comments. The transformation matrices $N_{\rm L}^{u}$, $N_{\rm L}^{d}$, $N_{\rm R}^{u}$, and $N_{\rm R}^{d}$ are not completely fixed, or $\tilde{V}_{\rm L}^{u} N_{\rm L}^{u}$, $\tilde{V}_{\rm L}^{d}N_{\rm L}^{d}$, $\tilde{V}_{\rm R}^{u} N_{\rm R}^{u}$, and $\tilde{V}_{\rm R}^{d} N_{\rm R}^{d}$ also offer the same kinetic coefficients $K^{(U_{\rm L})}$, $K^{(D_{\rm L})}$, $K^{(U_{\rm R})}$, and $K^{(D_{\rm R})}$, respectively. Here, $\tilde{V}_{\rm L}^{u}$, $\tilde{V}_{\rm L}^{d}$, $\tilde{V}_{\rm R}^{u}$, and $\tilde{V}_{\rm R}^{d}$ are arbitrary unitary matrices, and using this arbitrariness, $M^{(U)}$ and $M^{(D)}$ are diagonalized, as will be described below. Then, $V_{\rm KM}$ appears in $\displaystyle{\overline{q_{{\rm L}I}} i \SlashD q_{{\rm L}I}}$ on the mass eigenstates. The quark kinetic terms in ${\mathscr{L}'}_{\rm SM+\alpha}^{\rm quark}$ cannot be compactly written by using $U'_{{\rm L}}$, $U'_{{\rm R}}$, $D'_{{\rm L}}$, and $D'_{{\rm R}}$, because these variables contain fields with different quantum numbers under SU(2)$_{\rm L} \times$ U(1)$_{\rm Y}$ and do not treat $u'_{{\rm L}I}$, $d'_{{\rm L}I}$, $\displaystyle{\left(u'_{{\rm L(m)}}\right)^c}$, and $\displaystyle{\left(d'_{{\rm L(m)}}\right)^c}$ as SU(2)$_{\rm L}$ doublets.
Next, by the change of variables such as Eqs. (\[UL\]) and (\[DL\]), including suitable unitary matrices $\tilde{V}_{\rm L}^{u}$, $\tilde{V}_{\rm L}^{d}$, $\tilde{V}_{\rm R}^{u}$, and $\tilde{V}_{\rm R}^{d}$, $M^{(U)}$ and $M^{(D)}$ are transformed into $$\begin{aligned}
&~& \left({N_{\rm L}^{u}}^{\dagger}\right)^{-1} M^{(U)}
\left({{N}}_{\rm R}^{u}\right)^{-1}
= \left(
\begin{array}{ccccc}
y_{u}{\phi}^{0*} & 0 & 0 & 0 & 0 \\
0 & y_{c}{\phi}^{0*} & 0 & 0 & 0 \\
0 & 0 & y_{t}{\phi}^{0*} & 0 & 0 \\
0 & 0 & 0 & 0 & m_1^{(U)} \\
0 & 0 & 0 & m_2^{(U)} & 0
\end{array}
\right),
\label{MU-diag} \\
&~& \left({N_{\rm L}^{d}}^{\dagger}\right)^{-1} M^{(D)}
\left({{N}}_{\rm R}^{d}\right)^{-1}
= \left(
\begin{array}{ccccc}
y_{d}\phi^{0} & 0 & 0 & 0 & 0 \\
0 & y_{s}\phi^{0} & 0 & 0 & 0 \\
0 & 0 & y_{b}\phi^{0} & 0 & 0 \\
0 & 0 & 0 & 0 & m_1^{(D)} \\
0 & 0 & 0 & m_2^{(D)} & 0
\end{array}
\right),
\label{MD-diag}\end{aligned}$$ where, for simplicity, $\tilde{V}_{\rm L}^{u} N_{\rm L}^{u}$, $\tilde{V}_{\rm L}^{d}N_{\rm L}^{d}$, $\tilde{V}_{\rm R}^{u} N_{\rm R}^{u}$, and $\tilde{V}_{\rm R}^{d} N_{\rm R}^{d}$ are again denoted as $N_{\rm L}^{u}$, $N_{\rm L}^{d}$, $N_{\rm R}^{u}$, and $N_{\rm R}^{d}$, respectively. $m_1^{(U)}$, $m_2^{(U)}$, $m_1^{(D)}$, and $m_2^{(D)}$ are large masses of extra quarks. The experimental bounds on 4th generation quark masses require that an extra up-type quark is heavier than 1160 GeV from neutral-current decays and an extra down-type quark is heavier than 880 GeV from charged-current decays [@PDG].
Examination on conjectures
==========================
We carry out order estimations using $\lambda (\cong 0.225)$, and examine whether the conjectures on the Yukawa couplings hold or not, based on the extension of the SM described by ${\mathscr{L}'}_{\rm SM+\alpha}^{\rm quark}$. The analyses on the case with partial multiplets are carried out in Appendix A.
Seeking transformation matrices
-------------------------------
Using $\lambda$, the magnitude of the right-hand sides in Eqs. (\[MU-diag\]) and (\[MD-diag\]) is parametrized as $$\begin{aligned}
M^{(U)}_{\rm diag}
= \left(
\begin{array}{ccccc}
\lambda^7 & 0 & 0 & 0 & 0 \\
0 & \lambda^4 & 0 & 0 & 0 \\
0 & 0 & \lambda^0 & 0 & 0 \\
0 & 0 & 0 & 0 & \lambda^{-n_1} \\
0 & 0 & 0 & \lambda^{-n_2} & 0
\end{array}
\right) \frac{v}{\sqrt{2}},~~
M^{(D)}_{\rm diag}
= \left(
\begin{array}{ccccc}
\lambda^7 & 0 & 0 & 0 & 0 \\
0 & \lambda^5 & 0 & 0 & 0 \\
0 & 0 & \lambda^3 & 0 & 0 \\
0 & 0 & 0 & 0 & \lambda^{-n_3} \\
0 & 0 & 0 & \lambda^{-n_4} & 0
\end{array}
\right) \frac{v}{\sqrt{2}},
\label{M-diag-rep}\end{aligned}$$ where $n_1$, $n_2$, $n_3$, and $n_4$ are positive integers, as seen from the lower mass bounds 1160 GeV and 880 GeV, and $v/\sqrt{2}$ is the vacuum expectation value of a neutral component of the Higgs doublet. Note that $v$ is used for the sake of convenience, although $m_1^{(U)}$, $m_2^{(U)}$, $m_1^{(D)}$, and $m_2^{(D)}$ must be irrelevant to the breakdown of electroweak symmetry.
Using Eqs. (\[M(U)\]), (\[MU-diag\]), (\[MD-diag\]), and (\[M-diag-rep\]), we obtain the relations: $$\begin{aligned}
&~&
\left(
\begin{array}{cc}
y_{IJ}^{(U)} \langle \phi^{0*} \rangle & m_I^{(q_{\rm m})} \\
m_J^{(u_{\rm m})} & y^{(u_{\rm m})} \langle \phi^{0} \rangle
\end{array}
\right)
= {N_{\rm L}^{u}}^{\dagger}
\left(
\begin{array}{ccccc}
\lambda^7 & 0 & 0 & 0 & 0 \\
0 & \lambda^4 & 0 & 0 & 0 \\
0 & 0 & \lambda^0 & 0 & 0 \\
0 & 0 & 0 & 0 & \lambda^{-n_1} \\
0 & 0 & 0 & \lambda^{-n_2} & 0
\end{array}
\right) N_{\rm R}^{u} \frac{v}{\sqrt{2}},
\label{M(U)-diag}\\
&~&
\left(
\begin{array}{cc}
y_{IJ}^{(D)} \langle \phi^{0} \rangle & m_I^{(q_{\rm m})} \\
m_J^{(d_{\rm m})} & y^{(d_{\rm m})} \langle \phi^{0*} \rangle
\end{array}
\right)
= {N_{\rm L}^{d}}^{\dagger}
\left(
\begin{array}{ccccc}
\lambda^7 & 0 & 0 & 0 & 0 \\
0 & \lambda^5 & 0 & 0 & 0 \\
0 & 0 & \lambda^3 & 0 & 0 \\
0 & 0 & 0 & 0 & \lambda^{-n_3} \\
0 & 0 & 0 & \lambda^{-n_4} & 0
\end{array}
\right) N_{\rm R}^{d} \frac{v}{\sqrt{2}}.
\label{M(D)-diag}\end{aligned}$$
According to Dirac’s naturalness, we impose the following conditions on the kinetic coefficients, $$\begin{aligned}
&~& k_{IJ}^{(q)},~~ k_{IJ}^{(u)},~~ k_{IJ}^{(d)} = O(1)~~~ {\rm for}~I=J,~~~
(I, J = 1, \cdots, 4)
\nonumber \\
&~&
k_{IJ}^{(q)},~~ k_{IJ}^{(u)},~~ k_{IJ}^{(d)} \le O(1)~~~ {\rm for}~I \ne J,~~
\nonumber \\
&~& k^{(q_{\rm m})},~~ k^{(u_{\rm m})},~~ k^{(d_{\rm m})} = O(1).
\label{k=O(1)}\end{aligned}$$ The conjectures on the Yukawa couplings are written by $$\begin{aligned}
y^{(U)}_{IJ},~~ y^{(D)}_{IJ},~~ y^{(u_{\rm m})},~~ y^{(d_{\rm m})} = O(1)~~~
{\rm for~~\!some~~\!entries}
\label{y=O(1)}\end{aligned}$$ and $$\begin{aligned}
y^{(U)}_{IJ} = y^{(D)}_{IJ},
\label{yU=yD-IJ}\end{aligned}$$ respectively.
The examination on conjectures is carried out by studying whether $N_{\rm L}^u$, $N_{\rm L}^d$, $N_{\rm R}^u$, and $N_{\rm R}^d$ exist or not, to satisfy Eqs. (\[y=O(1)\]) and (\[yU=yD-IJ\]), based on Eqs. (\[K(UL)\]), (\[M(U)-diag\]), (\[M(D)-diag\]), and (\[k=O(1)\]), and specifying the form of those matrices.
Let us take the ansatzes:[^6] $$\begin{aligned}
&~& \left(N_{\rm L}^u\right)_{AB},~~ \left(N_{\rm L}^d\right)_{AB},~~
\left(N_{\rm R}^u\right)_{AB},~~ \left(N_{\rm R}^d\right)_{AB} = O(1)
~~~ {\rm for}~A=B,~~~ (A, B = 1, \cdots, 5)
\nonumber \\
&~& \left(N_{\rm L}^u\right)_{AB},~~ \left(N_{\rm L}^d\right)_{AB},~~
\left(N_{\rm R}^u\right)_{AB},~~ \left(N_{\rm R}^d\right)_{AB} \le O(1)
~~~ {\rm for}~A \ne B.
\label{N=O(1)}\end{aligned}$$
First, the relations (\[K(UL)\]) yield the conditions: $$\begin{aligned}
&~& K_{I5}^{(U_{\rm L})} = \left({N_{\rm L}^u}^{\dagger}\right)_{IA}
\left(N_{\rm L}^u\right)_{A5} = 0,~~
K_{5J}^{(U_{\rm L})} = \left({N_{\rm L}^u}^{\dagger}\right)_{5A}
\left(N_{\rm L}^u\right)_{AJ} = 0,~~
\label{KI5=0-UL}\\
&~& K_{I5}^{(D_{\rm L})} = \left({N_{\rm L}^d}^{\dagger}\right)_{IA}
\left(N_{\rm L}^d\right)_{A5} = 0,~~
K_{5J}^{(D_{\rm L})} = \left({N_{\rm L}^d}^{\dagger}\right)_{5A}
\left(N_{\rm L}^d\right)_{AJ} = 0,~~
\label{KI5=0-DL}\\
&~& K_{I5}^{(U_{\rm R})} = \left({N_{\rm R}^u}^{\dagger}\right)_{IA}
\left(N_{\rm R}^u\right)_{A5} = 0,~~
K_{5J}^{(U_{\rm R})} = \left({N_{\rm R}^u}^{\dagger}\right)_{5A}
\left(N_{\rm R}^u\right)_{AJ} = 0,~~
\label{KI5=0-UR}\\
&~& K_{I5}^{(D_{\rm R})} = \left({N_{\rm R}^d}^{\dagger}\right)_{IA}
\left(N_{\rm L}^d\right)_{A5} = 0,~~
K_{5J}^{(D_{\rm R})} = \left({N_{\rm R}^d}^{\dagger}\right)_{5A}
\left(N_{\rm R}^d\right)_{AJ} = 0.
\label{KI5=0-DR}\end{aligned}$$ These conditions are satisfied with suitable components, if the rank of $4 \times 4$ sub-matrices $\left(N_{\rm L}^u\right)_{IJ}$, $\left(N_{\rm L}^d\right)_{IJ}$, $\left(N_{\rm R}^u\right)_{IJ}$, and $\left(N_{\rm R}^d\right)_{IJ}$ is 4. In our case, they are automatically satisfied, because the transformation matrices are given in the form as $\tilde{V} N$ where $\tilde{V}$ is an arbitrary unitary matrix and $N$ is a block-diagonal matrix with $N_{I5} = 0$ and $N_{5J} = 0$.
Next, from the relations (\[M(U)-diag\]) and (\[M(D)-diag\]), the following relations for the Yukawa couplings are obtained, $$\begin{aligned}
&~& y_{IJ}^{(U)} = \lambda^7 \left({N_{\rm L}^u}^{\dagger}\right)_{I1}
\left(N_{\rm R}^u\right)_{1J}
+ \lambda^4 \left({N_{\rm L}^u}^{\dagger}\right)_{I2} \left(N_{\rm R}^u\right)_{2J}
+ \lambda^0 \left({N_{\rm L}^u}^{\dagger}\right)_{I3} \left(N_{\rm R}^u\right)_{3J}
\nonumber \\
&~& ~~~~~~~~~~~~~ + \lambda^{-n_1} \left({N_{\rm L}^u}^{\dagger}\right)_{I4}
\left(N_{\rm R}^u\right)_{5J}
+ \lambda^{-n_2} \left({N_{\rm L}^u}^{\dagger}\right)_{I5} \left(N_{\rm R}^u\right)_{4J},
\label{yuIJ}\\
&~& y^{(u_{\rm m})}
= \lambda^7 \left({N_{\rm L}^u}^{\dagger}\right)_{51} \left(N_{\rm R}^u\right)_{15}
+ \lambda^4 \left({N_{\rm L}^u}^{\dagger}\right)_{52} \left(N_{\rm R}^u\right)_{25}
+ \lambda^0 \left({N_{\rm L}^u}^{\dagger}\right)_{53} \left(N_{\rm R}^u\right)_{35}
\nonumber \\
&~& ~~~~~~~~~~~~~ + \lambda^{-n_1} \left({N_{\rm L}^u}^{\dagger}\right)_{54}
\left(N_{\rm R}^u\right)_{55}
+ \lambda^{-n_2} \left({N_{\rm L}^u}^{\dagger}\right)_{55} \left(N_{\rm R}^u\right)_{45},
\label{yu55}\\
&~& y_{IJ}^{(D)} = \lambda^7 \left({N_{\rm L}^d}^{\dagger}\right)_{I1}
\left(N_{\rm R}^d\right)_{1J}
+ \lambda^5 \left({N_{\rm L}^d}^{\dagger}\right)_{I2} \left(N_{\rm R}^d\right)_{2J}
+ \lambda^3 \left({N_{\rm L}^d}^{\dagger}\right)_{I3} \left(N_{\rm R}^d\right)_{3J}
\nonumber \\
&~& ~~~~~~~~~~~~~ + \lambda^{-n_3} \left({N_{\rm L}^d}^{\dagger}\right)_{I4}
\left(N_{\rm R}^d\right)_{5J}
+ \lambda^{-n_4} \left({N_{\rm L}^d}^{\dagger}\right)_{I5} \left(N_{\rm R}^d\right)_{4J},
\label{ydIJ}\\
&~& y^{(d_{\rm m})}
= \lambda^7 \left({N_{\rm L}^d}^{\dagger}\right)_{51} \left(N_{\rm R}^d\right)_{15}
+ \lambda^5 \left({N_{\rm L}^d}^{\dagger}\right)_{52} \left(N_{\rm R}^d\right)_{25}
+ \lambda^3 \left({N_{\rm L}^d}^{\dagger}\right)_{53} \left(N_{\rm R}^d\right)_{35}
\nonumber \\
&~& ~~~~~~~~~~~~~ + \lambda^{-n_3} \left({N_{\rm L}^d}^{\dagger}\right)_{54}
\left(N_{\rm R}^d\right)_{55}
+ \lambda^{-n_4} \left({N_{\rm L}^d}^{\dagger}\right)_{55} \left(N_{\rm R}^d\right)_{45}.
\label{yd55}\end{aligned}$$ From Eq. (\[yuIJ\]), we find that $y_{IJ}^{(U)} = O(1)$ can be realized if a large mixing occurs between $u'_{{\rm L}3}$ ($u'_{{\rm R}3}$) and other $u'_{{\rm L}}$s ($u'_{{\rm R}}$s), i.e., $\left({N_{\rm L}^u}^{\dagger}\right)_{I3} = O(1)$ and $\left(N_{\rm R}^u\right)_{3J} = O(1)$, and even if other contributions are not sizable. In contrast, for the down-type Yukawa couplings, we have the following no-go theorem.\
$[$Theorem$]$ [*If the magnitude of kinetic coefficients is at most $O(1)$, the magnitude of elements of down-type Yukawa coupling matrices cannot go beyond $\displaystyle{O\left(10^{-2}\right)}$ without sizable contributions from extra fermions, in the extension of the SM with vector-like fermions*]{}.
In the following, we study a case with sizable contributions from extra quarks. By imposing on the conditions $y_{IJ}^{(D)} = O(1)$ and $y^{(d_{\rm m})}=O(1)$, we obtain the relations: $$\begin{aligned}
&~& \left({N_{\rm L}^d}^{\dagger}\right)_{I4} \left(N_{\rm R}^d\right)_{5J}
= O\left(\lambda^{n_3}\right)~~~ {\rm and/or}~~~
\left({N_{\rm L}^d}^{\dagger}\right)_{I5} \left(N_{\rm R}^d\right)_{4J}
= O\left(\lambda^{n_4}\right),
\label{NdI4}\\
&~& \left({N_{\rm L}^d}^{\dagger}\right)_{54} \left(N_{\rm R}^d\right)_{55}
= O\left(\lambda^{n_3}\right)~~~ {\rm and/or}~~~
\left({N_{\rm L}^d}^{\dagger}\right)_{55} \left(N_{\rm R}^d\right)_{45}
= O\left(\lambda^{n_4}\right).
\label{Nd54}\end{aligned}$$ Hereafter, we consider the case with “and” in Eqs. (\[NdI4\]) and (\[Nd54\]). Then, from $\left({N_{\rm L}^d}^{\dagger}\right)_{44} = O(1)$, $\left(N_{\rm R}^d\right)_{44} = O(1)$, $\left(N_{\rm R}^d\right)_{55} = O(1)$, and $\left({N_{\rm L}^d}^{\dagger}\right)_{55} = O(1)$, we obtain $\displaystyle{\left(N_{\rm R}^d\right)_{5J} = O\left(\lambda^{n_3}\right)}$, $\displaystyle{\left({N_{\rm L}^d}^{\dagger}\right)_{I5} = O\left(\lambda^{n_4}\right)}$, $\displaystyle{\left({N_{\rm L}^d}^{\dagger}\right)_{54} = O\left(\lambda^{n_3}\right)}$, and $\displaystyle{\left(N_{\rm R}^d\right)_{45} = O\left(\lambda^{n_4}\right)}$, respectively. Then, the transformation matrices take the form: $$\begin{aligned}
{N_{\rm L}^d}^{\dagger}
= \left(
\begin{array}{ccccc}
1 & \star & \star & 1 & \lambda^{n_4} \\
\star & 1 & \star & 1 & \lambda^{n_4} \\
\star & \star & 1 & 1 & \lambda^{n_4} \\
\star & \star & \star & 1 & \lambda^{n_4} \\
\star & \star & \star & \lambda^{n_3} & 1
\end{array}
\right),~~~
N_{\rm R}^d
= \left(
\begin{array}{ccccc}
1 & \star & \star & \star & \star \\
\star & 1 & \star & \star & \star \\
\star & \star & 1 & \star & \star \\
1 & 1 & 1 & 1 & \lambda^{n_4} \\
\lambda^{n_3} & \lambda^{n_3} & \lambda^{n_3} & \lambda^{n_3} & 1
\end{array}
\right),
\label{NLd}\end{aligned}$$ where 1 means $O(1)$ and $\star$ stands for an unspecified one. From $K_{I5}^{(D_{\rm L})}
= \left({N_{\rm L}^d}^{\dagger}\right)_{IA}
\left(N_{\rm L}^d\right)_{A5} = 0$ and $K_{5J}^{(D_{\rm L})} = \left({N_{\rm L}^d}^{\dagger}\right)_{5A}
\left(N_{\rm L}^d\right)_{AJ} = 0$, we need $n_3 = n_4$ and $\displaystyle{\left({N_{\rm L}^d}^{\dagger}\right)_{5j}
= O\left(\lambda^{n_3}\right)}$ ($j=1,2,3$). From $K_{I5}^{(D_{\rm R})}
= \left({N_{\rm R}^d}^{\dagger}\right)_{IA}
\left(N_{\rm R}^d\right)_{A5} = 0$ and $K_{5J}^{(D_{\rm R})} = \left({N_{\rm R}^d}^{\dagger}\right)_{5A}
\left(N_{\rm R}^d\right)_{AJ} = 0$, we need $n_3 = n_4$ and $\displaystyle{\left({N_{\rm R}^d}\right)_{i5} = O\left(\lambda^{n_4}\right)}$ ($i=1,2,3$). Then, we obtain the transformation matrices such as $$\begin{aligned}
{N_{\rm L}^d},~~
N_{\rm R}^d
= \left(
\begin{array}{ccccc}
1 & \star & \star & \star & \lambda^{n_3} \\
\star & 1 & \star & \star & \lambda^{n_3} \\
\star & \star & 1 & \star & \lambda^{n_3} \\
1 & 1 & 1 & 1 & \lambda^{n_3} \\
\lambda^{n_3} & \lambda^{n_3} & \lambda^{n_3} & \lambda^{n_3} & 1
\end{array}
\right).
\label{NLd2}\end{aligned}$$ In this way, we find that [*an existence of a large mixing between $d'_{{\rm L}4}$ $(d'_{{\rm R}4})$ and other $d'_{{\rm L}}$s $(d'_{{\rm R}})$s, i.e., $\left({N_{\rm L}^d}^{\dagger}\right)_{I4} = O(1)$ and $\left(N_{\rm R}^d\right)_{4J} = O(1)$, is necessary to generate $y^{(D)}_{IJ} = O(1)$.*]{}
Next, we examine the case that a magnitude of both 4th and 5th terms in $y_{IJ}^{(U)}$ and $y^{(u_{\rm m})}$ can be $O(1)$. In this case, from $\left({N_{\rm L}^u}^{\dagger}\right)_{44} = O(1)$, $\left(N_{\rm R}^u\right)_{44} = O(1)$, $\left(N_{\rm R}^u\right)_{55} = O(1)$, and $\left({N_{\rm L}^u}^{\dagger}\right)_{55} = O(1)$, we obtain $\displaystyle{\left(N_{\rm R}^u\right)_{5J} = O\left(\lambda^{n_1}\right)}$, $\displaystyle{\left({N_{\rm L}^u}^{\dagger}\right)_{I5} = O\left(\lambda^{n_2}\right)}$, $\displaystyle{\left({N_{\rm L}^u}^{\dagger}\right)_{54} = O\left(\lambda^{n_1}\right)}$, and $\displaystyle{\left(N_{\rm R}^u\right)_{45} = O\left(\lambda^{n_2}\right)}$,respectively. From the conditions $K_{I5}^{(U_{\rm L})}
= \left({N_{\rm L}^u}^{\dagger}\right)_{IA}
\left(N_{\rm L}^u\right)_{A5} = 0$, $K_{5J}^{(U_{\rm L})} = \left({N_{\rm L}^u}^{\dagger}\right)_{5A}
\left(N_{\rm L}^u\right)_{AJ} = 0$, $K_{I5}^{(U_{\rm R})}
= \left({N_{\rm R}^u}^{\dagger}\right)_{IA}
\left(N_{\rm R}^u\right)_{A5} = 0$, and $K_{5J}^{(U_{\rm R})} = \left({N_{\rm R}^u}^{\dagger}\right)_{5A}
\left(N_{\rm R}^u\right)_{AJ} = 0$, we need $n_1 = n_2$, $\displaystyle{\left({N_{\rm L}^u}^{\dagger}\right)_{5j} = O\left(\lambda^{n_1}\right)}$, and $\displaystyle{\left({N_{\rm R}^u}\right)_{i5} = O\left(\lambda^{n_2}\right)}$. Then, we obtain the transformation matrices such as $$\begin{aligned}
{N_{\rm L}^u},~~
N_{\rm R}^u
= \left(
\begin{array}{ccccc}
1 & \star & \star & \star & \lambda^{n_1} \\
\star & 1 & \star & \star & \lambda^{n_1} \\
\star & \star & 1 & \star & \lambda^{n_1} \\
1 & 1 & 1 & 1 & \lambda^{n_1} \\
\lambda^{n_1} & \lambda^{n_1} & \lambda^{n_1} & \lambda^{n_1} & 1
\end{array}
\right).
\label{NLu}\end{aligned}$$
For large masses concerning extra quarks, we derive the relations: $$\begin{aligned}
&~& M_{I5}^{(U)}
= \left\{\lambda^7 \left({N_{\rm L}^u}^{\dagger}\right)_{I1} \left(N_{\rm R}^u\right)_{15}
+ \lambda^4 \left({N_{\rm L}^u}^{\dagger}\right)_{I2} \left(N_{\rm R}^u\right)_{25}
+ \lambda^0 \left({N_{\rm L}^u}^{\dagger}\right)_{I3} \left(N_{\rm R}^u\right)_{35}\right.
\nonumber \\
&~& ~~~~~~~~~~~~~~~
\left. + \lambda^{-n_1} \left({N_{\rm L}^u}^{\dagger}\right)_{I4} \left(N_{\rm R}^u\right)_{55}
+ \lambda^{-n_1} \left({N_{\rm L}^u}^{\dagger}\right)_{I5} \left(N_{\rm R}^u\right)_{45}\right\}
\frac{v}{\sqrt{2}},
\label{MUI5}\\
&~& M_{I5}^{(D)}
= \left\{\lambda^7 \left({N_{\rm L}^d}^{\dagger}\right)_{I1} \left(N_{\rm R}^d\right)_{15}
+ \lambda^5 \left({N_{\rm L}^d}^{\dagger}\right)_{I2} \left(N_{\rm R}^d\right)_{25}
+ \lambda^3 \left({N_{\rm L}^d}^{\dagger}\right)_{I3} \left(N_{\rm R}^d\right)_{35}\right.
\nonumber \\
&~& ~~~~~~~~~~~~~~~
\left. + \lambda^{-n_3} \left({N_{\rm L}^d}^{\dagger}\right)_{I4} \left(N_{\rm R}^d\right)_{55}
+ \lambda^{-n_3} \left({N_{\rm L}^d}^{\dagger}\right)_{I5} \left(N_{\rm R}^d\right)_{45}\right\}
\frac{v}{\sqrt{2}}.
\label{MDI5}\end{aligned}$$ From the fact that the 4th terms in the right hand side of Eqs. (\[MUI5\]) and (\[MDI5\]) dominate over others and the consequence of SU(2)$_{\rm L}$ symmetry, i. e., $M_{I5}^{(U)} = M_{I5}^{(D)} = m_I^{(q_{\rm m})}$, we obtain $n_1 = n_3$, and then the following relation holds $$\begin{aligned}
m_1^{(U)}
\left({N_{\rm L}^u}^{\dagger}\right)_{I4} \left(N_{\rm R}^u\right)_{55}
= m_1^{(D)}\left({N_{\rm L}^d}^{\dagger}\right)_{I4} \left(N_{\rm R}^d\right)_{55},
\label{MU=MD}\end{aligned}$$ up to $\displaystyle{O\left(\lambda^{n_1} v/\sqrt{2}\right)}$.
When taken together, we have arrived at the transformation matrices to realize the conjecture (\[y=O(1)\]), such as $$\begin{aligned}
{N_{\rm L}^u},~~ {N_{\rm L}^d},~~ N_{\rm R}^u,~~ N_{\rm R}^d
= \left(
\begin{array}{ccccc}
1 & \star & \star & \star & \lambda^n \\
\star & 1 & \star & \star & \lambda^n \\
\star & \star & 1 & \star & \lambda^n \\
1 & 1 & 1 & 1 & \lambda^n \\
\lambda^n & \lambda^n & \lambda^n & \lambda^n & 1
\end{array}
\right),
\label{Ns}\end{aligned}$$ under the parametrization (\[M-diag-rep\]) with $n = n_1 = n_2 = n_3 = n_4$. Inserting Eq. (\[Ns\]) into Eq. (\[K(UL)\]), we see that the magnitude of $k_{IJ}^{(q)}$, $k_{IJ}^{(u)}$, $k_{IJ}^{(d)}$, $k^{(q_{\rm m})}$, $k^{(u_{\rm m})}$, and $k^{(d_{\rm m})}$ can become $O(1)$.
Finally, we study whether the Yukawa couplings can be unified or not. As in the SM, we have the following no-go theorem.\
$[$Theorem$]$ [*Under the kinetic unification such as $k_{IJ}^{(q)} = k_{IJ}^{(u)} = k_{IJ}^{(d)}$ and $k^{(q_{\rm m})} = k^{(u_{\rm m})} = k^{(d_{\rm m})}$ and the mass unification such as $m_J^{(u_{\rm m})} = m_J^{(d_{\rm m})}$, the exact Yukawa coupling unification such as $y^{(U)}_{IJ} = y^{(D)}_{IJ}$ does not occur in the extension of the SM, described by ${\mathscr{L}'}_{\rm SM+\alpha}^{\rm quark}$*]{}. Here, we consider the case that the kinetic unification and the mass unification are spoiled. In this case, from Eqs. (\[yuIJ\]) and (\[ydIJ\]), we find that $y^{(U)}_{IJ} = y^{(D)}_{IJ}$ is realized with the relation: $$\begin{aligned}
&~& m_u \left({N_{\rm L}^u}^{\dagger}\right)_{I1} \left(N_{\rm R}^u\right)_{1J}
+ m_c \left({N_{\rm L}^u}^{\dagger}\right)_{I2} \left(N_{\rm R}^u\right)_{2J}
+ m_t \left({N_{\rm L}^u}^{\dagger}\right)_{I3} \left(N_{\rm R}^u\right)_{3J}
\nonumber \\
&~& ~~~~~~~ + m_1^{(U)} \left({N_{\rm L}^u}^{\dagger}\right)_{I4} \left(N_{\rm R}^u\right)_{5J}
+ m_2^{(U)} \left({N_{\rm L}^u}^{\dagger}\right)_{I5} \left(N_{\rm R}^u\right)_{4J}
\nonumber \\
&~& = m_d \left({N_{\rm L}^d}^{\dagger}\right)_{I1} \left(N_{\rm R}^d\right)_{1J}
+ m_s \left({N_{\rm L}^d}^{\dagger}\right)_{I2} \left(N_{\rm R}^d\right)_{2J}
+ m_b \left({N_{\rm L}^d}^{\dagger}\right)_{I3} \left(N_{\rm R}^d\right)_{3J}
\nonumber \\
&~& ~~~~~~~ + m_1^{(D)} \left({N_{\rm L}^d}^{\dagger}\right)_{I4} \left(N_{\rm R}^d\right)_{5J}
+ m_2^{(D)} \left({N_{\rm L}^d}^{\dagger}\right)_{I5} \left(N_{\rm R}^d\right)_{4J},
\label{yU=yD}\end{aligned}$$ where mass parameters are replaced with values at the unification scale. Here, we use quark masses defined by $m_u = y_u v/\sqrt{2}$, $m_c = y_c v/\sqrt{2}$, $m_t = y_t v/\sqrt{2}$, $m_d = y_d v/\sqrt{2}$, $m_s = y_s v/\sqrt{2}$, and $m_b = y_b v/\sqrt{2}$. In general, there appear breaking terms that contribute to the Yukawa coupling matrices, on the breakdown of a higher gauge symmetry. Then, the relation (\[yU=yD\]) is modified and their effects should be considered in a model-building.
Our results are summarized as follows. There is a possibility that a magnitude of elements of Yukawa coupling matrices is $O(1)$ and the Yukawa couplings are unified at some high-energy scale, if transformation matrices take a particular form such as Eq. (\[Ns\]).
Speculations
------------
We examine features on Yukawa coupling matrices and transformation matrices inferred from our conjectures.
First, we point out that our matrices possess different features from ordinary ones in the following points. One is that a small mixing of quark flavors between the weak and the mass eigenstates can occur as a result of a cancellation between a large mixing in $N_{\rm L}^u$ and $N_{\rm L}^d$. In our setup, the flavor mixing is defined by the matrix: $$\begin{aligned}
N_{\rm mix} = \left({N_{\rm L}^{u}}^{\dagger}\right)^{-1} {N_{\rm L}^{d}}^{\dagger},
\label{Nmix}\end{aligned}$$ and $N_{\rm mix}$ contains $V_{\rm KM}$ as the sub-matrix $\displaystyle{\left(N_{\rm mix}\right)_{ij}}$ ($i, j = 1, 2, 3$).
For reference, we explain the difference between consequences from a small and a large mixing in transformation matrices, based on the SM. In an ordinary case, from Eqs. (\[VKM\]) and (\[VKM-lambda\]), transformation matrices are also assumed to be the same form with a small mixing as $V_{\rm KM}$, $$\begin{aligned}
{V_{\rm L}^{(u)}}^{\dagger},~~ {V_{\rm L}^{(d)}}^{\dagger},~~
V_{\rm R}^{(u)},~~ V_{\rm R}^{(d)} =
\left(
\begin{array}{ccc}
1 & \lambda & \lambda^3 \\
\lambda & 1 & \lambda^2 \\
\lambda^3 & \lambda^2 & 1
\end{array}
\right).
\label{VL-para}\end{aligned}$$ In this case, using Eqs. (\[VKM\]), (\[VL-para\]), and the relations: $$\begin{aligned}
y^{(u)} = {V_{\rm L}^{(u)}}^{\dagger} y_{\rm diag}^{(u)} V_{\rm R}^{(u)},~~
y^{(d)} = {V_{\rm L}^{(d)}}^{\dagger} y_{\rm diag}^{(d)} V_{\rm R}^{(d)}
= {V_{\rm L}^{(u)}}^{\dagger} V_{\rm KM} y_{\rm diag}^{(d)} V_{\rm R}^{(d)},
\label{yuyd-SM}\end{aligned}$$ we find that the Yukawa coupling matrices take the forms: $$\begin{aligned}
y^{(u)} = \left(
\begin{array}{ccc}
\lambda^6 & \lambda^5 & \lambda^3 \\
\lambda^5 & \lambda^4 & \lambda^2 \\
\lambda^3 & \lambda^2 & 1
\end{array}
\right),~~
y^{(d)} = \left(
\begin{array}{ccc}
\lambda^7 & \lambda^6 & \lambda^6 \\
\lambda^6 & \lambda^5 & \lambda^5 \\
\lambda^6 & \lambda^5 & \lambda^3
\end{array}
\right).
\label{yuyd-lambda}\end{aligned}$$ They are often used in the Froggatt-Nielsen mechanism [@FN]. Using $y_u$, $y_c$, $y_t$, $y_d$, $y_s$, and $y_b$, $y^{(u)}$ and $y^{(d)}$ are also expressed as $$\begin{aligned}
&~&y^{(u)} = y_u \left(
\begin{array}{ccc}
1 & \lambda & \lambda^3 \\
\lambda & \lambda^2 & \lambda^4 \\
\lambda^3 & \lambda^4 & \lambda^6
\end{array}
\right)
+ y_c\left(
\begin{array}{ccc}
\lambda^2 & \lambda & \lambda^3 \\
\lambda & 1 & \lambda^2 \\
\lambda^3 & \lambda^2 & \lambda^4
\end{array}
\right)
+ y_t\left(
\begin{array}{ccc}
\lambda^6 & \lambda^5 & \lambda^3 \\
\lambda^5 & \lambda^4 & \lambda^2 \\
\lambda^3 & \lambda^2 & 1
\end{array}
\right),
\label{yu-lambda}\\
&~& y^{(d)} = y_d \left(
\begin{array}{ccc}
1 & \lambda & \lambda^3 \\
\lambda & \lambda^2 & \lambda^4 \\
\lambda^3 & \lambda^4 & \lambda^6
\end{array}
\right)
+ y_s\left(
\begin{array}{ccc}
\lambda^2 & \lambda & \lambda^3 \\
\lambda & 1 & \lambda^2 \\
\lambda^3 & \lambda^2 & \lambda^4
\end{array}
\right)
+ y_b \left(
\begin{array}{ccc}
\lambda^6 & \lambda^5 & \lambda^3 \\
\lambda^5 & \lambda^4 & \lambda^2 \\
\lambda^3 & \lambda^2 & 1
\end{array}
\right).
\label{yd-lambda}\end{aligned}$$
On the other hand, in the case with a large mixing among up-type quarks given by $$\begin{aligned}
{V_{\rm L}^{(u)}}^{\dagger},~~ V_{\rm R}^{(u)} =
\left(
\begin{array}{ccc}
1 & \star & 1 \\
\star & 1 & 1 \\
1 & 1 & 1
\end{array}
\right),
\label{VL-para-ours}\end{aligned}$$ the up-type Yukawa coupling matrix takes the form: $$\begin{aligned}
y^{(u)} = y_u \left(
\begin{array}{ccc}
1 & \lambda & 1 \\
\lambda & \lambda^2 & \lambda \\
1 & \lambda & 1
\end{array}
\right)
+ y_c\left(
\begin{array}{ccc}
\lambda^2 & \lambda & \lambda \\
\lambda & 1 & 1 \\
\lambda & 1 & 1
\end{array}
\right)
+ y_t\left(
\begin{array}{ccc}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{array}
\right),
\label{yu-lambda-ours}\end{aligned}$$ where we take $\star = \lambda$ in (\[VL-para-ours\]). In this case, a small mixing in $V_{\rm KM}$ originates from a cancellation between a large mixing in ${V_{\rm L}^{(u)}}$ and $V_{\rm L}^{(d)}$
The other is that small Yukawa couplings in the SM have a high-sensibility for a small variation of matrix elements, and it suggests that the quark mass hierarchy occurs as a consequence of a fine tuning among large parameters. This feature is, in general, different from that achieved from the top-down approach, and lies the other end of that obtained by a ‘stability’ principle [@St; @DR]. Here, the stability principle means that a tiny dimensionless parameter should not be sensitive to the change of matrix elements including it. In our model, the Yukawa coupling matrices are expressed by $$\begin{aligned}
y_{IJ}^{(U)} = y_{u_i} \left(T_i^{(u)}\right)_{IJ}
+ \xi_a^{(U)} \left(T_a^{(U)}\right)_{IJ},~~
y_{IJ}^{(D)} = y_{d_i} \left(T_i^{(d)}\right)_{IJ}
+ \xi_a^{(D)} \left(T_a^{(D)}\right)_{IJ},
\label{yudIJ}\end{aligned}$$ where $y_{u_i} = (y_u, y_c, y_t)$, $y_{d_i} = (y_d, y_s, y_b)$, $\xi_a^{(U)} = \sqrt{2} m_a^{(U)}/v$, $\xi_a^{(D)} = \sqrt{2} m_a^{(D)}/v$ ($a = 1, 2$) are dimensionless parameters, and $T_i^{(u)}$, $T_a^{(U)}$, $T_i^{(d)}$, and $T_a^{(D)}$ are $4 \times 4$ complex matrices given by $$\begin{aligned}
\hspace{-1.3cm}&~& \left(T_i^{(u)}\right)_{IJ} = \left({N_{\rm L}^u}^{\dagger}\right)_{Ii}
\left(N_{\rm R}^u\right)_{iJ},~~
\left(T_1^{(U)}\right)_{IJ} = \left({N_{\rm L}^u}^{\dagger}\right)_{I4}
\left(N_{\rm R}^u\right)_{5J},~~
\left(T_2^{(U)}\right)_{IJ} =
\left({N_{\rm L}^u}^{\dagger}\right)_{I5} \left(N_{\rm R}^u\right)_{4J},
\label{Tu}\\
\hspace{-1.3cm}&~& \left(T_i^{(d)}\right)_{IJ} = \left({N_{\rm L}^d}^{\dagger}\right)_{Ii}
\left(N_{\rm R}^d\right)_{iJ},~~
\left(T_1^{(D)}\right)_{IJ} = \left({N_{\rm L}^d}^{\dagger}\right)_{I4}
\left(N_{\rm R}^d\right)_{5J},~~
\left(T_2^{(D)}\right)_{IJ} =
\left({N_{\rm L}^d}^{\dagger}\right)_{I5} \left(N_{\rm R}^d\right)_{4J}.
\label{Td}\end{aligned}$$ The conditions that $y_{u_i}$ are stable under a change of the $(I, J)$ element are given by $$\begin{aligned}
\left|\frac{\delta y_{u_i}}{y_{u_i}}\right|
\lesssim \left|\frac{\delta y_{IJ}^{(U)}}{y_{IJ}^{(U)}}\right|,~~
\left|\frac{\delta y_{d_i}}{y_{d_i}}\right|
\lesssim \left|\frac{\delta y_{IJ}^{(D)}}{y_{IJ}^{(D)}}\right|,~~
({\rm no~~\!summation~~\!on}~~i,I,J)
\label{yudIJ-st}\end{aligned}$$ for $\delta y_{IJ}^{(U)} \ll y_{IJ}^{(U)}$ and $\delta y_{IJ}^{(D)} \ll y_{IJ}^{(D)}$. In other words, the sensibility of $y_{u_i}$ in $y_{IJ}^{(U)}$ and $y_{d_i}$ in $y_{IJ}^{(D)}$ is defined by $$\begin{aligned}
\left(\varDelta_{y_{u_i}}\right)_{IJ} \equiv
\frac{\left|{\delta y_{u_i}}/{y_{u_i}}\right|}
{\left|{\delta y_{IJ}^{(U)}}/{y_{IJ}^{(U)}}\right|},~~
\left(\varDelta_{y_{d_i}}\right)_{IJ} \equiv
\frac{\left|{\delta y_{d_i}}/{y_{d_i}}\right|}
{\left|{\delta y_{IJ}^{(D)}}/{y_{IJ}^{(D)}}\right|},~~ ({\rm no~~\!summation~~\!on}~~i,I,J)
\label{yu-sen}\end{aligned}$$ respectively, and then $\left(\varDelta_{y_{u_i}}\right)_{IJ} \le O(1)$ and $\left(\varDelta_{y_{d_i}}\right)_{IJ} \le O(1)$ are required from the stability condition. If $\left(\varDelta_{y_u}\right)_{IJ} \gg O(1)$, $y_u$ has a high-sensibility under the change of other parameters.
Using the Yukawa coupling matrices (\[yudIJ\]), we derive the relations: $$\begin{aligned}
&~& \frac{\delta y_{IJ}^{(U)}}{y_{IJ}^{(U)}}
= \frac{\delta y_{u_i}}{y_{u_i}} \frac{y_{u_i}\left(T_i^{(u)}\right)_{IJ}}{y_{IJ}^{(U)}}
+ \frac{\delta \xi_a^{(U)}}{\xi_a^{(U)}}
\frac{\xi_a^{(U)}\left(T_a^{(U)}\right)_{IJ}}{{y_{IJ}^{(U)}}},~~
\label{delta-yu}\\
&~& \frac{\delta y_{IJ}^{(D)}}{y_{IJ}^{(D)}}
= \frac{\delta y_{d_i}}{y_{d_i}} \frac{y_{d_i}\left(T_i^{(d)}\right)_{IJ}}{y_{IJ}^{(D)}}
+ \frac{\delta \xi_a^{(D)}}{\xi_a^{(D)}}
\frac{\xi_a^{(D)}\left(T_a^{(D)}\right)_{IJ}}{{y_{IJ}^{(D)}}},
\label{delta-yd}\end{aligned}$$ where no summations on $I$ and $J$ are done. Using the relations (\[yudIJ-st\]), (\[delta-yu\]), and (\[delta-yd\]), we obtain the conditions: $$\begin{aligned}
\left|y_{IJ}^{(U)}\right| \lesssim \left|y_{u_i}\left(T_i^{(u)}\right)_{IJ}\right|,~~
\left|y_{IJ}^{(D)}\right| \lesssim \left|y_{d_i}\left(T_i^{(d)}\right)_{IJ}\right|,
~~ ({\rm no~~\!summation~~\!on}~~i).
\label{|y|}\end{aligned}$$ To fulfill the conditions (\[|y|\]) for $y_u$ and $y_d$, the following inequalities are needed, $$\begin{aligned}
&~& \left(T_2^{(u)}\right)_{IJ} \le O\left(\lambda^3\right),~~
\left(T_3^{(u)}\right)_{IJ} \le O\left(\lambda^7\right),~~
\left(T_a^{(U)}\right)_{IJ} \le O\left(\lambda^{7+n}\right),
\label{Tu-cond}\\
&~& \left(T_2^{(d)}\right)_{IJ} \le O\left(\lambda^2\right),~~
\left(T_3^{(d)}\right)_{IJ} \le O\left(\lambda^4\right),~~
\left(T_a^{(D)}\right)_{IJ} \le O\left(\lambda^{7+n}\right),
\label{Td-cond}\end{aligned}$$ for the $(I, J)$ element with $\left(T_1^{(u)}\right)_{IJ}= O(1)$ and $\left(T_1^{(d)}\right)_{IJ}= O(1)$. In this way, we conjecture that the Yukawa couplings associated with a large mixing hardly satisfy the stability condition.
Let us illustrate this feature in the SM. Using Eq. (\[yu-lambda-ours\]), for the entries $(i, j) = (1, 1)$, $(1, 3)$, $(3, 1)$, $(3, 3)$, the sensibility of $y_{u}$ in $y_{ij}^{(u)}$ is roughly evaluated as $$\begin{aligned}
\frac{\left|{\delta y_{u}}/{y_{u}}\right|}
{\left|{\delta y_{ij}^{(u)}}/{y_{ij}^{(u)}}\right|}
= \frac{\left|{y_{ij}^{(u)}}\right|}{\left|y_u \left(T_{1}^{(u)}\right)_{ij}\right|}
= \frac{\left|y_u \left(T_{1}^{(u)}\right)_{ij}
+ y_c \left(T_{1}^{(c)}\right)_{ij}
+ y_t \left(T_{1}^{(t)}\right)_{ij}\right|}{\left|y_u \left(T_{1}^{(u)}\right)_{ij}\right|}
\cong \frac{y_t}{y_u} = O\left(\lambda^{-7}\right),
\label{yu-sen-ours}\end{aligned}$$ where no summations on $i$ and $j$ are done. Eq. (\[yu-sen-ours\]) shows that $y_u$ is highly sensitive to the change of other parameters.
In contrast, in the ordinary case, using Eq. (\[yu-lambda\]), for the entry $(i, j) = (1, 1)$, the sensibility of $y_{u}$ in $y_{ij}^{(u)}$ is roughly evaluated as $$\begin{aligned}
\frac{\left|{\delta y_{u}}/{y_{u}}\right|}
{\left|{\delta y_{11}^{(u)}}/{y_{11}^{(u)}}\right|}
\cong \frac{y_u + \lambda^2 y_c + \lambda^6 y_t}{y_u} = O\left(\lambda^{-1}\right),
\label{yu-sen-ordinary}\end{aligned}$$ and then the sensibility becomes much milder than that in Eq. (\[yu-sen-ours\]).
For the sake of completeness, we give an example to satisfy the condition $\left(\varDelta_{y_{u}}\right)_{ij} \le O(1)$. If we take the transformation matrices: $$\begin{aligned}
{V_{\rm L}^{(u)}}^{\dagger} =
\left(
\begin{array}{ccc}
1 & \lambda & \lambda^3 \\
\lambda & 1 & \lambda^2 \\
\lambda^3 & \lambda^2 & 1
\end{array}
\right),~~
V_{\rm R}^{(u)} =
\left(
\begin{array}{ccc}
1 & \lambda^p & \lambda^r \\
\lambda^p & 1 & \lambda^q \\
\lambda^r & \lambda^q & 1
\end{array}
\right),
\label{VLVRu-para}\end{aligned}$$ $y^{(u)}$ becomes as $$\begin{aligned}
y^{(u)} = y_u \left(
\begin{array}{ccc}
1 & \lambda^p & \lambda^r \\
\lambda & \lambda^{p+1} & \lambda^{r+1} \\
\lambda^3 & \lambda^{p+3} & \lambda^{r+3}
\end{array}
\right)
+ y_c\left(
\begin{array}{ccc}
\lambda^{p+1} & \lambda & \lambda^{q+1} \\
\lambda^p & 1 & \lambda^q \\
\lambda^{p+2} & \lambda^{2} & \lambda^{q+2}
\end{array}
\right)
+ y_t\left(
\begin{array}{ccc}
\lambda^{r+3} & \lambda^{q+3} & \lambda^{3} \\
\lambda^{r+2} & \lambda^{q+2} & \lambda^2 \\
\lambda^r & \lambda^q & 1
\end{array}
\right).
\label{yu-lambda-st}\end{aligned}$$ Then, we find that $\left(\varDelta_{y_{u}}\right)_{11} = O(1)$ for $p \ge 2$ and $r \ge 4$.
Finally, we give a speculation on the Yukawa coupling unification. After terms containing $m_u$, $m_c$, $m_d$, $m_s$, and $m_b$ are neglected, the relation (\[yU=yD\]) is written by $$\begin{aligned}
m_t \left(T_3^{(u)}\right)_{IJ}
+ m_1^{(U)} \left(T_1^{(U)}\right)_{IJ}
+ m_2^{(U)} \left(T_2^{(U)}\right)_{IJ}
= m_1^{(D)} \left(T_1^{(D)}\right)_{IJ}
+ m_2^{(D)} \left(T_2^{(D)}\right)_{IJ}.
\label{yU=yD2}\end{aligned}$$ For simplicity, we consider a case with $\left(T_1^{(U)}\right)_{IJ}=0$, $\left(T_2^{(U)}\right)_{IJ} = 0$, and $\left(T_2^{(D)}\right)_{IJ} = 0$. In this case, the relation reduces to $$\begin{aligned}
m_t \left({N_{\rm L}^u}^{\dagger}\right)_{I3} \left(N_{\rm R}^u\right)_{3J}
= m_1^{(D)} \left({N_{\rm L}^d}^{\dagger}\right)_{I4} \left(N_{\rm R}^d\right)_{5J},
\label{yU=yD-1}\end{aligned}$$ and this is realized by transformation matrices which satisfy $$\begin{aligned}
\left({N_{\rm L}^u}^{\dagger}\right)_{I3}
=\left({N_{\rm L}^d}^{\dagger}\right)_{I4},~~
m_t \left(N_{\rm R}^u\right)_{3J}
= m_1^{(D)} \left(N_{\rm R}^d\right)_{5J}.
\label{yU=yD-case1-sol}\end{aligned}$$ They show that $d'_{{\rm L}4}$ and $\left(d'_{\rm L(m)}\right)^{\rm c}$ in the down-type quark sector play a role of $u'_{{\rm L}3}$ and $u'_{{\rm R}3}$ in the up-type one, respectively.
In a case with $\left(T_1^{(U)}\right)_{IJ}=0$, $\left(T_2^{(U)}\right)_{IJ} = 0$, and $\left(T_1^{(D)}\right)_{IJ} = 0$, the relation reduces to $$\begin{aligned}
m_t \left({N_{\rm L}^u}^{\dagger}\right)_{I3} \left(N_{\rm R}^u\right)_{3J}
= m_2^{(D)} \left({N_{\rm L}^d}^{\dagger}\right)_{I5} \left(N_{\rm R}^d\right)_{4J}.
\label{yU=yD-2}\end{aligned}$$ and this is realized by transformation matrices which satisfy $$\begin{aligned}
m_t \left({N_{\rm L}^u}^{\dagger}\right)_{I3}
= m_2^{(D)} \left({N_{\rm L}^d}^{\dagger}\right)_{I5},~~
\left(N_{\rm R}^u\right)_{3J}
= \left(N_{\rm R}^d\right)_{4J}.
\label{yU=yD-2-sol}\end{aligned}$$ They show that $\left(d'_{\rm R(m)}\right)^{\rm c}$ and $d'_{{\rm R}4}$ in the down-type quark sector play a role of $u'_{{\rm L}3}$ and $u'_{{\rm R}3}$ in the up-type one, respectively.
Conclusions and discussions
===========================
We have studied an origin of fermion mass hierarchy based on an extension of the SM with vector-like fermions, using a bottom-up approach. It is shown that the magnitude of elements of Yukawa coupling matrices can be sizable of $O(1)$ and the Yukawa coupling unification can be realized at a high-energy scale, if vector-like fermions mix with three families. Through study on the extension with partial multiplets of vector-like fermions (see Appendix A), we have a no-go theorem such that [*the magnitude of elements of down-type Yukawa coupling matrices cannot go beyond $\displaystyle{O\left(10^{-2}\right)}$, in the extension of the SM with vector-like up-type quarks alone, if the magnitude of kinetic coefficients is at most $O(1)$*]{}.
Our results are obtained, independent of a theory beyond our setup, and hence they could hold in various models. In fact, large masses of extra fermions are free parameters, and it is difficult to determine their magnitude from theoretical considerations alone. Vector-like fermions including down-type ones can exist at a terascale as remnants of unification and supersymmetry [@YK3]. Conversely, if vector-like fermions are discovered and their masses are precisely measured, they would provide useful information about our setup and take hints for HETs.
We explain preceding works on the fermion masses based on models with vector-like fermions. Vector-like fermions are used to generate small quark masses through a see-saw type mechanism [@B; @CM; @DW; @R]. The flavor structure has been studied in unified theories with vector-like fermions [@B2; @BB; @BB2; @M; @MYE6; @BBT; @DFM; @BC]. It would be meaningful to reexamine various models with our conclusions in mind. For a realistic model-building based on a grand unification, we must consider leptons as well as quarks. Our method is also applicable to the lepton sector and models with several Higgs doublets.
In our case with a large mixing in transformation matrices, Yukawa coupling matrices contain tiny parameters such as $y_u$ and $y_d$ possessing a high-sensibility for a small variation of matrix elements, and hence the quark mass hierarchy can occur as a consequence of accidental cancellations among large parameters. Because this feature is different from that of the top-down approach, and lies the other end of that by the stability principle, it seems to be unnatural. However, if it becomes evident that a small flavor mixing in $V_{\rm KM}$ stems from a cancellation of large mixing angles among various fermions, it must surge one to reconsider implications of fine tuning or naturalness, in connection with a naturalness problem relating to the Higgs boson mass.
Finally, we point out a limitation of our approach and problems left behind. From the bottom up, it is possible to show a possible existence (if not an existence proof) and to offer some suggestions for HETs, but it is difficult to specify the structure of HETs from our findings alone. It would be a next task to answer the following questions by exploiting to various methods. Are transformation matrices with a large mixing realistic? Is there any circumstantial evidence to support them? Or is there a model, theory or mechanism to realize them?
At present, a theory beyond the SM has not been yet known, and hence it is worth pursuing every possibility including a large mixing or a fine tuning. Then, our approach would be useful as a complimentary one to solve riddles in the SM.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author thanks Prof. T. Yamashita for valuable discussions. This work was supported in part by scientific grants from the Ministry of Education, Culture, Sports, Science and Technology under Grant No. 17K05413.
Cases with partial multiplets
=============================
Case with $q'_{{\rm L}4}$, $q'_{{\rm L(m)}}$, $u'_{{\rm R}4}$, and $u'_{{\rm R(m)}}$
------------------------------------------------------------------------------------
In the absence of $d'_{{\rm R}4}$ and $d'_{{\rm R(m)}}$, the quark kinetic coefficients are written by $$\begin{aligned}
&~& K^{(U_{\rm L})} =
\left(
\begin{array}{cc}
k_{IJ}^{(q)} & 0 \\
0 & k^{(u_{\rm m})}
\end{array}
\right) = {N_{\rm L}^{u}}^{\dagger}N_{\rm L}^{u},~~
K^{(D_{\rm L})} = k_{IJ}^{(q)}
= {N_{\rm L}^{d}}^{\dagger}N_{\rm L}^{d},~~
\nonumber \\
&~& K^{(U_{\rm R})} =
\left(
\begin{array}{cc}
k_{IJ}^{(u)} & 0 \\
0 & k^{(q_{\rm m})}
\end{array}
\right) = {N_{\rm R}^{u}}^{\dagger}N_{\rm R}^{u},~~
K^{(D_{\rm R})} =
\left(
\begin{array}{cc}
k_{ij}^{(d)} & 0 \\
0 & k^{(q_{\rm m})}
\end{array}
\right) = {N_{\rm R}^{d}}^{\dagger}N_{\rm R}^{d},
\label{K(UL)-case1}\end{aligned}$$ where $N_{\rm L}^{u}$ and $N_{\rm R}^{u}$ are $5 \times 5$ complex matrices, and $N_{\rm L}^{d}$ and $N_{\rm R}^{d}$ are $4 \times 4$ complex matrices.
Relations of mass matrices are written by $$\begin{aligned}
&~&
\left(
\begin{array}{cc}
y_{IJ}^{(U)} \langle \phi^{0*} \rangle & m_I^{(q_{\rm m})} \\
m_J^{(u_{\rm m})} & y^{(u_{\rm m})} \langle \phi^{0} \rangle
\end{array}
\right)
= {N_{\rm L}^{u}}^{\dagger}
\left(
\begin{array}{ccccc}
m_u & 0 & 0 & 0 & 0 \\
0 & m_c & 0 & 0 & 0 \\
0 & 0 & m_t & 0 & 0 \\
0 & 0 & 0 & 0 & m_1^{(U)} \\
0 & 0 & 0 & m_2^{(U)} & 0
\end{array}
\right) N_{\rm R}^{u},
\label{M(U)-case1}\\
&~&
\left(
\begin{array}{cc}
y_{Ij}^{(D)} \langle \phi^{0} \rangle & m_I^{(q_{\rm m})}
\end{array}
\right)
= {N_{\rm L}^{d}}^{\dagger}
\left(
\begin{array}{cccc}
m_d & 0 & 0 & 0 \\
0 & m_s & 0 & 0 \\
0 & 0 & m_b & 0 \\
0 & 0 & 0 & m^{(D)}
\end{array}
\right) N_{\rm R}^{d},
\label{M(D)-case1}\end{aligned}$$ where the magnitude of $m_1^{(U)}$, $m_2^{(U)}$, and $m^{(D)}$ is given by $\displaystyle{O\left(\lambda^{-n} v/\sqrt{2}\right)}$. Then, we obtain the following transformation matrices which realize $\displaystyle{y_{IJ}^{(U)}, y_{Ij}^{(D)}, y^{(u_{\rm m})} =O(1)}$, $$\begin{aligned}
&~& {N_{\rm L}^u},~~ N_{\rm R}^u
= \left(
\begin{array}{ccccc}
1 & \star & \star & \star & \lambda^{n} \\
\star & 1 & \star & \star & \lambda^{n} \\
\star & \star & 1 & \star & \lambda^{n} \\
1 & 1 & 1 & 1 & \lambda^{n} \\
\lambda^n & \lambda^n & \lambda^n & \lambda^{n} & 1
\end{array}
\right),~~~
\label{Nu-case1}\\
&~& {N_{\rm L}^d}^{\dagger}
= \left(
\begin{array}{cccc}
1 & \star & \star & 1 \\
\star & 1 & \star & 1 \\
\star & \star & 1 & 1 \\
\star & \star & \star & 1
\end{array}
\right),~~~
N_{\rm R}^d
= \left(
\begin{array}{cccc}
1 & \star & \star & \lambda^n \\
\star & 1 & \star & \lambda^n \\
\star & \star & 1 & \lambda^n \\
\lambda^n & \lambda^n & \lambda^n & 1
\end{array}
\right),
\label{Nd-case1}\end{aligned}$$ using Eqs. (\[K(UL)-case1\]), (\[M(U)-case1\]), and (\[M(D)-case1\]).
The relation $\displaystyle{y_{Ij}^{(U)} = y_{Ij}^{(D)}}$ is realized if the following relation holds, $$\begin{aligned}
&~& m_t \left({N_{\rm L}^u}^{\dagger}\right)_{I3} \left(N_{\rm R}^u\right)_{3j}
+ m_1^{(U)} \left({N_{\rm L}^u}^{\dagger}\right)_{I4} \left(N_{\rm R}^u\right)_{5j}
+ m_2^{(U)} \left({N_{\rm L}^u}^{\dagger}\right)_{I5} \left(N_{\rm R}^u\right)_{4j}
\nonumber \\
&~& = m^{(D)} \left({N_{\rm L}^d}^{\dagger}\right)_{I4} \left(N_{\rm R}^d\right)_{4j},
\label{yU=yD-case1}\end{aligned}$$ where we neglect tiny contributions including $m_u$, $m_c$, $m_d$, $m_s$, and $m_b$.
Case with $q'_{{\rm L}4}$, $q'_{{\rm L(m)}}$, $d'_{{\rm R}4}$, and $d'_{{\rm R(m)}}$
------------------------------------------------------------------------------------
In the absence of $u'_{{\rm R}4}$ and $u'_{{\rm R(m)}}$, the quark kinetic coefficients are written by $$\begin{aligned}
&~& K^{(U_{\rm L})} = k_{IJ}^{(q)}
= {N_{\rm L}^{u}}^{\dagger}N_{\rm L}^{u},~~
K^{(D_{\rm L})} = \left(
\begin{array}{cc}
k_{IJ}^{(q)} & 0 \\
0 & k^{(d_{\rm m})}
\end{array}
\right)
= {N_{\rm L}^{d}}^{\dagger}N_{\rm L}^{d},~~
\nonumber \\
&~& K^{(U_{\rm R})} =
\left(
\begin{array}{cc}
k_{ij}^{(u)} & 0 \\
0 & k^{(q_{\rm m})}
\end{array}
\right) = {N_{\rm R}^{u}}^{\dagger}N_{\rm R}^{u},~~
K^{(D_{\rm R})} =
\left(
\begin{array}{cc}
k_{IJ}^{(d)} & 0 \\
0 & k^{(q_{\rm m})}
\end{array}
\right) = {N_{\rm R}^{d}}^{\dagger}N_{\rm R}^{d},
\label{K(UL)-case2}\end{aligned}$$ where $N_{\rm L}^{u}$ and $N_{\rm R}^{u}$ are $4 \times 4$ complex matrices, and $N_{\rm L}^{d}$ and $N_{\rm R}^{d}$ are $5 \times 5$ complex matrices.
Relations of mass matrices are written by $$\begin{aligned}
&~&
\left(
\begin{array}{cc}
y_{Ij}^{(U)} \langle \phi^{0*} \rangle & m_I^{(q_{\rm m})}
\end{array}
\right)
= {N_{\rm L}^{u}}^{\dagger}
\left(
\begin{array}{cccc}
m_u & 0 & 0 & 0 \\
0 & m_c & 0 & 0 \\
0 & 0 & m_t & 0 \\
0 & 0 & 0 & m^{(U)}
\end{array}
\right) N_{\rm R}^{u},
\label{M(U)-case2}\\
&~&
\left(
\begin{array}{cc}
y_{IJ}^{(D)} \langle \phi^{0} \rangle & m_I^{(q_{\rm m})} \\
m_J^{(d_{\rm m})} & y^{(d_{\rm m})} \langle \phi^{0*} \rangle
\end{array}
\right)
= {N_{\rm L}^{d}}^{\dagger}
\left(
\begin{array}{ccccc}
m_d & 0 & 0 & 0 & 0 \\
0 & m_s & 0 & 0 & 0 \\
0 & 0 & m_b & 0 & 0 \\
0 & 0 & 0 & 0 & m_1^{(D)} \\
0 & 0 & 0 & m_2^{(D)} & 0
\end{array}
\right) N_{\rm R}^{d},
\label{M(D)-case2}\end{aligned}$$ where the magnitude of $m^{(U)}$, $m_1^{(D)}$, and $m_2^{(D)}$ is given by $\displaystyle{O\left(\lambda^{-n} v/\sqrt{2}\right)}$. Then, we obtain the following transformation matrices which realize $\displaystyle{y_{Ij}^{(U)}, y_{IJ}^{(D)}, y^{(d_{\rm m})}=O(1)}$, $$\begin{aligned}
&~& {N_{\rm L}^u}^{\dagger}
= \left(
\begin{array}{cccc}
1 & \star & \star & 1 \\
\star & 1 & \star & 1 \\
\star & \star & 1 & 1 \\
\star & \star & \star & 1
\end{array}
\right),~~~
N_{\rm R}^u
= \left(
\begin{array}{cccc}
1 & \star & \star & \lambda^n \\
\star & 1 & \star & \lambda^n \\
\star & \star & 1 & \lambda^n \\
\lambda^n & \lambda^n & \lambda^n & 1
\end{array}
\right),
\label{Nu-case2}\\
&~& {N_{\rm L}^d},~~ N_{\rm R}^d
= \left(
\begin{array}{ccccc}
1 & \star & \star & \star & \lambda^{n} \\
\star & 1 & \star & \star & \lambda^{n} \\
\star & \star & 1 & \star & \lambda^{n} \\
1 & 1 & 1 & 1 & \lambda^{n} \\
\lambda^n & \lambda^n & \lambda^n & \lambda^{n} & 1
\end{array}
\right),
\label{Nd-case2}\end{aligned}$$ using Eqs. (\[K(UL)-case2\]), (\[M(U)-case2\]), and (\[M(D)-case2\]).
The relation $\displaystyle{y_{Ij}^{(U)} = y_{Ij}^{(D)}}$ is realized if the following relation holds, $$\begin{aligned}
&~& m_t \left({N_{\rm L}^u}^{\dagger}\right)_{I3} \left(N_{\rm R}^u\right)_{3j}
+ m^{(U)} \left({N_{\rm L}^u}^{\dagger}\right)_{I4} \left(N_{\rm R}^u\right)_{4j}
\nonumber \\
&~& = m_1^{(D)} \left({N_{\rm L}^d}^{\dagger}\right)_{I4} \left(N_{\rm R}^d\right)_{5j}
+ m_2^{(D)} \left({N_{\rm L}^d}^{\dagger}\right)_{I5} \left(N_{\rm R}^d\right)_{4j},
\label{yU=yD-case2}\end{aligned}$$ where we neglect tiny contributions including $m_u$, $m_c$, $m_d$, $m_s$, and $m_b$.
Case with $u'_{{\rm R}4}$, $u'_{{\rm R(m)}}$, $d'_{{\rm R}4}$, and $d'_{{\rm R(m)}}$
------------------------------------------------------------------------------------
In the absence of $q'_{{\rm L}4}$ and $q'_{{\rm L(m)}}$, the quark kinetic coefficients are written by $$\begin{aligned}
&~& K^{(U_{\rm L})} =
\left(
\begin{array}{cc}
k_{ij}^{(q)} & 0 \\
0 & k^{(u_{\rm m})}
\end{array}
\right) = {N_{\rm L}^{u}}^{\dagger}N_{\rm L}^{u},~~
K^{(D_{\rm L})} =
\left(
\begin{array}{cc}
k_{ij}^{(q)} & 0 \\
0 & k^{(d_{\rm m})}
\end{array}
\right)
= {N_{\rm L}^{d}}^{\dagger}N_{\rm L}^{d},~~
\nonumber \\
&~& K^{(U_{\rm R})} = k_{IJ}^{(u)}
= {N_{\rm R}^{u}}^{\dagger}N_{\rm R}^{u},~~
K^{(D_{\rm R})} = k_{IJ}^{(d)}
= {N_{\rm R}^{d}}^{\dagger}N_{\rm R}^{d},
\label{K(UL)-case3}\end{aligned}$$ where $N_{\rm L}^{u}$, $N_{\rm L}^{d}$, $N_{\rm R}^{u}$, and $N_{\rm R}^{d}$ are $4 \times 4$ complex matrices.
Relations of mass matrices are written by $$\begin{aligned}
&~&
\left(
\begin{array}{c}
y_{iJ}^{(U)} \langle \phi^{0*} \rangle \\
m_J^{(u_{\rm m})}
\end{array}
\right)
= {N_{\rm L}^{u}}^{\dagger}
\left(
\begin{array}{cccc}
m_u & 0 & 0 & 0 \\
0 & m_c & 0 & 0 \\
0 & 0 & m_t & 0 \\
0 & 0 & 0 & m^{(U)}
\end{array}
\right) N_{\rm R}^{u},
\label{M(U)-case3}\\
&~&
\left(
\begin{array}{c}
y_{iJ}^{(D)} \langle \phi^{0} \rangle \\
m_J^{(d_{\rm m})}
\end{array}
\right)
= {N_{\rm L}^{d}}^{\dagger}
\left(
\begin{array}{cccc}
m_d & 0 & 0 & 0 \\
0 & m_s & 0 & 0 \\
0 & 0 & m_b & 0 \\
0 & 0 & 0 & m^{(D)}
\end{array}
\right) N_{\rm R}^{d},
\label{M(D)-case3}\end{aligned}$$ where the magnitude of $m^{(U)}$ and $m^{(D)}$ is given by $\displaystyle{O\left(\lambda^{-n} v/\sqrt{2}\right)}$. Then, we obtain the following transformation matrices which realize $\displaystyle{y_{iJ}^{(U)}, y_{iJ}^{(D)} =O(1)}$, $$\begin{aligned}
{N_{\rm L}^u}^{\dagger},~~ {N_{\rm L}^d}^{\dagger}
= \left(
\begin{array}{cccc}
1 & \star & \star & \lambda^n \\
\star & 1 & \star & \lambda^n \\
\star & \star & 1 & \lambda^n \\
\lambda^n & \lambda^n & \lambda^n & 1
\end{array}
\right),~~~
N_{\rm R}^u,~~ N_{\rm R}^d
= \left(
\begin{array}{cccc}
1 & \star & \star & 1 \\
\star & 1 & \star & 1 \\
\star & \star & 1 & 1 \\
1 & 1 & 1 & 1
\end{array}
\right),
\label{Ns-case3}\end{aligned}$$ using (\[K(UL)-case3\]), (\[M(U)-case3\]), and (\[M(D)-case3\]).
The relation $\displaystyle{y_{iJ}^{(U)} = y_{iJ}^{(D)}}$ is realized if the following relation holds, $$\begin{aligned}
m_t \left({N_{\rm L}^u}^{\dagger}\right)_{i3} \left(N_{\rm R}^u\right)_{3J}
+ m^{(U)} \left({N_{\rm L}^u}^{\dagger}\right)_{i4} \left(N_{\rm R}^u\right)_{4J}
= m^{(D)} \left({N_{\rm L}^d}^{\dagger}\right)_{i4} \left(N_{\rm R}^d\right)_{4J},
\label{yU=yD-case3}\end{aligned}$$ where we neglect tiny contributions including $m_u$, $m_c$, $m_d$, $m_s$, and $m_b$.
Case with $q'_{{\rm L}4}$ and $q'_{{\rm L(m)}}$
-----------------------------------------------
In the absence of $u'_{{\rm R}4}$, $u'_{{\rm R(m)}}$, $d'_{{\rm R}4}$, and $d'_{{\rm R(m)}}$, the quark kinetic coefficients are written by $$\begin{aligned}
&~& K^{(U_{\rm L})} = k_{IJ}^{(q)}
= {N_{\rm L}^{u}}^{\dagger}N_{\rm L}^{u},~~
K^{(D_{\rm L})} = k_{IJ}^{(q)}
= {N_{\rm L}^{d}}^{\dagger}N_{\rm L}^{d},~~
\nonumber \\
&~& K^{(U_{\rm R})} =
\left(
\begin{array}{cc}
k_{ij}^{(u)} & 0 \\
0 & k^{(q_{\rm m})}
\end{array}
\right) = {N_{\rm R}^{u}}^{\dagger}N_{\rm R}^{u},~~
K^{(D_{\rm R})} =
\left(
\begin{array}{cc}
k_{ij}^{(d)} & 0 \\
0 & k^{(q_{\rm m})}
\end{array}
\right) = {N_{\rm R}^{d}}^{\dagger}N_{\rm R}^{d},
\label{K(UL)-case4}\end{aligned}$$ where $N_{\rm L}^{u}$, $N_{\rm L}^{d}$, $N_{\rm R}^{u}$, and $N_{\rm R}^{d}$ are $4 \times 4$ complex matrices.
Relations of mass matrices are written by $$\begin{aligned}
&~&
\left(
\begin{array}{cc}
y_{Ij}^{(U)} \langle \phi^{0*} \rangle & m_I^{(q_{\rm m})}
\end{array}
\right)
= {N_{\rm L}^{u}}^{\dagger}
\left(
\begin{array}{cccc}
m_u & 0 & 0 & 0 \\
0 & m_c & 0 & 0 \\
0 & 0 & m_t & 0 \\
0 & 0 & 0 & m^{(U)}
\end{array}
\right) N_{\rm R}^{u},
\label{M(U)-case4}\\
&~&
\left(
\begin{array}{cc}
y_{Ij}^{(D)} \langle \phi^{0} \rangle & m_I^{(q_{\rm m})}
\end{array}
\right)
= {N_{\rm L}^{d}}^{\dagger}
\left(
\begin{array}{cccc}
m_d & 0 & 0 & 0 \\
0 & m_s & 0 & 0 \\
0 & 0 & m_b & 0 \\
0 & 0 & 0 & m^{(D)}
\end{array}
\right) N_{\rm R}^{d},
\label{M(D)-case4}\end{aligned}$$ where the magnitude of $m^{(U)}$ and $m^{(D)}$ is given by $\displaystyle{O\left(\lambda^{-n} v/\sqrt{2}\right)}$. Then, we obtain the following transformation matrices which realize $\displaystyle{y_{Ij}^{(U)}, y_{Ij}^{(D)} = O(1)}$, $$\begin{aligned}
{N_{\rm L}^u}^{\dagger},~~ {N_{\rm L}^d}^{\dagger}
= \left(
\begin{array}{cccc}
1 & \star & \star & 1 \\
\star & 1 & \star & 1 \\
\star & \star & 1 & 1 \\
\star & \star & \star & 1
\end{array}
\right),~~~
N_{\rm R}^u,~~ N_{\rm R}^d
= \left(
\begin{array}{cccc}
1 & \star & \star & \lambda^n \\
\star & 1 & \star & \lambda^n \\
\star & \star & 1 & \lambda^n \\
\lambda^n & \lambda^n & \lambda^n & 1
\end{array}
\right),
\label{Ns-case4}\end{aligned}$$ using (\[K(UL)-case4\]), (\[M(U)-case4\]), and (\[M(D)-case4\]).
The relation $\displaystyle{y_{Ij}^{(U)} = y_{Ij}^{(D)}}$ is realized if the following relation holds, $$\begin{aligned}
m_t \left({N_{\rm L}^u}^{\dagger}\right)_{I3} \left(N_{\rm R}^u\right)_{3j}
+ m^{(U)} \left({N_{\rm L}^u}^{\dagger}\right)_{I4} \left(N_{\rm R}^u\right)_{4j}
= m^{(D)} \left({N_{\rm L}^d}^{\dagger}\right)_{I4} \left(N_{\rm R}^d\right)_{4j},
\label{yU=yD-case4}\end{aligned}$$ where we neglect tiny contributions including $m_u$, $m_c$, $m_d$, $m_s$, and $m_b$.
Case with $u'_{{\rm R}4}$ and $u'_{{\rm R(m)}}$
-----------------------------------------------
In the absence of $q'_{{\rm L}4}$, $q'_{{\rm L(m)}}$, $d'_{{\rm R}4}$, and $d'_{{\rm R(m)}}$, the quark kinetic coefficients are written by $$\begin{aligned}
&~& K^{(U_{\rm L})} =\left(
\begin{array}{cc}
k_{ij}^{(q)} & 0 \\
0 & k^{(u_{\rm m})}
\end{array}
\right)
= {N_{\rm L}^{u}}^{\dagger}N_{\rm L}^{u},~~
K^{(D_{\rm L})} = k_{ij}^{(q)}
= {N_{\rm L}^{d}}^{\dagger}N_{\rm L}^{d},~~
\nonumber \\
&~& K^{(U_{\rm R})} = k_{IJ}^{(u)}
= {N_{\rm R}^{u}}^{\dagger}N_{\rm R}^{u},~~
K^{(D_{\rm R})} = k_{ij}^{(d)}
= {N_{\rm R}^{d}}^{\dagger}N_{\rm R}^{d},
\label{K(UL)-case5}\end{aligned}$$ where $N_{\rm L}^{u}$ and $N_{\rm R}^{u}$ are $4 \times 4$ complex matrices, and $N_{\rm L}^{d}$ and $N_{\rm R}^{d}$ are $3 \times 3$ complex matrices.
Relations of mass matrices are written by $$\begin{aligned}
&~&
\left(
\begin{array}{c}
y_{iJ}^{(U)} \langle \phi^{0*} \rangle \\
m_J^{(u_{\rm m})}
\end{array}
\right)
= {N_{\rm L}^{u}}^{\dagger}
\left(
\begin{array}{cccc}
m_u & 0 & 0 & 0 \\
0 & m_c & 0 & 0 \\
0 & 0 & m_t & 0 \\
0 & 0 & 0 & m^{(U)}
\end{array}
\right) N_{\rm R}^{u},
\label{M(U)-case5}\\
&~&
y_{ij}^{(D)} \langle \phi^{0} \rangle
= {N_{\rm L}^{d}}^{\dagger}
\left(
\begin{array}{ccc}
m_d & 0 & 0 \\
0 & m_s & 0 \\
0 & 0 & m_b
\end{array}
\right) N_{\rm R}^{d},
\label{M(D)-case5}\end{aligned}$$ where the magnitude of $m^{(U)}$ is given by $\displaystyle{O\left(\lambda^{-n} v/\sqrt{2}\right)}$. Then, there is no transformation matrices ${N_{\rm L}^d}^{\dagger}$ and $N_{\rm R}^d$ of $O(1)$ to realize $y_{ij}^{(D)} = O(1)$, as in the SM.
Case with $d'_{{\rm R}4}$ and $d'_{{\rm R(m)}}$
-----------------------------------------------
In the absence of $q'_{{\rm L}4}$, $q'_{{\rm L(m)}}$, $u'_{{\rm R}4}$, and $u'_{{\rm R(m)}}$, the quark kinetic coefficients are written by $$\begin{aligned}
&~& K^{(U_{\rm L})} =k_{ij}^{(q)}
= {N_{\rm L}^{u}}^{\dagger}N_{\rm L}^{u},~~
K^{(D_{\rm L})} = \left(
\begin{array}{cc}
k_{ij}^{(q)} & 0 \\
0 & k^{(d_{\rm m})}
\end{array}
\right)
= {N_{\rm L}^{d}}^{\dagger}N_{\rm L}^{d},~~
\nonumber \\
&~& K^{(U_{\rm R})} = k_{ij}^{(u)}
= {N_{\rm R}^{u}}^{\dagger}N_{\rm R}^{u},~~
K^{(D_{\rm R})} = k_{IJ}^{(d)}
= {N_{\rm R}^{d}}^{\dagger}N_{\rm R}^{d},
\label{K(UL)-case6}\end{aligned}$$ where $N_{\rm L}^{u}$ and $N_{\rm R}^{u}$ are $3 \times 3$ complex matrices, and $N_{\rm L}^{d}$ and $N_{\rm R}^{d}$ are $4 \times 4$ complex matrices.
Relations of mass matrices are written by $$\begin{aligned}
&~&
y_{ij}^{(U)} \langle \phi^{0*} \rangle
= {N_{\rm L}^{u}}^{\dagger}
\left(
\begin{array}{ccc}
m_u & 0 & 0 \\
0 & m_c & 0 \\
0 & 0 & m_t
\end{array}
\right) N_{\rm R}^{u},
\label{M(U)-case6}\\
&~&
\left(
\begin{array}{c}
y_{iJ}^{(D)} \langle \phi^{0} \rangle \\
m_J^{(u_{\rm m})}
\end{array}
\right)
= {N_{\rm L}^{d}}^{\dagger}
\left(
\begin{array}{cccc}
m_d & 0 & 0 & 0 \\
0 & m_s & 0 & 0 \\
0 & 0 & m_b & 0 \\
0 & 0 & 0 & m^{(D)}
\end{array}
\right) N_{\rm R}^{d},
\label{M(D)-case6}\end{aligned}$$ where the magnitude of $m^{(D)}$ is given by $\displaystyle{O\left(\lambda^{-n} v/\sqrt{2}\right)}$. Then, we obtain the following transformation matrices which realize $\displaystyle{y_{ij}^{(U)}, y_{iJ}^{(D)} =O(1)}$, $$\begin{aligned}
&~& {N_{\rm L}^u}^{\dagger}
= \left(
\begin{array}{ccc}
1 & \star & 1 \\
\star & 1 & 1 \\
\star & \star & 1
\end{array}
\right),~~~ N_{\rm R}^u
= \left(
\begin{array}{ccc}
1 & \star & \star \\
\star & 1 & \star \\
1 & 1 & 1
\end{array}
\right),
\label{Nu-case6}\\
&~& {N_{\rm L}^d}^{\dagger}
= \left(
\begin{array}{cccc}
1 & \star & \star & \lambda^{n} \\
\star & 1 & \star & \lambda^{n} \\
\star & \star & 1 & \lambda^{n} \\
\lambda^{n} & \lambda^{n} & \lambda^{n} & 1
\end{array}
\right),~~~ N_{\rm R}^d
= \left(
\begin{array}{cccc}
1 & \star & \star & \star \\
\star & 1 & \star & \star \\
\star & \star & 1 & \star \\
1 & 1 & 1 & 1
\end{array}
\right),
\label{Nd-case6}\end{aligned}$$ using (\[K(UL)-case6\]), (\[M(U)-case6\]), and (\[M(D)-case6\]).
The relation $\displaystyle{y_{ij}^{(U)} = y_{ij}^{(D)}}$ is realized if the following relation holds, $$\begin{aligned}
m_t \left({N_{\rm L}^u}^{\dagger}\right)_{i3} \left(N_{\rm R}^u\right)_{3j}
= m^{(D)} \left({N_{\rm L}^d}^{\dagger}\right)_{i4} \left(N_{\rm R}^d\right)_{4j},
\label{yU=yD-case6}\end{aligned}$$ where we neglect tiny contributions including $m_u$, $m_c$, $m_d$, $m_s$, and $m_b$. As a simple case, the above relation holds with $$\begin{aligned}
m_t \left({N_{\rm L}^u}^{\dagger}\right)_{i3}
= m^{(D)} \left({N_{\rm L}^d}^{\dagger}\right)_{i4},~~
\left(N_{\rm R}^u\right)_{3j}
= \left(N_{\rm R}^d\right)_{4j}.
\label{yU=yD-sol}\end{aligned}$$ In this case, $\left(d'_{\rm R(m)}\right)^{\rm c}$ and $d'_{{\rm R}4}$ in the down-type quark sector play a role of $u'_{{\rm L}3}$ and $u'_{{\rm R}3}$ in the up-type one, respectively.
[99]{} M.S. Chanowitz, J. Ellis, and M.K. Gailard, Nucl. Phys. B **129**, 506 (1977).
H. Fritzsch, Phys. Lett. B **73**, 317 (1978).
H. Harari, H. Haut, and J. Weyers, Phys. Lett. B **78**, 459 (1978).
H. Fritzsch, Nucl. Phys. B **155**, 189 (1979).
H. Georgi and C. Jarlskog, Phys. Lett. B **86**, 297 (1979).
C.D. Froggatt and H.B. Nielsen, Nucl. Phys. B **147**, 277 (1979).
T. Maehara and T. Yanagida, Prog. Theor. Phys. **60**, 822 (1978).
K. Inoue, Prog. Theor. Phys. **93**, 403 (1995).
P.F. Harrison, D.H. Perkins, and W.G. Scott, Phys. Lett. B **530**, 167 (2002).
P.F. Harrison and W.G. Scott, Phys. Lett. B **535**, 163 (2002).
G. Altareli and F. Feruglio, Rev. Mod. Phys. **82**, 2701 (2010).
H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada, and M. Tanimoto, Prog. Theor. Phys. Suppl. **183**, 1 (2010).
H. Ishimori, T. Kobayashi, H. Ohki, H. Okada, Y. Shimizu, and M. Tanimoto, Lect. Note. Phys. **858**, 1 (2012) and references therein.
M. Leurer, Y. Nir, and N. Seiberg, Nucl. Phys. B **398**, 319 (1993).
Y. Koide, Phys. Rev. D **71**, 016010 (2005).
A. Kusenko, Phys. Lett. B **284**, 390 (1992).
C. H. Albright and S. Nandi, Phys. Rev. Lett. **73**, 930 (1994).
D. Marzocca and A. Romanino, J. High Energy Phys. **11**, 159 (2014).
V. Domeke and A. Romanino, J. High Energy Phys. **06**, 031 (2016).
Y. Kawamura, Prog. Theor. Exp. Phys. **2019**, 043B05 (2019).
Y. Kawamura, Prog. Theor. Exp. Phys. **2019**, 073B03 (2019).
H. Georgi and D. V. Nanopoulos, Nucl. Phys. B **155**, 52 (1979).
R. Barbieri and D. V. Nanopoulos, Phys. Lett. B **91**, 369 (1980).
M. Kobayashi and T. Maskawa, Prog. Theor. Phys. **49**, 652 (1973).
C. Patrignani et al. \[Particle Date Group\], Chin. Phys. C **40**, 100001 (2016).
N. Cabibbo, Phys. Rev. Lett. **10**, 531 (1963).
L. Wolfenstein, Phys. Rev. Lett. **51**, 1945 (1983).
P. Binétruy and E. Dudas, Nucl. Phys. B **442**, 21 (1995).
P. Binétruy and E. Dudas, Nucl. Phys. B **451**, 31 (1995).
P. Binétruy, S. Lavignac, and P. Ramond, Nucl. Phys. B **477**, 353 (1996).
M. Kakizaki and M. Yamaguchi, Phys. Lett. B **573**, 123 (2003).
K. Hamaguchi, M. Kakizaki, and M. Yamaguchi, Phys. Rev. D **68**, 056007 (2003).
J.R. Espinosa and A. Ibarra, J. High Energy Phys. **08**, 010 (2004).
C. Liu, Comm. Theor. Phys. **47**, 1088 (2007).
S. Davidson, G. Isidori, and S. Uhlig, Phys. Lett. B **663**, 73 (2008).
Y. Kawamura, Prog. Theor. Exp. Phys. **2013**, 081B01 (2013).
Z. G. Berezhiani, Phys. Lett. B **129**, 99 (1983).
D. Chang and R. N. Mohapatra, Phys. Rev. Lett. **58**, 1600 (1987).
A. Davidson and K. C. Wali, Phys. Rev. Lett. **59**, 393 (1987).
S. Rajpoot, Phys. Rev. D **39**, 351 (1989).
Z. G. Berezhiani, Phys. Lett. B **355**, 178 (1995).
K. S. Babu and S. M. Barr, Phys. Lett. B **381**, 202 (1996).
K. S. Babu and S. M. Barr, Phys. Rev. D **56**, 2614 (1997).
N. Maekawa, Prog. Theor. Phys. **106**, 401 (2001).
N. Maekawa and T. Yamashita, Prog. Theor. Phys. **107**, 1201 (2002).
K. S. Babu, B. Bajc, and Z. Tavartkiladze, Phys. Rev. D **86**, 075005 (2012).
I. Dor$\breve{\rm s}$ner, S. Fajfer, and I. Mustac, Phys. Rev. D **89**, 115004 (2014).
S. M. Barr and H. Y. Chen, Phys. Rev. D **95**, 053009 (2016).
[^1]: E-mail: [email protected]
[^2]: The flavor structure of quarks and leptons has been studied intensively, based on various flavor symmetries [@FN; @MY; @I; @HPS; @HS; @AF; @IKOSOT; @IKOOST]. It is shown that there are no exact flavor-dependent symmetries in the SM [@LNS; @Koide].
[^3]: Although the magnitude of $\displaystyle{\left(V_{\rm KM}\right)_{13}}$ is $0.00365 \pm 0.00012$ and is regarded as $\displaystyle{O\left(\lambda^4\right)}$, we treat it as $\displaystyle{O\left(\lambda^3\right)}$ with respect for the Wolfenstein parametrization [@Wolf].
[^4]: Several works on the flavor physics have been carried out based on non-canonical matter kinetic terms [@BD1; @BD2; @BLR; @KY; @HKY; @EI; @Liu; @DIU].
[^5]: In our terminology, grand unified theories belong to HET, and the minimal supersymmetric extension of the SM (MSSM) belong to SM + $\alpha$.
[^6]: Although a generality is lost by adopting them, it is enough if $N_{\rm L}^u$, $N_{\rm L}^d$, $N_{\rm R}^u$, and $N_{\rm R}^d$ are found with this choice, from the viewpoint of a possible existence.
|
---
abstract: 'The Hamiltonian structure of spin generalization of the rational Ruijsenaars-Schneider model is found by using the Hamiltonian reduction technique. It is shown that the model possesses the current algebra symmetry. The possibility of generalizing the found Poisson structure to the trigonometric case is discussed and degeneration to the Euler-Calogero-Moser system is examined.'
author:
- |
G.E.Arutyunov [^1]\
and\
S.A.Frolov [^2]\
title: 'On Hamiltonian structure of the spin Ruijsenaars-Schneider model'
---
-20mm -10mm
\#1
Appendix \#1 {#appendix-1 .unnumbered}
============
Introduction
============
Recently a spin generalization [@KrZ] of the elliptic Ruijsenaars-Schneider model [@RS; @R] (spin RS model) was introduced as a dynamical system describing the pole evolution of the elliptic solutions of the non-abelian 2D Toda chain. Equations of motion proposed for the model generalize the ones for the Euler-Calogero-Moser system (ECM) [@GH]-[@KBBT], which is an integrable system of $N$ particles with internal degrees of freedom interacting by a special pairwise potential.
An important tool for dealing with classical integrable systems and especially for quantizing them is the Hamiltonian formalism. Although equations of motion defining the spin RS model can be integrated in terms of Riemann theta-functions, the question about their Hamiltonian form remains open. The aim of the present paper is to give a partial answer to this question, which lies in constructing the explicit Hamiltonian formulation for the rational spin RS model.
Our construction is based on the Hamiltonian reduction procedure acknowledged as the unifying approach to dynamical systems of Calogero or Ruijsenaars type [@KKS]-[@ACF]. In this approach one starts with a large initial phase space and a simple Hamiltonian possessing a symmetry group. Then factorizing the corresponding motion by this symmetry one is left with a nontrivial dynamical system defined on a reduced phase space. In particular, the rational RS model and the trigonometric Calogero-Moser system appear in this way if one uses the cotangent bundle $T^{*}G$ over a Lie group $G$ as the initial phase space [@GNe].
A natural generalization of this approach allowing us to include spin variables consists in replacing $T^{*}G$ by a more general phase space $\cal P$ that we choose to be $T^{*}G\times {\cal J}^*$, where ${\cal J}^*$ is a dual space to the Lie algebra ${\cal J}$ of $G$. Considering on $\cal P$ a special Hamiltonian $H_R$ and performing the Hamiltonian reduction by $G$-action, we obtain the Poisson structure of the rational spin RS model.
Let us briefly describe the content of the paper and the results obtained. For simplicity, we restrict ourselves to the case of $G=GL(N,{\bf C})$. In Section 2 we define on $\cal P$ two dynamical systems governed by Hamiltonians $H_C$ and $H_R$ and show that the corresponding integrals of motion combine into generators of the Yangian and the current algebra respectively. Since all these integrals are gauge-invariant, the corresponding symmetries will survive after the reduction.
As is known [@Ne] the dynamical system on the reduced phase space corresponding to $H_C$ is the trigonometric ECM model. This immediately reproduces the result found in [@BGHP; @BAB] that the model possesses the Yangian symmetry.
Section 3 is devoted to the rational spin RS model. First we introduce $G$-invariant spin variables that after solving the moment map equation can be identified with coordinates on the reduced phase space ${\cal P}_r$. Equations of motion for dynamical variables of ${\cal P}_r$ produced by $H_R$ coincide with the ones introduced in [@KrZ] for the rational case. That is the way we obtain an explicit Hamiltonian formulation of the spin RS model. The Poisson structure of the model is found to be rather nontrivial and admits at least two equivalent descriptions in terms of different phase variables. Moreover, it depends on a parameter $\gamma$ being a coupling constant of the model.
It turns out that the spin RS model admits such (spectral-independent) $L$-operator (Lax matrix) that satisfies the same $L$-operator algebra as for the corresponding spinless model. We also show that the Hamiltonian reduction provides an alternative way of solving equations of motion without using spectral curves. A similar method of integrating equations of motion of the spin RS model was used in [@RaS].
Finally, we present an explicit expression for generators of the current algebra via phase variables of the spin RS model and define the gauge-invariant momentum variables.
In Section 4 degeneration of the rational spin RS model to the ECM system is examined. An interesting feature we come here is the appearance of spin variables obeying the defining relations of the Frobenius Lie algebra. We observe that the general elliptic ECM system can be also formulated in terms of Frobenius spin variables.
Current and Yangian symmetries
==============================
In this section we construct representations of the Yangian and current algebras related to the cotangent bundle $T^*G$ over the matrix group $G=GL(N,{\bf C})$ and describe their connection to the ECM and the spin RS models respectively.
Consider the following manifold ${\cal P}=T^*G\times {\cal G}^*$, where ${\cal G}^*$ is a dual space to the Lie algebra ${\cal G}=\mbox{Mat}(N,{\bf C})$ of $G$. We parametrize an element from ${\cal G}^*$ by a matrix $S\in {\cal G}$ due to the isomorphism ${\cal G}^{*}\approx {\cal G}$. The space $T^*G$ is naturally isomorphic to ${\cal G}^*\times G$ and we parametrize it by pairs $(A,g)$, where $A\in {\cal G}$ and $g\in G$. The algebra of regular functions on ${\cal P}$ is supplied with a Poisson structure, which can be written in terms of variables $(A,g,S)$ as follows $$\begin{aligned}
\{ A_1,A_2\} &=&\frac 1 2 [C,A_1
-A_2] \label{AA}\\
\{ A_1,g_2\}
&=&g_2 C,~~~~\{g_1,g_2\}=0, \label{pb} \\
\{S_1,g_2\}&=&\{S_1,A_2\}=0
\label{gg} \\
\{ S_1,S_2\} &=&-\frac 1 2 [C,S_1 -S_2].
\label{SS}\end{aligned}$$ Here we use the standard tensor notation and $C=\sum_{i,j}E_{ij}\otimes E_{ji}$ is the permutation operator.
The Poisson structure is invariant under the following action of the group $G$: AhAh\^[-1]{}, ghgh\^[-1]{}, ShSh\^[-1]{}. We refer to (\[sym\]) as to gauge transformations. The moment map of this action is of the form =gAg\^[-1]{}-A+S. The simplest gauge-invariant Hamiltonians are $H_C=\mbox{tr}A^2$ and $H_R=\mbox{tr}g$.
The Poisson bracket of the variables $S_{ij}$ can be realized by using $2N$ $l$-dimensional vectors $a_i,b_i$ which form $lN$-pairs of canonically conjugated variables: $$\{a^{\alpha}_i,b^{\beta}_j\}=-\delta_{ij}\delta^{\alpha\beta},$$ where $i,j=1,\cdots ,N$ and $\alpha ,\beta=1,\cdots , l$. Supposing the matrix elements of $S$ to be S\_[ij]{}=\_a\^\_ib\^\_j one recovers the Poisson bracket (\[SS\]). Obviously, under gauge transformations the variables $a$ and $b$ transform in the following way $$a^{\alpha}\to ha^{\alpha},~~~b^{\alpha}\to b^{\alpha}h^{-1},$$ where we regard $a^{\alpha}$ as a column and $b^{\alpha}$ as a row.
The variables $a$ and $b$ allow one to construct a lot of gauge-invariants Poisson commuting with $H_C$ or with $H_R$.
First we consider a family of integrals of motion for $H_C$: $I^{\alpha\beta}_n=\mbox{tr}A^nS^{\alpha\beta}$, where for any $\alpha$ and $\beta$ the matrix $S^{\alpha\beta}$ has the entries $S^{\alpha\beta}_{ij}=a^{\alpha}_ib^{\beta}_j$. In fact, the integrals $I_n$ form a representation of the classical Yangian. To see this, one can introduce the following generating function $T^{\alpha\beta}(z)$ of $I_n$: $$T^{\alpha\beta}(z)=\delta^{\alpha\beta}+
\mbox{tr}\frac{1}{z-A}S^{\alpha\beta}.$$ By using the Poisson bracket for the variables $S^{\alpha\beta}$: $$\{S_1^{\alpha\beta},S_2^{\mu\nu}\}=
C_{12}(\delta^{\beta\mu}S_2^{\alpha\nu}-\delta^{\alpha\nu}S_1^{\mu\beta})$$ and performing simple calculations, one obtains the Yangian algebra {T\_1(z),T\_2(w)}=\[r(z-w),T\_1(z)T\_2(w)\]. Here we regard $T(z)$ as an $l\times l$-matrix with entries $T^{\alpha\beta}(z)$, and $r(z-w)$ is a rational solution of the classical Yang-Baxter equation: $$r(z-w)=\frac{K}{z-w},$$ where $K$ is the permutation operator acting in ${\bf C}^l\otimes {\bf C}^l$.
A well-known property of the Yangian is the existence of the involutive subalgebra generated by $I_k(z)=\mbox{tr}T(z)^k$.
For the Hamiltonian $H_R$ one can choose the following family of integrals of motion $J^{\alpha\beta}_n=\mbox{tr}g^nS^{\alpha\beta}$. Introducing the formal generating function $J(z)$: $$J^{\alpha\beta}(z)=\sum_{n=-\infty}^{\infty}J^{\alpha\beta}_nz^{-n-1},$$ one can easily show that $J(z)$ satisfies the current algebra relations: {J\_1(z),J\_2(w)}=\[K,J\_2(w)\](), where $\delta(\frac{z}{w})=\frac{1}{z}
\sum_{n=-\infty}^{\infty}(\frac{z}{w})^{n}$ is the formal $\delta$-function.
It is obvious that $\mbox{tr}J(z)^n$ are central elements of the current algebra. In addition, the current algebra admits an involutive family of integrals of motion polynomial in $g$ and $S$. It is constructed as $J_n^{+}(z)=\mbox{tr}J^+(z)^n$, where $J^+(z)=\sum_{n=0}^{\infty}J_nz^{-n-1}$. The involutivity is a consequence of the algebra satisfied by $J^+(z)$: {J\^+\_1(z),J\^+\_2(w)}=\[r(z-w),J\^+\_1(z)+J\^+\_2(w)\]. It is well known that the Yangian (\[Y\]) is a deformation of (\[J+\]).
The dynamical systems governed by the Hamiltonians $H_C$ and $H_R$ are trivial. However, factorizing the initial phase space $\cal P$ by the action of a symmetry group, one gets nontrivial systems defined on the reduced phase space ${\cal P}_r$. In particular, to get the ECM and the spin RS models one should fix the moment map as [@GN] gAg\^[-1]{}-A+S=I, where $\gamma$ is a complex number being identified with a coupling constant. Then solving this equation modulo the action of the gauge group $G$ ($G$ coincides with the isotropy group of (\[momf\]), i.e. (\[momf\]) is a set of first-class constraints), one obtains the reduced phase space. The dynamical systems on ${\cal P}_r$ corresponding to the Hamiltonians $H_C$ and $H_R$ are identified with the ECM and the spin RS models respectively.
Since the generators of the Yangian and current algebras are gauge-invariant, we conclude that the ECM model possesses the Yangian symmetry whereas the spin RS model has the current symmetry. As was mentioned in the Introduction this result for the ECM model was obtained in [@BGHP; @BAB] by exploiting an explicit $L$-operator describing the model.
Rational spin RS model
======================
In this section we present the Hamiltonian formulation of the spin RS model by considering the reduction of the phase space $\cal P$ by the action of $G$. Given the moment map, the space of functions on the reduced phase space ${\cal P}_{r}$ can be identified with the space $\mbox{Fun}^G{\cal P}$ of $G$-invariant functions on $\cal P$ restricted to the surface (\[momf\]) of the constant moment level. A choice of an appropriate basis in $\mbox{Fun}^G{\cal P}$ and calculation of the induced Poisson structure make the description of ${\cal P}_{r}$ explicit.
To construct a basis in $\mbox{Fun}^G{\cal P}$ we first note that any semisimple element of $\cal G$ can be diagonalized by a gauge transformation: A=TQT\^[-1]{}, where $Q$ is a diagonal matrix with gauge-invariant entries $q_i\neq q_j$. By using the action of the Weyl group we fix the order of $q_i$. For given $A$, the matrix $T$ in (\[d\]) is uniquely defined by requiring to be an element of the Frobenius group, i.e. it satisfies the condition Te=e, where $e$ is an $N$-dimensional column with all $e_i=1$. Such a choice for $T$ is known [@AF] to be relevant for the description of the RS model.
Given $A$ and $g$, we can diagonalize a matrix $A'=gAg^{-1}=UQU^{-1}$ with the help of an element $U$ such that $Ue=e$. This introduces a useful parametrization for $g$: $g=UPT^{-1}$, where $P$ is some diagonal matrix.
Under gauge transformation (\[sym\]) matrices $T$ and $U$ transform as follows $T\to hTh[T]$, $U\to hUh[U]$ where $h[T]$, $h[U]$ are diagonal matrices $h[T]_i=(T^{-1}h^{-1}e)_i$, $h[U]_i=(U^{-1}h^{-1}e)_i$.
Introduce diagonal matrices $t_{ij}=t_i\delta_{ij}$ and $u_{ij}=u_i\delta_{ij}$ with entries t\_i=\_(T\^[-1]{}a\^)\_i, u\_i=\_(U\^[-1]{}a\^)\_i which transform under gauge transformation (\[sym\]) in the following way $$t_i\to h[T]_i^{-1}t_i,~~~
u_i\to h[U]_i^{-1}u_i.$$ We use $t$ to define $G$-invariant spin variables $${\bf a}^{\alpha}_i=t_i^{-1}(T^{-1}a^{\alpha})_i,~~~
{\bf c}^{\alpha}_i=t_i(b^{\alpha}UP)_i.$$ Note that ${\bf a}^{\alpha}_i$ are not arbitrary but satisfy constraints $\sum_{\alpha}{\bf a}^{\alpha}_i=1$ for any $i$. The relevance of this definition will be clarified later.
To calculate the Poisson algebra of ${\bf a}$ and ${\bf c}$ one needs to use the one for $(T,U,P,Q)$-variables. In [@AF] it was proved that the standard Poisson structure (\[AA\],\[pb\]) on $T^*G$ rewritten in terms of $(T,U,P,Q)$-variables has the form {T\_1,T\_2}&=&T\_1T\_2r\_[12]{}, {U\_1,U\_2}=-U\_1U\_2r\_[12]{},\
{T\_1,P\_2}&=&T\_1P\_2|[r]{}\_[12]{}, {U\_1,P\_2}=U\_1P\_2|[r]{}\_[12]{},\
{T\_1,Q\_2}&=&{U\_1,Q\_2}={P\_1,P\_2}={T\_1,U\_2}=0,\
{Q\_1,Q\_2}&=&0, {Q\_1,P\_2}=P\_2\_i E\_[ii]{}E\_[ii]{}. Here $r_{12}$ is an $N$-parametric solution of the classical Yang-Baxter equation: r\_[12]{}=\_[ij]{}F\_[ij]{}F\_[ji]{}, where $F_{ij}=E_{ii}-E_{ij}$ is a basis of the Frobenius Lie algebra and the matrix $\bar{r}_{12}$ is given by |[r]{}\_[12]{}=\_[ij]{}F\_[ij]{}E\_[jj]{}. Note that (\[QP\]) implies that $q_i$ and $p_i=\log P_i$ are canonically conjugated variables.
With formulae (\[TT\])-(\[QP\]) at hand we first calculate $$\{t_i,t_j\}=-\frac{1}{q_{ij}}(t_i-t_j)^2,~~
\{u_i,u_j\}=\frac{1}{q_{ij}}(u_i-u_j)^2,~~\{t_i,u_j\}=0$$ and then the Poisson brackets of the invariant spins: {[**a**]{}\^\_i,[**a**]{}\^\_j}&=& ([**a**]{}\^\_i[**a**]{}\^\_j+[**a**]{}\^\_j[**a**]{}\^\_i -[**a**]{}\^\_i[**a**]{}\^\_i-[**a**]{}\^\_j[**a**]{}\^\_j) ,\
{[**c**]{}\^\_i,[**c**]{}\^\_j}&=& ([**c**]{}\^\_i[**c**]{}\^\_j+[**c**]{}\^\_j[ **c**]{}\^\_i)+ [**c**]{}\^\_j [**L**]{}\_[ji]{}-[**L**]{}\_[ij]{}[**c**]{}\^\_i,\
{[**c**]{}\^\_i,[**a**]{}\^\_j}&=& \^[**L**]{}\_[ji]{}- [**a**]{}\^\_j[**L**]{}\_[ji]{} +\^\_i([**a**]{}\^\_i -[**a**]{}\^\_j),\
{[**a**]{}\^\_i,[**c**]{}\^\_j}&=& -\^[**L**]{}\_[ij]{}+ [**a**]{}\^\_i[**L**]{}\_[ij]{} +\^\_j([**a**]{}\^\_j -[**a**]{}\^\_i). The Poisson structure of invariant spins is not closed since it involves another gauge invariant object ${\bf L}$: ${\bf L}_{ij}=t_i^{-1}L_{ij}t_j$, where $L=T^{-1}gT$. In [@AF] $L$ was identified with the $L$-operator of the rational RS model. Analogously, ${\bf L}$ will be called the $L$-operator of the spin RS model. The relevance of this definition will be justified later. Note that in eqs.(\[aa\])-(\[ac\]) and in formulas below it is assumed that if some denominator becomes zero, the corresponding fraction is omitted.
Calculating the Poisson algebra for ${\bf L}$ with the help of eqs.(\[TT\])-(\[QP\]), we obtain that it coincides with the one for $L$, namely {[**L**]{}\_1,[**L**]{}\_2}&=&r\_[12]{}[**L**]{}\_1[**L**]{}\_2+[**L**]{}\_1[**L**]{}\_2 \_[12]{} +[**L**]{}\_1|[r]{}\_[21]{}[**L**]{}\_2-[**L**]{}\_2|[r]{}\_[12]{}[**L**]{}\_1, where $\hat{r}_{12}=\bar{r}_{12}-\bar{r}_{21}-r_{12}$ is a constant solution of the Gervais-Neveu-Felder equation [@GN; @Fel]. Therefore, the $L$-operator algebra for the spin RS model and the one for the RS model without spins are the same.
To complete the description of the Poisson algebra of invariant spins we find the Poisson brackets of ${\bf L}$ with ${\bf a}$ and ${\bf c}$: {[**a**]{}\^\_i,[**L**]{}\_[kl]{}}&=& ([**a**]{}\^\_i-[**a**]{}\^\_k)[**L**]{}\_[kl]{}- ([**a**]{}\^\_i-[**a**]{}\^\_k)[**L**]{}\_[il]{}- ([**a**]{}\^\_i-[**a**]{}\^\_l)[**L**]{}\_[kl]{}, \
{[**c**]{}\^\_i,[**L**]{}\_[kl]{}}&=& \^\_i[**L**]{}\_[kl]{}+ \^\_l[**L**]{}\_[ki]{}- \^\_i[**L**]{}\_[kl]{}+ \^\_i[**L**]{}\_[il]{}\
&+&[**L**]{}\_[li]{}[**L**]{}\_[kl]{}-[**L**]{}\_[ki]{}[**L**]{}\_[kl]{}. Thus, the Poisson algebra of gauge-invariant variables ${\bf a},{\bf c}$ and ${\bf L}$ is closed.
Remark that the choice of gauge-invariant spins and the $L$-operator is not unique. In particular, one could use $\omega_i=\sum_{\alpha}(b^{\alpha}T)_i$ to define other gauge-invariant spins $\hat{a}^{\alpha}_i=\omega_i (T^{-1}a^{\alpha})_i$, $\hat{c}^{\alpha}_i=\omega_i^{-1} (b^{\alpha}UP)_i$ and the $L$-operator: $\hat{L}_{ij}=\omega_iL_{ij}\omega_j^{-1}$. One can verify that this set of gauge-invariant variables satisfies a different algebra.
The next step consists in restricting the Poisson algebra (\[aa\])-(\[cL\]) to the surface (\[momf\]) of the constant moment level. Diagonalizing the variable $A$, we find that eq.(\[momf\]) is equivalent to LQ-QL-L = - T\^[-1]{}STL. Multiplying (\[red\]) by the diagonal matrix $t$ from the right and by $t^{-1}$ from the left, and taking into account that $L=T^{-1}gT=T^{-1}UP$, we rewrite (\[red\]) in terms of gauge-invariant variables Q-Q[**L**]{}- = - t\^[-1]{}T\^[-1]{}SUPt, since $(t^{-1}T^{-1}SUPt)_{ij}=
\sum_{\alpha}{\bf a}^{\alpha}_i{\bf c}^{\alpha}_j$. Thus, we can solve eq.(\[red1\]) with respect to ${\bf L}$. The solution is given by =\_[ij]{}E\_[ij]{}, where we introduced $f_{ij}=\sum_{\alpha}{\bf a}^{\alpha}_i{\bf c}^{\alpha}_j$. Now the reduction of the Poisson structure (\[aa\])-(\[cL\]) on the surface (\[red\]) amounts to the substitution in the r.h.s. of (\[aa\])-(\[cL\]) the entries of the $L$-operator (\[Li\]). The consistency of the reduced Poisson structure can be also checked by direct calculations. To this end one first find the Poisson algebra of $f_{ij}$-variables: {f\_[ij]{},f\_[kl]{}}&=& (+++)f\_[ij]{}f\_[kl]{} +(+)f\_[ij]{}f\_[il]{}\
&+& (++-)f\_[il]{}f\_[kj]{} +(-)f\_[ij]{}f\_[jl]{}\
&+& (-)f\_[kj]{}f\_[kl]{}+ (+)f\_[lj]{}f\_[kl]{} and $\{f_{ij},q_k\}=-f_{ij}\delta_{kj}$. Then using the representation (\[Li\]) for ${\bf L}$ one does recover the $L$-operator algebra (\[L\]).
Now we proceed with describing dynamics on ${\cal P}_r$. The invariant Hamiltonian $H_R$ acquires on ${\cal P}_r$ a form $H_R=\mbox{tr}{\bf L}=\frac{1}{\gamma}\sum_{i}f_{ii}$. This Hamiltonian and the Poisson structure on ${\cal P}_r$ produce the following equations of motion &&\_i=L\_[ii]{}=f\_[ii]{},\
&&\^\_i=-\_[ji]{} ([**a**]{}\^\_i-[**a**]{}\^\_j) [**L**]{}\_[ij]{}= - \_[ji]{}([**a**]{}\^\_i-[**a**]{}\^\_j)f\_[ij]{}V(q\_[ij]{}),\
&&\^\_i=\_[ji]{} ([**c**]{}\^\_i[**L**]{}\_[ij]{}+ [**c**]{}\^\_j [**L**]{}\_[ji]{})= \_[ji]{}([**c**]{}\^\_if\_[ij]{}V(q\_[ij]{})- [**c**]{}\^\_jf\_[ji]{}V(q\_[ji]{})), where we introduced the potential $V(q_{ij})=\frac{1}{q_{ij}}-\frac{1}{q_{ij}+\gamma}$. Differentiating $\dot{q}_i$ and taking into account eqs.(\[ad\]),(\[cd\]), one gets \_i=2\_[ji]{}L\_[ij]{}L\_[ji]{}= 2\_[ji]{}\_[ij]{}[**L**]{}\_[ji]{}= \_[ji]{}f\_[ij]{}f\_[ji]{}(V(q\_[ij]{})-V(q\_[ji]{})), and equations of motion for $f_{ij}$: =\_[ki,j]{} V(q\_[kj]{})f\_[ik]{}f\_[kj]{}-V(q\_[ik]{})f\_[ik]{}f\_[kj]{}+V(q\_[ik]{})f\_[ik]{}f\_[ij]{}- V(q\_[jk]{})f\_[jk]{}f\_[ij]{}. It follows from eqs.(\[qdd\]) and (\[eqf\]) that the equation of motion for ${\bf L}$ can be written in the Lax form $\dot{{\bf L}}=[{\bf L},{\bf M}]$ with ${\bf M}=\sum_{i\neq
j}\frac{1}{q_{ij}}{\bf L}_{ji}F_{ij}$. However, this equation is not equivalent to eqs.(\[qdd\]) and (\[eqf\]).
In the paper [@KrZ] the spin generalization of the elliptic RS model was introduced. The generalized model is a system of $N$ particles with coordinates $q_i$, each particle has internal degrees of freedom described by $l$-dimensional vector $a^{\alpha}_i$ and $l$-dimensional vector $c^{\alpha}_i$. The equations of motion generalize the ones for the ECM system: \_i&=&\_[j i]{}f\_[ij]{}f\_[ji]{}(V(q\_[ij]{})-V(q\_[ji]{})),\
\_i&=&\_[j i]{}[a]{}\_j f\_[ij]{}V(q\_[ij]{})-\_ia\_i,\
\_i&=&-\_[j i]{}[c]{}\_j f\_[ji]{}V(q\_[ji]{})+\_ic\_i, where $V(q)=\zeta(q)-\zeta(q+\gamma)$, $\lambda_i(t)$ are arbitrary functions of $t$ and $\zeta(q)$ denotes the Weierstrass zeta-function. Equations of motion (\[qsK\]-\[bK\]) are invariant under rescaling: $$a_i\to k_ia_i,~~~c_i\to \frac{1}{k_i}c_i.$$ Introducing the invariant variables $\hat{a}_i^{\alpha}=(\sum_{\alpha}a_i^{\alpha})^{-1}a_i^{\alpha}$ and $\hat{c}_i^{\alpha}=(\sum_{\alpha}a_i^{\alpha})c_i^{\alpha}$, and calculating from (\[aK\]) and (\[bK\]) equations of motion for $\hat{a}_i^{\alpha}$ and $\hat{c}_i^{\alpha}$, one discovers that all $\lambda_i$ drop out and equations of motion coincide with (\[ad\]) and (\[cd\]) with the change $\hat{a}^{\alpha}_i\to {\bf a}_i^{\alpha}$, $\hat{c}^{\alpha}_i\to \frac{1}{\gamma}{\bf c}_i^{\alpha}$ and with the substitution $V(q)$ for its rational analog $\frac{1}{q}-\frac{1}{q+\gamma}$. To present eqs.(\[qsK\]) in the Lax form, in the paper [@KrZ] a spectral-dependent $L$-operator $L(z)$ was suggested. One can see that in the rational case $L(z)$ coincides with $\bf L$ in the limit $z\to \infty$. Thus, we obtain the Hamiltonian formulation of the spin generalization of the rational RS model.
Now we show how equations of motion (\[qd\])-(\[cd\]) can be solved in terms of the factorization problem (see also [@OP; @RS; @RaS]). The Hamiltonian $H_R$ induces on ${\cal P}$ equations of motion: $$\dot{g}=0,~~~\dot{A}=g,~~~\dot{S}=0$$ that can be easily integrated: $A(t)=gt+A_0$, $g(t)=const$, $S(t)=const$. We suppose that the positions of particles at $t=0$ are given by $q_i$ lying on ${\cal P}_r$. It means that $A(t)=gt+Q$. Since for any $t$ the point $(A(t),g(t),S(t))$ satisfies the constraint (\[momf\]) one gets that $g$ should be identified with the $L$-operator $L_0$ at $t=0$.
Let us show that the solution of (\[qd\]) is given by the diagonal factor $Q(t)$ in the decomposition of $A(t)$: A(t)=L\_0t+Q=T(t)Q(t)T\^[-1]{}(t), T(t)e=e. In [@AF] it was proved that &=&\_[aj]{} (T\_[ia]{}T\_[nj]{}T\_[am]{}\^[-1]{}+T\_[ij]{}T\_[na]{}T\_[jm]{}\^[-1]{}),\
&=&T\_[ni]{}T\_[im]{}\^[-1]{}. Using these formulae, we find $$\dot{q}_i=\sum_{mn}\frac{\delta q_i}{\delta A_{mn}}\frac{dA_{mn}}{dt}=
(T^{-1}gT)_{ii}(t)=L_{ii}(t).$$ Differentiating $\dot{q}_i$ once again, one gets \_i=\_[ii]{}=\[T\^[-1]{}gT,T\^[-1]{}\]\_[ii]{}, where (T\^[-1]{})\_[ij]{}=-L\_[ij]{}+\_[ij]{}\_[aj]{} L\_[ja]{}. Substituting eq.(\[TdT\]) in eq.(\[dqd\]), we obtain eq.(\[qd\]).
As to the spin variables their equations of motion are automatically solved if one knows the factor $T(t)$ in the decomposition (\[eqm\]). Indeed, if we define $\tilde{a}_i^{\alpha}(t)=(T^{-1}(t)a^{\alpha})_i$ then $$\dot{\tilde{a}}_i^{\alpha}=
-\sum_{j\neq i}
\frac{1}{q_{ij}}(\tilde{a}_i^{\alpha}-\tilde{a}_j^{\alpha})L_{ij}$$ and for the invariant spin ${\bf a}_i^{\alpha}=\frac{1}{t_i}\tilde{a}_i^{\alpha}$ we get eq.(\[ad\]). Solution of the equation of motion for ${\bf c}_i^{\alpha}$ is given by ${\bf c}_i^{\alpha}(t)=t_i(t)(b^{\alpha}L_0T(t))_i$.
The integrals of motion $J_n^{\alpha\beta}=\mbox{tr}(g^nS^{\alpha\beta})$ introduced in Section 2 take on ${\cal P}_r$ the following form $$J_n^{\alpha\beta}=\sum_{ij}({\bf L}^{n-1})_{ij}{\bf a}^{\alpha}_j
{\bf c}^{\beta}_i.$$ Substituting here the explicit form (\[Li\]) of the $L$-operator, one can recast $J_n^{\alpha\beta}$ for $n\geq 1$ in the form J\_1\^&=&\_[i]{}S\_i\^,\
J\_n\^&=&\_[i\_1,…,i\_n]{} , where we use $l\times l$-matrices $S_i^{\alpha\beta}={\bf c}_i^{\alpha}{\bf a}_i^{\beta}$ ($i=1,\ldots N$).
An important property of $S_i$-variables is that they form a set of gauge-invariant variables equivalent to $({\bf a},{\bf c})$. In fact, one can see that $${\bf c}_i^{\alpha}=\sum_{\beta}S_i^{\alpha\beta}, ~~~
{\bf a}_i^{\alpha}=\frac{S_i^{\beta\alpha}}{\sum_{\gamma}S_i^{\beta\gamma}}.$$ The Poisson structure of the model can be conveniently rewritten in terms of $S_i^{\alpha\beta}$: {S\_i\^,S\_j\^}&=& (S\_i\^S\_j\^+S\_i\^S\_j\^) -(S\_iS\_j)\^ +(S\_jS\_i)\^,\
{q\_i,S\_j\^}&=&S\_j\^\_[ij]{}. Since the Hamiltonian $H_R$ can be expressed as $H_R=\sum_i\mbox{Tr}S_i$, eq.(\[qd\]) acquires the form \_i=\_[ji]{} (S\_iS\_j)(V(q\_[ij]{})-V(q\_[ji]{})), where $\mbox{Tr}$ is used to denote the trace of an $l\times l$-matrix. Analogously, eqs.(\[ad\]) and (\[cd\]) produce the equations of motion for $S_i$: \_i=\_[ji]{}(S\_iS\_jV(q\_[ij]{})-S\_jS\_iV(q\_[ji]{})).
Observe that the Poisson structure (\[qS\]) and the Hamiltonian $H_R$ are invariant under the transformations $S_i\to \Omega^{-1}S_i\Omega$, where $\Omega\in GL(l,{\bf C})$. These transformations are generated by $J_0^{\alpha\beta}$.
Thus, we see that the Hamiltonian formalism of the rational spin RS model can be equivalently presented in terms of either $({\bf a},{\bf c})$- or $S_i$-variables.
The definition of ${\bf c}_i^{\alpha}=t_i(b^{\alpha}UP)_i$ implies that they contain the variables conjugated to $q_i$. However, we can not identify them with $P_i$ since the latter are not gauge-invariant. The gauge-invariant momentum ${\bf P}_i$ can be defined as ${\bf P}_i=u_i^{-1}P_it_i$. Computing the Poisson brackets of ${\bf P}_i$, one gets that $\{{\bf P}_i,{\bf P}_j\}=0$ and $\{q_i,{\bf P}_j\}=\delta_{ij}{\bf P}_j$.
Recall that the invariant $L$-operator has the form ${\bf L}=t^{-1}T^{-1}UPt$. Thus, it can be written as ${\bf L}={\bf W}{\bf P}$, where ${\bf W}$ is a gauge-invariant variable: $${\bf W}=t^{-1}T^{-1}Uu.$$ Then it is easy to see that ${\bf W}$ belongs to the Frobenius group, i.e. it obeys the condition ${\bf W}e=e$: ([**W**]{}e)\_i=\_[k,m]{}(t\^[-1]{}T\^[-1]{})\_[ik]{}U\_[km]{}(U\^[-1]{}a\^)\_m= (T\^[-1]{}a\^)\_i=1. Just as it was for the spinless RS model, the Poisson bracket for ${\bf W}$ coincides with the Sklyanin bracket: {[ **W**]{}\_1,[**W**]{}\_2}=\[r\_[12]{},[**W**]{}\_1[**W**]{}\_2\].
On ${\cal P}_r$ the variable ${\bf W}$ acquires the form \_[ij]{}=\_, where ${\bf b}_i^{\alpha}={\bf c}_i^{\alpha}{\bf P}_i^{-1}$.
Remind that the variables ${\bf a}_i^{\alpha}$ obey the constraints $\sum_{\alpha}{\bf a}_i^{\alpha}=1$ for any $i$. The condition (\[We\]) implies that ${\bf b}_i^{\alpha}$ are also not arbitrary but subject to the constraints: \_[**a**]{}\_i\^ \_j=1 for any $i$. Therefore, the number of independent spin variables is $2N(l-1)$. In terms of these variables the Poisson structure of ${\cal P}_r$ looks as follows: {q\_i,[**P**]{}\_j}&=&\_[ij]{}[**P**]{}\_j, {q\_i,[**a**]{}\_j\^}=0={q\_i,[**b**]{}\_j\^},\
{[**P**]{}\_i,[**a**]{}\^\_j}&=& ([**a**]{}\^\_i-[**a**]{}\^\_j)[**P**]{}\_i,\
{[**P**]{}\_i,[**b**]{}\^\_j}&=&(\_[ij]{}-[**W**]{}\_[ij]{} +\^\_j+ \_[ij]{}\_[ni]{}\^\_n )[**P**]{}\_i,\
{[**a**]{}\^\_i,[**a**]{}\^\_j}&=& ([**a**]{}\^\_i[**a**]{}\^\_j+[**a**]{}\^\_j[**a**]{}\^\_i -[**a**]{}\^\_i[**a**]{}\^\_i-[**a**]{}\^\_j[**a**]{}\^\_j),\
{[**b**]{}\_i\^,[**b**]{}\_j\^}&=& \_[ij]{}([**b**]{}\_i\^-[**b**]{}\_i\^)+ ([**b**]{}\_j\^[**b**]{}\_i\^-[**b**]{}\_j\^[**b**]{}\_i\^) +\_[ij]{}\_[ni]{} ([**b**]{}\_n\^[**b**]{}\_i\^- [**b**]{}\_n\^[**b**]{}\_i\^),\
{[**a**]{}\_i\^,[**b**]{}\_j\^}&=&-\^ [**W**]{}\_[ij]{}+[**a**]{}\_i\^[**W**]{}\_[ij]{}, where ${\bf W}$ is given by (\[Wr\]). One should point out that the structure (\[ps\]) is Poisson only due to the constraints imposed on the spin variables. Therefore, the rational spin RS model provides a new realization of the Poisson relations (\[WW\]) as well as the $L$-operator algebra (\[L\]).
Now we discuss the problem of generalizing the found Poisson structure for the spin rational RS model to the trigonometric case.
Relaying on the fact that both in the spin and spinless cases the $L$-operator algebras may be the same, one can easily derive the trigonometric analog of the Poisson bracket (\[ff\]) for the variables $f_{ij}$. It follows from the results of [@AFM] that the trigonometric RS model can be described by the $L$-operator algebra (\[L\]), where this time the $r$-matrices $r$, $\bar{r}$ and $\hat{r}$ are given by &&r=\_[ij]{}E\_[ij]{}E\_[ji]{}+ \_[ij]{}E\_[ii]{}E\_[jj]{}+ \_[ij]{}E\_[ij]{}E\_[ji]{}\
&& - \_[ij]{}E\_[ij]{}E\_[jj]{}+ \_[ij]{}E\_[jj]{} E\_[ij]{},\
&& |[r]{}=-\_[i]{}E\_[ii]{}E\_[ii]{}+ \_[ij]{}E\_[ii]{}E\_[jj]{}- \_[ij]{}E\_[ij]{} E\_[jj]{}, \
&&=-\_[ij]{}E\_[ij]{}E\_[ji]{}+ \_[ij]{}(E\_[ii]{}E\_[jj]{}-E\_[ij]{} E\_[ji]{}). For the spinless trigonometric RS model the $L$-operator satisfying (\[L\]) is of the form $$L=\sum_{ij}\frac{e^{q_{ij}+\gamma}}{\sinh{(q_{ij}+\gamma)}}c_j E_{ij},$$ where $c_j$ are some functions of dynamical variables. It is natural to assume that in the spin case the corresponding $L$-operator has the form $$L=\sum_{ij}\frac{e^{q_{ij}+\gamma}}{\sinh{(q_{ij}+\gamma)}}f_{ij}
E_{ij}.$$ Then the Poisson relations for $f_{ij}$ follow immediately from the $L$-operator algebra (\[L\]): {f\_[ij]{},f\_[kl]{}}&=& (+++)f\_[ij]{}f\_[kl]{}\
&+& (++-))f\_[il]{}f\_[kj]{}\
&+& (+))f\_[ij]{}f\_[il]{} +(-)f\_[ij]{}f\_[jl]{}\
&+& (-)f\_[kj]{}f\_[kl]{}+ (+)f\_[lj]{}f\_[kl]{} and they look like a trigonometric generalization of (\[ff\]).
Now one can easily verify that equations of motion for $f_{ij}$ are given by (\[eqf\]) with the potential $V(q)=\coth(q)-\coth(q+\gamma)$ and with the change of the overall factor $\frac{1}{\gamma}$ by $\frac{e^{\gamma}}{\sinh(\gamma)}$. On the other hand, these equations follow from eqs.(\[aK\]) and (\[bK\]). The problem of describing the Poisson structure of the trigonometric spin RS model would be completely solved if one finds such Poisson brackets for variables ${\bf a}$ and ${\bf c}$ that induce the ones (\[trig\]) for $f_{ij}$. The straightforward generalization of eqs.(\[aa\]-\[ac\]) to the case at hand by replacing $\frac{1}{q_{ij}}$ by $\coth(q_{ij})$ is failed. At the moment we can not offer a solution of the problem.
Euler-Calogero-Moser model
==========================
We start this section with discussing the degeneration of the spin RS system to the rational ECM model. For this purpose we rescale $\log {\bf P}_i={\bf p}_i\to \varepsilon {\bf p}_i$ and $q_i\to
\frac{1}{\varepsilon}q_i$, and consider the limit $\varepsilon\to 0$. The constraint (\[We\]) implies that in this limit \_[**a**]{}\_i\^[**b**]{}\_i\^=[**S**]{}\_[ii]{}= and ${\bf W}_{ij}$ has the following expansion $${\bf W}_{ij}=\delta_{ij}+\varepsilon(1-\delta_{ij})
\frac{{\bf S}_{ij}}{q_{ij}}-
\varepsilon \delta_{ij}\sum_{k\neq i}
\frac{{\bf S}_{ik}}{q_{ik}}+o(\varepsilon),$$ where ${\bf S}_{ij}=\sum_{\alpha}{\bf a}_i^{\alpha}{\bf b}_j^{\alpha}$. The corresponding expansion of the $L$-operator produces in the first order in $\varepsilon$ the $L$-operator ${\cal L}$ of the rational ECM model: \_[ij]{}=\_[ij]{}( [**p**]{}\_i- \_[k i]{})+(1-\_[ij]{}) . In the limit $\varepsilon\to 0$ the Poisson structure (\[ps\]) reduces to {q\_i,[**p**]{}\_j}&=&\_[ij]{}, {q\_i,[**a**]{}\_j\^}=0={q\_i,[**b**]{}\_j\^},\
{[**p**]{}\_i,[**a**]{}\^\_j}&=& ([**a**]{}\^\_i-[**a**]{}\^\_j),\
{[**p**]{}\_i,[**b**]{}\^\_j}&=& \_[ij]{}\_[ki]{}- (1-\_[ij]{})+ \^\_j+ \_[ij]{} \_[ni]{}\^\_n ,\
{[**a**]{}\^\_i,[**a**]{}\^\_j}&=& 0,\
{[**b**]{}\_i\^,[**b**]{}\_j\^}&=& \_[ij]{}([ **b**]{}\_i\^-[**b**]{}\_i\^),\
{[ **a**]{}\_i\^,[**b**]{}\_j\^}&=&-\^ \_[ij]{}+[**a**]{}\_i\^\_[ij]{}, Introducing new momenta $$p_i={\bf p}_i- \sum_{k\neq i}\frac{{\bf S}_{ik}}{q_{ik}},$$ one can check that they have vanishing Poisson brackets with ${\bf a}^{\alpha}_i$ and ${\bf b}^{\alpha}_i$. The $L$-operator ${\cal L}$ turns into the standard one used in description of the ECM system [@GH]-[@BAB]: \_[ij]{}=\_[ij]{}p\_i+(1-\_[ij]{}) . To make a contact with the usual description of the ECM system we introduce $lN$ pairs of canonical variables: $\{a_i^{\alpha},b_j^{\beta}\}=-\delta_{ij}\delta^{\alpha\beta}$. Then the invariant variables ${\bf a}_i^{\alpha}$ and ${\bf b}_j^{\beta}$ with the Poisson algebra (\[psdeg\]) can be realized as $${\bf a}_i^{\alpha}=\frac{a_i^{\alpha}}{\sum_{\beta}a_i^{\beta}},~~~
{\bf b}_i^{\alpha}=b_i^{\alpha} \sum_{\beta}a_i^{\beta}.$$
It is interesting to note that the Poisson algebra of the variables ${\bf S}_{ij}$ coincides with the defining relations of the Frobenius Lie algebra: {[**S**]{}\_[ij]{},[ **S**]{}\_[kl]{}}=\_[il]{}([**S**]{}\_[ij]{}-[**S**]{}\_[kj]{})+ \_[jl]{}([ **S**]{}\_[kj]{}-[**S**]{}\_[ij]{})+ \_[jk]{}([**S**]{}\_[il]{}-[**S**]{}\_[kl]{}). These relations are compatible with the constraint ${\bf S}_{ii}=\gamma$.
The appearance of the Frobenius spin variables in the rational ECM model is not accidental. In fact, the same phenomenon takes place for the general elliptic ECM model. To elucidate this fact we recall that the elliptic ECM system is described by the Hamiltonian $H=\frac{1}{2}\sum_ip_i^2-\frac{1}{2}\sum_{i\neq j}S_{ij}S_{ji}V(q_{ij})$, where $V(q_{ij})={\cal P}(x)$ is the Weierstrass ${\cal P}$-function and $S_{ij}$ are the spin variables defined by (\[Ss\]) and having the Poisson bracket: $$\{S_{ij},S_{kl}\}=\delta_{jk}S_{il}-\delta_{il}S_{kj}.$$ The model is described by the $L$-operator [@BAB] $$L=\sum_{i}(p_i+\zeta(z)S_{ii})E_{ii}+\sum_{i\neq
j}\Phi(z,q_{ij})S_{ij}E_{ij},$$ where $\Phi(z,q)=\frac{\sigma(z+q)}{\sigma(z)\sigma(q)}$. This $L$-operator satisfies the Poisson algebra {L\_1(z),L\_2(w)}&=&\[r\_[12]{}(z,w),L\_1(z)\]-\[r\_[21]{}(w,z),L\_2(w)\]\
&+&\_[ij]{}(z-w,q\_[ij]{}) (S\_[ii]{}-S\_[jj]{})E\_[ij]{}E\_[ji]{}, with the dynamical $r$-matrix r\_[12]{}(z,w)=(z-w)\_[i]{}E\_[ii]{}E\_[ii]{}+ \_[ij]{}(z-w,q\_[ij]{}) E\_[ij]{}E\_[ji]{}. Due to the last term in (\[Lc\]) the model is not integrable. However, the Hamiltonian is invariant under the symmetry $a_i\to k_ia_i$, $b_i\to \frac{1}{k_i}b_i$ generated by $S_{ii}$. The integrability is obtained on the reduced space $S_{ii}=const$. As in the case of the RS system, to perform the reduction we define the gauge-invariant $L$-operator ${\bf L}=tLt^{-1}$ with $t_{ij}=\delta_{ij}\sum_{\alpha}a^{\alpha}_i$ or explicitly =\_[i]{}(p\_i+(z))E\_[ii]{}+ \_[ij]{}(z,q\_[ij]{})[**S**]{}\_[ij]{}E\_[ij]{}, where the gauge-invariant spin variables appeared $${\bf S}_{ij}=S_{ij}\frac{\sum_{\alpha}a^{\alpha}_j}
{\sum_{\alpha}a^{\alpha}_i}.$$ Computing the Poisson bracket of ${\bf S}_{ij}$, we obtain that it precisely coincides with (\[Sin\]).
Now it is easy to establish that the Poisson algebra of the $L$-operator (\[Cin\]) has the form $$\{{\bf L}_1(z),{\bf L}_2(w)\}=[{\bf r}_{12}(z,w),{\bf L}_1(z)]-[{\bf
r}_{21}(w,z), {\bf L}_2(w)],$$ where a matrix ${\bf r}$ literally coincides with the $r$-matrix of the elliptic Calogero-Moser model [@Skl; @BS]: \_[12]{}(z,w)&=&((z-w)+(w))\_[i]{}E\_[ii]{}E\_[ii]{}\
&+& \_[ij]{}(z-w,q\_[ij]{}) E\_[ij]{}E\_[ji]{}+ \_[ij]{}(w,q\_[ij]{}) E\_[jj]{}E\_[ij]{}. Thus, the ECM model corresponds to a representation of the $L$-operator algebra of the Calogero-Moser model that depends not only on $q_i$ and $p_i$ but also on the additional spin variables ${\bf S}_{ij}$ with the bracket (\[Sin\]). As to the spectral-dependent $L$-operator of the spin RS model, it does not satisfy the $L$-operator algebra found for the spinless case [@NKSR; @Sur1]. This algebra is quadratic what fixes the form of the corresponding $L$-operator almost uniquely.
Conclusion
==========
In this paper we presented a detailed description of the Poisson structure for the rational spin RS model by using the Hamiltonian reduction technique. The results obtained can not be extended to the trigonometric spin RS model in a straightforward manner. It was shown in [@GNe] that the trigonometric RS model can be obtained by means of the Poisson reduction technique applied to the Heisenberg double $D$ associated to $G=GL(N,{\bf C})$. Therefore, one may hope to describe the Poisson structure of the trigonometric spin RS model in the same fashion starting from the phase space $D\times G^*$, where $G^*$ is a Poisson-Lie group dual to $G$.
As is known [@GH] the rational ECM model possesses the current algebra symmetry and, as we have established, the same symmetry occurs in the rational spin RS model. On the other hand, the trigonometric ECM model has the Yangian symmetry. Thus, for the trigonometric spin RS model it is natural to expect the appearance of the Yangian symmetry.
The elliptic case is much more involved since at the moment a reduction procedure leading to the elliptic spin RS model remains to be unknown.
Another interesting open problem is to quantize the spin RS models. In the rational case one could use the quantum Hamiltonian reduction procedure developed in [@AF]. [**ACKNOWLEDGMENT**]{} The authors are grateful to A.Zabrodin for valuable discussions. This work is supported in part by the RFBI grants N96-01-00608, N96-01-00551 and by the ISF grant a96-1516.
[99]{} [ I.M.Krichever, A.V.Zabrodin, Spin generalizations of the Ruijsenaars-Schneider model, non-abelian 2D Toda chain and representations of Sklyanin algebra, hep-th/9505039. S.N.M.Ruijsenaars and H.Schneider, Ann.Phys. 170 (1986) 370. S.N.M.Ruijsenaars, Comm.Math.Phys. 115 (1988) 127. J.Gibbons and T.Hermsen, Physica D 11 (1984) 337. S.Wojciechowski, Phys.Lett. A 111 (1985) 101. D.Bernard, M.Gaudin, F.D.M.Haldane and V.Pasquier, J.Phys.A 26 (1993) 5219. E.Billey, J.Avan, O.Babelon, Phys.Lett. A 188 (1994) 263-271. I.M.Krichever, O.Babelon, E.Billey, M.Talon, Spin generalization of Calogero-Moser system and the matrix KP equation, hep-th/9411160. D.Kazhdan, B.Kostant and S.Sternberg, Comm.Pure Appl.Math. 31 (1978) 481. M.A.Olshanetsky, A.M.Perelomov, Phys. Reps. 71 (1981) 313. Gorsky A. and Nekrasov N., Nucl.Phys. B414 (1994) 213; Nucl.Phys. B436 (1995) 582; A.Gorsky, Integrable many body systems in the field theories, Prep. UUITP-16/94, (1994). J.Avan, O.Babelon and M.Talon, Alg.Anal. 6(2) (1994) 67. N.Nekrasov, Holomorphic bundles and many-body systems, hep-th/9503157. G.E.Arutyunov and P.B.Medvedev, Phys.Lett.A 223 (1996) 66-74. G.E.Arutyunov, S.A.Frolov, Quantum dynamical $R$-matrices and quantum Frobenius group, q-alg/9610009, to appear in CMP. G.E.Arutyunov, S.A.Frolov and P.B.Medvedev, Elliptic Ruijsenaars-Schneider model via the Poisson reduction of the affine Heisenberg double, hep-th/9607170. G.E.Arutyunov, S.A.Frolov and P.B.Medvedev Elliptic Ruijsenaars-Schneider model from the cotangent bundle over the two-dimensional current group, hep-th/9608013. G.E.Arutyunov, L.O.Chekhov and S.A.Frolov, $R$-matrix quantization of the Elliptic Ruijsenaars-Schneider model, q-alg/9612032. O.Ragnisco and Yu.B.Suris, Integrable discretizations of the spin Ruijsenaars-Schneider models, solv-int/9605001. J.L.Gervais and A.Neveu, Nucl.Phys. B238 (1984) 125. G.Felder, Conformal field theory and integrable systems associated to elliptic curves, hep-th/9407154. E.K.Sklyanin, Alg.Anal., 6(2) (1994) 227. H.W.Braden and T.Suzuki, Lett.Math.Phys. 30 (1994) 147. F.W.Nijhoff, V.B.Kuznetsov, E.K.Sklyanin and O.Ragnisco, Dynamical $r$-matrix for the elliptic Ruijsenaars-Schneider model, solv-int/9603006. Yu.B.Suris, Elliptic Ruijsenaars-Schneider and Calogero-Moser hierarchies are governed by the same $r$-matrix, solv-int/9603011. ]{}
[^1]: Steklov Mathematical Institute, Gubkina 8, GSP-1, 117966, Moscow, Russia; [email protected]
[^2]: Steklov Mathematical Institute, Gubkina 8, GSP-1, 117966, Moscow, Russia; [email protected]
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---
abstract: 'We develop a correction to the density matrix used in density matrix renormalization group calculations to take into account the incompleteness of the environment block. The correction allows successful calculations using only a single site in the center of the system, rather than the standard two sites, improving typical computation times by a factor of two to four. In addition, in many cases where ordinary DMRG can get stuck in metastable configurations, the correction eliminates the sticking. We test the new method on the Heisenberg $S=1$ chain.'
author:
- 'Steven R. White'
title: Density matrix renormalization group algorithms with a single center site
---
Since the density matrix renormalization group (DMRG) was devloped[@dmrg; @schollwock], it has gradually been applied to more and more difficult systems, such as wide ladders and 2D clusters, and systems with long-range interactions. One of the problems arising in these systems is the possibility that the simulation gets stuck far from the ground state[@stripe]. Several approaches have been developed to overcome this problem, such as controlling the starting wavefunction through potentials or quantum numbers, with the controls later removed. Nevertheless, there has remained much room for improvement. In 1D short-range systems, the standard DMRG finite system algorithm avoids convergence problems remarkably well because of the presence of the second center site in the block configuration. However, the extra site increases the computation time and memory requirements. An alternative to utilizing the extra site, which works better in the more difficult cases, has not been available. In this paper we describe such an alternative method, which relies on a correction to the reduced density matrix in order to retain a broader variety of states.
In the top panel of Fig. 1 we show the “superblock” configuration for the standard finite-system algorithm, where the lattice is divided into two large blocks, the system and the envirionment blocks, both with truncated bases, with two sites between them. The algorithm for a single DMRG step consists of finding the ground state for this “superblock”; obtaining the density matrix for the system block plus site; diagonalizing this density matrix; and then changing basis to the most probable eigenvectors of the density matrix. This step replaces the system block, described by $m$ states, by a block one site larger, but also described (approximately) by $m$ states. One then shifts the dividing line between the system and environment by one site, in order to add another site to the system block, and repeats the process. When the system block encompasses the whole system, the direction is reversed and the roles of system and environment blocks are reversed. A sweep consists of one pass back and forth through the system. In a simple 1D spin system one often obtains convergence to very high accuracy, e.g. an accuracy in the energy of order $10^{-10}$, with one or two sweeps through the lattice.
![ Standard two-site DMRG method (top) and the single site method. []{data-label="figone"}](fig1.eps){width="4cm"}
In this description, it is apparent that one of the two sites in the center is crucial to the algorithm. The role of the other site is to increase the dimension and also the accuracy of the environment, particularly at the point where it connects to the system. One can leave out this extra site, i.e. use an environment block with the site already a part of it, as shown in the bottom panel of Fig. 1. This decreases the computation time for a step by roughly the number of states in a single site. However, one finds that even in 1D systems, the progress towards the ground state is much slower, and often stops altogether far from the true ground state. This can be understood in various ways. For example, suppose the ground state has total $z$ component of spin $S_z=0$, and also suppose the environment block is poor and only has states with $S_z=0$. Then the renormalized system block will only have states with $S_z=0$, and no fluctuations in the spin will develop between the two blocks. In fact, any limitation on the quantum numbers present in the environment translates into a restriction on the states appearing in the renormalized system block. The distribution of states between various quantum numbers in the environment also translates directly to the renormalized system block. Note that if the environment block has $m$ states, then the maximum number of nonzero eigenvalues of the density matrix is also $m$, and the number of states never increases unless states are added “artificially” despite having density matrix eigenvalues of zero. Simple fixes, such as adding extra random states with a larger range of quantum numbers, improve but do not fix the very poor convergence of the single site algorithm.
The essential problem here arises when a particular fluctuation between the system and environment which should be present is not because the environment block does not have the relevant states. Hence, the fluctuation is not represented in the density matrix and the new system block will not possess its relevant states for that fluctuation. Later, when the roles of system and environment are reversed, the relevant states again do not appear. In a 1D system with short range interactions the extra environment site does a very good job of ensuring that the most relevant fluctuations are at least approximately present the environment, so that subsequent sweeps can build in the fluctuations to high accuracy.
In wide ladders or systems with longer range interactions, the addition of a single site to the environment is not always adequate. There may be missing fluctuations which are far from the extra site, and so are never built in. Even in these cases the extra site allows $m$ to increase sensibly and as one lets $m \to \infty$ one obtains exact results. However, for practical values of $m$ one may find unacceptably slow convergence.
In this paper we describe an approximate correction to the density matrix to describe the key states which have been left out because the environment block is inadequate. With this correction, the single site superblock configuration converges well. In addition, convergence in more difficult systems is dramatically improved, in either the single site or two site configurations. We present two different derivations of the correction, and give examples using the $S=1$ Heisenberg chain.
We first give a simple, rough argument. Consider the power method for finding the ground state: iterate $\psi_{n+1} = (1 - \varepsilon H) \psi_n$, where $\varepsilon$ is a small constant. As long as $\psi_0$ is not orthogonal to the exact ground state, and $\varepsilon$ is small enough, the power method is guaranteed to converge to the ground state. Consequently, if the basis represents both $\psi$ and $H \psi$ exactly, and we minimize the energy within this basis, we expect exact convergence. The crucial point is the need to enlarge the basis to represent $H \psi$. Within the standard DMRG basis obtained from $\psi$, after solving for the ground state, $H \psi = E \psi$, and nothing is changed by adding $H \psi$ to the basis. To go beyond the basis, we need to construct the parts of $H \psi$ as the basis is built up. The crucial terms of $H \psi$ come from the terms of $H$ which connect the system and environment blocks.
For the current superblock configuration, write the Hamiltonian in the form $$\begin{aligned}
H = \sum_\alpha t_\alpha \hat A^\alpha \hat B^\alpha.\end{aligned}$$ Here the $\hat A^\alpha$ act only on the system block (including the site to be added to it), and the $\hat B^\alpha$ act only on the environment block (plus its site). All the terms which do not connect the blocks are contained in two terms of the sum which have either $A$ or $B$ equal to the identity operator, so that this form is completely general. (The other term in each case is the block Hamiltonian.) In order to put $H \psi$ into the basis, we need to target, in addition to $\psi$, the terms ${\hat A^\alpha \psi}$ for all $\alpha$. Let the states of the system have indices $s$, $p$, and $q$, and the states of the environment $e$. The state $\hat A^\alpha \psi$ can be written as $$\begin{aligned}
\sum_{se} \sum_{p} A^\alpha_{sp} \psi_{pe} | s{\rangle}| e{\rangle}.\end{aligned}$$ Targetting this wavefunction means adding into the density matrix a term $$\begin{aligned}
\Delta \rho^\alpha_{ss'} = a_\alpha \sum_{epq} A^\alpha_{sp} \psi_{pe} \psi_{qe}^* {A^\alpha_{s'q}}^*\end{aligned}$$ where $a_\alpha$ is an arbitrary constant determining how much weight to put into this additional state. The total contribution of all the terms is $$\begin{aligned}
\Delta \rho = \sum_{\alpha} a_\alpha \hat A^\alpha \rho \hat A^{\alpha\dagger}\end{aligned}$$ where $\rho$ is the density matrix determined in the usual way, only from $\psi$. This is the form of the correction that we use, with $a_\alpha = a \sim 10^{-3}-10^{-4}$.
As a second derivation, we utilize perturbation theory. First, imagine that the environment block, but not the system block, is complete. We obtain the ground state exactly for this superblock, and then transform to the basis of density matrix eigenstates for the system block, and then also do the same for the environment block. Then the wavefunction can be written in the form $$\begin{aligned}
|\psi {\rangle}= \sum_s \psi_s |L_s{\rangle}|R_s{\rangle}.
\label{psieqn}\end{aligned}$$ The reduced density matrix is $$\begin{aligned}
\label{rhoeqn}
\rho = \sum_s{\langle}R_s|\psi {\rangle}{\langle}\psi | R_s {\rangle}= \sum_s |\psi_s|^2 |L_s{\rangle}{\langle}L_s|\end{aligned}$$
Now consider the realistic case where the environment block is not complete. Assume the incompleteness takes the simple form that some of the $|R_s{\rangle}$ are missing, labeled $\bar s$, whereas $s$ are present. Let $P$ be a projection operator for the environment block $P = \sum_s |s{\rangle}{\langle}s|$, and take $\bar P=1-P$. Let the unperturbed ground state, with energy $E_0$ and density matrix $\rho_0$, be obtained using the incomplete environment basis. We take as a perturbation the terms in the Hamiltonian which couple to the states $\bar s$, namely $$\begin{aligned}
H' = \sum_\alpha t_\alpha \hat A^\alpha (\bar P \hat B^\alpha P + P \hat B^\alpha \bar P).\end{aligned}$$ The first order perturbative correction to the wavefunction due to $H'$ is $$\begin{aligned}
\label{psipeqn}
|\psi' {\rangle}= \sum_\alpha t_\alpha (E_0-H_0)^{-1}
\hat A^\alpha \bar P \hat B^\alpha |\psi {\rangle}\end{aligned}$$ where $H_0 = H-H'$.
In order to make progress we assume that each perturbation term $A^\alpha \bar P \hat B^\alpha$ acting on the ground state creates a set of nearly degenerate excited states, with average energy $E_\alpha$. This assumption is equivalent to saying that the spectral function associated with each term is dominated by a narrow peak at $E_\alpha$. This significant approximation is reasonable because the correction to the density matrix is only used to enlarge the basis, to improve DMRG convergence. Correspondingly, we approximate $(E_0-H_0)^{-1}$ as $(E_0-E_\alpha)^{-1} \equiv 1/\varepsilon_\alpha$. This gives $$\begin{aligned}
\label{psipeqntwo}
|\psi' {\rangle}\approx \sum_s \psi_s \sum_\alpha \frac{t_\alpha }{\varepsilon_\alpha}
\hat A^\alpha \bar P \hat B^\alpha |L_s{\rangle}|R_s{\rangle}.\end{aligned}$$ There are no first order corrections to the density matrix from $|\psi'{\rangle}$, since $\bar P |\psi {\rangle}= 0$. The lowest order correction to $\rho$ can be written as $$\begin{aligned}
\label{rhoeqntwo}
\Delta \rho = \sum_{ss'} \psi_s \psi^*_{s'} \sum_{\alpha{\alpha'}}
\frac{t_\alpha}{\varepsilon_\alpha}
\frac{t_{\alpha'}}{\varepsilon_{\alpha'}}
\hat A^\alpha |L_s{\rangle}{\langle}L_{s'}| \hat A^{\alpha \dagger}
M_{s'\alpha'\alpha s}\end{aligned}$$ where $$\begin{aligned}
\label{rhoeqnthree}
M_{s'\alpha'\alpha s} =
{\langle}R_{s'} | \hat B^{\alpha'\dagger} \bar P \hat B^\alpha |R_s{\rangle}\end{aligned}$$
Here if $A$ is the unit operator, the term adds nothing to the basis. If $B$ is the unit operator, $M_{s'\alpha'\alpha s}$ vanishes. For the nontrivial pairs of operators $A$ and $B$, this matrix element somewhat resembles a correlation function and it is natural to assume that the diagonal terms are dominant, where $\alpha={\alpha'}$ and $s=s'$. We expect the off diagonal terms $\alpha\ne{\alpha'}$ to describe coherence between different perturbation terms which would tend to reduce the number of basis functions needed to describe the system block; therefore, ignoring the offdiagonal terms is a conservative assumption. Accordingly, we take $$\begin{aligned}
M_{s'\alpha'\alpha s}
\approx \delta_{s{s'}} \delta_{\alpha{\alpha'}} b_\alpha\end{aligned}$$ This gives Eq. (4) with $a_\alpha = b_\alpha |t_\alpha|^2/\varepsilon_\alpha^2$, and where we omit block-Hamiltonian terms.
In practice, we take $a_\alpha$ to be a small constant $a$ independent of $\alpha$. Construction of the correction to $\rho$ take a calculation time for a single step proportional to $m^3$ times the number of connecting terms, which is typically significantly smaller than the other parts of the DMRG calculation, although the scaling is the same. Larger values of $a$ introduce more “noise” into the basis, speeding convergence, but also limiting the final accuracy. Note that it is just as easy to apply the correction within the two-site method as the single-site method, which may be useful in some very difficult cases. We do not present results for this combination here.
![ Error in the total energy for a 100 site Heisenberg spin-one chain, keeping $m=50$ states per block, and using open boundaries terminated with $S=1/2$ spins to remove the $S=1/2$ end states (98 $S=1$ sites + 2 $S=1/2$’s). The results are displayed for each half-sweep corresponding to reaching either the left or right end of the system. The two site method is the standard DMRG approach. The numerically exact energy was determined with the two-site method, using $m=200$. []{data-label="figtwo"}](fig2.eps){width="5cm"}
As a first test calculation, we consider the $S=1$ Heisenberg model $$H = \sum_j \vec{S}_j \vec{S}_{j+1} \;,$$ where we have set the exchange coupling $J$ to unity. The correction consist of the following: for each boundary site $i$ of a block, i.e. a site directly connected to the other block, we add into the density matrix $$\begin{aligned}
\label{rhopeqn}
\Delta \rho = a (S^+_i \rho S^-_i + S^-_i \rho S^+_i + S^z_i \rho S^z_i).\end{aligned}$$ For a chain with open boundaries, there is one site $i$; for periodic boundaries, there are two. One could argue that this expression should be adjusted with factors of 2 between the $z$ term and the other two terms, but this is not likely to make a significant difference. Note that the $S^+$, $S^-$ terms automatically increase the range of quantum numbers (i.e. total $S^z$) with nonzero density matrix eigenvalues. Figure 2 shows the convergence of the energy for a 100 site chain with open boundaries as a function of the sweep, keeping $m=50$ states, relative to the numerically exact result obtained with $m=200$ and 10 sweeps. One can see the excellent convergence of the standard approach. The single-site method without corrections does not do too badly in this case, but still gets stuck significantly above the two-site energy. Adding the corrections, in this case with $a=10^{-4}$, dramatically improves the convergence, making the single site method converge nearly as fast as the two site method. The two site method is roughly a factor of three slower than the single site method. Thus, even in this simple 1D case where the standard approach works extremely well, there are advantages to using the corrected single site method.
![ Error in the total energy for a 100 site Heisenberg spin-one chain, with periodic boundaries. The number of states kept per block is indicated, and is the same for all three methods; four sweeps were made for each $m$. The correction parameter $a$ was taken to be $10^{-4}$ for sweeps 1 - 8, and $10^{-6}$ for later sweeps. A somewhat slower convergence is visible for $a=10^{-6}$. The reference energy used was 100 times the infinite energy per site, -1.401484038971(4)[@whitehuse]. The corrected single site method using $m=4000$ states gives a slightly lower total energy, due to exponentially small finite size effects, of -140.14840390392. []{data-label="figthree"}](fig3.eps){width="5cm"}
The results change significantly if we consider periodic boundary conditions. Here we consider the same superblock configuration as with open boundary conditions, but simply add in the connection to the Hamiltonian between the first and last sites. There are better configurations for periodic boundaries, such as considering it to be a ladder with the interchain couplings turned off except at the ends. These other configurations are superior only in the sense of improved convergence with the number of sweeps, not improved with respect to the number of states for a large number of sweeps. This naive configuration thus provides a difficult test for the single site method with corrections. In Fig. 3, we show the results for the same three cases as in Fig. 2. In this case, in the early sweeps, both uncorrected methods are stuck, ignoring the extra link between the first and last sites. The extra link eventually appears in the basis, but there is still sticking two or three times in higher energy states. In contrast, the corrected single site method never gets stuck and shows excellent convergence.
A very useful DMRG technique is the extrapolation of the energy with the truncation error, i.e. the weight in the states which are thrown out. If the truncation error were measured exactly, with a complete basis for the environment, then the energy error would be proportional to the truncation error, allowing a linear extrapolation to zero truncation error. In practice, the apparent truncation error from the two site method may often be an underestimate, but one often finds that it is very consistent and still allows excellent extrapolation, even on fairly wide ladders. The truncation error within the corrected single-site method depends on $a$: as $a \to 0$, the apparent truncation error goes to zero and is unrelated to the exact truncation error. However, if $a$ is not too small, linearity and excellent extrapolation are possible.
Figure 4 shows results for the 100 site periodic system with a larger value of $a$, $10^{-2}$, suitable for extrapolation. The results show excellent linearity. The extrapolation gives -140.148416, off by $1.2\times10^{-5}$, whereas the sweep with $m=340$ gave -140.148279, off by $1.2\times10^{-4}$. We have found that typically an order of magnitude improvement in the estimate for the energy is obtained by extrapolation in good cases; here we see similar improvement. In performing these extrapolations one always need to check the linearity for the system being studied.
![ Error in the total energy for the system of Fig. 3 versus the truncation error, with $a=10^{-2}$. In this run two sweeps for each value of $m$, were made. The points shown are for $m = $80, 100, 120, 160, 200, 260, and 340. The line is a linear extrapolation, weighted with a standard deviation for each point assumed to be proportional to the truncation error at that point. []{data-label="figfour"}](fig4.eps){width="5cm"}
In summary, we have demonstrated a correction to the density matrix which allows the single-site DMRG method to converge well, and which improves the convergence dramatically for hard-to-converge systems.
We thank J. Rissler and F. Verstraete for helpful conversations. We acknowledge the support of the NSF under grant DMR03-11843.
S.R. White, Phys. Rev. Lett. [**69**]{}, 2863 (1992); Phys.Rev. B [**48**]{}, 10345 (1993).
U. Schollwöck, Rev. Mod. Phys. 77, 259 (2005).
For example, see S.R. White and D.J. Scalapino, , 1272 (1998).
S.R. White and D.A. Huse, , 3844 (1993).
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---
abstract: |
In this paper, a class of nonlinear driftless control-affine systems satisfying the bracket generating condition is considered. A gradient-free optimization algorithm is developed for the minimization of a cost function along the trajectories of the controlled system. The algorithm comprises an approximation scheme with fast oscillating controls for the nonholonomic dynamics and a model-free extremum seeking component with respect to the output measurements. Exponential convergence of the trajectories to an arbitrary neighborhood of the optimal point is established under suitable assumptions on time scale parameters of the extended system. The proposed algorithm is tested numerically with the Brockett integrator for different choices of generating functions.
KEYWORDS: nonholonomic systems, extremum seeking, stability of nonlinear systems, output feedback control, Lyapunov methods.
author:
- 'Victoria Grushkovskaya$^{1,4}$'
- 'Alexander Zuyev$^{2,3,4}$'
bibliography:
- 'biblio\_es.bib'
title: |
**Extremum Seeking Approach for Nonholonomic Systems\
with Multiple Time Scale Dynamics[^1]**
---
Introduction
============
Extremum seeking theory aims at designing universal control algorithms which steer the trajectories of dynamical systems with uncertainties to the minimum (or maximum) of a cost function whose analytical representation may be partially or completely unknown. The first results in this direction date back to the twenties of the last century, while the first thorough analysis of the stability properties of extremum seeking systems has been carried out only in the early 2000s, cf. [@Kr00]. Since then, many new extremum seeking algorithms and their applications have been developed (see, e.g., [@Krst03; @Tan06; @Nes10; @Fu11; @Liu12; @Durr13; @Har13; @Guay15; @Ben16; @GE16; @EMGG17; @Pov17; @SK17; @SD17; @GZE18; @Gu18; @Lab19; @Mand19]). A special place in these extremum seeking studies is given to nonlinear systems with dynamic input-output maps of the form $$\label{sys_gen}
\begin{aligned}
&\dot x=f(x,\xi),\quad x\in\mathbb R^n, \xi\in\mathbb R^m,\,f:\mathbb R^n\times\mathbb R^m\to \mathbb R^n,\\
&y=h(x,\xi),\quad y\in\mathbb R^p,\,h:\mathbb R^n\times\mathbb R^m\to\mathbb R^p.
\end{aligned}$$ The classical extremum seeking problem statement for system is to define the input $\xi$ in such a way that the output of system is optimized in the sense of minimization (or maximization) of an output-dependent cost function $J:\mathbb R^p\to\mathbb R$. In this direction one can mention, e.g., the papers by [@Kr00; @Tan06; @Ghaf12; @Guay15; @Har16; @Durr17; @Gu18]. Typically, extremum seeking approaches for are based on the construction of a dynamic extension $\dot \xi =g\big(J(y),t\big)$, where $g:\mathbb R\times [0,\infty)\to \mathbb R^m$ is chosen to ensure the desired vicinity of the trajectories of to an optimal point. The analysis of the resulting system relies on singular perturbation theory and requires that system admits a steady-state $x=\ell(\xi)$, which is asymptotically stable for each fixed value of $\xi$. Furthermore, a crucial assumption in such studies is the existence of certain Lyapunov function for system . However, there are many important classes of systems which do not admit a control Lyapunov function with desired properties.
In this paper, we consider a class of nonholonomic systems governed by driftless control-affine systems, in which the number of inputs can be significantly smaller than the number of state variables. In general, the linearization of these systems is not controllable. Moreover, as it was proved in the famous work by [@Bro83], such nonholonomic systems cannot be stabilized by a continuous feedback law. To stabilize such systems one can use, e.g., discontinuous (e.g., [@ast94; @Clar97]) or time-varying feedback laws (e.g., [@ZuSIAM; @GZ18]). Consequently, the resulting closed-loop system becomes discontinuous or non-autonomous and, in general, does not admit a regular Lyapunov function of the form $V(x)$.
The goal of our paper is to construct extremum seeking controls for a class of nonholonomic systems with time-varying inputs adapted from [@GZ18]. We propose a novel solution of the extremum seeking problem for nonholonomic systems based on combination of stabilizing strategies for nonholonomic systems and gradient-free extremum seeking controllers. Although the main idea of our control design approach is inspired by singular perturbation techniques, we do not apply them directly in the proof. Instead, we propose a novel approach for dynamic stabilization of nonholonomic systems and generalize the techniques introduced in [@GZE18] to systems with multiple time scales.
The rest of this paper is organized as follows. In Section \[sec\_prel\], we introduce basic notations, formulate the problem statement, and describe the main idea of our control design approach. Section \[sec\_main\] provides the main results of the paper, which are illustrated with an example in Section \[sec\_exmpl\]. Section \[sec\_concl\] contains concluding remarks. Some auxiliary statements are given in Appendix A, and the proof of the main result is contained in Appendix B.
Preliminaries
=============
\[sec\_prel\]
Notations and Definitions
--------------------------
$\delta_{ij}$ is the Kronecker delta; ${\rm dist}(x,S)$ is the Euclidian distance between an $x\in\mathbb R^{n}$ and an $S\subset\mathbb R^{n}$;
$B_\delta(x^*)$ is a $\delta$-neighborhood of an $x^*\in \mathbb R^n$;
$\partial M$, $\overline M$ is the boundary and the closure of a set $M\subset\mathbb R^n$, respectively; $\overline M= M\cup \partial M$;
$|S|$ is the cardinality of a set $S$;
$\mathcal K$ is the class of continuous strictly increasing functions $\varphi:\mathbb R^+\to\mathbb R^+$ such that $\varphi(0)=0$;
$[f,g](x)$ is the Lie bracket of vector fields $f,g:\mathbb R^n\to\mathbb R^n $ at a point $x\in\mathbb R^n$, $[f,g](x)=L_fg(x)- L_gf(x)$, where $ L_gf(x)=\lim_{s\to0}\dfrac{f(x+sg(x))-f(x)}{s}$.
Similarly to [@Clar97; @ZuSIAM], we exploit the sampling approach for the stabilization of nonholonomic systems. Given an $\varepsilon{>}0$, we define the partition $\pi_\varepsilon$ of $[0,+\infty)$ into the intervals $$I_j=[t_j,t_{j+1}),\;t_j=\varepsilon j, \ j\in\mathbb N{\cup}\{0\}.$$
Assume given a feedback $u=\varphi(x,\xi,t)$, $\varphi:D\times D\times[0,+\infty)\to\mathbb R^m$, $\varepsilon>0$, and $x^0,\xi^0\in D\subseteq \mathbb R^n$. *A $\pi_\varepsilon$-solution* of the system $$\begin{aligned}
&\dot x=f(x,u),\ \dot \xi =g(x,\xi,t),\, x,\xi\in D\subseteq \mathbb R^n, u\in\mathbb R^m,
\end{aligned}
\label{extended}$$ corresponding to $(x^0,\xi^0,\varphi)$, is an absolutely continuous function $(x^\top(t),\xi^\top(t))^\top\in D\times D$, defined for $t\in[0,+\infty)$, which satisfies the initial conditions $x(0)=x^0$, $\xi(0)=\xi^0$ and the differential equations $$\begin{aligned}
&\dot x(t)=f\big(x(t), \varphi(x(t_j),\xi(t_j),t)\big), \quad t\in I_j=[t_j,t_{j+1}),\\
&\dot \xi(t) = g(x(t),\xi(t),t)\text{ for each }j=0,1,2,\dots\ .
\end{aligned}$$
The above definition will be applied for the stabilization of nonholonomic systems using the approach of [@ZuSIAM; @GZ18]. However, the extremum seeking scheme proposed in this paper can also be used for output stabilization of systems with well-defined classical solutions.
Problem statement & Main idea
------------------------------
Consider a class of nonholonomic systems governed by driftless control-affine equations with single output: $$\label{sys1}
\begin{aligned}
&\dot x=\sum_{i=1}^mu_if_i(x),\\
& y=J(x),
\end{aligned}$$ where $x=(x_1,\dots,x_n)^\top\in D{\subseteq} \mathbb R^n$ is the state, $x(0)=x^0\in D$, $u=(u_1,\dots,u_m)^\top\in\mathbb R^m$ is the control, $m<n$, $y\in\mathbb R$ is the output of the system, $J:D\to\mathbb R$ is the cost function, and the vector fields $f_i:D\to \mathbb R^n$ are linearly independent. Let the following rank condition be satisfied in $D$: $$\label{rank}
\begin{aligned}
{\rm span}\big\{f_{i}(x), [f_{j_1},f_{j_2}](x) \,|\,i\in S_1,(j_1,j_2)\in S_2\big\}=\mathbb{R}^n,
\end{aligned}$$ where $S_1\subseteq \{1,2,...,m\}$ and $S_2\subseteq \{1,2,...,m\}^2$ are some sets of indices, $|S_1|+|S_2|=n$. We study the following extremum seeking problem:
Let $J\in C^2( D;\mathbb R)$ be a strongly convex function, and let $x^*\in D$ be such that $J(x)>J(x^*)$ for all $x\in D\setminus\{x^*\}$. The goal is to construct a control law $u=u(t,x,J(x))$ such that the trajectories $x(t)$ of system with the initial conditions from $D$ tend asymptotically to an arbitrary small neighborhood of $x^*$.
The main idea of the control algorithm proposed in this paper can be described in two stages:
\(1) For each value $\xi\in D$, we construct time-periodic fast oscillating control laws with *state-dependent* coefficients to ensure that the corresponding steady-state $x=\xi$ of is asymptotically (and even exponentially) stable. Further we assume that $\xi(t)$ evolves according to certain differential equations, so the result of [@ZuSIAM; @GZ18] cannot be directly applied for establishing stability properties of the extended system . Note that, in general, does not admit a control Lyapunov function. Instead, we will prove that with the proposed choice of the control $u$ the trajectory $x(t)$ remains in a sufficiently small neighborhood of $\xi(t)$ for $t\in[0,\infty)$. These controls are model-based, i.e. the dynamics (control vector fields) and the coordinates of the system are assumed to be known, but not the analytical expression of $J$ and the optimal point $x^*$. We will apply sampling controllers, that is the solutions of will be defined in the sense of Definition 1.
\(2) To optimize the state $x=\xi$ with respect to minimizing the cost function $J(x)$ along the trajectories of , we construct a dynamic extension $\dot \xi=g(y,t)$, where $g(y,t)$ is taken in the form of fast oscillating time-periodic functions with *output-dependent* coefficients from ([@GZE18]). Thus, this part of the controller is model-free.
[In Problem 1, we assume that the cost function $J$ depends only on the state variable $x$, but not on the control input $u$. This assumption is not crucial and is made in order to simplify the proof. Besides, if $J$ depends only on $u$, the stability properties directly follow from ([@GZE18]) and ([@GZ18]) with the same proof techniques. ]{}
Main results
============
\[sec\_main\]
Control design
--------------
In this section, we formalize the control algorithm announced in Subsection 2.2. Namely, the overall system has the following form:
$$\begin{aligned}
&\dot x=\sum_{i=1}^mu_if_i(x),\, x(0)=x^0\nonumber,\\
& u_i=\varphi_i^\varepsilon(x,\xi,t),\label{nonhA}\\
&y=J(x),\nonumber\\
&\dot \xi=g(y,t),\, g(y,t)=\sum_{j=1}^{2n}g_{j}(y)v_{j}^\mu(t)e_j,\;\quad\;\,\xi(0)=x^0.\label{nonhB}\end{aligned}$$
\[nonh\]
In , the stabilizing component $u_i=\varphi_i^\varepsilon(x,\xi,t)$ is [$$\begin{aligned}
&\varphi_i^\varepsilon(x,\xi,t)=\sum_{i_1\in S_1}a_{i_1}(x,\xi)\delta_{i i_1}\label{cont}\\
&+\sqrt{\dfrac{4\pi}{\varepsilon}}\sum_{(i_1,i_2)\in S_2}{\sqrt{\kappa_{i_1i_2}|a_{i_1i_2}(x,\xi)|}} \Big(\delta_{ii_1}{\rm sign}(a_{i_1,i_2}(x,\xi))\cos{\dfrac{2\pi \kappa_{i_1i_2}}{\varepsilon}}t+\delta_{ii_2}\sin{\dfrac{2\pi \kappa_{i_1i_2}}{\varepsilon}}t\Big).\nonumber\end{aligned}$$ ]{} Here $\kappa_{i_1i_2}\in\mathbb N$, $\kappa_{i_1i_2}\ne\kappa_{i_3i_4}$ for all $(i_1,i_2)\ne(i_3,i_4)$, and $$a(x,\xi)=\Big((a_{i_1}(x,\xi))_{i_1\in S_1}\ ( a_{i_1i_2}(x,\xi))_{(i_1,i_2)\in S_2}\Big)^\top\in\mathbb R^n$$ is defined as [$$\label{a}
a(x,\xi)=- \gamma_1 \mathcal F^{-1}(x) (x-\xi)$$]{} with $\mathcal F^{-1}(x)$ being the $n\times n$ matrix inverse to $$\mathcal F(x)= \Big(\big(f_{j_1}(x)\big)_{j_1\in S_1}\ \ \big([f_{j_1},f_{j_2}](x)\big)_{(j_1,j_2)\in S_2}\Big),$$ and the control gain $\gamma_1{>}0$ to be defined later in the proof of the main result.
Such a choice of $u_i$ is aimed to ensure that the trajectories $x(t)$ are close enough to $\xi(t)$ for all $t\ge 0$ and all initial conditions $x(0)$. Note that the rank condition implies nonsingularity of $\mathcal F(x)$ for any $x\in D$.
In , $g(y,t)$ is the extremum seeking component. Here $e_j$ denotes the unit vector in $\mathbb R^n$ with non-zero $j$-th entry if $j\le n$, and non-zero $(j-n)$-th entry if $n+1\le j\le 2n$, the functions $g_j,g_{j+n}$ have to satisfy the relation $$[g_j(z),g_{j+n}(z)]=-\gamma_2,\;\gamma_2>0, \ j=\overline{1,n}.$$ For example, the choice $g_{j+n}(z)=-\gamma_2 g_{j}(z)\int{\dfrac{dz}{g_{j}(z)^2}}$ was proposed in [@GZE18]. In this paper, we propose to parameterize the functions $g_j,g_{j+n}$ as $$\label{class}
\begin{aligned}
&g_j(z)=r_j(z)\sin\phi_j(z),\,g_{j+n}(z)=r_j(z)\cos\phi_j(z),\\
&\text{with }r_j,\phi_j\text{ such that }r_j^2(z)\phi'_j(z)\equiv \gamma_2.
\end{aligned}$$ The discrete-time version of the above parametrization has also been used by [@FG19].
Next, the inputs $v_{j}^\mu(t)$ are given by $$v_{j}^\mu(t)=\left\{
\begin{aligned}
\sqrt{\dfrac{4\pi k_j}{\mu}}\cos\dfrac{2\pi k_jt}{\mu} &\text{ for }j=\overline{1,n},\\
\sqrt{\dfrac{4\pi k_{j-n}}{\mu}}\sin\dfrac{2\pi k_{j-n}t}{\mu} &\text{ for }j=\overline{n+1,2n},
\end{aligned}
\right.$$ where $\mu>0$, $k_j\in\mathbb N$, $k_{j_1}\ne k_{j_2}$ for all $j_1\ne j_2$.
[Although the choice of $g_j,g_{j+n}$ in may look rather artificial, there are many extremum seeking systems whose control vector fields satisfy this relation. For example, the functions $g_{j}(z)=z$, $g_{j+n}(z)=1$ have been exploited by [@Durr13a; @Durr17]; $g_{j}(z)=\sin\ z$, $g_{j+n}(z)=\cos\ z$ by [@SK17]; $g_{j}(z)=\sqrt z\sin(\ln z)$, $g_{j+n}(z)=\sqrt z\cos(\ln z)$ by [@SD17]; $g_j(z)=\sqrt{\dfrac{1-e^{-z}}{1+e^{z}}}\sin(e^{z}+2\ln(e^{z}-1))$, $g_{j+n}(z)=\sqrt{\dfrac{1-e^{-z}}{1+e^{z}}}\cos(e^{z}+2\ln(e^{z}-1))$ by [@GZE18]. One more example will be given in Section \[sec\_exmpl\].]{}
Stability conditions
---------------------
Assume that the cost function $J\in C^2(D;\mathbb R)$ satisfies the following properties in $D$: [ $$\begin{aligned}
\sigma_{11}\|x-x^*\|^2\le J(x)-J^* &\le\sigma_{12}\|x-x^*\|^2,\nonumber\\
\sigma_{21}(J(x)-J^*)\le \|\nabla J(x)\|^2&\le\sigma_{22}(J(x)-J^*),\label{J}\\
&\Big\|\dfrac{\partial^2J(x)}{\partial x^2}\Big\|\le\sigma_3,\nonumber
\end{aligned}$$]{} with $x^*\in D$ and some positive constants $\sigma_{11}$, $\sigma_{12}$, $\sigma_{21}$, $\sigma_{22}$, $\sigma_3$. The main result of this paper is as follows.
\[thm\_step1\]
Given system and a function $J\in C^2(D;\mathbb R)$ satisfying , assume that:
- the vector fields $f_i\in C^2(D;\mathbb R^n)$ in satisfy in $D$, and there is an $\alpha{>}0$ such that $\|\mathcal F^{-1}(x)\|\le \alpha \text{ for all }x\in D;$\
- $g_{j}(J(\cdot))\in C^2(D\setminus\{x^*\};\mathbb R)$, $L_{g_{j}}g_{i}(J(\cdot))$, $L_{g_{l}}L_{g_{j}}g_{i}(J(\cdot))\in C(D;\mathbb R)$ for all $i,j,l=\overline{1,2n}$;\
- for any compact $D'\subseteq D$, there are $L_g, L_{2g}, M_{3g}\ge 0$ s.t. $$\begin{aligned}
&\|g_{i}\big(J(x)\big)-g_{i}\big(J(\xi)\big)\|\le L_g\|x-\xi\|,\\
&\|L_{\big(g_{j_2}(J(x))-g_{j_2}(J(\xi))\big)}g_{j_1}\big(J(\xi)\big)\|\le L_{2g}\|x-\xi\|,\\
&\|L_{g_{j_3}(J(x))} L_{g_{j_2}(J(\xi))}g_{j_1}\big(J(\xi)\big)\|\le M_{3g},\quad x,\xi\in D',\,i,j,l=\overline{1,2n}.
\end{aligned}$$
Then, for any $\delta\in\Big(0,\sqrt{{\sigma_{11}}/{\sigma_{12}}}{\rm dist}(x^*,\partial D)\Big)$ and any $\rho{>}0$, there exist $\bar\mu>0,\bar\gamma_1(\mu)>0$, and $\bar\varepsilon(\gamma_1,\mu)>0$ such that, for any $\mu\in(0,\bar\mu]$, $\gamma_1\in[\bar \gamma_1(\mu),\infty)$, and any $\varepsilon\in(0,\bar\varepsilon(\gamma_1,\mu)]$, each $\pi_\varepsilon$ solution of with $u_i=\varphi_i^\varepsilon(x,\xi,t)$ defined by and the initial conditions from $\overline{B_\delta(x^*)}$ satisfies $$\label{decay}
\|x(t)-x^*\|\le \beta\|x^0-x^*\|e^{-\lambda t}+\rho\text{ for all }t\in[0,\infty),$$ with some $\beta,\lambda>0$.
The proof of this theorem is given in Appendix B.
[ The proof of Theorem \[thm\_step1\] represents a constructive procedure for choosing $\bar\mu$, $\bar\gamma_1(\mu)$, $\bar\varepsilon(\gamma_1,\mu)$, and clarifies the relation between these parameters and the coefficients $\beta$ and $\lambda$. We would like to underline that the proposed bounds are quite conservative. The crucial assumption is $\varepsilon<\mu$, which means that subsystem oscillates faster than subsystem . To simplify the proof, we also suppose that $\dfrac{\mu}{\varepsilon}\in\mathbb N$ and $x(0)=\xi(0)$, however the assertion of Theorem 1 can also be obtained without these assumptions.]{}
In order to have $\gamma_1$ independent on $\mu$, one may introduce an additional parameter $\eta$ which will ensure a “slow” dynamics of (similarly to, e.g., [@Durr17]). This, however, will result in a slower convergence rate of the overall system to the optimal point. Namely, by taking $\tilde v_j^\mu(t):=\dfrac{1}{\eta}v_j^\mu\Big(\dfrac{t}{\eta}\Big)$ in and keeping the conditions of Theorem \[thm\_step1\], one can prove the following statement:
*For any $\delta\in\Big(0,\sqrt{\dfrac{\sigma_{11}}{\sigma_{12}}}{\rm dist}(x^*,\partial D)\Big)$ and any $\rho>0$, there exist $\bar\mu>0$, $\bar\varepsilon(\mu)>0$, and $\bar\eta(\varepsilon,\mu)>0$ such that, for any $\mu\in(0,\bar\mu]$, $\varepsilon\in(0,\bar\varepsilon(\mu)]$ and $\eta\in[\bar \eta(\varepsilon,\mu),\infty)$, each $\pi_\varepsilon$-solution of with $u_i=\varphi_i^\varepsilon(x,\xi,t)$ defined by and the initial conditions from $\overline{B_\delta(x^*)}$ satisfies $
\|x(t)-x^*\|\le \beta\|x^0-x^*\|e^{-\dfrac{\lambda t}{\eta}}+\rho\text{ for all }t\in[0,\infty),\,\beta,\lambda>0.
$*
Similarly to [@GZE18], the behavior of the solutions of can be improved by generating $g_j$ vanishing at $x^*$. We will illustrate this feature with an example in the next section.
Example
=======
\[sec\_exmpl\] As an example, consider the well-known Brockett integrator ([@Bro83]): $$\label{bro_x}
\begin{aligned}
& \dot x_1 =u_1,\ \dot x_2=u_2,\ \dot x_3=u_1x_2-u_2x_1.
\end{aligned}$$ It is easy to see that, for all $x\in\mathbb R^3$, the vector fields $f_1=(1,0,x_2)^\top$ and $f_2=(0,1,-x_1)^\top$ of system satisfy the rank condition with $S_1=\{1,2\}$, $S_2=\{(1,2)\}$: $
{\rm span}\big\{f_1(x),\,f_2(x),\, [f_1,f_2](x)\big\}=\mathbb{R}^3\text{ for all }x\in\mathbb R^3;
$ thus, we may apply the control algorithm proposed in Section 3.1. Namely, we take [ $$\begin{aligned}
&u_1=a_1(x,\xi)+\sqrt{{4\pi\kappa_{12}\big|a_{12}(x)\big|}/{\varepsilon}}\ {\rm sign}\big({a_{12}(x,\xi)}\big)\cos({2\pi\kappa_{12}t}/{\varepsilon}),\nonumber\\
&u_2=a_2(x,\xi)+\sqrt{{4\pi\kappa_{12}\big|a_{12}(x)\big|}/{\varepsilon}}\ \sin({2\pi\kappa_{12}t}/{\varepsilon}),\label{bro_u}\\
&a(x,\xi)=\big(a_1(x,\xi),a_2(x,\xi),a_{12}(x,\xi)\big)^\top=- \gamma_1 \mathcal F^{-1}(x) (x-\xi)\nonumber\\
&\quad = -\gamma_1\left(
x_1-\xi_1,
x_2-\xi_2,
\dfrac{1}{2}\big(-x_2\xi_1+x_1\xi_2-x_3+\xi_3\big)^\top
\right), \nonumber\\
&\dot \xi_j=\sqrt{{4\pi k_j}/{\mu}}\Big(g_1(y)\cos({2\pi k_jt}/{\mu})+g_2(y)\sin({2\pi k_jt}/{\mu})\Big)e_j,\label{bro_xi}\end{aligned}$$ $j=1,2,3$]{}. In this example, we take $y=J(x)=\|x\|^2$, $\gamma_1=20$, $\gamma_2=1$, $\kappa_{12}=4$, $k_1=1$, $k_2=2$, $k_3=3$, and consider two types of functions $g_1,g_2$. The results of numerical simulations with the functions from [@Durr17], $$\label{class_durr}
g_1(z)=z,\,g_2(z)=1,$$ are depicted on Fig. \[fig\_bro\] (left). Here $\varepsilon=0.1$ and $\mu=0.5$.
To improve the qualitative behavior of –, we can apply another pair of the generating functions satisfying , which vanish when $J$ takes its minimal value, e.g., [ $$\begin{aligned}
& g_1(z)=\sqrt{\tanh{z}/{2}}\sin\big(2\ln(e^z-1)-z\big),\label{class_vv}\\
& g_2(z)=\sqrt{\tanh{z}/{2}}\cos\big(2\ln(e^z-1)-z\big)\text{ if }z>0,g_1(0)=g_2(0)=0.\nonumber\end{aligned}$$ ]{}
{width="1\linewidth"} {width="1\linewidth"} {width="1\linewidth"}
{width="1\linewidth"} {width="1\linewidth"} {width="1\linewidth"}
In this case, we took $\varepsilon=0.25$, $\mu=1$. Note that, unlike the results of [@GZE18], the trajectories of – exhibit non-vanishing oscillations in a neighborhood of the extremum point (which are, however, considerably smaller than with the functions ) (see Fig. \[fig\_bro\], right). Thus, an interesting question is whether it is possible to achieve asymptotic stability in the sense of Lyapunov with the proposed control algorithm.
In both case, we take the initial conditions $x(0)=(1,-1,1)^\top$, $\xi(0)=(-1,1,1)^\top$ to illustrate that the proposed approach can be applied also for $x^0\ne \xi^0$.
Conclusions & Future work
=========================
\[sec\_concl\] To simplify the presentation, we consider only the class of nonholonomic systems satisfying one-step bracket generating condition in this paper, i.e. we assume that the vector fields together with their Lie brackets span the whole $n$-dimensional space at each state $x\in D\subseteq {\mathbb R}^n$. Another hypothesis is put in , so that the cost $J$ possesses properties of a quadratic function. This hypothesis is introduced in order not to overcomplicate the proof of the main results. It should be emphasized that information about the analytical expression of $J$ and its minimizer $x^*$ is not required for the control design. Furthermore, all the constants in may also be unknown. In future work, we expect to address broader classes of cost functions possessing polynomial convergence properties, similarly to the results of [@GZE18]. We also plan to extend the proposed control design approach to nonholonomic systems under higher order controllability conditions with iterated Lie brackets.
Auxiliary results
=================
This section contains several technical results which be used for the proof of Theorem \[thm\_step1\].
\[lemma\_x\] [ Let $D{\subseteq}\mathbb R^n$, $\xi(t){\in }D$, $t\in[0,\tau]$, be a solution of the system $$\dot \xi=\sum_{i=1}^l h_i(\xi)w_i(t),$$ and let the vector fields $h_i$ be Lipschitz continuous in $D$ with the Lipschitz constant $L$. Then $$\|\xi(t)-\xi(0)\|\le t\nu\max_{1\le i\le l}\|h_i(\xi(0))\|e^{\nu Lt},\;t\in[0,\tau],$$ with $\nu=\max\limits_{t\in[0,\tau]}\sum_{i=1}^{l}|w_{i}(t)|$.]{}
Lemma \[lemma\_x\] follows from the Grönwall–Bellman inequality.
\[lemma\_volt\] [Let vector fields $h_i$ be Lipschitz continuous in a domain $D{\subseteq}\mathbb R^n$, and $h_i\in C^2(D\setminus\Xi;\mathbb R)$, where $\Xi=\{\xi{\in } D{:}h_i(\xi)=0\text{ for }1{\le }i{\le }l\}$, and $ L_{h_j}h_i, L_{h_l}L_{h_j}h_i\in
C(D;\mathbb R^n)$ for all $i,j,l=\overline{1,l}$. If $\xi(t)\in D$, $t\in[0,\tau]$, is a solution of $\dot \xi=\sum_{i=1}^l h_i(\xi)w_i(t)$ with $u\in C([0,\tau];\mathbb R^m)$ and $x(0)=x^0\in D$, then $\xi(t)$ can be represented by the Chen–Fliess series: $$\label{volt1}
\begin{aligned}
\xi(t)=\xi^0&+{\sum_{{i_1}=1}^{l}}h_{i_1}(\xi^0)\int\limits_0^t w_{{i_1}}(v)dv+\sum_{\hspace{-0.75em}{i_1},{i_2}=1}^{l} L_{h_{i_2}}h_{i_1}(\xi^0)\int\limits_0^t\int\limits_0^v w_{{i_1}}(v) w_{{i_2}}(s)dsdv\\
&+R(t),\qquad\qquad\qquad\qquad\qquad t\in[0,\tau],\\
R(t)=&\sum\limits_{\hspace{-0.75em}{i_1},{i_2},{i_3}=1}^{l}{\int\limits_0^t}{\int\limits_0^v}{\int\limits_0^s} L_{h_{i_3}} L_{h_{i_2}}h_{i_1}(\xi(p)) w_{{i_1}}(v)w_{{i_2}}(s)w_{{i_3}}(p)dpdsdv \end{aligned}$$ is the remainder of the Chen–Fliess series expansion.]{}
\[lemma\_rem\]\
[ Let the conditions of Lemma \[lemma\_x\] be satisfied and let $\xi^*\in D$. Assume that there exist $M_1,M_3{\ge}0$, $m{\ge} 1$, $\varpi\in\{0\}\cup[1,\infty)$ such that [ $$\begin{aligned}
\max\limits_{1\le {i_1}\le l}&\|h_{i_1}(\xi(0))\|\le M_1\|\xi(0)-\xi^*\|^m,\\
\max\limits_{1\le{i_1},{i_2},{i_3}\le l}&\|L_{h_{i_3}}L_{h_{i_2}}h_{i_1}(\xi)\|\le M_3\|\xi-\xi^*\|^{\varpi}\text{ for all } \xi\in D.
\end{aligned}$$]{} Then, for all $t\in[0,\tau]$, the remainder $R(t)$ of the Chen–Fliess expansion of $x(t)$ satisfies the estimate $$\begin{aligned}
\|R(t)\|\le &\dfrac{2^{\varpi-2}}{3}(t\nu)^3\|\xi(0)-\xi^*\|^\varpi M_3 \\
&\times\Big(1+M_1(\tau\nu)^\varpi e^{\nu L\varpi\tau}\|\xi(0)-\xi^*\|^{\varpi(m-1)}\Big).
\end{aligned}$$]{}
\[lemma\_decay\] [ Let $D\subseteq\mathbb R^n$ be a bounded convex domain, $W\in C^2(D;\mathbb R)$, $x^*\in D$, and let the following inequalities hold: $$\begin{aligned}
& \sigma_{11}\|x-x^*\|^{2m} \le W(x)\le \gamma_{12}\|x-x^* \|^{2m},\\
& \sigma_{21} W(x)^{2-\dfrac{1}{m}}\le \|\nabla W(x)\|^2\le\sigma_{22} W(x)^{2-\dfrac{1}{m}},\\
&\left\|\dfrac{\partial^2 W(x)}{\partial x^2}\right\|\le\sigma_3 W(x)^{1-\dfrac{1}{m}},\\
\end{aligned}$$ where $m\ge1$ and $\sigma_{11},\sigma_{12},\sigma_{21},\sigma_{22},\sigma_3$ are positive constants. Then, for any $x^0=x(0)\in D\setminus\{x^*\}$ and any function $x:[0,\varepsilon]\to D$ satisfying the conditions [ $$ x(0)=x^0,\; x(\varepsilon)=x^0-\gamma\varepsilon\nabla W(x^0)+r_\varepsilon,\,\gamma>0,\,r_\varepsilon\in\mathbb R^n,$$]{} the function $W$ satisfies the estimate: [$$\begin{aligned}
W(x(\varepsilon))\le W(x^0)\Big(1-\dfrac{\varepsilon\varkappa_1}{m}W^{1-\dfrac{1}{m}}(x^0)+\dfrac{\varepsilon^2\varkappa_2}{2m^2}W^{2-\dfrac{2}{m}}(x^0)\Big)^{m},
\end{aligned}$$]{} where [ $\varkappa_1=\gamma\sigma_{21}-{\sqrt{\sigma_{22}}\|r_\varepsilon\|}{ W^{\dfrac{1}{2m}-1}(x^0)}/\varepsilon$, $\varkappa_2=((m-1)\sigma_{22}+m\sigma_{12})\bigg(\gamma\sqrt{\sigma_{22}}+{\|r_\varepsilon\|}{ W^{\dfrac{1}{2m}-1}(x^0)}/\varepsilon\bigg)^2$.]{}]{}
Proof of Theorem \[thm\_step1\]
===============================
For the sake of clarity, we divide the proof into several steps resulting in intermediate statements.\
*Notations and preliminary constructions.* To practically stabilize system at $(0,x^*)$, we will focus on three parameters: $\gamma_1$, $\varepsilon$, and $\mu$, assuming that $\varepsilon<\mu$. In the proof, we will determine big enough $\gamma_1=\gamma_1(\mu)$, small enough $\varepsilon=\varepsilon(\gamma_1,\mu)$, and small enough $\mu$. It can be seen from the proof that such a choice is always possible. We use the following notations in the proof: for any $\tau\in[0,\varepsilon]$, [ $$\begin{aligned}
&U(x(\varepsilon),\xi(\varepsilon),\tau)=\max_{\varepsilon\le t\le \varepsilon+\tau}\sum_{i=1}^{m} |\varphi_i^\varepsilon(x(\varepsilon),\xi(\varepsilon),t)|,\\
& W(\tau)=\max_{0\le t\le \tau}\sum_{j=1}^{2n} |v_j^\tau(t)|\le \dfrac{c_{w}}{\sqrt\mu},\;c_{w}=2\sum_{j=1}^{n}\sqrt{2\pi k_j}.\end{aligned}$$]{} Recall that the state-dependent control coefficients are defined by , which implies that, for any $x(0)=x^0\in D$, $\xi(0)=\xi^0\in D$, $$\|a(x^0,\xi^0)\|\le \gamma_1\alpha\|x^0-\xi^0\|.$$ The Hölder inequality implies that, for any $\varepsilon>0$ and all $\tau\in[0,\varepsilon]$, [$$\label{est_u}
\begin{aligned}
&U(x^0,\xi^0,\tau)\varepsilon\le \varepsilon \sum_{i\in S_1}|a_i(x^0,\xi^0)|
+2\sqrt{2\pi\varepsilon}\\
&\times\sum_{(i_1,i_2)\in S_2}\sqrt{\kappa_{i_1i_2}|a_{i_1i_2}(x^0,\xi^0)|}\le {c_u}\sqrt {\gamma_1\varepsilon\|x^0-\xi^0\|},
\end{aligned}$$ where $c_{u}= \sqrt{\alpha}(\sqrt{\gamma_1\alpha\varepsilon\|x^0-\xi^0\||S_1|} + 2\sqrt{2\pi})\Big(\sum_{(j_1,j_2)\in S_2}{\kappa^{2/3}_{j_1j_2}}\Big)^{3/4}$]{} is strictly monotonically increasing w.r.t. $\varepsilon$.
For any [$\delta\in\Big(0,\sqrt{\dfrac{\sigma_{11}}{\sigma_{12}}}{\rm dist}(x^*,\partial D)\Big)$]{}, let [ $\delta_x\in\Big(\sqrt{\dfrac{\sigma_{12}}{\sigma_{11}}}\delta,{\rm dist}(x^*,\partial D)\Big),$]{} and let $D'$ be compact, $D_x=\overline{B_{\delta_x}(x^*)}{\subset} D'{\subseteq} D.$ If $D$ is compact, then we take $D'=D$. By the conditions of Theorem 1, there exist $M_f,M_g,M_{3g}>0$ such that, for all $x,\xi\in D_x$, [ $$\begin{aligned}
&\|f_i(x)\|\le M_f,\,\|g_{j}( J(x))\|\le M_g,\,i=\overline{1,m},\ j=\overline{1,2n}\label{M}\\
&\| L_{f_{j_1}}f_{j_2}(x)\|\le M_{2f},\ \big\| L_{f_{j_3}} L_{f_{j_2}}f_{j_1}(x)\big\|\le 6M_{3f},\,j_1,j_2,j_3=\overline{1,m}.\nonumber
$$ ]{} If and inequalities from the fourth condition of Theorem 1 hold globally in $D$, then we take $D'=D$.\
*At this step we construct some a priori estimates which will be exploited further in the proof.*\
It is easy to see that the $\pi_\varepsilon$-solutions of system satisfy [$$\begin{aligned}
&\| x(t)- x(0)\|\le M_f{c_u}\sqrt {\gamma_1\varepsilon\|x(0)-\xi(0)\|}\text{ for all }t\in[0,\varepsilon].\label{barx_est}
$$]{} Let $\rho_1{>}0$ be given, $\nu=c_wM_g$, $\varsigma{>}0$, $\rho_0={\rho_1\mu^\varsigma\sqrt\mu}/{3}$, $
\delta_\xi=\delta_x+\nu\sqrt\mu,$ $D_\xi=\overline{B_{\delta_\xi}(x^*)}$, $d={\rm dist}(x^*,\partial D')-\delta_x>0, $ and let $\mu_0$ be the smallest positive root of the equation [$$\label{eps0}
\sqrt\mu\Big({2\rho_1\mu^\varsigma}/{3}+\nu\Big)=d.$$]{} Obviously, for any $\mu\in(0,\mu_0]$, $\nu\sqrt\mu< d$, so that $\delta_\xi<\delta_x+d$ and $D_\xi\subseteq D'$. We will also assume that [$$\label{gamma0}
\gamma_1>\bar\gamma_1(\mu)=\dfrac{3\nu}{\rho_1\mu^{\varsigma+1}}.$$ ]{} Such a choice of $\gamma_1$ will be motivated in Step 2.\
Next, we take [$$\label{vareps0}
\varepsilon_0(\gamma_1,\mu)=\dfrac{1}{\gamma_1}\min\Big\{1,\dfrac{\rho_1\mu^{\varsigma}\sqrt\mu}{3M_f^2c_u^2}\Big\},$$]{} and observe that $$\varepsilon_0(\gamma_1,\mu)\le \dfrac{\rho_1\mu^{\varsigma+1}}{3\nu}$$ because of .
From and we obtain that, for each $\mu\in(0,\mu_0]$, $\gamma_1\in(\bar\gamma_1,\infty)$, $\varepsilon\in(0,\varepsilon_0(\gamma_1,\mu)]$, and for any $x(0)\in D_x$, $\xi(0)\in \overline{B_{\rho_0}(x(0))}$, if $\|\xi(t)-\xi(0)\|\le \dfrac{\nu\varepsilon}{\sqrt\mu}$ with $t\in[0,\varepsilon]$ then [$$\begin{aligned}
\| x(t)- x(0)\|&\le M_f{c_u}\sqrt {\gamma_1\varepsilon\|x(0)-\xi(0)\|}\le M_fc_u\sqrt {\gamma_1\varepsilon\rho_0}\\
&\le M_fc_u\sqrt {\dfrac{\gamma_1\varepsilon\rho_1\mu^\varsigma\sqrt\mu}{3}}\le M_fc_u\sqrt {\dfrac{\gamma_1\varepsilon\rho_1\mu^\varsigma\sqrt\mu}{3}}\le\dfrac{\rho_1\mu^\varsigma\sqrt\mu}{3},\\
\|x(t)-\xi(t)\|&\le \|x(t)-x^0\|+\|x^0-\xi^0\|+\|\xi(t)-\xi^0\| \\
&\le M_f{c_u}\sqrt {\dfrac{\gamma_1\varepsilon\rho_1\mu^\varsigma\sqrt\mu}{3}}+\dfrac{\rho_1\mu^\varsigma\sqrt\mu}{3}+\dfrac{\nu\varepsilon}{\sqrt\mu}\le \rho_1\mu^\varsigma\sqrt\mu.
\end{aligned}$$]{} If, additionally, $\|\xi^0-\xi^*\|\in D_\xi$ then [$$\begin{aligned}
\|x(t)-x^*\|&\le \|x(t)-x^0\|+\|x^0-\xi^0\|+\|\xi^0-\xi^*\|\\
&\le \dfrac{2\rho_1\mu^\varsigma\sqrt\mu}{3}+\delta_\xi\le {\rm dist}(x^*,\partial D').
\end{aligned}$$]{} This proves the following intermediate statement.
[ For any $\mu\in(0,\mu_0]$, $\gamma_1\in(\bar\gamma_1,\infty)$, $\varepsilon\in(0,\varepsilon_0(\gamma_1,\mu)]$, $x^0\in D_x$, the $\pi_\varepsilon$-solutions of system with the initial conditions $x(0)=x^0,\xi(0)=\xi^0$ satisfy the following property: $$\|x^0-\xi^0\|\le\dfrac{\rho_1\mu^\varsigma\sqrt\mu}{3}{\Rightarrow}\|x(t)-\xi(t)\|\le\rho_1\mu^\varsigma\sqrt\mu\text{ for }t\in[0,\varepsilon].$$ Furthermore, if $\xi^0\in D_\xi\subseteq D'$ then $x(t)$ is well-defined in $D'$ for $t\in[0,\mu]$. ]{}
*Our next goal is to ensure that the $x$-component of the $\pi_\varepsilon$ solution of system is in a sufficiently small neighborhood of the $\xi$-component.*
For this, we apply Lemma \[volt1\]. Namely, assume that $x(t)\in D_x$ for $t\in[0,\varepsilon]$, $\xi(0)=\xi^0\in B_{\rho_1}(x^0)$. Then [$$\begin{aligned}
\| \xi(t)- \xi(0)\|\le \dfrac{\nu\varepsilon}{\sqrt\mu},\label{bary_est}\end{aligned}$$]{} and the $\pi_\varepsilon$-solution $ x(t)$ of system with controls can be represented my means of the Chen–Fliess series: [$$\label{xeps}
\begin{aligned}
x(\varepsilon)= x^0- \varepsilon \gamma_1(x^0-\xi^0)+R_1(\varepsilon)+ R_2(x^0,\xi^0,\varepsilon),
\end{aligned}$$]{} where $R_1(\varepsilon)$ is defined from Lemma \[lemma\_volt\], and [$$\begin{aligned}
R_2(x^0,&\xi^0,\varepsilon)={\varepsilon^{3/2}}\sum_{j_1\in S_1}\sum_{j_2=1}^m[f_{j_1},f_{j_2}](x^0) a_{j_1}(x^0,\xi^0)\sum_{q:(q,j_2)\in S_2}
\sqrt{\dfrac{|a_{qj_2}(x^0,\xi^0)|}{\pi K_{qj_2}}}
\\
&+\dfrac{\varepsilon^2}{2}\sum_{j_1,j_2\in S_1} L_{f_{j_2}}f_{j_1}(x^0) a_{j_1}(x^0,\xi^0)a_{j_2}(x^0,\xi^0).
\end{aligned}$$]{} Denote $R(x^0,\xi^0,\varepsilon)=R_1(\varepsilon)+R_2(x^0,\xi^0,\varepsilon)$. Using and notations from , we get $$\|R_1(\varepsilon)\|\le M_{3f}c_u^3\big(\varepsilon\|x^0-\xi^0\|\big)^{3/2}\text{ for all }t\in[0,\varepsilon],$$ and $$\begin{aligned}
\| R(x^0,\xi^0,\varepsilon)\|\le \zeta_1\big(\varepsilon\|x^0-\xi^0\|\big)^{3/2},\label{est_R} $$ $$\zeta_1= M_{3f}c_u^3+\dfrac{ M_{2f}}{2}\sqrt{\nu\varsigma\alpha}(\gamma_1\alpha)^{3/2} +2(\gamma_1\alpha)^{3/2}M_{2f} \sqrt{|S_1|}\sum_{j_1=1}^m\Big(\sum_{(j_2,j_1)\in S_2}\kappa_{j_2j_1}^{-2/3}\Big)^{3/4}.$$ Combining , , and , we come to the following estimate: [$$\|x(\varepsilon)-\xi(\varepsilon)\|\le(1-\varepsilon\gamma_1)\|x^0-\xi^0\|+\zeta_1\big(\varepsilon\|x^0-\xi^0\|\big)^{3/2}+\dfrac{\nu\varepsilon}{\sqrt\mu}.$$]{} For any $\gamma_1>\bar\gamma_1$, let $\lambda_1\in[\bar\gamma_1,\gamma_1)$ and define [$$\label{vareps1}
\varepsilon_1(\gamma_1,\mu)=\min\Big\{\varepsilon_0(\mu),\Big(\dfrac{\gamma_1-\lambda_1}{\zeta_1\sqrt\delta_x}\Big)^2\Big\}.$$]{} Recall that $\varepsilon\lambda_1<\varepsilon\gamma_1<1$. Then, for any $\varepsilon\in(0,\varepsilon_1(\gamma_1,\mu_1))$, [$$\begin{aligned}
\|x(\varepsilon)-\xi(\varepsilon)\|&< (1-\varepsilon\bar\gamma_1)\|x^0-\xi^0\|+\dfrac{\nu\varepsilon}{\sqrt\mu}.
\end{aligned}$$]{} Recall that $\bar\gamma_1$ is given by , which implies $
\dfrac{\nu\varepsilon}{\sqrt\mu}=\dfrac{\gamma_1\rho_1\mu^\varsigma\sqrt\mu}{3}.
$ This together with Statement 1 gives us the next intermediate result.
[Assume that $x(t)\in D'$ for all $t\in[0,\varepsilon_0]$, $x(0)\in D_x$. Then, for any $\mu\in(0,\mu_0]$, $\gamma_1\in(\bar\gamma_1,\infty)$, $\varepsilon\in(0,\varepsilon_1(\gamma_1,\mu)]$, the following properties hold: $$\begin{aligned}
\text{if }&\|x^0-\xi^0\|\le \dfrac{\rho_1\mu^\varsigma\sqrt\mu}{3}\text{ then }\|x(\varepsilon)-\xi(\varepsilon)\|\le\dfrac{\rho_1\mu^\varsigma\sqrt\mu}{3},\\
\text{ and }&\|x(t)-\xi(t)\|\le \rho_1\mu^\varsigma\sqrt\mu\text{ for all }t\in[0,2\varepsilon].
\end{aligned}$$]{}
Now let us put $x(0)=x^0=\xi^0=\xi(0)$, $x^0\in D_x$. Then $x(t)\equiv x^0\in D_x$ for all $t\in[0,\varepsilon]$, and [$$\|\xi(t)-\xi^0\|\le \|\xi(t)-\xi^0\|+\|\xi^0-\xi^*\|\le\nu \sqrt\mu +\delta_x=\delta_\xi,$$]{} i.e. $\xi(t)\in D_\xi\subset D'$ for $t\in[0,\varepsilon]$. Besides, Statement 2 implies $$\|x(\varepsilon)-\xi(\varepsilon)\|<\dfrac{\rho_1\mu^\varsigma\sqrt\mu}{3}.$$ From Statements 1 and 2, the $x$-component of the $\pi_\varepsilon$-solution of system is also well-defined in $D'$ for $t\in[\varepsilon,2\varepsilon]$. Again, it is easy to see that $$\|\xi(t)-\xi^0\|\le \nu \sqrt\mu +\delta_x\text{ for }t\in[0,2\varepsilon],$$ i.e. $\xi(2\varepsilon)\in D_\xi$ and $$\|x(2\varepsilon)-\xi(2\varepsilon)\|<\dfrac{\rho_1\mu^\varsigma\sqrt\mu}{3}.$$ Without loss of generality, we may assume that $\dfrac{\mu}{\varepsilon}=\mathcal N_1,\text{ with some }\mathcal N_1\in\mathbb N.$ Repeating Steps 1–2 until $t=\mathcal N\varepsilon$, we come to the following statement.
For any $\mu\in(0,\mu_0]$, $\gamma_1\in(\bar\gamma_1,\infty)$, $\varepsilon\in(0,\varepsilon_1(\gamma_1,\mu)]$, the $\pi_\varepsilon$-solutions $(x(t),\xi(t))$ of system with the initial conditions $x(0)=\xi(0)\in D_x$ are well-defined in $D'\times D'$ for all $t\in[0,(\mathcal N_1+1)\varepsilon]$, $$\|x(t)-\xi(t)\|\le \rho_1\mu^\varsigma\sqrt\mu\text{ for all }t\in[0,\mu],\ \|x(\mu)-\xi(\mu)\|\le \dfrac{\rho_1\mu^\varsigma\sqrt\mu}{3}.$$
Thus, for any $\mu\in(0,\mu_0]$, we can take $\gamma_1(\mu)$, $\varepsilon(\gamma_1(\mu),\mu)$, such that $x(t),\xi(t)\in D'$ for $t\in[0,\mu]$. In the next steps, we will find sufficiently small $\mu$ independently on $\varepsilon$ and $\gamma_1$.
*The goal of this step is to ensure the decay of the cost function $J(x)$ along the trajectories of system by choosing sufficiently small $\mu$.*
For this purpose we apply again Lemma \[volt1\]. Since $x(t),\xi(t)\in D'$ for $t\in[0,\mu]$, we may consider the Chen–Fliess series expansion of the $\xi$-component of solution of system on the interval $[0,\mu]$: [ $$\begin{aligned}
\label{volt_y}
\xi(\mu)&=\xi^0-\mu\gamma_2\nabla J(\xi^0)+R_3(\mu),\end{aligned}$$]{} where [$$\begin{aligned}
R_3(\mu)=&\sum_{j=1}^{2n}\int_0^{\mu}\Big(g_{j}( J(x(s_1)))-g_{j}( J(\xi(s_1)))\Big)e_jv_{j}^\mu({s_1})ds_1\\
&+\sum_{j_1,j_2=1}^{2n}\int_0^{\mu}\int_0^{s_1} L_{e_{j_2}\big(g_{j_2}\circ J(x(s_2))-g_{j_2}( J(\xi(s_2)))\big)} g_{j_1}( J(\xi(s_2)))e_{j_1}v_{j_2}^\mu ({s_2})v_{j_1}^\mu({s_1})ds_2ds_1\\
& +\sum_{j_1,j_2,j_3=1}^{2n}\int_0^{\mu}\int_0^{s_1}\int_0^{s_2} v_{j_3}^\mu ({s_3})v_{j_3}^\mu ({s_2})v_{j_1}^\mu({s_1}) \\
&\times L_{e_{j_3}g_{j_3}( J(x(s_3)))} L_{e_{j_2}g_{j_2}( J(\xi(s_3)))}g_{j_1}( J(\xi(s_3)))e_{j_1}ds_3ds_2ds_1.
\end{aligned}$$]{} Under the assumptions of Theorem 1, we conclude that [ $$\begin{aligned}
\|R_3(\mu)\|\le c_w{\sqrt\mu}(L_g+\sqrt\mu L_{2g}c_w)\max\limits_{0\le t\le\mu}\|x(t)&-\xi(t)\|+\mu^{3/2}M_{3g}.
\end{aligned}$$]{} Thus, applying Statement 3 we get $$\|R_3(\mu)\| \le \zeta_2 \mu^{1+\tilde\varsigma},$$ where $\tilde\varsigma=\min\{\varsigma,1/2\}$, $\zeta_2=c_w\mu^{\max\{0,\varsigma-1/2\}}\rho_1(L_g+\sqrt\mu L_{2g}c_w)+\mu^{\max\{0,1/2-\varsigma\}}M_{3g}$.\
Using Taylor’s formula for the function $J(\xi)$, [$$\begin{aligned}
J&(\xi(t))=J(\xi^0)+\nabla J(\xi^0)(\xi(t)-\xi^0)+\dfrac{1}{2}\sum_{i,j=1}^{2n}\dfrac{\partial^2 J(x)}{\partial x_i\partial x_j}\Big|_{x=\xi^0+\theta \xi(t)}(\xi_i-\xi_i^0)(\xi_j-\xi_j^0),
\end{aligned}$$]{} and exploiting , we obtain [$$\begin{aligned}
J(\xi(\mu))&\le J(\xi^0)-\mu\gamma_2\sigma_{21}J(\xi^0)+\mu^{1+\varsigma}\zeta_2\sqrt{\sigma_{22}J(\xi^0)}+\sigma_3\big(\mu^2\gamma_2^2\sigma_{22}^2J(\xi^0)+\zeta_2^2\mu^{2+2\varsigma}\big)\\
&=J(\xi^0)\Big(1-\mu\gamma_2\big(\sigma_{21}-\mu\gamma_2\sigma_3\sigma_{22}^2\big)\Big)+\mu^{1+\varsigma}\zeta_2\Big(\sqrt{\sigma_{22}J(\xi^0)}+\sigma_3\zeta_2\mu^{1+\varsigma}\Big).
\end{aligned}$$]{} Let $\mathcal L_c=\{x\in D:J(x)\le c\}$, $c_J=\sigma_{11}\delta_x^2$. Then [$$\label{lset}
\overline{B_\delta(x^*)}\subseteq \mathcal L_{c_J} \subseteq D_x.$$]{} For any $\rho_2\in(0,c_J]$, $\lambda_2\in(0,\gamma_2\sigma_{21})$, we define [$$\label{eps1}
\mu_1=\min\{\mu_0,{1}/{\lambda_2},\hat\mu_1\},$$]{} where $\hat\mu_1$ is the smallest positive root of the equation [$$\rho_2\mu\gamma_2\sigma_3\sigma_{22}^2+{\mu^{\varsigma}}\zeta_2\Big(\sqrt{\sigma_{22}\rho_2}+\sigma_3\zeta_2\mu^{1+\varsigma}\Big)=\rho_2(\gamma_2\sigma_{21}-\lambda_2).$$]{} Then, for any $\mu\in(0,\mu_1)$, the following two scenarios are possible:
S1) If [ $J(\xi^0)\le\rho_2$]{} then [$J(\xi(\mu))\le \rho_2(1-\mu\lambda_2){<}\rho_2$.]{} In this case, $\xi(\mu)\in D_x$, Additionally, Statement 3 implies that $\|x(\mu)-\xi(\mu)\|\le \dfrac{\rho_1\mu^\varsigma\sqrt\mu}{3}$. Repeating the above argumentation, we get $\xi(N\mu)\in D_x$ for all natural numbers $N$.
S2) If $J(\xi^0)>\rho_2$ then $J(\xi(\mu))<J(\xi^0)(1-\mu\lambda_2)<J(\xi^0)$.
Consider S2). If $\xi^0=x^0\in \overline{B_\delta(x^*)}$ then $\xi(\mu)\in \mathcal L_{c_J} \subseteq D_x$. Again, Statement 3 gives $\|x(\mu)-\xi(\mu)\|\le \dfrac{\rho_1\mu^\varsigma\sqrt\mu}{3}$. Thus, we may repeat all the steps for $t\in[\mu,2\mu]$.
Summarizing all the above, we arrive at the following conclusion: there exists an $\mathcal N_2\in\mathbb N\cup\{0\}$ such that [$$\begin{aligned}
&J(\xi(t))\le J(\xi^0)e^{-\lambda_2t}\text{ for }t=0,\mu,\dots, (\mathcal N_2-1)\mu,\\
&J(\xi(t))\le \rho_2\text{ for }t=\mathcal N_2\mu,(\mathcal N_2+1)\mu,\dots \, .
\end{aligned}$$]{} Consequently, $$\|\xi(t)-x^*\|\le \sqrt{\dfrac{\sigma_{12}}{\sigma_{11}}}\|x^0-x^*\|e^{-\lambda_2t}\text{ for }t=0,\mu,\dots,(\mathcal N_2-1)\mu,$$ $$\|\xi(\mathcal N_2\mu)-x^*\|\le \sqrt{\dfrac{\rho_2}{\sigma_{11}}}\le \delta_x\text{ for }t=\mathcal N_2\mu,(\mathcal N_2+1)\mu,\dots .$$ For an arbitrary $t\in[0,\mathcal N_2\mu]$, we denote the integer part of $\dfrac{t}{\mu}$ as $\Big[\dfrac{t}{\mu}\Big]$ and observe that $0<t-\Big[\dfrac{t}{\mu}\Big]\mu<\mu$. Then [$$\begin{aligned}
\|\xi(t)-x^*\|&\le \Big\|\xi\Big(\Big[\dfrac{t}{\mu}\Big]\mu\Big)-x^*\Big\|+\Big\|\xi(t)-\xi\Big[\dfrac{t}{\mu}\Big]\Big\|\\
&\le \sqrt{\dfrac{\sigma_{12}}{\sigma_{11}}}\|x^0-x^*\|e^{-\lambda_2\Big[\dfrac{t}{\mu}\Big]\mu}+\nu\sqrt\mu\le \beta\|x^0-x^*\|e^{-\lambda_2 t}+\nu\sqrt\mu,
\end{aligned}$$]{} where $\beta=\sqrt{\dfrac{\sigma_{12}}{\sigma_{11}}}e^{\lambda_2\mu}$. This yields the following result.
[For any $\mu\in(0,\mu_1]$, $\gamma_1\in(\bar\gamma_1,\infty)$, $\varepsilon\in(0,\varepsilon_0(\gamma_1,\mu)]$, the $\pi_\varepsilon$-solutions $(x(t),\xi(t))$ of system with the initial conditions $x(0)=\xi(0)\in D_x$ are well-defined in $D'\times D'$ for all $t\in[0,\infty)$, and the following estimates hold: $$\begin{aligned}
&\|\xi(t)-x^*\|\le \beta\|x^0-x^*\|e^{-\lambda t}+\nu\sqrt\mu\text{ for }t\in[0,\mathcal N_2\mu],\\
&\|\xi(t)-x^*\|\le \sqrt{\dfrac{\rho_2}{\sigma_{11}}}+\nu\sqrt\mu\text{ for }t\in[\mathcal N_2\mu,\infty).
\end{aligned}$$ Furthermore, $$\|x(t)-\xi(t)\|\le \rho_1\mu^\varsigma\sqrt\mu\text{ for all }t\in[0,\infty).$$]{}
*Finally, we estimate $\|x(t)-x^*\|$ for $t\in[0,\infty)$*.
Applying the triangle inequality together with Statement 4, we get the following: [$$\begin{aligned}
&\|x(t)-x^*\|\le \beta\|x^0-x^*\|e^{-\lambda t}+\rho_1\mu^\varsigma\sqrt\mu+\nu\sqrt\mu\text{ for }t\in[0,\mathcal N_2\mu],\\
&\|x(t)-x^*\|\le \rho_1\mu^\varsigma\sqrt\mu+\sqrt{\dfrac{\rho_2}{\sigma_{11}}}+\nu\sqrt\mu\text{ for }t\in[\mathcal N_2\mu,\infty).
\end{aligned}$$]{} Since $\rho_1$, $\rho_2$ are arbitrary and $\mu$ can be chosen small enough, the above inequalities imply the assertion of Theorem 1. In particular, for an arbitrary $\rho>0$, one can take $\rho_1>0$ and $\mu>0$ such that $$\label{eps2}
\rho_1\mu^\varsigma\sqrt\mu+\nu\sqrt\mu\le\dfrac{\rho}{2},$$ and $\rho_2\le\dfrac{1}{4}\rho^2\sigma_{11}$. Then $$\|x(t)-x^*\|\le \beta\|x^0-x^*\|e^{-\lambda t}+\rho\text{ for all }t\in[0,\infty).$$ Note that the choice of $\mu$ does not depend on $\varepsilon,\eta$, and the choice of $\gamma_1$ does not depend on $\varepsilon$. Namely, given $\delta,\rho,\rho_1,\rho_2$, one can choose a $\bar\mu>0$ satisfying , and , and take any $\hat\mu\in(0,\bar\mu]$. The next step is to determine $\bar\gamma_1(\hat\mu)$ satisfying , and take any $\hat\gamma_1\in(\bar\gamma_1,\infty)$. Finally, $\bar\varepsilon(\hat\gamma_1,\hat\mu)$ has to be specified according to and .
[^1]: This work was supported in part by the German Research Foundation (projects GR 5293/1-1 and ZU 359/2-1).$^1$Institute of Mathematics, Alpen-Adria University of Klagenfurt, Austria $^2$Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany $^3$Otto von Guericke University Magdeburg, Germany $^4$Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine
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\#1\#2 -32pt 1ex
**Search for the Supersymmetric\
Partner of the Top-Quark\
in $p \overline{p}$ Collisions at $\sqrt{s} = 1.8 \, {\rm TeV}$**
=cmti8 \#1[$^{#1}$]{}
T. Affolder, H. Akimoto, A. Akopian, M. G. Albrow, P. Amaral,8 S. R. Amendolia, D. Amidei, K. Anikeev, J. Antos,1 G. Apollinari, T. Arisawa, T. Asakawa, W. Ashmanskas,8 F. Azfar, P. Azzi-Bacchetta, N. Bacchetta, M. W. Bailey, S. Bailey, P. de Barbaro, A. Barbaro-Galtieri, V. E. Barnes, B. A. Barnett, S. Baroiant,5 M. Barone, G. Bauer, F. Bedeschi, S. Belforte, W. H. Bell, G. Bellettini, J. Bellinger, D. Benjamin, J. Bensinger,4 A. Beretvas, J. P. Berge, J. Berryhill,8 B. Bevensee, A. Bhatti, M. Binkley, D. Bisello, M. Bishai, R. E. Blair,2 C. Blocker,4 K. Bloom, B. Blumenfeld, S. R. Blusk, A. Bocci, A. Bodek, W. Bokhari, G. Bolla, Y. Bonushkin,6 D. Bortoletto, J. Boudreau, A. Brandl, S. van den Brink, C. Bromberg, M. Brozovic, N. Bruner, E. Buckley-Geer, J. Budagov,9 H. S. Budd, K. Burkett, G. Busetto, A. Byon-Wagner, K. L. Byrum,2 P. Calafiura, M. Campbell, W. Carithers, J. Carlson, D. Carlsmith, W. Caskey,5 J. Cassada, A. Castro, D. Cauz, A. Cerri, A. W. Chan,1 P. S. Chang,1 P. T. Chang,1 J. Chapman, C. Chen, Y. C. Chen,1 M. -T. Cheng,1 M. Chertok, G. Chiarelli, I. Chirikov-Zorin,9 G. Chlachidze,9 F. Chlebana, L. Christofek, M. L. Chu,1 Y. S. Chung, C. I. Ciobanu, A. G. Clark, A. Connolly, J. Conway, M. Cordelli, J. Cranshaw, D. Cronin-Hennessy, R. Cropp, R. Culbertson, D. Dagenhart, S. D’Auria, F. DeJongh, S. Dell’Agnello, M. Dell’Orso, L. Demortier, M. Deninno,3 P. F. Derwent, T. Devlin, J. R. Dittmann, S. Donati, J. Done, T. Dorigo, N. Eddy, K. Einsweiler, J. E. Elias, E. Engels, Jr., D. Errede, S. Errede, Q. Fan, R. G. Feild, J. P. Fernandez, C. Ferretti, R. D. Field, I. Fiori,3 B. Flaugher, G. W. Foster, M. Franklin, J. Freeman, J. Friedman, Y. Fukui, I. Furic, S. Galeotti, M. Gallinaro, T. Gao, M. Garcia-Sciveres, A. F. Garfinkel, P. Gatti, C. Gay, D. W. Gerdes, P. Giannetti, P. Giromini, V. Glagolev,9 M. Gold, J. Goldstein, A. Gordon, I. Gorelov, A. T. Goshaw, Y. Gotra, K. Goulianos, C. Green, G. Grim,5 P. Gris, L. Groer, C. Grosso-Pilcher,8 M. Guenther, G. Guillian, J. Guimaraes da Costa, R. M. Haas, C. Haber, E. Hafen, S. R. Hahn, C. Hall, T. Handa, R. Handler, W. Hao, F. Happacher, K. Hara, A. D. Hardman, R. M. Harris, F. Hartmann, K. Hatakeyama, J. Hauser,6 J. Heinrich, A. Heiss, M. Herndon, C. Hill,5 K. D. Hoffman, C. Holck, R. Hollebeek, L. Holloway, R. Hughes, J. Huston, J. Huth, H. Ikeda, J. Incandela, G. Introzzi, J. Iwai, Y. Iwata, E. James, H. Jensen, M. Jones, U. Joshi, H. Kambara, T. Kamon, T. Kaneko, K. Karr, H. Kasha, Y. Kato, T. A. Keaffaber, K. Kelley, M. Kelly, R. D. Kennedy, R. Kephart, D. Khazins, T. Kikuchi, B. Kilminster, B. J. Kim, D. H. Kim, H. S. Kim, M. J. Kim, S. H. Kim, Y. K. Kim, M. Kirby, M. Kirk,4 L. Kirsch,4 S. Klimenko, P. Koehn, A. Köngeter, K. Kondo, J. Konigsberg, K. Kordas, A. Korn, A. Korytov, E. Kovacs,2 J. Kroll, M. Kruse, S. E. Kuhlmann,2 K. Kurino, T. Kuwabara, A. T. Laasanen, N. Lai,8 S. Lami, S. Lammel, J. I. Lamoureux,4 J. Lancaster, M. Lancaster, R. Lander,5 G. Latino, T. LeCompte,2 A. M. Lee IV, K. Lee, S. Leone, J. D. Lewis, M. Lindgren,6 T. M. Liss, J. B. Liu, Y. C. Liu,1 N. Lockyer, J. Loken, M. Loreti, D. Lucchesi, P. Lukens, S. Lusin, L. Lyons, J. Lys, R. Madrak, K. Maeshima, P. Maksimovic, L. Malferrari,3 M. Mangano, M. Mariotti, G. Martignon, A. Martin, J. A. J. Matthews, J. Mayer, P. Mazzanti,3 K. S. McFarland, P. McIntyre, E. McKigney, M. Menguzzato, A. Menzione, C. Mesropian, A. Meyer, T. Miao, R. Miller, J. S. Miller, H. Minato, S. Miscetti, M. Mishina, G. Mitselmakher, N. Moggi,3 E. Moore, R. Moore, Y. Morita, T. Moulik, M. Mulhearn, A. Mukherjee, T. Muller, A. Munar, P. Murat, S. Murgia, J. Nachtman,6 S. Nahn, H. Nakada, T. Nakaya,8 I. Nakano, C. Nelson, T. Nelson, C. Neu, D. Neuberger, C. Newman-Holmes, C.-Y. P. Ngan, H. Niu,4 L. Nodulman,2 A. Nomerotski, S. H. Oh, T. Ohmoto, T. Ohsugi, R. Oishi, T. Okusawa, J. Olsen, W. Orejudos, C. Pagliarone, F. Palmonari, R. Paoletti, V. Papadimitriou, S. P. Pappas, D. Partos,4 J. Patrick, G. Pauletta, M. Paulini, C. Paus, L. Pescara, T. J. Phillips, G. Piacentino, K. T. Pitts, A. Pompos, L. Pondrom, G. Pope, M. Popovic, F. Prokoshin,9 J. Proudfoot,2 F. Ptohos, O. Pukhov,9 G. Punzi, K. Ragan, A. Rakitine, D. Reher, A. Reichold, A. Ribon, W. Riegler, F. Rimondi,3 L. Ristori, M. Riveline, W. J. Robertson, A. Robinson, T. Rodrigo,7 S. Rolli, L. Rosenson, R. Roser, R. Rossin, A. Roy, A. Safonov, R. St. Denis, W. K. Sakumoto, D. Saltzberg,6 C. Sanchez, A. Sansoni, L. Santi, H. Sato, P. Savard, P. Schlabach, E. E. Schmidt, M. P. Schmidt, M. Schmitt, L. Scodellaro, A. Scott,6 A. Scribano, S. Segler, S. Seidel, Y. Seiya, A. Semenov,9 F. Semeria,3 T. Shah, M. D. Shapiro, P. F. Shepard, T. Shibayama, M. Shimojima, M. Shochet,8 J. Siegrist, G. Signorelli, A. Sill, P. Sinervo, P. Singh, A. J. Slaughter, K. Sliwa, C. Smith, F. D. Snider, A. Solodsky, J. Spalding, T. Speer, P. Sphicas, F. Spinella, M. Spiropulu, L. Spiegel, J. Steele, A. Stefanini, J. Strologas, F. Strumia, D. Stuart, K. Sumorok, T. Suzuki, T. Takano, R. Takashima, K. Takikawa, P. Tamburello, M. Tanaka, B. Tannenbaum,6 W. Taylor, M. Tecchio, P. K. Teng,1 K. Terashi, S. Tether, A. S. Thompson, R. Thurman-Keup,2 P. Tipton, S. Tkaczyk, K. Tollefson, A. Tollestrup, H. Toyoda, W. Trischuk, J. F. de Troconiz, J. Tseng, N. Turini, F. Ukegawa, T. Vaiciulis, J. Valls, E. Vataga-Pagliarone, S. Vejcik III, G. Velev, R. Vidal, R. Vilar,7 I. Volobouev, D. Vucinic, R. G. Wagner,2 R. L. Wagner, J. Wahl,8 N. B. Wallace, A. M. Walsh, C. Wang, M. J. Wang,1 T. Watanabe, D. Waters, T. Watts, R. Webb, H. Wenzel, W. C. Wester III, A. B. Wicklund,2 E. Wicklund, T. Wilkes,5 H. H. Williams, P. Wilson, B. L. Winer, D. Winn, S. Wolbers, D. Wolinski, J. Wolinski, S. Wolinski, S. Worm, X. Wu, J. Wyss, A. Yagil, W. Yao, G. P. Yeh, P. Yeh,1 J. Yoh, C. Yosef, T. Yoshida, I. Yu, S. Yu, Z. Yu, A. Zanetti, F. Zetti, and S. Zucchelli3
.026in
(CDF Collaboration)
.026in
1 [Institute of Physics, Academia Sinica, Taipei, Taiwan 11529, Republic of China]{}\
2 [Argonne National Laboratory, Argonne, Illinois 60439]{}\
3 [Istituto Nazionale di Fisica Nucleare, University of Bologna, I-40127 Bologna, Italy]{}\
4 [Brandeis University, Waltham, Massachusetts 02254]{}\
5 [University of California at Davis, Davis, California 95616]{}\
6 [University of California at Los Angeles, Los Angeles, California 90024]{}\
7 [Instituto de Fisica de Cantabria, CSIC-University of Cantabria, 39005 Santander, Spain]{}\
8 [Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637]{}\
9 [Joint Institute for Nuclear Research, RU-141980 Dubna, Russia]{}\
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We report on a search for the supersymmetric partner of the top quark (stop) produced in $t \overline{t}$ events using $110 \, {\rm pb}^{-1}$ of $p \overline{p}$ collisions at $\sqrt{s} = 1.8 \, {\rm TeV}$ recorded with the Collider Detector at Fermilab. In the case of a light stop squark, the decay of the top quark into stop plus the lightest supersymmetric particle (LSP) could have a significant branching ratio. The observed events are consistent with Standard Model $t \overline{t}$ production and decay. Hence, we set limits on the branching ratio of the top quark decaying into stop plus LSP, excluding branching ratios above 45% for a LSP mass up to $40$ ${\rm GeV/c}^{2}$.
With the observation in 1995 of a heavy top quark [@cdf_top; @d0_top], an important prerequisite was met for low energy supersymmetry (SUSY) [@general_susy] to explain electroweak symmetry breaking. In the Minimal Supersymmetric extension of the Standard Model (MSSM), all known particles of the Standard Model (SM) acquire supersymmetric partners, or superpartners. For fermions the superpartners are bosons, while for bosons the superpartners are fermions. We assume conservation of a multiplicative quantum number $\mathcal{R}$-parity, which requires these new particles to be produced in pairs and prevents decays of the lightest supersymmetric particle (LSP). From cosmological considerations [@ellis_hagelin], the LSP is normally assumed to be the lightest neutralino.
The large Yukawa coupling of the top quark opens up the possibility of a large mass splitting in the third generation of fermionic superpartners (the squarks and sleptons). The superpartners of the right-handed and left-handed top quark (the stop squarks) combine to form the mass eigenstates. The lightest stop squark ($\tilde{t}_{1}$) could then be lighter than the superpartners of all other squarks. Most limits on squark masses [@cdf_d0_squark] do not apply to the stop squark because they are usually based on a model of five degenerate squarks. Current stop squark mass limits are significantly lower [@lep_slc_stop] than these limits or based on the assumption of a very heavy chargino ($\tilde{\chi}^{\pm}_{1}$) [@d0_clsp]. The latter searches are complementary to the analysis presented here since the stop decay mode $c + \mbox{LSP}$ does not coexist with the decay mode $b + \tilde{\chi}^{\pm}_{1}$. If the stop squark is light, decays of the top quark into stop plus the lightest neutralino could be kinematically allowed. If this neutralino is the LSP it will be stable and only weakly interacting. Such a particle would pass through the detector without interaction, causing a considerable energy imbalance. For the stop squark we assume decays analogous to the Standard Model top quark decay, i.e. into chargino and $b$-quark. The chargino could then decay into a LSP plus either a quark-antiquark pair or a lepton and neutrino.
Branching ratios as large as $40$ to 50% for the top decay into stop have been suggested [@stop_branching]. In such scenarios about one half of $t \overline{t}$ events would have one SM and one supersymmetric top decay. If the SM top decay to a W and a $b$-quark is followed by the leptonic decay of the W, then the selection criteria of the online leptonic trigger, the offline dataset selection and the $t \overline{t}$ event identification [@cdf_top] will all be satisfied. The decay of the second top quark can then be used to search for decays into stop.
The CDF detector [@cdf_1a_detector] is well suited to search for supersymmetric top quark decays. The following components are relevant to this analysis: the central tracking chamber, which is inside a $1.4 \,{\rm T}$ superconducting solenoidal magnet, measures the momentum of charged particles with a resolution of $\delta p_{\rm T} / p_{\rm T} = 0.001 * p_{\rm T}$ ($p_{\rm T}$ in ${\rm GeV/c}$) [@cdf_coord]. The silicon vertex detector, with an inner radius of $3 \, {\rm cm}$ and an outer of $8 \, {\rm cm}$, identifies secondary vertices with a resolution of $130 \, \mu {\rm m}$ in the transverse plane. The electromagnetic and hadronic calorimeters cover the pseudorapidity region $| \eta | < 4.2$ and are used to identify jets and electrons, and to measure the missing transverse energy $\;\not\!\!\!E_{\rm T}$ [@cdf_coord]. An outer layer of drift chambers provides muon identification in the region $| \eta | < 1.0$.
The search reported here is based on $110 \, {\rm pb}^{-1}$ of $p \overline{p}$ collisions at $\sqrt{s} = 1.8 \, {\rm TeV}$ recorded with the Collider Detector at Fermilab during the 1992-93 and 1994-95 collider periods. The analysis is a combination of a CDF single lepton plus jet top analysis [@cdf_top] and a kinematic analysis [@cdf_top_kin]. The analysis cuts are slightly revised to improve sensitivity.
The leptonic W decay from the SM top decay yields an energetic lepton. Events are selected as in the single lepton plus $b$-jet top analysis by requiring a central electron ($| \eta | \leq 1.1$) or central muon ($| \eta | \leq 1.0$) with transverse momentum $p_{\rm T} \geq 20 \, {\rm GeV/c}$.
The neutrino from the W decay, as well as any LSP’s, will escape the apparatus without detection, resulting in an energy imbalance. The $\,\not\!\!\!E_{\rm T}$ measured by the calorimeter is corrected if the lepton is a muon. We have increased the $\,\not\!\!\!E_{\rm T}$ requirement, from $\,\not\!\!\!E_{\rm T} \geq 25 \, {\rm GeV}$ in the single lepton plus $b$-jet top analysis to $\not\!\!E_{\rm T} \geq 45 \, {\rm GeV}$, as our signal is expected to have a harder $\not\!\!\!E_{\rm T}$ spectrum. To reject non-W background, we require the transverse mass $M_{\rm T}$ of the lepton and $\,\not\!\!\!E_{\rm T}$ system to be larger than $40 \, {\rm GeV}$. While the top analyses are concerned with separating $t \overline{t}$ from W plus jet production, in this analysis we are interested in minimizing W plus multijet background and then focusing on separating SUSY top decays from SM top decays. In both SUSY and SM top decays the W has a substantial transverse momentum. An additional requirement that the lepton$-\not\!\!\!E_{\rm T}$ system have $p_{\rm T} \geq 50 \,{\rm GeV/c}$ has therefore been made.
In addition to the $b$-jet from the SM top decay, we have three additional jets from the other top decay when the W or the chargino decays hadronically. We require two jets [@cdf_jetclus] with transverse energy $E_{\rm T} \ge$ $20 \,{\rm GeV}$ and a third jet with $E_{\rm T} \ge$ $15 \, {\rm GeV}$ all within $| \eta | \leq 2.0$. Jet energy is corrected according to an average response function prior to event selection [@cdf_top_kin]. We require one of the jets to be identified as a $b$-jet candidate. We use a secondary vertex tagging method \[1\], based on SVX information, that reconstructs secondary vertices from $B$ hadron decays. As in the kinematic top analysis [@cdf_top_kin], we require the three jets to have large polar angles in the rest frame of the lepton, $\not\!\!E_{\rm T}$, and jets: $| \cos ( \theta ^{*} (\mbox{jet}_{i}) ) | < 0.9$, $| \cos ( \theta ^{*} (\mbox{jet}_{j}) ) | < 0.8$, and $| \cos ( \theta ^{*} (\mbox{jet}_{k}) ) | < 0.7$, where the jets [*i,j,k*]{} are ordered according to $| \cos ( \theta ^{*} )|$. These requirements were relaxed from the requirement of $| \cos ( \theta ^{*} (\mbox{jet}_{i,j,k}) ) | < 0.7$ in [@cdf_top_kin] in order to obtain a good acceptance for the SM-SUSY top decays. For simplicity, these polar angles are calculated assuming a null longitudinal component for the neutrino. In order to insure that the three jets are well separated in $\eta$-$\phi$ space, we require $\Delta R (\mbox{jet, jet}) \geq 0.9$, where $\Delta R (\mbox{jet, jet})=\,\sqrt{\Delta \eta^{2} + \Delta \phi ^{2}}$ is the minimum distance between the jets.
With these jet requirements our sample is defined. In the data $9$ events pass the above cuts. For the theoretical $t \overline{t}$ production cross section of $ 5\, {\rm pb}$ [@cdf_cross] we expect $9.5$ events. Non-top Standard Model background contributes less than half an event.
Jets from a SUSY top decay are less energetic than jets from a SM top decay due to the presence of LSP’s and possible small chargino mass. In order to best distinguish between SM and SUSY top decays, we combine the transverse energy information of the second and third most energetic jet in a likelihood variable defined as: $R_{L}=
\frac{{\mathcal{P}}^{SM-SM}(E_{\rm T}^{jet2})\times {\mathcal{P}}^{SM-SM}(E_{\rm T}^{jet3})}
{ {\mathcal{P}}^{SM-SUSY}(E_{\rm T}^{jet2})\times {\mathcal{P}}^{SM-SUSY}(E_{\rm T}^{jet3})}$, where $\mathcal{P}$ are the expected differential transverse energy distributions $\frac{1}{\sigma} \frac{d\sigma}{dE_{\rm T}}$ evaluated from the Monte Carlo.
3.6in 3.6in
Figure 1 shows those $E_{\rm T}$ distributions while Figure 2 shows the distribution of the likelihood variable for SM and SUSY top decays. Events with one top quark decaying into stop plus LSP are clustered in the region of negative values of ${\rm ln}(R_{L})$, whereas events with two SM top decays are at positive values [@susysusy]. The region ${\rm ln}(R_{L})<-1$ defines our SUSY search region. The region ${\rm ln}(R_{L})>-1$ is dominated by double SM top decays and will be used to normalize the expected number of these decays. Our search is then independent of the $t \overline{t}$ production cross section [@footnote1].
To study the distributions of the kinematic variables in the supersymmetric top decays, the signals are produced with the [ISAJET]{} Monte Carlo generator [@isajet] and passed through detector simulation programs [@QFL]. We have fixed the top quark mass to $175 \, {\rm GeV/c}^{2}$ and varied stop, chargino, and LSP masses. Different branching ratios for top decaying into stop plus LSP are obtained by analyzing the three possible combinations of top decays (SM-SM, SM-SUSY, and SUSY-SUSY) individually and recombining them with appropriate weights.
Systematic uncertainties in the simulation of $t \overline{t}$ events can impact both the expected number of events and the shape of the two jet $E_{\rm T}$ distributions and thus the likelihood distribution.
3.6in 3.6in
We have evaluated the systematic uncertainties for multiple points in the parameter space of the stop and chargino masses. The systematic uncertainties are expected to become significant close to the kinematic bounds. In the region of small chargino mass the uncertainties due to gluon radiation and the calorimeter energy scale are important. The procedure used to evaluate these systematic uncertainties is the same as that of Ref. [@cdf_top_prd]. The uncertainty due to the $t \overline{t}$ production cross section and to the integrated luminosity is negligible since we normalize the Monte Carlo predictions to the data in the SM dominated region of large likelihood. We note that this normalization has a large statistical uncertainty due to the small number of events in this region. The uncertainty in the mass of the top quark becomes important in the region of large stop mass, close to the kinematic limit. The effect of using parton distribution functions other than CTEQ-3 (LO) [@cteq] is within the statistical uncertainty of the Monte Carlo samples. The total systematic uncertainty for both expected number of events and fraction of events with ${\rm ln}(R_{L})<-1$ from SM-SUSY top quark decays is typically around 40%. The major systematic uncertainty comes from the calorimeter energy scale and it varies between $\pm$15% and $\pm$25% in the region of the analyzed parameter space. The uncertainty on the top quark mass contributes about $\pm$15% for most of the parameter space. The uncertainty due to the gluon radiation, on both number of events and shape, is always less than $\pm$10%.
All nine events observed in the data cluster in the SM-like region of large ${\rm ln}(R_{L})$. To set a limit on the branching ratio of top decaying into stop plus LSP, we calculate the branching ratios that would yield at least one event in the SUSY-like region 95% of the time as a function of stop, chargino and LSP mass. The method used is essentially a Bayesian-style integration over the systematic and statistical uncertainties in the SM-SM, SM-SUSY, and SUSY-SUSY contributions, where the uncertainties are assumed to be Gaussian distributed. Figure \[f:stop\_br\_limit\] shows the 95% confidence level top decaying into stop plus LSP branching ratio limit as a function of stop and chargino mass, for a LSP mass of $20$ and $40 \, {\rm GeV/c}^{2}$. For larger LSP masses the kinematically allowed region shrinks. The sensitivity of this analysis, however, stays rather constant.
In conclusion, we have looked for the top decay into stop plus the LSP. In the case of a light stop squark, the top is allowed to decay into stop plus a LSP. The number of events observed in the data is consistent with the Standard Model top decay expectation. We exclude branching ratios for top decaying into stop above 45% for an LSP mass up to $40$ $ {\rm GeV/c}^{2}$. The upcoming run at the Tevatron, which will feature an approximately 20-fold increase in the total integrated luminosity as well as a significantly improved CDF detector, should allow these results to be greatly extended.
[**Acknowledgements**]{}
We thank the Fermilab staff and the technical staff of the participating institutions for their vital contributions. This work was supported by the U.S. Department of Energy and National Science Foundation; the Italian Istituto Nazionale di Fisica Nucleare; the Ministry of Education, Science and Culture of Japan; the Natural Sciences and Engineering Research Council of Canada; the National Science Council of the Republic of China; and the A. P. Sloan Foundation.
[99]{}
F. Abe [*et al.*]{}, Phys. Rev. Lett. [**74**]{}, 2626 (1995).
S. Abachi [*et al.*]{}, Phys. Rev. Lett. [**74**]{}, 2632 (1995).
For reviews of the MSSM and supergravity, see H. P. Nilles, Phys. Rep. [**110**]{}, 1 (1984); P. Nath, R. Arnowitt, and A. Chamseddine, Applied N=1 Supergravity, ICTP Series in Theoretical Physics Vol. I, World Scientific, Singapore, (1984); H. Haber and G. Kane, Phys. Rep. [**117**]{}; X. Tata, The Standard Model and Beyond, edited by J. E. Kim, World Scientific, Singapore, 304 (1991).
J. Ellis [*et al.*]{}, Nucl. Phys. [**B 238**]{}, 453 (1984).
S. Abachi [*et al.*]{}, Phys. Rev. Lett. [**75**]{}, 618 (1995);\
F. Abe [*et al.*]{}, Phys. Rev. Lett. [**76**]{}, 2006 (1996);\
T. Affolder [*et al.*]{}, Phys. Rev. Lett. [**84**]{}, 5273 (2000);\
T. Affolder [*et al.*]{}, Phys. Rev. Lett. [**84**]{}, 5704 (2000).
G. Abbiendi [*et al.*]{}, Phys. Lett. [**B456**]{}, 95 (1999);\
K. Ackerstaff [*et al.*]{}, Eur. Phys. J. [**C6**]{}, 225 (1999);\
K. Ackerstaff [*et al.*]{}, Z. Phys. [**C75**]{}, 409 (1997);\
M. Acciarri [*et al.*]{}, Phys. Lett. [**B445**]{}, 428 (1999);\
R. Barate [*et al.*]{}, Phys. Lett. [**B413**]{}, 431 (1997).
S. Abachi [*et al.*]{}, Phys. Rev. [**D57**]{}, 589 (1998);\
S. Abachi [*et al.*]{}, Phys. Rev. Lett. [**76**]{}, 2222 (1996).
H. L. Lai [*et al.*]{}, Phys. Rev. [**D55**]{}, 1280 (1997).
J. Guasch [*et al.*]{}, Proceedings of the 2nd Joint ECFA/DESY Study on Physics and Detectors for a Linear Electron Positron Collider, Frascati, Italy, 8-10 Nov 1998;\
M. Hosch [*et al.*]{}, Phys. Rev. [**D58**]{}, 034002 (1998);\
G. Mahlon and G. L. Kane, Phys. Rev. [**D55**]{}, 2779 (1997);\
S. Ambrosanio [*et al.*]{}, Phys. Rev. [**D54**]{}, 5395 (1996);\
S. Mrenna and C. P. Yuan, Phys. Lett. [**B367**]{}, 188 (1996);\
J. Sender, Phys. Rev. [**D54**]{}, 3271 (1996);\
J. D. Wells and G. L. Kane, Phys. Rev. Lett. [**76**]{}, 869 (1996).
F. Abe [*et al.*]{}, Nucl. Instrum. Methods Phys. Res., Sect. [**A271**]{}, 387 (1988).
In the CDF coordinate system, $\phi$ is the azimuthal angle and $\theta $ is the polar angle with respect to the proton beam direction. The pseudorapidity $\eta $ is defined as $\eta=- \ln \tan (\theta / 2)$. The transverse momentum of a particle is $p_{\rm T} = p \sin \theta $. If the magnitude of this vector is obtained using the calorimeter energy rather than the spectrometer momentum, it becomes the transverse energy $E_{\rm T}$. Jets are defined as clusters of energy in $\eta-\phi $ space with a fixed cone of $0.4$. The missing transverse energy ($\not\!\!\!E_{\rm T}$) is defined as the difference between the vector sum of all the transverse energies and zero.
F. Abe [*et al.*]{}, Phys. Rev. [**D51**]{}, 4623 (1995);\
F. Abe [*et al.*]{}, Phys. Rev. [**D52**]{}, 2605 (1995).
F. Abe [*et al.*]{}, Phys. Rev. [**D45**]{}, 1448 (1992).
P. Nason, S. Dawson and R. K. Ellis, Nucl. Phys. [**B303**]{}, 607 (1988);\
S. Catani, M. L. Mangano, P. Nason and L. Trentadue, Phys. Lett. [**B378**]{}, 329 (1996).
Events with two SUSY top decays have small acceptance due to the W requirement in our selection.
The $t \overline{t}$ production cross section measured by CDF is based on the assumption of only SM top decays. Theoretical calculations may omit $t \overline{t}$ contributions from processes such as gluino production.
We used CTEQ-3(LO) as the Parton Distribution Function.\
H. Baer, [*et al.*]{}, Proceedings of the Workshop on Physics at Current Accelerators and the Supercollider, Argonne, 703 (1993).
T. Affolder [*et al.*]{}, FERMILAB-PUB-00-127-E, Jun 2000. F. Abe [*et al.*]{}, Phys. Rev. [**D50**]{}, 2966 (1994).
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abstract: 'The decay $B^0_s\to\mu^+\mu^-$ is a key probe for the search of physics beyond the Standard Model. While the current measurements of the corresponding branching ratio agree with the Standard Model within the uncertainties, significant New-Physics effects may still be hiding in $B^0_s\to\mu^+\mu^-$. In order to reveal them, the observable $\mathcal{A}^{\mu\mu}_{\Delta \Gamma_s}$, which is provided by the decay width difference $\Delta\Gamma_s$ of the $B^0_s$-meson system, plays a central role. We point out that a measurement of a CP-violating observable ${\cal S}_{\rm \mu\mu}$, which is induced through interference between $B^0_s$–$\bar B^0_s$ mixing and $B_s\to\mu^+\mu^-$ decay processes, is essential to obtain the full picture, in particular to establish new scalar contributions and CP-violating phases. We illustrate these findings with future scenarios for the upgrade(s) of the LHC, exploiting also relations which emerge within an effective field theory description of the Standard Model, complemented with New Physics entering significantly beyond the electroweak scale.'
---
= 1
Nikhef-2017-038
[**Robert Fleischer${}^{a,b}$, Daniela Gal' arraga Espinosa${}^{a,c}$, Ruben Jaarsma${}^{a}$\
and Gilberto Tetlalmatzi-Xolocotzi${}^{a}$**]{}
${}^a$[*Nikhef, Science Park 105, NL-1098 XG Amsterdam, Netherlands*]{}
${}^b$[*Department of Physics and Astronomy, Vrije Universiteit Amsterdam,\
NL-1081 HV Amsterdam, Netherlands*]{}
${}^c$[*Physics Department, Université Paris-Sud, F-91405 Orsay, France*]{}
September 2017
Introduction
============
The decay $B^0_s\to \mu^+\mu^-$ is one of the most interesting processes offered by Nature, allowing us to test the Standard Model (SM) and probe New Physics (NP). In the SM, this channel has no contributions at the tree level and shows a helicity suppression [@Bsmumu-SM]. Consequently, the SM branching ratio is enormously suppressed, and only about three out of one billion $B^0_s$ mesons decay into the $\mu^+\mu^-$ final state. Another key feature of $B^0_s\to \mu^+\mu^-$ is related to the impact of strong interactions. As gluons do not couple to the leptonic final state, only the $B^0_s$ decay constant $f_{B_s}$ enters the theoretical description, which can be calculated by means of lattice QCD [@lattice-rev].
As NP effects may enhance the branching ratio of $B^0_s\to \mu^+\mu^-$ significantly, experiments have searched for this channel for decades [@rev]. It has been a highlight of the results of the Large Hadron Collider (LHC) that $B^0_s\to \mu^+\mu^-$ could eventually be observed by the CMS and LHCb collaborations and is now experimentally well established [@CMS-LHCb], with a measured branching ratio in the ballpark of the SM prediction. In addition to the branching ratio, $B^0_s\to \mu^+\mu^-$ offers another observable, $\mathcal{A}^{\mu\mu}_{\Delta \Gamma_s}$, which is accessible thanks to the sizeable decay width difference $\Delta\Gamma_s$ of the mass eigenstates of the $B^0_s$-meson system [@Bsmumu-ADG]. This observable is theoretically clean and plays an important role in the search for NP effects [@BFGK; @ANS; @FJTX]. A pioneering measurement of $\mathcal{A}^{\mu\mu}_{\Delta \Gamma_s}$ has recently been reported by the LHCb collaboration [@LHCb-2017]. This analysis requires, in contrast to the measurement of the branching ratio, time information for untagged $B_s$ data samples.
If also tagging information is available, a CP-violating observable $\mathcal{S}_{\mu\mu}$ can be measured which arises from the interference between $B^0_s$–$\bar B^0_s$ mixing and decay processes. Should it be possible to determine the helicity of the final-state muons, yet another CP asymmetry ${\cal C}_{\mu\mu}$ can be measured, as discussed in detail in Refs. [@Bsmumu-ADG; @BFGK]. It is not independent from $\mathcal{A}^{\mu\mu}_{\Delta \Gamma_s}$ and $\mathcal{S}_{\mu\mu}$, as the observables satisfy the following relation: $$\label{CP-rel}
\left(\mathcal{A}^{\mu\mu}_{\Delta \Gamma_s}\right)^2+
\left(\mathcal{S}_{\mu\mu}\right)^2
+\left({\cal C}_{\mu\mu}\right)^2=1\,.$$ In these observables, as in the case of $\mathcal{A}^{\mu\mu}_{\Delta \Gamma_s}$, the decay constant $f_{B_s}$ cancels. Consequently, they are theoretically clean. Within the SM, the CP asymmetries vanish. However, in the presence of physics beyond the SM, we may in general encounter new sources of CP violation, generating non-vanishing CP asymmetries and affecting also the observable $\mathcal{A}^{\mu\mu}_{\Delta \Gamma_s}$.
In analyses of rare $B_{(s)}$ decays, it is usually – for simplicity – assumed that CP-violating NP phases vanish. Within specific models, such assumptions can be made, where an important example is given by scenarios with “Minimal Flavour Violation" [@MFV]. However, we would rather like to learn from experimental data whether new CP-violating phases enter the dynamics of the decay $B^0_s\to \mu^+\mu^-$.
In this paper, we explore this question. Interestingly, we find that $\mathcal{S}_{\mu\mu}$ is an essential observable to reveal the nature of possible NP effects. The sign of the CP asymmetry ${\cal C}_{\mu\mu}$ would allow us to resolve certain ambiguities. We shall illustrate these findings with various examples, showing in particular how we may establish new (pseudo)-scalar contributions to $B^0_s\to \mu^+\mu^-$ and further resolve their structure and dynamics. These considerations are completely general and can also be applied to the rare $B^0_s\to \tau^+\tau^-$ and $B^0_s\to e^+e^-$ decays [@FJTX].
The outline of this paper is as follows: in Section \[sec:TH\], we discuss the theoretical description of $B^0_s\to \mu^+\mu^-$ and introduce the corresponding observables. In Section \[sec:CP-gen\], we explore then the situation with general CP-violating NP contributions. Assuming relations between short-distance coefficients, which are motivated by considerations within effective field theory, we analyze the interplay between the $B^0_s\to \mu^+\mu^-$ observables in Section \[sec:CP-rel\]. In Section \[sec:Exp\], we shall address experimental aspects by discussing scenarios and illustrating their physics reach by making assumptions about the experimental precision. Finally, we summarize our key results and give a brief outlook in Section \[sec:concl\].
Theoretical Description and Observables {#sec:TH}
=======================================
Decay Amplitude
---------------
The theoretical framework to describe the decay $\bar B^0_s\to \mu^+\mu^-$ is given by effective quantum field theory, which allows the calculation of a low-energy effective Hamiltonian of the following general structure [@Bsmumu-SM; @Bsmumu-ADG; @ANS]: $$\label{Heff}
{\cal H}_{\rm eff}=-\frac{G_{\rm F}}{\sqrt{2}\pi} V_{ts}^\ast V_{tb} \alpha
\bigl[C_{10} O_{10} + C_{S} O_S + C_P O_P+ C_{10}' O_{10}' + C_{S}' O_S' + C_P' O_P' \bigr].$$ Here $G_{\rm F}$ is Fermi’s constant, $V_{ts}^\ast V_{tb}$ is a factor with elements of the Cabibbo–Kobayashi–Maskawa (CKM) matrix, and $\alpha$ denotes the QED fine structure constant. The Wilson coefficients $C_{10}^{(')}$, $C_{P}^{(')}$ and $C_{S}^{(')}$ describe heavy degrees of freedom, which have been integrated out from appearing as explicit fields, and are associated with the four-fermion operators $$\label{ops}
\begin{array}{rclcrcl}
O_{10}&=&(\bar s \gamma_\mu P_L b) (\bar\mu\gamma^\mu \gamma_5\mu), & \mbox{}\qquad &
O_{10}'&=&(\bar s \gamma_\mu P_R b) (\bar\mu\gamma^\mu \gamma_5\mu), \\
O_S&=&m_b (\bar s P_R b)(\bar \mu \mu), & \mbox{}\qquad & O_S'&=&m_b (\bar s P_L b)(\bar \mu \mu),\\
O_P&=&m_b (\bar s P_R b)(\bar \mu \gamma_5 \mu), & \mbox{}\qquad
& O_P'&=&m_b (\bar s P_L b)(\bar \mu \gamma_5 \mu),
\end{array}$$ with $m_b$ denoting the $b$-quark mass, and $$P_{L}\equiv\frac{1}{2}\left(1-\gamma_5\right), \quad P_{R}\equiv\frac{1}{2}\left(1+\gamma_5\right).$$ In general, the Wilson coefficients are different for $b \to s$ and $b \to d$ transitions, and depend on the flavour of the final-state leptons [@FJTX]. For simplicity, we do not give the corresponding labels explicitly in the following discussion. In the SM, we have only to deal with the $O_{10}$ operator, having a real coefficient $C_{10}^{\rm SM}$.
Introducing the combinations of Wilson coefficients $$\begin{aligned}
P &\equiv &\frac{C_{10} - C'_{10}}{C_{10}^{\rm SM}} + \frac{M_{B_s}^2}{2m_\mu}
\left(\frac{m_b}{m_b + m_s}\right) \left( \frac{C_{P} - C'_{P}}{C_{10}^{\rm SM}}\right)\equiv
|P|e^{i\varphi_P}, \label{P-def} \\
S &\equiv &\sqrt{1-\frac{4m_\mu^2}{M_{B_s}^2}}\frac{M_{B_s}^2}{2m_\mu}\left(\frac{m_b}{m_b + m_s}\right)
\left( \frac{C_{S} - C'_{S}}{C_{10}^{\rm SM}}\right)\equiv |S|e^{i\varphi_S},
\label{S-def}\end{aligned}$$ where $M_{B_s}$, $m_\mu$, $m_b$, $m_s$ are the corresponding particle masses and $\varphi_P$, $\varphi_S$ denote CP-violating phases, we obtain the following expression for the decay amplitude [@Bsmumu-ADG]: $$A(\bar B^0_s\to\mu_\lambda^+\mu_\lambda^-)\propto V_{ts}^\ast V_{tb} f_{B_s} M_{B_s}
m_\mu C_{10}^{\rm SM} \left [\eta_\lambda P + S \right].$$ Here $\lambda={\rm L, R}$ describes the helicity of the final-state leptons with $\eta_{\rm L}=+1$ and $\eta_{\rm R}=-1$.
In the SM, we have $$\label{SM_PS}
P|_{\rm SM}=1, \quad S|_{\rm SM}=0,$$ and the relevant Wilson coefficient is given as [@BFGK] $$\label{C10-SM}
C^{\rm SM}_{10}=-\eta_Y \sin^{-2}\theta_W Y_{0}(x_t) = -4.134,$$ where $\eta_Y$ describes QCD corrections, $\theta_W$ is the weak mixing angle, $Y(x_t)$ represents one of the Inami–Lim functions, and $x_t\equiv m_t^2/M_W^2$ parametrizes the top-quark and $W$ mass dependence [@Buras:1996]. We would like to emphasize that, by convention, $C^{\rm SM}_{10}$ does not have a complex phase. However, it takes a negative value, such that $$\label{C10-neg}
C^{\rm SM}_{10}= - |C^{\rm SM}_{10}|.$$
In the following discussion, the CP-violating phases $\varphi_P$ and $\varphi_S$ play a central role. While the latter is directly related to the phase of the short-distance coefficient $C_{S} - C'_{S}$ of new scalar contributions, the former may get contributions both from $C_{10} - C'_{10}$ and from the coefficient $C_{P} - C'_{P}$, which arises from new pseudo-scalar operators.
Branching Ratio and Effective Lifetime
--------------------------------------
Due to $B^0_s$–$\bar B^0_s$ mixing, an initially, i.e. at time $t=0$, present $B^0_s$ meson evolves into a time-dependent linear combination of $B^0_s$ and $\bar B^0_s$ states. For the “untagged" rate $$\langle \Gamma(B_s(t)\to \mu_\lambda^+\mu_\lambda^-)\rangle\equiv
\Gamma(B^0_s(t)\to \mu_\lambda^+\mu_\lambda^-)+ \Gamma(\bar B^0_s(t)\to
\mu_\lambda^+\mu_\lambda^-)$$ $$\label{untagged}
\propto e^{-t/\tau_{B_s}}\bigl[\cosh(y_st/ \tau_{B_s})+ {\cal A}_{\Delta\Gamma_s}^\lambda
\sinh(y_st/ \tau_{B_s})\bigr]=R_{\rm H}^\lambda e^{-\Gamma_{\rm H}^{(s)}t}+
R_{\rm L}^\lambda e^{-\Gamma_{\rm L}^{(s)}t},$$ no “tagging" of the initially present $B_s$ meson is needed. This quantity depends only on two exponentials and involves the parameter $$\label{ys-exp}
y_s\equiv \frac{\Delta\Gamma_s}{2\Gamma_s}= 0.0645\pm0.0045,$$ which characterizes the decay width difference of the $B_s$ mass eigenstates, with $\tau_{B_s}\equiv1/\Gamma_s$ denoting the $B_s$ mean lifetime [@DFN; @BR-Bs]; for the experimental value, see Ref. [@Amh16]. The decay dynamics enters through the following observable [@Bsmumu-ADG; @BFGK]: $$\label{ADG-expr}
{\cal A}^{\lambda}_{\Delta\Gamma_s}=\frac{R_{\rm H}^\lambda-R_{\rm L}^\lambda}{R_{\rm H}^\lambda
+R_{\rm L}^\lambda}
= \frac{|P|^2\cos(2\varphi_P-\phi_s^{\rm NP}) - |S|^2\cos(2\varphi_S-\phi_s^{\rm NP})}{|P|^2 + |S|^2}\equiv
{\cal A}^{\mu\mu}_{\Delta\Gamma_s},$$ which is independent of the muon helicity, as reflected by the definition of ${\cal A}^{\mu\mu}_{\Delta\Gamma_s} $. Within the SM, we have $${\cal A}^{\mu\mu}_{\Delta\Gamma_s}|_{\rm SM}=+1.$$
The phase $\phi_s^{\rm NP}$ originates from possible CP-violating NP contributions to the $B_s^0$–$\bar B_s^0$ mixing phase $$\phi_s = -2\beta_s + \phi_s^{\rm NP},$$ which is already strongly constrained by experimental data for CP-violating effects in $B^0_s\to J/\psi \phi$ and decays with similar dynamics, yielding the following results [@peng-anat; @Amh16; @CKMFitter:2016]: $$\label{phi_s}
\phi_s = -0.030\pm 0.033 = -(1.72\pm 1.89)^{\circ}$$ $$\label{phis-NP}
\phi_s^{\rm NP}=0.007 \pm 0.033 =(0.4\pm 1.9)^{\circ},$$ where we have used the SM value $\phi_s^{\rm SM}=-2\beta_s=-(2.12\pm0.04)^\circ$ in Eq. (\[phis-NP\]).
Since it is challenging to measure the muon helicity, we consider the helicity-summed rates $$\Gamma({B}_s^0(t)\to \mu^+\mu^-)\equiv \sum_{\lambda={\rm L,R}}
\Gamma({B}_s^0(t)\to \mu^+_\lambda \mu^-_\lambda)$$ $$\Gamma(\bar{B}_s^0(t)\to \mu^+\mu^-)\equiv \sum_{\lambda={\rm L,R}}
\Gamma(\bar{B}_s^0(t)\to \mu^+_\lambda \mu^-_\lambda),$$ and use them to define an untagged rate $\langle\Gamma(B_s(t)\to \ell^+\ell^-)\rangle$ in analogy to Eq. (\[untagged\]). The branching ratio reported by experiments actually corresponds to the following time-integrated untagged rate [@Bsmumu-ADG; @DFN]: $$\overline{\mathcal{B}}(B_s\to\ell^+\ell^-) \equiv \frac{1}{2}\int_0^\infty \langle
\Gamma(B_s(t)\to \ell^+\ell^-)
\rangle\,dt.\label{BRexp}$$ Combining the CMS result from 2013 [@CMSmumu] with the most recent LHCb analysis [@LHCb-2017] yields $$\label{eq:BsmumuExpComb}
\overline{\mathcal{B}}(B_s \to \mu^+\mu^-)_\text{LHCb'17+CMS} = (3.0 \pm 0.5 ) \times 10^{-9}.$$ This average was calculated by means of the Particle Data Group (PDG) procedure [@PDG:2016]. For comparison, we give also the constraint $\overline{\mathcal{B}}(B_s \to \mu^+\mu^-)_\text{ATLAS'16}=(0.9^{+1.1}_{-0.8})
\times 10^{-9}$ reported by the ATLAS collaboration [@ATLAS:2016].
In the SM, we have the following expression [@Bsmumu-SM]: $$\overline{\mathcal{B}}(B_s\rightarrow \mu^+\mu^-)_{\rm SM}=
\frac{\tau_{B_s}G^4_F M^4_W \sin^4\theta_W}{8\pi^5}
\frac{\left|C^{\rm SM}_{10}V_{ts} V^{*}_{tb} \right|^2 }{(1-y_s)}f^2_{B_s}M_{B_s}m^2_{\mu}
\sqrt{1-4\frac{m^2_{\mu}}{M^2_{B_s}}},$$ where special care has to be taken concerning the use of renormalization schemes to properly include next-to-leading-order electroweak corrections (for details, see Ref. [@Bsmumu-SM]). Using current state-of-the-art input parameters yields the following result [@FJTX]: $$\label{BsmumuSM}
\overline{\mathcal{B}}(B_s \to \mu^+\mu^-)_\text{SM} = (3.57 \pm 0.16) \times 10^{-9}.$$ In a very recent analysis [@BBS], QED corrections from dynamics below the renormalization scale $\mu = m_b$ were calculated, affecting the branching ratio by almost $1\%$.
In order to search for NP effects by means of the branching ratio of $B^0_s\to\mu^+\mu^-$, the following ratio plays the key role [@Bsmumu-ADG; @BFGK]: $$\label{Rbar}
\overline{R} \equiv \frac{\overline{\mathcal{B}}(B_s\to\mu^+\mu^-)}{\overline{\mathcal{B}}
(B_s\to\mu^+\mu^-)_{\rm SM}},$$ taking by definition the SM value $$\overline{R}|_{\rm SM}=1.$$ Using the expressions given above yields $$\label{R-expr}
\overline{R} = \left[\frac{1+{\cal A}^{\mu\mu}_{\Delta\Gamma_s}\,y_s}{1+y_s} \right]
(|P|^2 + |S|^2)= \Upsilon_{P} |P|^2 + \Upsilon_{S} |S|^2$$ with $$\label{y-pm}
\Upsilon_{P}\equiv \left[\frac{1 + y_s\cos(2\varphi_P - \phi_s^{\rm NP})}{1+y_s} \right],
\quad
\Upsilon_{S}\equiv \left[\frac{1 - y_s\cos(2\varphi_S - \phi_s^{\rm NP})}{1+y_s} \right].$$ The numerical results in Eqs. (\[eq:BsmumuExpComb\]) and (\[BsmumuSM\]) give $$\left.\overline{R}\right|_\text{LHCb'17+CMS} = 0.84 \pm 0.16.
\label{eq:Rbar}$$
The effective lifetime of the decay $B^0_s\to\mu^+\mu^-$, which is defined through $$\tau^s_{\mu\mu} \equiv \frac{\int^\infty_0 t\,
\langle\Gamma(B_s(t)\to \mu^+ \mu^-)\rangle\, dt}{\int_0^\infty \langle
\Gamma(B_s(t)\to \mu^+ \mu^-)\rangle\, dt},$$ contains the same physics information as the observable ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$ [@Bsmumu-ADG]: $$\label{eq:aDGLifetime}
{\cal A}^{\mu\mu}_{\Delta\Gamma_s} = \frac{1}{y_s}\left[\frac{(1-y_s^2)\tau^s_{\mu\mu}-(1+
y_s^2)\tau_{B_s}}{2\tau_{B_s}-(1-y_s^2)\tau^s_{\mu\mu}}\right].$$ A pioneering measurement of the effective lifetime of $B^0_s\to\mu^+\mu^-$ was recently reported by the LHCb collaboration [@LHCb-2017]: $$\label{eq:tauEffmumu}
\tau^s_{\mu\mu} = \left[2.04 \pm 0.44 ({\rm stat}) \pm 0.05 ({\rm syst}) \right]\hbox{ps}.$$ Using Eq. (\[eq:aDGLifetime\]), this result can be converted into $$\label{ADG-exp}
\mathcal{A}_{\Delta\Gamma_s}^{\mu\mu} = 8.24 \pm 10.72,$$ where the error is fully dominated by the uncertainty of $\tau^s_{\mu\mu}$. In view of the general model-independent range $$\label{ADG-range}
-1\leq\mathcal{A}_{\Delta\Gamma_s}^{\mu\mu}\leq+1,$$ it will be crucial to improve the experimental precision for this observable at the LHC upgrade(s) in order to use this quantity for testing the flavour sector of the SM.
CP Asymmetries
--------------
In contrast to the untagged $B_s$ rate in Eq. (\[untagged\]), the tagged, time-dependent rates involve oscillatory $\sin(\Delta M_st)$ and $\cos(\Delta M_st)$ terms, where $\Delta M_s$ is the mass difference between the heavy and light $B_s$ mass eigenstates. We obtain a CP-violating rate asymmetry of the following form [@Bsmumu-ADG; @BFGK]: $$\frac{\Gamma(B^0_s(t)\to \mu_\lambda^+\mu^-_\lambda)-
\Gamma(\bar B^0_s(t)\to \mu_\lambda^+
\mu^-_\lambda)}{\Gamma(B^0_s(t)\to \mu_\lambda^+\mu^-_\lambda)+
\Gamma(\bar B^0_s(t)\to \mu_\lambda^+\mu^-_\lambda)}
=\frac{{\cal C}_{\mu\mu}^\lambda\cos(\Delta M_st)+{\cal S}_{\mu\mu}^\lambda
\sin(\Delta M_st)}{\cosh(y_st/\tau_{B_s}) +
{\cal A}_{\Delta\Gamma_s}^\lambda \sinh(y_st/\tau_{B_s})},$$ with the observables $$\begin{aligned}
{\cal C}_{\mu\mu}^\lambda &= & -\eta_\lambda\left[\frac{2|PS|\cos(\varphi_P-\varphi_S)}{|P|^2+|S|^2}
\right] \equiv -\eta_\lambda {\cal C}_{\rm \mu\mu} \overset{\text{SM}}{\longrightarrow} 0 ,\label{Cobs}\\
{\cal S}_{\mu\mu}^\lambda
&=&\frac{|P|^2\sin(2\varphi_P-\phi_s^{\rm NP})-|S|^2\sin(2\varphi_S-\phi_s^{\rm NP})}{|P|^2+|S|^2}
\equiv {\cal S}_{\mu\mu} \overset{\text{SM}}{\longrightarrow} 0 ,\label{S-expr}\end{aligned}$$ where $\eta_{\rm L}=+1$ and $\eta_{\rm R}=-1$ for left- and right-handed muon helicity, respectively. It should be noted that the CP asymmetry ${\cal S}_{\mu\mu}^\lambda$, which is caused by interference between $B^0_s$–$\bar B^0_s$ mixing and $B_s\to\mu^+\mu^-$ decay processes, does actually not depend on the muon helicity, just as the observable $\mathcal{A}_{\Delta\Gamma_s}^{\mu\mu} \equiv
{\cal A}_{\Delta\Gamma_s}^\lambda$. Using the helicity-summed rates introduced above yields $$\label{CP-asym}
\frac{\Gamma(B^0_s(t)\to \mu^+\mu^-)-
\Gamma(\bar B^0_s(t)\to \mu^+\mu^-)}{\Gamma(B^0_s(t)\to \mu^+\mu^-)+
\Gamma(\bar B^0_s(t)\to \mu^+\mu^-)}
=\frac{{\cal S}_{\mu\mu}\sin(\Delta M_st)}{\cosh(y_st/ \tau_{B_s}) +
{\cal A}^{\mu\mu}_{\Delta\Gamma_s} \sinh(y_st/ \tau_{B_s})},$$ where the ${\cal C}^\lambda_{\mu\mu}$ terms cancel because of the $\eta_\lambda$ factor. It should be noted that a non-vanishing ${\cal C}_{\rm \mu\mu}$ would be a smoking-gun signal for a new scalar contribution $S$. CP-violating asymmetries of this kind in $B_{s,d}\to\ell^+\ell^-$ decays were also considered for various NP scenarios in Refs. [@HL; @DP; @CKWW], neglecting the effects of $\Delta\Gamma_s$ and the associated observable $\mathcal{A}^{\mu\mu}_{\Delta \Gamma_s}$. For a more recent study, including the untagged observable, see Ref. [@BFGK].
It should be stressed that the non-perturbative decay constant $f_{B_s}$ cancels in $\mathcal{A}^{\mu\mu}_{\Delta \Gamma_s}$ as well as in ${\cal S}_{\mu\mu}$ and ${\cal C}_{\mu\mu}$, thereby making these observables theoretically clean probes for the search of NP signals [@Bsmumu-ADG; @BFGK]. In the SM, a tiny residual uncertainty arises from QED corrections, which lead to effects at the $10^{-5}$ and $10^{-3}$ levels for $\mathcal{A}^{\mu\mu}_{\Delta \Gamma_s}$ and the CP asymmetries ${\cal S}_{\mu\mu}$, ${\cal C}_{\mu\mu}$, respectively [@BBS].
In the following discussion, we will explore the interplay of ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$ and ${\cal S}_{\mu\mu}$ with the observable $\overline{R}$ to search for NP and reveal its nature, in particular whether it involves new (pseudo)-scalar contributions. Experimental feasibility studies of measurements of the CP asymmetry in Eq. (\[CP-asym\]) have not yet been performed to the best of our knowledge. However, we envision that an effort should be made to perform such a measurement at the LHC upgrade(s). In view of the relation in Eq. (\[CP-rel\]), a measurement of ${\cal C}_{\rm \mu\mu}$ would not provide independent information. As such an analysis would require the reconstruction of the muon helicity, it is much more challenging than the asymmetry in Eq. (\[CP-asym\]) involving the helicity-averaged rates. However, we will show that already information on just the sign of ${\cal C}_{\rm \mu\mu}$ would be sufficient to resolve certain ambiguities affecting the determination of $P$ and $S$. We encourage experimentalists to explore avenues to eventually measure the sign of the ${\cal C}_{\rm \mu\mu}$ observable.
General CP-Violating New Physics {#sec:CP-gen}
================================
Theoretical Description {#ssec:theo-des}
-----------------------
Let us start the general discussion of the CP-violating coefficients $P$ and $S$ in Eqs. (\[P-def\]) and (\[S-def\]), respectively, with the ratio $\overline{R}$ in Eq. (\[Rbar\]). Using the expression in Eq. (\[R-expr\]), we obtain $$\label{rr-def}
r\equiv \left[\frac{1+y_s}{1+{\cal A}^{\mu\mu}_{\Delta\Gamma_s}\,y_s} \right]\overline{R} = |P|^2 + |S|^2.$$ If we had a precise measurement of ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$, we could straightforwardly convert $\overline{R}$ into $r$. In view of the large uncertainty in Eq. (\[ADG-exp\]), we use the general range in Eq. (\[ADG-range\]) to calculate $$\label{r-val}
0.69 \leq r \leq 1.13 ,$$ where we have also taken into account the $1\sigma$ uncertainty of $\overline{R}$, given in Eq. (\[eq:Rbar\]). This observable fixes a circular band with radius $\sqrt{r}$ in the $|P|$–$|S|$ plane, which we show in Fig. \[fig:absSabsP\_ex\]. Using the observable ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$, we can calculate a straight line in this plane through $$\frac{|S|}{|P|}=\sqrt{\frac{\cos\Phi_P-{\cal A}^{\mu\mu}_{\Delta\Gamma_s}}{\cos\Phi_S+
{\cal A}^{\mu\mu}_{\Delta\Gamma_s}}},$$ where we have introduced the abbreviations $$\Phi_P\equiv 2\varphi_P-\phi_s^{\rm NP}, \quad
\Phi_S\equiv 2\varphi_S-\phi_s^{\rm NP}.$$ If we assume that the CP-violating phases $\varphi_P$ and $\varphi_S$ take trivial values, i.e. $0^\circ$ or $180^\circ$, $\overline{R}$ allows us to fix a circle in the $|P|$–$|S|$ plane through Eq. (\[R-expr\]), and the intersection with the straight line following from $$\frac{|S|}{|P|}=\sqrt{\frac{\cos\phi_s^{\rm NP}-{\cal A}^{\mu\mu}_{\Delta\Gamma_s}}{\cos\phi_s^{\rm NP}
+{\cal A}^{\mu\mu}_{\Delta\Gamma_s}}}=\sqrt{\frac{1-{\cal A}^{\mu\mu}_{\Delta\Gamma_s}}{1+
{\cal A}^{\mu\mu}_{\Delta\Gamma_s}}}$$ fixes $|P|$ and $|S|$, as discussed in detail in Refs. [@Bsmumu-ADG; @BFGK; @ANS; @FJTX]; note that we use the result for $\phi_s^{\rm NP}$ in Eq. (\[phis-NP\]). However, if we allow for general CP-violating phases, any point on the circle with radius $\sqrt{r}$ is allowed since we obtain $|S|=0$ for $\cos\Phi_P={\cal A}^{\mu\mu}_{\Delta\Gamma_s}$ and $|P|=0$ for $\cos\Phi_S=-{\cal A}^{\mu\mu}_{\Delta\Gamma_s}$.
![Allowed region in the $|P|$–$|S|$ plane following from $r$, which is obtained by varying ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$ between $-1$ and $+1$ and taking the $1\,\sigma$ uncertainty of the current $\overline{R}$ measurement into account. The black star indicates the SM values given in Eq. (\[SM\_PS\]).\[fig:absSabsP\_ex\]](fig1.pdf){width="50.00000%"}
The measurement of a non-vanishing CP asymmetry ${\cal S}_{\mu\mu}$ would immediately establish the presence of non-trivial CP-violating phases. This observable fixes another straight line in the $|P|$–$|S|$ plane: $$\frac{|S|}{|P|}=\sqrt{\frac{\sin\Phi_P-{\cal S}_{\mu\mu}}{\sin\Phi_S+{\cal S}_{\mu\mu}}}.$$ However, as the CP-violating phases are in general unknown, the slope of this straight line is not determined, in analogy to the constraint following from ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$.
We have three independent observables at our disposal, $r$ as well as ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$ and ${\cal S}_{\mu\mu}$, which depend on the four unknown parameters $|P|$, $\Phi_P$ and $|S|$, $\Phi_S$. Using the general expressions for ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$ and ${\cal S}_{\mu\mu}$ in Eqs. (\[ADG-expr\]) and (\[S-expr\]), respectively, yields $$A\cos\Phi_P - B\sin\Phi_P = C$$ with $$\begin{aligned}
A & \equiv & {\cal S}_{\mu\mu} + \sin\Phi_S \\
B & \equiv & {\cal A}^{\mu\mu}_{\Delta\Gamma_s} + \cos\Phi_S \\
C & \equiv & {\cal A}^{\mu\mu}_{\Delta\Gamma_s}\sin\Phi_S - {\cal S}_{\mu\mu} \cos\Phi_S.\end{aligned}$$ This equation allows us to determine $\Phi_P$ as a function of $\Phi_S$ with the help of $$\begin{aligned}
\sin\Phi_P&=&-\left(\frac{BC}{A^2+B^2}\right)\pm\sqrt{\left(\frac{BC}{A^2+B^2}\right)^2+
\left(\frac{A^2-C^2}{A^2+B^2}\right)}\\
&=&\frac{-BC\pm |A|\sqrt{A^2+B^2-C^2}}{A^2+B^2}.
\label{eq:sinPhiP}\end{aligned}$$ The expression under the square root is actually factorizable, thereby yielding $$\begin{aligned}
\sqrt{A^2+B^2-C^2}&=&|1+{\cal A}^{\mu\mu}_{\Delta\Gamma_s}\cos\Phi_S+{\cal S}_{\mu\mu}\sin\Phi_S|.\end{aligned}$$
![Flowchart to illustrate the general strategy to determine $|P|$ and $|S|$ as functions of the the CP-violating phase $\varphi_S$ from the $B^0_s\to\mu^+\mu^-$ observables.[]{data-label="fig:strategy_general"}](fig2.pdf){width="2.8in"}
Using then the observables $r$ and ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$, we may determine $$|P|=\sqrt{\left(\frac{\cos\Phi_S+{\cal A}^{\mu\mu}_{\Delta\Gamma_s}}{\cos\Phi_S+\cos\Phi_P}\right)r},
\quad
|S|=\sqrt{\left(\frac{\cos\Phi_P-{\cal A}^{\mu\mu}_{\Delta\Gamma_s}}{\cos\Phi_P+\cos\Phi_S}\right)r}
\label{eq:PSADG}$$ as functions of the CP-violating phase $\Phi_S$. Using instead of ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$ the CP asymmetry ${\cal S}_{\mu\mu}$ yields $$|P|=\sqrt{\left(\frac{\sin\Phi_S+{\cal S}_{\mu\mu}}{\sin\Phi_S+\sin\Phi_P}\right)r},
\quad
|S|=\sqrt{\left(\frac{\sin\Phi_P-{\cal S}_{\mu\mu}}{\sin\Phi_P+\sin\Phi_S}\right)r}.
\label{eq:PSSmumu}$$ The expression in Eq. (\[eq:sinPhiP\]) leaves us with a twofold ambiguity for $\varphi_P$ for every value of $\varphi_S$. Information on the sign of ${\cal C}_{\mu\mu}$ allows us to determine the correct branch and thus obtain a single solution for $\varphi_P$ as a function of $\varphi_S$. However, both branches have the same dependence of $|P|$ and $|S|$ on $\varphi_S$, so a single solution for $|P|$ and $|S|$ as a function of $\varphi_S$ can be obtained even when no information on the sign of ${\cal C}_{\mu\mu}$ is available. In the flowchart in Fig. \[fig:strategy\_general\], we illustrate this general method for analyzing the observables provided by the $B^0_s\to\mu^+\mu^-$ decay, and we will provide an example of this formalism in the next subsection.
Discussion and Illustration
---------------------------
### Vanishing Mixing-Induced CP Violation
An interesting situation arises for ${\cal S}_{\mu\mu}=0$. Although one may naively conclude that the CP-violating phases take then simply trivial values, this is actually not the case because of the structure of the expression in Eq. (\[S-expr\]). In fact, we obtain the following extremal values on the circle with radius $\sqrt{r}$ in the $|P|$–$|S|$ plane: $$|P_\pm|=\sqrt{\left(\frac{1\mp{\cal A}^{\mu\mu}_{\Delta\Gamma_s}}{2}\right)r},
\quad
|S_\pm|=\sqrt{\left(\frac{1\pm{\cal A}^{\mu\mu}_{\Delta\Gamma_s}}{2}\right)r},$$ where the region between these points can be accessed by varying $\Phi_S$. In the case of ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}=\pm1$, we have $$|S|=0, \quad |P|=\sqrt{r}, \quad \sin\Phi_P=0,$$ yielding ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}=+\cos\Phi_P=\pm1$, or $$|P|=0, \quad |S|=\sqrt{r}, \quad \sin\Phi_S=0,$$ yielding ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}=-\cos\Phi_S=\pm1$. For $|{\cal A}^{\mu\mu}_{\Delta\Gamma_s}|<1$, we get $$\left.\frac{|S|}{|P|}\right.=\sqrt{\frac{(1-{\cal A}^{\mu\mu}_{\Delta\Gamma_s})(1+
{\cal A}^{\mu\mu}_{\Delta\Gamma_s})}{1+2\,{\cal A}^{\mu\mu}_{\Delta\Gamma_s}\cos\Phi_S+
({\cal A}^{\mu\mu}_{\Delta\Gamma_s})^2}}\,.$$ A particularly interesting situation arises for ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}=0$, corresponding to the following point in the $|P|$–$|S|$ plane: $$|P|=|S|=\sqrt{\frac{r}{2}}\,.$$
### Sizeable Mixing-Induced CP Violation {#sssec:genex}
Let us now turn to mixing-induced CP violation in $B^0_s\to\mu^+\mu^-$, and discuss a scenario with a large value of ${\cal S}_{\mu\mu}$, which requires significant CP-violating phases originating from physics beyond the SM. In order to illustrate this situation and the formalism discussed in Subsection \[ssec:theo-des\], we consider an example which is characterized by $$\label{S-input}
|S|= 0.30, \quad \varphi_S = 20^\circ.$$ Assuming furthermore $$\varphi_P = 30^\circ,$$ the central value of the observable $ \overline{R}$ in Eq. (\[eq:Rbar\]) yields $$|P|= 0.89.$$ These values of $|P|$ and $|S|$ fall well within the currently allowed region in the $|P|$–$|S|$ plane shown in Fig. \[fig:absSabsP\_ex\]. We obtain the following set of observables: $$\overline{R}=0.84, \quad {\cal A}^{\mu\mu}_{\Delta\Gamma_s}= 0.37, \quad {\cal S}_{\mu\mu} = 0.71,
\quad {\cal C}_{\mu\mu} = 0.60,$$ and assume that they were measured at a future experiment.
Let us now illustrate how we may obtain insights into NP effects using these observables. The deviation of ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$ from the SM prediction $+1$ would indicate NP effects. Having the measured ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$ at hand, we may use Eq. (\[rr-def\]) to convert $\overline{R}$ into $r$, yielding $$r = 0.87.$$ Moreover, the precision of the measured $B^0_s\to\mu^+\mu^-$ branching ratio will then have significantly increased (see Section \[sec:Exp\] for a more detailed discussion), allowing us to reduce the width of the circular band in Fig. \[fig:absSabsP\_ex\]. However, without any information on ${\cal S}_{\mu\mu}$, we could not narrow down further $|S|$ and $|P|$ in a model-independent way, i.e. we would still be left with the whole circular region, and could in particular not establish a non-vanishing scalar contribution $S$.
![Correlation between $\varphi_P$ and $\varphi_S$ for ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}=0.37$ and ${\cal S}_{\mu\mu}=0.71$. The red and grey curves correspond to ${\cal C}_{\mu\mu}>0$ and ${\cal C}_{\mu\mu}<0$, respectively. The green dot marks the input parameters in Eq. (\[S-input\]).\[fig:PhiPvsPhiS\]](fig3.pdf){width="55.00000%"}
The measurement of the observable ${\cal S}_{\mu\mu}$ different from zero would signal new sources of CP violation. Using then Eq. (\[eq:sinPhiP\]), we could determine $\varphi_P$ as a function of $\varphi_S$, as illustrated in Fig. \[fig:PhiPvsPhiS\]. The information on the sign of ${\cal C}_{\mu\mu}$ would allow us to resolve the ambiguity, as indicated in the figure. Note that the points $(\varphi_S,\varphi_P)=(0^\circ,0^\circ)$ and $(180^\circ,180^\circ)$ would be excluded through the contours. Using Eqs. (\[eq:PSADG\]) or (\[eq:PSSmumu\]), we obtain $|S|$ and $|P|$ as functions of $\varphi_S$, as shown in Fig. \[fig:SphiS\_ex-PphiS\_ex\]. Here, information about the sign of ${\cal C}_{\mu\mu}$ plays no further role. Interestingly, we would now be able to put a lower bound on $|S|$, i.e. could conclude that we have new scalar contributions. We insist on the fact that in order to obtain this highly non-trivial information, a measurement of the CP asymmetry ${\cal S}_{\mu\mu}$ is required.
Although we can only determine the $B^0_s\to\mu^+\mu^-$ parameters as functions of $\varphi_S$, this analysis would have profound implications, establishing in particular new scalar and pseudo-scalar contributions with CP-violating phases. In order to obtain further insights, more information is needed and assumptions about short-distance coefficients have to be made.
![The coefficients $|S|$ and $|P|$ determined as functions of $\varphi_S$ for the example discussed in the text. The corresponding input parameters are marked by the green dot. \[fig:SphiS\_ex-PphiS\_ex\]](fig4a.pdf "fig:"){width="44.00000%"} ![The coefficients $|S|$ and $|P|$ determined as functions of $\varphi_S$ for the example discussed in the text. The corresponding input parameters are marked by the green dot. \[fig:SphiS\_ex-PphiS\_ex\]](fig4b.pdf "fig:"){width="44.00000%"}
Relations Between (Pseudo)-Scalar Coefficients {#sec:CP-rel}
==============================================
General Framework {#ssec:gen-frame}
-----------------
The effects of new particles enter the coefficients in Eqs. (\[P-def\]) and (\[S-def\]) through the short-distance coefficients $C_P$, $C_P'$ and $C_S$, $C_S'$, which describe new pseudo-scalar and scalar contributions, respectively, and $C_{10}$, $C_{10}'$. As the constraints from the ATLAS and CMS experiments at the LHC for direct searches of new particles support the picture of a NP scale $\Lambda_{\rm NP}$ which is much larger than the electroweak scale $\Lambda_{\rm EW}$, the corresponding NP effects can be described in a model-independent way through an effective Lagrangian where the heavy degrees of freedom, i.e. the NP particles, have been integrated out at $\Lambda_{\rm NP}$. If we require then invariance under the SM gauge group $SU(2)_L\times U(1)_Y$ for the renormalization group evolution between $\Lambda_{\rm NP}$ and $\Lambda_{\rm EW}$, a “SM Effective Field Theory" (SMEFT) can be set up [@BW; @GIMR] and matched to the effective Hamiltonian in Eq. (\[Heff\]) describing $B^0_s\to\mu^+\mu^-$ decays. Following these lines and applying the machinery of effective quantum field theory, the following relations among the corresponding short-distance coefficients can be derived [@AGMC]: $$\label{rel-PS}
C_P=-C_S$$ $$\label{rel-PS-prime}
C_P'=C_S'.$$ A further application of these relations – assuming no new sources of CP violation – can be found in Ref. [@ANS], while a fit of data to the SMEFT scenario with complex coefficients was performed in Ref. [@BBJ]. For a discussion within specific models, see Ref. [@BFGK].
In this section, we explore the implication of Eqs. (\[rel-PS\]) and (\[rel-PS-prime\]) for the general analysis of CP violation discussed in Section \[sec:CP-gen\]. To this end, we express the relevant quantities in terms of the scalar short-distance coefficients $$C_S\equiv |C_S|e^{i\tilde\varphi_S}, \quad
C_S'\equiv |C_S'|e^{i\tilde\varphi_S'},$$ which yields $$\label{conv-1}
P\equiv |P|e^{i\varphi_P}=|P|\cos\varphi_P + i |P| \sin\varphi_P=
{\cal C}_{10}-\frac{1}{w}\left[\frac{1+|x|e^{i\Delta}}{1-|x|e^{i\Delta}}\right]|S|e^{i\varphi_S}$$ with $$w\equiv\sqrt{1-\frac{4m_\mu^2}{M_{B_s}^2}}, \qquad {\cal C}_{10} \equiv \frac{C_{10} - C'_{10}}{C_{10}^{\rm SM}}$$ and $$x\equiv|x|e^{i\Delta}\equiv\left|\frac{C_S'}{C_S}\right|e^{i(\tilde\varphi_S'-\tilde\varphi_S)}.$$ We will refer to this notation as the SMEFT parametrization. It it useful to write Eq. (\[conv-1\]) in the following form: $$\label{gen-rel-PS}
wP+\left[\frac{1+|x|e^{i\Delta}}{1-|x|e^{i\Delta}}\right]S = w \, {\cal C}_{10}.$$ In the Appendix, we present expressions that allow us to obtain the $B^0_s\to\mu^+\mu^-$ observables in terms of the parametrization introduced above. As $P$ requires input for $C_{10}$, $C_{10}'$, we shall now first discuss these coefficients.
Closer Look at $C_{10}$ and $C'_{10}$ {#sec:C10}
-------------------------------------
The Wilson coefficients $C_{10}$ and $C_{10}'$ enter in $P$ through the following combination: $$\label{eq:CalC10}
{\cal C}_{10}\equiv |{\cal C}_{10}|e^{i\varphi_{10}} \equiv \frac{C_{10} - C'_{10}}{C_{10}^{\rm SM}}
=1+{\cal C}_{10}^{\rm NP},$$ where $\varphi_{10}$ is a CP-violating phase and $${\cal C}_{10}^{\rm NP}\equiv|{\cal C}_{10}^{\rm NP}|e^{i\varphi_{10}^{\rm NP}}
=\frac{C_{10}^{\rm NP} - C'_{10}}{C_{10}^{\rm SM}}$$ parametrizes NP effects. The relations $$|{\cal C}_{10}|=\sqrt{1+2|{\cal C}_{10}^{\rm NP}|\cos\varphi_{10}^{\rm NP}+|{\cal C}_{10}^{\rm NP}|^2},$$ $$|{\cal C}_{10}|\cos\varphi_{10} = 1+|{\cal C}_{10}^{\rm NP}|\cos\varphi_{10}^{\rm NP}, \quad
|{\cal C}_{10}|\sin\varphi_{10} = |{\cal C}_{10}^{\rm NP}|\sin\varphi_{10}^{\rm NP},$$ $$\label{tanphi10}
\tan\varphi_{10} =\frac{|{\cal C}_{10}^{\rm NP}|\sin\varphi_{10}^{\rm NP}}{1+
|{\cal C}_{10}^{\rm NP}|\cos\varphi_{10}^{\rm NP}}$$ allow us to express ${\cal C}_{10}$ in terms of the – in general – complex NP coefficient ${\cal C}_{10}^{\rm NP}$.
In order to reveal the substructure of $P$, information on ${\cal C}_{10}$ is required. In specific models, we may calculate ${\cal C}_{10}^{\rm NP}$ (see, for instance, Ref. [@BFGK]). Alternatively, using experimental data for $B\to K^{(*)}\ell^+\ell^-$ decays, we may determine $C_{10} - C'_{10}$ from experiment (see Ref. [@ANSS] and references therein). In practice, the corresponding NP contributions are extracted through involved global fits to sets of large numbers of observables. We use the results from Ref. [@ANSS], where different scenarios for NP in real Wilson coefficients are discussed. Considering NP in individual Wilson coefficients, the authors find that the data is best explained by a contribution to the short-distance coefficient $C_9$ of the four-fermion operator $O_9=(\bar s \gamma_\mu P_L b) (\bar\mu\gamma^\mu\mu)$, which does not contribute to $B_s^0 \to \mu^+\mu^-$, yielding ${\cal C}_{10}^\text{NP} = 0$ and thus ${\cal C}_{10} = 1$. However, a similarly good fit is obtained by assuming the relation $C_9^\text{NP} = -C_{10}^\text{NP}$ for real coefficients, which appears in models with new particles that couple only to left-handed leptons. In this case, we find $$\label{C10-range}
{\cal C}_{10}^{\rm NP}=-0.16^{+0.04}_{-0.04},$$ where the minus sign follows from $C_{10}^\text{SM}$ taking a negative value, as given in Eq. (\[C10-SM\]), resulting in $$\label{C10-range-Fit}
{\cal C}_{10} = 0.84_{-0.04}^{+0.04}.$$
In Ref. [@ANSS], CP-violating phases are neglected. However, the short-distance coefficients are in general complex, and the phases can be included in the fit. In Ref. [@Altmannshofer:2014rta], such an analysis is performed. The results are presented as 2D confidence contours in the complex plane of the coefficients $C_{10}$ and $C'_{10}$. To probe for the possible size of $\varphi_{10}$ and $ |\mathcal{C}_{10}|$, we assume that $C'_{10}=0$ and convert the $1\sigma$ allowed regions for the complex Wilson coefficient $C_{10}$ shown in Ref. [@Altmannshofer:2014rta] into $\mathcal{C}_{10}$ using Eq. (\[eq:CalC10\]), yielding $$\begin{aligned}
-40^{\circ} < \varphi_{10} < -14^{\circ} \quad& \lor& \quad 14^{\circ} < \varphi_{10} < 40^{\circ},\\
0.79< &|\mathcal{C}_{10}|&<0.98.\end{aligned}$$ Due to the structure of Eq. (\[tanphi10\]), we obtain a rather constrained range for the CP-violating phase $ \varphi_{10}$. It is also interesting to note that the range for the absolute value $|\mathcal{C}_{10}|$ is consistent with the result in Eq. (\[C10-range-Fit\]).
In the future, analyses of CP-violating effects in $B\to K^{(*)}\ell^+\ell^-$ and $B_s\to \phi\mu^+\mu^-$ observables, as introduced in Refs. [@CP-SL-rare-1; @CP-SL-rare-2], will allow us to get a much sharper picture of $|{\cal C}_{10}|$ and a possible complex phase $\varphi_{10}$. It would be very useful to add the complex coefficient ${\cal C}_{10}$ as a default output to the corresponding sophisticated fits to the semileptonic rare $B_{(s)}$ decay data.
For the numerical illustrations below, we will either use the range in Eq. (\[C10-range-Fit\]) for real Wilson coefficients $C_{10}$ and $C_{10}'$, or we will consider the case $|{\cal C}_{10}|=1$, $\varphi_{10} = 0^\circ$, where NP effects would enter exclusively through (pseudo)-scalar contributions.
An interesting situation arises if we consider a scenario where NP effects enter only through ${\cal C}_{10}$, with vanishing coefficients $C_P$, $C_P'$ and $C_S$, $C_S'$, yielding $P={\cal C}_{10}$ and $S=0$. Specific examples are given by models with extra $Z'$ bosons (see, for instance, Ref. [@BFGK]) and scenarios with modified $Z$ couplings (such as in models with vector-like quarks [@BBCJ]). We would then have the simple expressions $${\cal A}^{\mu\mu}_{\Delta\Gamma_s}=\cos(2\varphi_{10}-\phi_s^{\rm NP}), \quad
{\cal S}_{\mu\mu} =\sin(2\varphi_{10}-\phi_s^{\rm NP})
\label{eq:ObsCsCp0}$$ with ${\cal C}_{\mu\mu} = 0$. Consequently, the observables would lie on a circle with radius one in the ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$–${\cal S}_{\mu\mu}$ plane.
Extraction of $|x|$ and $\Delta$ {#ssec:extractionxDelta}
--------------------------------
Applying the method presented in Subsection \[ssec:theo-des\], we may determine $|S|$, $|P|$ and $\varphi_P$ from the $B^0_s\to\mu^+\mu^-$ observables as functions $\varphi_S$. Using Eq. (\[conv-1\]), we may convert these parameters into the ratio $x$ of the – in general – complex scalar short-distance coefficients: $$|x|e^{i\Delta} = \frac{w(P-{\cal C}_{10})+S}{w(P-{\cal C}_{10})-S},$$ with $$|x|=\sqrt{\frac{w^2\left|P-{\cal C}_{10}\right|^2+\left|S\right|^2+
2 \,w\,\Re\left[(P^*-{\cal C}_{10}^*)S\right]}{w^2\left|P-{\cal C}_{10}\right|^2+\left|S\right|^2-
2 \,w\,\Re\left[(P^*-{\cal C}_{10}^*)S\right]}}$$ and $$\cos\Delta\propto w^2|P-{\cal C}_{10}|^2-|S|^2, \quad
\sin\Delta\propto 2 \,w\, \Im\left[(P^*-{\cal C}_{10}^*)S\right],$$ yielding $$\tan\Delta=\frac{2 \,w\, \Im\left[(P^*-{\cal C}_{10}^*)S\right]}{w^2|P-{\cal C}_{10}|^2-|S|^2}.$$ The quantities entering these expression can be expressed in terms of the absolute values and phases of the relevant complex coefficients as $$|P-{\cal C}_{10}|=\sqrt{|P|^2-2|P||{\cal C}_{10}|\cos(\varphi_{10}-\varphi_P)+|{\cal C}_{10}|^2}$$ and $$\Im\left[(P^*-{\cal C}_{10}^*)S\right]=|S|\Bigl[|P|\sin(\varphi_S-\varphi_P)-|{\cal C}_{10}|\sin(\varphi_S-\varphi_{10}) \Bigr]$$ $$\Re\left[(P^*-{\cal C}_{10}^*)S\right]=|S|\Bigl[|P|\cos(\varphi_S-\varphi_P)-|{\cal C}_{10}|\cos(\varphi_S-\varphi_{10}) \Bigr].$$
![Flowchart to illustrate the use of the SMEFT relations in Subsection \[ssec:gen-frame\] for the analysis of the $B^0_s\to\mu^+\mu^-$ observables as described in the text.[]{data-label="fig:strategy_general_relations"}](fig5.pdf){width="5.0in"}
It is instructive to consider the example in Subsection \[sssec:genex\], where $|S|=0.30$ and $\varphi_S=20^\circ$. Using the expressions given above, we can convert the corresponding values of $|P|=0.89$ and $\varphi_P=30^\circ$ into $$\label{X-Del-Illu}
|x| = 0.89, \qquad \Delta = -62^\circ,$$ where we have assumed no NP in ${\cal C}_{10}$, so $|{\cal C}_{10}| = 1$ and $\varphi_{10} = 0^\circ$. In Fig. \[fig:strategy\_general\_relations\], we give a flowchart for this strategy, and show in Fig. \[fig:X-Del-Applic\] the situation corresponding to Eq. (\[X-Del-Illu\]). Using information on the sign of ${\cal C}_{\mu\mu}$, we would only be left with the red contours. We observe that $|x|e^{i\Delta}$ could be constrained in a very non-trivial way. The resulting contours depend strongly on the associated $B^0_s\to\mu^+\mu^-$ decay observables.
In order to constrain the parameters more stringently, it is useful to make assumptions about scenarios, as we will illustrate in the next section. Following these lines, we may rule out a given scenario or confirm it, allowing us then to extract the corresponding parameters. By the time we may have measurements of CP violation in $B^0_s\to\mu^+\mu^-$ available, we should have a much better picture of the physics beyond the SM, thanks to the interplay between model building and data coming both from the high-energy and the high-precision frontiers. In particular, we should then also have some preferred scenarios, including specific patterns for the CP-violating phases, which could be confronted with experimental data and the new strategies presented in this paper.
![Implementation of Fig. \[fig:strategy\_general\_relations\] for the example in Subsection \[sssec:genex\], corresponding to Eq. (\[X-Del-Illu\]), which is illustrated by the green dots. In the left panel, we give the resulting dependence of $|x|$ on $\varphi_S$, while in the right panel, we show $|x|e^{i\Delta}$ in the complex plane. The grey contours could be excluded through sign information for the observable ${\cal C}_{\mu\mu}$.[]{data-label="fig:X-Del-Applic"}](fig6a.pdf "fig:"){width="2.8in"} ![Implementation of Fig. \[fig:strategy\_general\_relations\] for the example in Subsection \[sssec:genex\], corresponding to Eq. (\[X-Del-Illu\]), which is illustrated by the green dots. In the left panel, we give the resulting dependence of $|x|$ on $\varphi_S$, while in the right panel, we show $|x|e^{i\Delta}$ in the complex plane. The grey contours could be excluded through sign information for the observable ${\cal C}_{\mu\mu}$.[]{data-label="fig:X-Del-Applic"}](fig6b.pdf "fig:"){width="3.0in"}
Illustration {#ssec:illu}
------------
As experimental data have already constrained the NP contribution $\phi_s^{\rm NP}$ to the $B^0_s$–$\bar B^0_s$ mixing phase to be tiny, as given in Eq. (\[phis-NP\]), we may simplify the discussion by neglecting this quantity. Moreover, for the decay $B^0_s\to\mu^+\mu^-$, we have with excellent precision $w=1$. Let us now illustrate the formalism and strategy discussed above through various examples. Here we shall choose values for the input parameters to calculate the decay observables. Assuming then that these quantities have been measured at the future LHC upgrade(s), we discuss the pictures emerging from the strategy discussed above. For simplicity, we do not consider experimental aspects in this section but will illustrate scenarios assuming uncertainties of future measurements in Section \[sec:Exp\].
### $x=0$ and $|x|\to\infty$ {#sec:x0xinfty}
The case $x=0$, which corresponds to $C_S'=C_P'=0$, is frequently considered in the literature for vanishing CP-violating phases (see, for instance, Ref. [@ANS]). It is interesting to note that the relation in Eq. (\[gen-rel-PS\]) gives $$\label{Rel-p}
wP+S=w \, {\cal C}_{10},$$ which reduces to $P+S=1$ for $w=1$ and ${\cal C}_{10}=1$. Allowing for possible CP violation, using the expressions in the Appendix we obtain $$\label{rx0-calc}
r|_{x=0}=|{\cal C}_{10}|^2-2\cos(\varphi_{10}-\varphi_S)|{\cal C}_{10}||S|+2|S|^2$$ as well as $$\label{ADGx0-calc}
{\cal A}^{\mu\mu}_{\Delta\Gamma_s}|_{x=0}=
\frac{|{\cal C}_{10}|^2\cos2\varphi_{10}-2\cos(\varphi_{10}+\varphi_S)|{\cal C}_{10}||S|}{|{\cal C}_{10}|^2-
2\cos(\varphi_{10}-\varphi_S)|{\cal C}_{10}||S| + 2 |S|^2}$$ $$\label{Smumux0-calc}
{\cal S}_{\mu\mu}|_{x=0}=
\frac{|{\cal C}_{10}|^2\sin2\varphi_{10}-2\sin(\varphi_{10}+\varphi_S)|{\cal C}_{10}||S|}{|{\cal C}_{10}|^2-
2\cos(\varphi_{10}-\varphi_S)|{\cal C}_{10}||S| + 2 |S|^2}$$ $$\label{Cmumux0-calc}
{\cal C}_{\mu\mu}|_{x=0}=\frac{2|S|\left[|{\cal C}_{10}|\cos(\varphi_{10}-\varphi_S)-
|S|\right]}{|{\cal C}_{10}|^2-2\cos(\varphi_{10}-\varphi_S)|{\cal C}_{10}||S| + 2 |S|^2}.$$
Using Eq. (\[rr-def\]), and substituting $r$ and ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$ according to Eqs. (\[rx0-calc\]) and (\[ADGx0-calc\]), we may determine $|S|$ as a function of $\varphi_{10}-\varphi_S$ from the measured value of $\overline{R}$: $$\hspace*{-2.0truecm}
|S| = \frac{|{\cal C}_{10}|}{2}\Bigg\{\left[\cos(\varphi_{10}-\varphi_S)+y_s\cos(\varphi_{10}+\varphi_S)\right]$$ $$\pm\sqrt{\left[\cos(\varphi_{10}-\varphi_S)+y_s\cos(\varphi_{10}+\varphi_S)\right]^2-
2\left[1+y_s\cos2\varphi_{10}-\frac{\overline{R}}{|{\cal C}_{10}|^2}(1+y_s)\right]}\Bigg\}.
\label{eq:S0}$$ Note that the discriminant must have a value greater than or equal to zero, which implies the following upper bound: $$|{\cal C}_{10}| \leq \sqrt{\left(\frac{2}{1-y_s}\right)\overline{R}}.$$ The current experimental value of $\overline{R}$ in Eq. (\[eq:Rbar\]) yields $$|{\cal C}_{10}| \leq 1.3 \pm 0.1,$$ which is obviously consistent with ${\cal C}_{10}=1$.
![Functional dependences between $|S|$, ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$, ${\cal S}_{\mu\mu}$, ${\cal C}_{\mu\mu}$ and the CP-violating phase $\varphi_S$ for $|\mathcal{C}_{10}|=1, \ \varphi_{10}=0^\circ$. The blue and red contours correspond to the scenarios $x=0$ and $|x|\rightarrow \infty$, respectively. The allowed regions are determined within the $1\,\sigma$ range for $\overline{R}$ given in Eq. (\[eq:Rbar\]), where the dashed curve is associated with the central value for this observable. Notice that for each value of $\varphi_S$, we have in general two possible solutions for the observables, leading to closed loops in the parameter space. The black dot refers to the input parameters of the scenario in Eq. (\[eq:example0infinity\]), whereas the green line shows the value of the observables in Eq. (\[eq:obsx0\]).\[fig:absSvsPhisC1\]](fig8a.pdf "fig:"){width="50.00000%"} ![Functional dependences between $|S|$, ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$, ${\cal S}_{\mu\mu}$, ${\cal C}_{\mu\mu}$ and the CP-violating phase $\varphi_S$ for $|\mathcal{C}_{10}|=1, \ \varphi_{10}=0^\circ$. The blue and red contours correspond to the scenarios $x=0$ and $|x|\rightarrow \infty$, respectively. The allowed regions are determined within the $1\,\sigma$ range for $\overline{R}$ given in Eq. (\[eq:Rbar\]), where the dashed curve is associated with the central value for this observable. Notice that for each value of $\varphi_S$, we have in general two possible solutions for the observables, leading to closed loops in the parameter space. The black dot refers to the input parameters of the scenario in Eq. (\[eq:example0infinity\]), whereas the green line shows the value of the observables in Eq. (\[eq:obsx0\]).\[fig:absSvsPhisC1\]](fig8b.pdf "fig:"){width="50.00000%"} ![Functional dependences between $|S|$, ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$, ${\cal S}_{\mu\mu}$, ${\cal C}_{\mu\mu}$ and the CP-violating phase $\varphi_S$ for $|\mathcal{C}_{10}|=1, \ \varphi_{10}=0^\circ$. The blue and red contours correspond to the scenarios $x=0$ and $|x|\rightarrow \infty$, respectively. The allowed regions are determined within the $1\,\sigma$ range for $\overline{R}$ given in Eq. (\[eq:Rbar\]), where the dashed curve is associated with the central value for this observable. Notice that for each value of $\varphi_S$, we have in general two possible solutions for the observables, leading to closed loops in the parameter space. The black dot refers to the input parameters of the scenario in Eq. (\[eq:example0infinity\]), whereas the green line shows the value of the observables in Eq. (\[eq:obsx0\]).\[fig:absSvsPhisC1\]](fig8c.pdf "fig:"){width="50.00000%"} ![Functional dependences between $|S|$, ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$, ${\cal S}_{\mu\mu}$, ${\cal C}_{\mu\mu}$ and the CP-violating phase $\varphi_S$ for $|\mathcal{C}_{10}|=1, \ \varphi_{10}=0^\circ$. The blue and red contours correspond to the scenarios $x=0$ and $|x|\rightarrow \infty$, respectively. The allowed regions are determined within the $1\,\sigma$ range for $\overline{R}$ given in Eq. (\[eq:Rbar\]), where the dashed curve is associated with the central value for this observable. Notice that for each value of $\varphi_S$, we have in general two possible solutions for the observables, leading to closed loops in the parameter space. The black dot refers to the input parameters of the scenario in Eq. (\[eq:example0infinity\]), whereas the green line shows the value of the observables in Eq. (\[eq:obsx0\]).\[fig:absSvsPhisC1\]](fig8d.pdf "fig:"){width="50.00000%"}
![ Functional dependences between $|S|$, ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$, ${\cal S}_{\mu\mu}$, ${\cal C}_{\mu\mu}$ and the CP-violating phase $\varphi_S$ for $|\mathcal{C}_{10}|=0.84$ and $\varphi_{10}=0^\circ$. The blue and red contours correspond to the scenarios $x=0$ and $|x|\rightarrow \infty$, respectively. The allowed regions are determined within the $1\,\sigma$ range for $\overline{R}$ given in Eq. (\[eq:Rbar\]), where the dashed curve is associated with the central value for this observable. The black dot refers to the input parameters of the scenario in Eq. (\[eq:example0infinity2\]), whereas the green line shows the value of the observables in Eq. (\[eq:obsx02\]). \[fig:absSvsPhisC84\]](fig9a.pdf "fig:"){width="50.00000%"} ![ Functional dependences between $|S|$, ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$, ${\cal S}_{\mu\mu}$, ${\cal C}_{\mu\mu}$ and the CP-violating phase $\varphi_S$ for $|\mathcal{C}_{10}|=0.84$ and $\varphi_{10}=0^\circ$. The blue and red contours correspond to the scenarios $x=0$ and $|x|\rightarrow \infty$, respectively. The allowed regions are determined within the $1\,\sigma$ range for $\overline{R}$ given in Eq. (\[eq:Rbar\]), where the dashed curve is associated with the central value for this observable. The black dot refers to the input parameters of the scenario in Eq. (\[eq:example0infinity2\]), whereas the green line shows the value of the observables in Eq. (\[eq:obsx02\]). \[fig:absSvsPhisC84\]](fig9b.pdf "fig:"){width="50.00000%"} ![ Functional dependences between $|S|$, ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$, ${\cal S}_{\mu\mu}$, ${\cal C}_{\mu\mu}$ and the CP-violating phase $\varphi_S$ for $|\mathcal{C}_{10}|=0.84$ and $\varphi_{10}=0^\circ$. The blue and red contours correspond to the scenarios $x=0$ and $|x|\rightarrow \infty$, respectively. The allowed regions are determined within the $1\,\sigma$ range for $\overline{R}$ given in Eq. (\[eq:Rbar\]), where the dashed curve is associated with the central value for this observable. The black dot refers to the input parameters of the scenario in Eq. (\[eq:example0infinity2\]), whereas the green line shows the value of the observables in Eq. (\[eq:obsx02\]). \[fig:absSvsPhisC84\]](fig9c.pdf "fig:"){width="50.00000%"} ![ Functional dependences between $|S|$, ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$, ${\cal S}_{\mu\mu}$, ${\cal C}_{\mu\mu}$ and the CP-violating phase $\varphi_S$ for $|\mathcal{C}_{10}|=0.84$ and $\varphi_{10}=0^\circ$. The blue and red contours correspond to the scenarios $x=0$ and $|x|\rightarrow \infty$, respectively. The allowed regions are determined within the $1\,\sigma$ range for $\overline{R}$ given in Eq. (\[eq:Rbar\]), where the dashed curve is associated with the central value for this observable. The black dot refers to the input parameters of the scenario in Eq. (\[eq:example0infinity2\]), whereas the green line shows the value of the observables in Eq. (\[eq:obsx02\]). \[fig:absSvsPhisC84\]](fig9d.pdf "fig:"){width="50.00000%"}
The number of allowed solutions for a given angle $\varphi_S$ depends on the value of the Wilson coefficient ${\cal C}_{10}$. In order illustrate this feature, we consider two scenarios for $\mathcal{C}_{10}$. Let us first assume that there is a vanishing NP contribution $\mathcal{C}^{\rm NP}_{10}=0$, which yields $|{\cal C}_{10}|=1, \ \varphi_{10}=0^\circ$. In this case, Eq. (\[eq:S0\]) results in two solutions for $|S|$ as a function of $\varphi_S$, as can be seen in the top-left plot in Fig. \[fig:absSvsPhisC1\]. Using Eqs. (\[ADGx0-calc\]), (\[Smumux0-calc\]) and (\[Cmumux0-calc\]), we can determine the observables $\mathcal{A}^{\mu\mu}_{\Delta \Gamma_s}$, $\mathcal{S}^{\mu\mu}$ and $\mathcal{C}_{\mu\mu}$ as functions of $\varphi_S$, respectively, as shown in Fig. \[fig:absSvsPhisC1\]. In particular, once $\mathcal{A}^{\mu\mu}_{\Delta \Gamma_s}$ has been measured, the value of $\mathcal{S}_{\mu\mu}$ can be predicted. Should this CP asymmetry be measured correspondingly, this scenario would be confirmed, allowing us to determine the corresponding NP parameters. On the other hand, should the measurement of $\mathcal{S}_{\mu\mu}$ be in conflict with the prediction, the NP scenario would be ruled out by experimental data.
Let us now consider a scenario with NP contributions to ${\cal C}_{10}$. If we follow the analysis of Ref. [@ANSS] and use the central value of ${\cal C}_{10}$ in Eq. (\[C10-range-Fit\]), we obtain the functional dependence of $|S|$ and the corresponding observables on $\varphi_S$ shown Fig. \[fig:absSvsPhisC84\]. Interestingly, for a given value of $\varphi_S$, Eq. (\[eq:S0\]) gives now a single solution for $|S|$. Consequently, unlike their counterparts in Fig. \[fig:absSvsPhisC1\], the contours no longer form closed loops, thereby indicating that the degeneracy with respect to $\varphi_S$ has disappeared. In Fig. \[fig:Strategy\_relations\], we illustrate this strategy, which is actually more general, i.e. does not only apply to the case of $x=0$.
![Flowchart to illustrate the use of the relations in Subsection \[ssec:gen-frame\] with information on ${\cal C}_{10}$ to convert the measured value of $\overline{R}$ into predictions of the $B^0_s\to\mu^+\mu^-$ observables. Once these are measured in accordance with the pattern characterizing the NP scenario, $|S|$ and $\varphi_S$ can be extracted from the data.[]{data-label="fig:Strategy_relations"}](fig7.pdf){width="5.5in"}
A closer look at the expressions in the Appendix shows that the case of $x=0$ is connected with $|x|\to \infty$, where the scalar and pseudo-scalar coefficients $C_S$ and $C_P$ vanish while $C_P'=C_S'$ takes a non-vanishing value. The expression in Eq. (\[gen-rel-PS\]) takes then the form $$\label{Rel-m}
wP - S=w \, {\cal C}_{10},$$ which reduces to $P-S=1$ for $w=1$ and ${\cal C}_{10}=1$. For the observables $r$ as well as ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$ and ${\cal S}_{\mu\mu}$, we have the symmetry relation $$\varphi_S\to \pi+ \varphi_S,
\label{eq:symmetry_trans}$$ which is equivalent to $|S|\to -|S|$, and yields $$\label{rxinf-calc}
r|_{|x|\to \infty}=|{\cal C}_{10}|^2+2\cos(\varphi_{10}-\varphi_S)|{\cal C}_{10}||S|+2|S|^2$$ $$\label{ADGxinf-calc}
{\cal A}^{\mu\mu}_{\Delta\Gamma_s}|_{|x|\to \infty}=
\frac{|{\cal C}_{10}|^2\cos2\varphi_{10}+2\cos(\varphi_{10}+\varphi_S)|{\cal C}_{10}||S|}{|{\cal C}_{10}|^2+
2\cos(\varphi_{10}-\varphi_S)|{\cal C}_{10}||S| + 2 |S|^2}$$ $$\label{eq:SmumuInfty}
{\cal S}_{\mu\mu}|_{|x|\to \infty}=
\frac{|{\cal C}_{10}|^2\sin2\varphi_{10}+2\sin(\varphi_{10}+\varphi_S)|{\cal C}_{10}||S|}{|{\cal C}_{10}|^2+
2\cos(\varphi_{10}-\varphi_S)|{\cal C}_{10}||S| + 2 |S|^2}.$$ In the case of ${\cal C}_{\mu\mu}$, the symmetry is broken by an overall minus sign: $$\label{Cmumuxinf-calc}
{\cal C}_{\mu\mu}|_{|x|\to \infty}=\frac{2|S|\left[|{\cal C}_{10}|\cos(\varphi_{10}-\varphi_S)+
|S|\right]}{|{\cal C}_{10}|^2+2\cos(\varphi_{10}-\varphi_S)|{\cal C}_{10}||S| + 2 |S|^2}.$$ More explicitly, we have $$\begin{aligned}
\label{eq:x0xinf-symm-r-ADG-smumu}
r|_{x=0}(\varphi_S + \pi)&=&r|_{|x|\rightarrow \infty}(\varphi_S)\nonumber \\
{\cal A}^{\mu\mu}_{\Delta\Gamma_s}|_{x=0}(\varphi_S + \pi)
& =&{\cal A}^{\mu\mu}_{\Delta\Gamma_s}|_{|x|\rightarrow \infty}(\varphi_S) \\
{\cal S}_{\mu\mu}|_{x=0}(\varphi_S + \pi)&=&{\cal S}_{\mu\mu}|_{|x|\rightarrow \infty}(\varphi_S),\nonumber\end{aligned}$$ while $$\begin{aligned}
\label{eq:x0xinf-symm-cmumu}
{\cal C}_{\mu\mu}|_{x=0}(\varphi_S+\pi)&=&-{\cal C}_{\mu\mu}|_{|x|\rightarrow \infty}(\varphi_S).\end{aligned}$$ As we will see below, this feature has interesting phenomenological implications.
In order to illustrate the expressions given above, we consider two examples with different values of the coefficient $\mathcal{C}_{10}$:\
[**Example (a):**]{}\
We first assume a situation with vanishing NP contributions ${\cal C}_{10}^{\rm NP}=0$, and employ the following setup: $$\label{eq:example0infinity}
\overline{R}=0.84\pm 0.16, \quad x=0, \quad \varphi_S=54^{\circ}, \quad |\mathcal{C}_{10}|=1,
\quad \varphi_{10}=0^{\circ}.$$ Using Eq. (\[eq:S0\]), we determine $|S|$ as a function of $\varphi_S$. As discussed above, for $|\mathcal{C}_{10}|=1, \ \varphi_{10}=0^\circ$ and the central value of $\overline{R}$ in Eq. (\[eq:example0infinity\]), we obtain a twofold solution. For the sake of illustration, we consider only the solution with the plus sign in front of the square root, yielding $$|S|=0.43.$$ With the help of Eq. (\[conv-1\]), we may now calculate $$|P|=0.82, \quad \varphi_P=-25^{\circ}.$$ The corresponding values for the observables read as follows: $$\label{eq:obsx0}
{\cal A}^{\mu\mu}_{\Delta\Gamma_s}=0.58 ,
\quad {\cal S}_{\mu\mu} = -0.80, \quad {\cal C}_{\mu\mu} = 0.16.$$
Let us now assume that these observables have been measured, and discuss how we may then – with the help of the strategy discussed above – reveal the dynamics of the $B^0_s\to\mu^+\mu^-$ decay and distinguish between the $x=0$ and $|x|\rightarrow \infty$ cases:
- It is plausible to expect that $\mathcal{A}^{\mu\mu}_{\Delta \Gamma_s}$ is the next observable to be measured. With the help of the top-right plot in Fig. \[fig:absSvsPhisC1\], we identify four possible values for $\varphi_S$ which are compatible with the “experimental" result of $\mathcal{A}^{\mu\mu}_{\Delta \Gamma_s}=0.58$ in Eq. (\[eq:obsx0\]): $\varphi^{(1)}_S=-126^{\circ}$, $\varphi^{(2)}_S=-54^{\circ}$, $\varphi^{(3)}_S=54^{\circ}$ and $\varphi^{(4)}_S=126^{\circ}$.
- We may now predict the observable $\mathcal{S}_{\mu\mu}$. Using the bottom-left plot in Fig. \[fig:absSvsPhisC1\] or the expressions in Eqs. (\[Smumux0-calc\]) and (\[eq:SmumuInfty\]), we obtain $\mathcal{S}_{\mu\mu}=-0.80$ for $\varphi^{(1)}_S=-126^{\circ}$ (branch $|x|\rightarrow \infty$) and $\varphi^{(3)}_S=54^{\circ}$ (branch $x=0$). Moreover, we find $\mathcal{S}_{\mu\mu}=0.80$ for $\varphi^{(2)}_S=-54^{\circ}$ (branch $x=0$) and $\varphi^{(4)}_S=126^{\circ}$ (branch $|x|\rightarrow \infty$).
- The measurement $\mathcal{S}_{\mu\mu}=-0.80$ would then allow us to narrow down the four solutions for $\varphi_S$ to only two at $\varphi^{(1)}_S=-126^{\circ}$ and $\varphi^{(3)}_S=54^{\circ}$, corresponding to $|x|\rightarrow \infty$ and $x=0$, respectively. It should be emphasized that both solutions would be valid at this stage of the analysis, i.e. we would have confirmed a CP-violating NP scenario with either $|x|\rightarrow \infty$ or $x=0$.
- This ambiguity can be resolved through information on the sign of $\mathcal{C}_{\mu\mu}$, which is given by $\mathcal{C}_{\mu\mu}=-0.16$ and $\mathcal{C}_{\mu\mu}=+0.16$ for $|x|\rightarrow \infty$ and $x=0$, respectively, as can be seen in Fig. \[fig:absSvsPhisC1\]. Consequently, the fact that $\mathcal{C}_{\mu\mu}$ breaks the symmetry in Eq. (\[eq:symmetry\_trans\]) gives us a powerful tool to distinguish between $x=0$ and $|x|\rightarrow \infty$.
[**Example (b):**]{}\
Now we have a look at a scenario with NP contributions to $\mathcal{C}_{10}$, which is characterized as follows: $$\label{eq:example0infinity2}
\overline{R}=0.84\pm 0.16, \quad x=0, \quad \varphi_S=-70^{\circ}, \quad |\mathcal{C}_{10}|=0.84,
\quad \varphi_{10}=0^{\circ}.$$ Here the value of ${\cal C}_{10}$ follows from Eq. (\[C10-range-Fit\]), and is discussed in more detail in Subsection \[sec:C10\]. In contrast to Example (a), we obtain now a single solution for $|S|$ from Eq. (\[eq:S0\]), which is given by $$|S|=0.46.$$ Using Eq. (\[conv-1\]), we find $$|P|=0.81, \quad \varphi_P=33^{\circ},$$ resulting in the following values of the observables: $$\label{eq:obsx02}
{\cal A}^{\mu\mu}_{\Delta\Gamma_s}=0.50 ,
\quad {\cal S}_{\mu\mu} = 0.84, \quad {\cal C}_{\mu\mu} = -0.19.$$ In analogy to Example (a), using the plots in Fig. \[fig:absSvsPhisC84\], we may again show the compatibility of the “measured" observables with the scenario $x=0$, and rule out the case of $|x|\rightarrow \infty$ through the sign of the ${\cal C}_{\mu\mu}$ asymmetry. For the convenience of the reader, we summarize the main features of these examples in Table \[tab:Tab1\].
![Allowed region in the ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$–${\cal S}_{\mu\mu}$ plane following from the current experimental value of $\overline{R}$ for $x=0$; the same correlation is obtained for $|x|\rightarrow \infty$. The circular region corresponds to the $1\sigma$ uncertainty of $\overline{R}$ in Eq. (\[eq:Rbar\]). The black star indicates the SM point. \[fig:aDGsmumu\_corr-x0-xinf\]](fig10.pdf){width="50.00000%"}
In Fig. \[fig:aDGsmumu\_corr-x0-xinf\], we show the correlation between ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$ and ${\cal S}_{\mu\mu}$ through the CP-violating phase $\varphi_S$. It should be noted that the corresponding regions for $|\mathcal{C}_{10}|=0.84, \ \varphi_{10} = 0^\circ$ and $|\mathcal{C}_{10}|=1, \ \varphi_{10}=0^\circ$ do not differ substantially and are included in a single plot. Due to the symmetry transformation in Eq. (\[eq:symmetry\_trans\]), the scenarios $x=0$ and $|x|\rightarrow\infty$ cover the same region once we make a scan over the full range of $\varphi_S$. The allowed region in Fig. \[fig:aDGsmumu\_corr-x0-xinf\] exhibits the following interesting features:
1. The currently available measurement of $\overline{R}$ implies a remarkably constrained circular region in the ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$–${\cal S}_{\mu\mu}$ plane for CP-violating NP scenarios characterized by $x=0$ and $|x|\rightarrow \infty$.
2. A future measurement of the observable combination ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$ and ${\cal S}_{\mu\mu}$ lying outside the allowed region would rule out the $x=0$ and $|x|\rightarrow \infty$ scenarios.
3. The allowed region in the ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$–${\cal S}_{\mu\mu}$ plane is close to the unit circle. Consequently, due to Eq. (\[CP-rel\]), the observable $\mathcal{C}_{\mu\mu}$ is constrained to take a smallish value.
4. The allowed region is similar to the one arising for the scenario described in Section \[sec:C10\]. While here $\varphi_{10}=0^{\circ}$ would imply the SM results ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}=1$ and ${\cal S}_{\mu\mu}=0$, in the case of $x=0$ or $|x|\rightarrow \infty$ we may still deviate substantially from the SM even in spite of having a vanishing phase $\varphi_{10}$.
In a complementary way, if we can obtain the value of the phase $\varphi_S$ from external information or theoretical considerations, we will be able to predict the observables ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$ and ${\cal S}_{\mu\mu}$ compatible with vanishing short distance contributions $C_{P,S}$ or $C'_{P,S}$. Strong deviations from these determinations will indicate that the corresponding scenarios are not realized in Nature. A discussion of NP scenarios characterized by the relations $P\pm S=1$ (see Eqs. (\[Rel-p\]) and (\[Rel-m\])) can be found in Ref. [@BFGK].
[ |c|c|c|]{}\
\
\
Observables & Solutions & Scenario\
& $\varphi^{(1)}_S=-126^{\circ}$,$\varphi^{(4)}_S=126^{\circ}$& $|x|\rightarrow \infty$ $(C_S=C_P=0)$\
&$\varphi^{(2)}_S=-54^{\circ}$, $\varphi^{(3)}_S=54^{\circ}$ & $x=0$ $(C'_S=C'_P=0)$\
& $\varphi^{(1)}_S=-126^{\circ}$ &$|x|\rightarrow \infty$ $(C_S=C_P=0)$\
& $\varphi^{(3)}_S=54^{\circ}$ & $x=0$ $(C'_S=C'_P=0)$\
$\mathcal{A}^{\mu\mu}_{\Delta \Gamma_s}$, $\mathcal{S}_{\mu\mu}$, $\mathcal{C}_{\mu\mu}$ &$\varphi^{(3)}_S=54^{\circ}$ & $x=0$ $(C'_S=C'_P=0)$\
\
\
\
& $\varphi^{(1)}_S=-110^{\circ}$,$\varphi^{(4)}_S=110^{\circ}$& $|x|\rightarrow \infty$ $(C_S=C_P=0)$\
&$\varphi^{(2)}_S=-70^{\circ}$, $\varphi^{(3)}_S=70^{\circ}$ & $x=0$ $(C'_S=C'_P=0)$\
& $\varphi^{(4)}_S=110^{\circ}$ &$|x|\rightarrow \infty$ $(C_S=C_P=0)$\
& $\varphi^{(2)}_S=-70^{\circ}$ & $x=0$ $(C'_S=C'_P=0)$\
$\mathcal{A}^{\mu\mu}_{\Delta \Gamma_s}$, $\mathcal{S}_{\mu\mu}$, $\mathcal{C}_{\mu\mu}$ &$\varphi^{(2)}_S=-70^{\circ}$ & $x=0$ $(C'_S=C'_P=0)$\
### $\Delta=0^\circ$ {#sec:Delta0}
Another interesting case arises if $C_S'$ and $C_S$ have the same CP-violating phases, i.e. $\Delta=0^\circ$, which yields $$\label{r-Del0}
r|_{\Delta=0^\circ}=|{\cal C}_{10}|^2-2\left(\frac{1+|x|}{1-|x|}\right)\cos(\varphi_{10}-\varphi_S)|{\cal C}_{10}||S|
+2\left[\frac{1+|x|^2}{(1-|x|)^2}\right]|S|^2$$ $${\cal A}^{\mu\mu}_{\Delta\Gamma_s}|_{\Delta=0^\circ}=
\frac{(1-|x|)^2|{\cal C}_{10}|^2\cos2\varphi_{10}-2\left(1-|x|^2\right)
\cos(\varphi_{10}+\varphi_S)|{\cal C}_{10}||S| +4 |x||S|^2\cos2\varphi_S}{(1-|x|)^2|{\cal C}_{10}|^2-2\left(1-|x|^2\right)
\cos(\varphi_{10}-\varphi_S)|{\cal C}_{10}||S| + 2 (1+|x|^2)|S|^2}
\label{eq:ADGx1}$$ $${\cal S}_{\mu\mu}|_{\Delta=0^\circ}=
\frac{(1-|x|)^2|{\cal C}_{10}|^2\sin2\varphi_{10}-2\left(1-|x|^2\right)
\sin(\varphi_{10}+\varphi_S)|{\cal C}_{10}||S| +4 |x||S|^2\sin2\varphi_S}{(1-|x|)^2|{\cal C}_{10}|^2-2\left(1-|x|^2\right)
\cos(\varphi_{10}-\varphi_S)|{\cal C}_{10}||S| + 2 (1+|x|^2)|S|^2}
\label{eq:Smumux1}$$ $${\cal C}_{\mu\mu}|_{\Delta=0^\circ}=\frac{2|S|\left[(1-|x|)^2|{\cal C}_{10}|\cos(\varphi_{10}-\varphi_S)-
(1-|x|^2)|S|\right]}{(1-|x|)^2|{\cal C}_{10}|^2-2\left(1-|x|^2\right)
\cos(\varphi_{10}-\varphi_S)|{\cal C}_{10}||S| + 2 (1+|x|^2)|S|^2}.
\label{eq:Cmumux1}$$ In analogy to the scenarios $x=0$ and $|x|\to \infty$ discussed in Subsection \[sec:x0xinfty\], the expressions in Eqs. (\[r-Del0\])–(\[eq:Smumux1\]) are invariant under the symmetry transformation $$\label{sym-Del-0}
|x| \to 1/|x|, \quad \varphi_S \to \varphi_S +\pi,$$ leading to $$\begin{aligned}
\label{eq:Del-0-symm-r-ADG-smumu}
r|_{\Delta=0^\circ}(|x|,\varphi_S)&=&r|_{\Delta=0^\circ}(1/|x|,\varphi_S+\pi) \nonumber \\
{\cal A}^{\mu\mu}_{\Delta\Gamma_s}|_{\Delta=0^\circ}(|x|,\varphi_S)&=&
{\cal A}^{\mu\mu}_{\Delta\Gamma_s}|_{\Delta=0^\circ}(1/|x|,\varphi_S+\pi)\nonumber\\
{\cal S}_{\mu\mu}|_{\Delta=0^\circ}(|x|,\varphi_S)&=&{\cal S}_{\mu\mu}|_{\Delta=0^\circ}(1/|x|,\varphi_S+\pi),\end{aligned}$$ while the symmetry is again broken by the observable ${\cal C_{\mu\mu}}$ through an overall sign change: $$\begin{aligned}
{\cal C}_{\mu\mu}|_{\Delta=0^\circ}(|x|,\varphi_S)&=&-{\cal C}_{\mu\mu}|_{\Delta=0^\circ}(1/|x|,\varphi_S+\pi).\end{aligned}$$
The three observables $r$, $\mathcal{A}^{\mu\mu}_{\Delta \Gamma_s}$ and $\mathcal{S}_{\mu\mu}$ in Eqs. (\[r-Del0\])–(\[eq:Smumux1\]) depend on the three unknowns $x$, $|S|$ and $\varphi_S$. Consequently, if the observables are measured, we may determine these parameters. The twofold ambiguity following from the symmetry transformation in Eq. (\[sym-Del-0\]) can be resolved through the measurement of the sign of ${\cal C_{\mu\mu}}$. Unfortunately, in view of the highly non-linear structure of the equations, we cannot give simple analytic solutions. However, the parameters can be determined numerically. In Section \[sec:Exp\], we will illustrate this determination through fits to scenarios of future measurements.
Alternatively, we can apply the strategy depicted in the flowchart in Fig. \[fig:Strategy\_relations\]. We start with the experimental value of $\overline{R}$ given in Eq. (\[eq:Rbar\]). Furthermore, we assume that $|x|=0.5$ and $|{\cal C}_{10}| = 1, \ \varphi_{10}=0^\circ$. This allows us to solve for $|S|$ as a function of $\varphi_S$, and to subsequently determine ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$, ${\cal S}_{\mu\mu}$ and ${\cal C}_{\mu\mu}$ as functions of $\varphi_S$. The results are shown as the blue contours in Fig. \[fig:asymmvsPhis\_Delta0\_x05\]. Here, also the symmetric situation with $|x|=2$ is shown in red, illustrating nicely how ${\cal C}_{\mu\mu}$ breaks the symmetry.
Finally, in Fig. \[fig:x-contours\], we show the correlation between ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$ and ${\cal S}_{\rm \mu\mu}$ for $|x|=0.5$ and $|x| = 3$. Contrary to the situation for $x=0, \ |x|\to\infty$, we are not constrained to a contour close to the unit circle, but can also obtain values in the interior region. For the scenario $|x|=3$, the relations $|S|$, ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$, ${\cal S}_{\mu\mu}$ and ${\cal C}_{\mu\mu}$ as functions of $\varphi_S$ are similar to the ones shown in Fig. \[fig:asymmvsPhis\_Delta0\_x05\].
![Functional dependences between $|S|$, ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$, ${\cal S}_{\mu\mu}$, ${\cal C}_{\mu\mu}$ and the CP-violating phase $\varphi_S$ for $|\mathcal{C}_{10}|=1, \ \varphi_{10}=0^\circ$ and $\Delta=0^\circ$. The blue and red contours correspond to the scenarios $|x|=0.5$ and the associated value $|x| = 2$, respectively. The allowed regions are determined within the $1\,\sigma$ range for $\overline{R}$ given in Eq. (\[eq:Rbar\]), where the dashed curve corresponds to the central value for this observable. Notice that for each value of $\varphi_S$ we have in general two possible solutions for the observables, leading to closed loops in the parameter space. \[fig:asymmvsPhis\_Delta0\_x05\]](fig11a.pdf "fig:"){width="50.00000%"} ![Functional dependences between $|S|$, ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$, ${\cal S}_{\mu\mu}$, ${\cal C}_{\mu\mu}$ and the CP-violating phase $\varphi_S$ for $|\mathcal{C}_{10}|=1, \ \varphi_{10}=0^\circ$ and $\Delta=0^\circ$. The blue and red contours correspond to the scenarios $|x|=0.5$ and the associated value $|x| = 2$, respectively. The allowed regions are determined within the $1\,\sigma$ range for $\overline{R}$ given in Eq. (\[eq:Rbar\]), where the dashed curve corresponds to the central value for this observable. Notice that for each value of $\varphi_S$ we have in general two possible solutions for the observables, leading to closed loops in the parameter space. \[fig:asymmvsPhis\_Delta0\_x05\]](fig11b.pdf "fig:"){width="50.00000%"} ![Functional dependences between $|S|$, ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$, ${\cal S}_{\mu\mu}$, ${\cal C}_{\mu\mu}$ and the CP-violating phase $\varphi_S$ for $|\mathcal{C}_{10}|=1, \ \varphi_{10}=0^\circ$ and $\Delta=0^\circ$. The blue and red contours correspond to the scenarios $|x|=0.5$ and the associated value $|x| = 2$, respectively. The allowed regions are determined within the $1\,\sigma$ range for $\overline{R}$ given in Eq. (\[eq:Rbar\]), where the dashed curve corresponds to the central value for this observable. Notice that for each value of $\varphi_S$ we have in general two possible solutions for the observables, leading to closed loops in the parameter space. \[fig:asymmvsPhis\_Delta0\_x05\]](fig11c.pdf "fig:"){width="50.00000%"} ![Functional dependences between $|S|$, ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$, ${\cal S}_{\mu\mu}$, ${\cal C}_{\mu\mu}$ and the CP-violating phase $\varphi_S$ for $|\mathcal{C}_{10}|=1, \ \varphi_{10}=0^\circ$ and $\Delta=0^\circ$. The blue and red contours correspond to the scenarios $|x|=0.5$ and the associated value $|x| = 2$, respectively. The allowed regions are determined within the $1\,\sigma$ range for $\overline{R}$ given in Eq. (\[eq:Rbar\]), where the dashed curve corresponds to the central value for this observable. Notice that for each value of $\varphi_S$ we have in general two possible solutions for the observables, leading to closed loops in the parameter space. \[fig:asymmvsPhis\_Delta0\_x05\]](fig11d.pdf "fig:"){width="50.00000%"}
![Correlations between ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$ and ${\cal S}_{\rm \mu\mu}$ in the case of $\Delta=0^\circ$ for $|x|=0.5$ and $|x|=3$ in the left and right panels, respectively. The region corresponds to the $1\sigma$ uncertainty of $\overline{R}$ in Eq. (\[eq:Rbar\]). The black star indicates the SM point.[]{data-label="fig:x-contours"}](fig12a.pdf "fig:"){width="2.6in"} ![Correlations between ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$ and ${\cal S}_{\rm \mu\mu}$ in the case of $\Delta=0^\circ$ for $|x|=0.5$ and $|x|=3$ in the left and right panels, respectively. The region corresponds to the $1\sigma$ uncertainty of $\overline{R}$ in Eq. (\[eq:Rbar\]). The black star indicates the SM point.[]{data-label="fig:x-contours"}](fig12b.pdf "fig:"){width="2.6in"}
In the expression for $r$ given in Eq. (\[r-Del0\]), a pole seems to arise for $|x|=1$, which corresponds to $$\label{CSCS-p}
C_S=C'_S.$$ However, this is a spurious divergence, which is cancelled by the $C_S-C_S'$ term in the expression for $S$ in Eq. (\[S-def\]), implying $$S|_{|x|=1, \ \Delta=0^\circ}=0.
\label{eq:Sx1Delta0}$$ Using the relations in Eqs. (\[rel-PS\]) and (\[rel-PS-prime\]), we obtain $$C_P'=C_S'=C_S=-C_P.$$ Consequently, Eq. (\[P-def\]) yields $$\begin{aligned}
\hspace*{-0.5truecm}|P|e^{i\varphi_P}|_{|x|=1, \ \Delta=0^\circ}=
\mathcal{C}_{10} -
\frac{M^2_{B_s}}{m_{\mu}}\Bigl(\frac{m_b}{m_b + m_s}\Bigl)\frac{C_S}{C^{\rm SM}_{10}} \nonumber \\
= |\mathcal{C}_{10}|e^{i\varphi_{10}} - \frac{M^2_{B_s}}{m_{\mu}}\Bigl(\frac{m_b}{m_b + m_s}\Bigl)
\frac{|C_S|}{C^{\rm SM}_{10}}e^{i\tilde{\varphi}_S}\end{aligned}$$ and shows that also the divergence in Eq. (\[conv-1\]) for $|x|=1, \ \Delta=0^\circ$ is spurious. If we neglect, for simplicity, again the tiny NP contribution $\phi_s^{\rm NP}$ to the $B^0_s$–$\bar B^0_s$ mixing phase, we obtain $$\begin{aligned}
r|_{|x|=1, \ \Delta=0^\circ}&=&|P|^2\\
\mathcal{A}^{\mu\mu}_{\Delta \Gamma_s}|_{|x|=1, \ \Delta=0^\circ}&=&\cos(2\varphi_P)\\
\mathcal{S}_{\mu\mu}|_{|x|=1, \ \Delta=0^\circ}&=&\sin(2\varphi_P)\\
\mathcal{C}_{\mu\mu}|_{|x|=1, \ \Delta=0^\circ}&=&0.
\label{eq:obs_x_1_delta0}\end{aligned}$$ For a discussion of NP models describing this situation, see Ref. [@BFGK]. Obviously, also extensions of the SM with scalars, which couple in a left-right-symmetric way to quarks (see the operators in Eq. (\[ops\]) and the relations in Eq. (\[CSCS-p\])), fall into this category.
------------------ ----------------------- ----------- ----------------------------------------- ------------------- --
Subsection
$\Delta=0^{\circ}$, $|x|=0.5$ $\tilde{\varphi}_S=\tilde{\varphi}'_S,$ $|C_S|=2|C'_S|$
$\Delta=0^{\circ}$, $|x|=2$ $\tilde{\varphi}_S=\tilde{\varphi}'_S,$ $|C_S|=0.5|C'_S|$
\[sec:Delta180\] $\Delta=180^{\circ}$, $|x|=1$ $C_S=-C'_S$, $C_P=C'_P$
------------------ ----------------------- ----------- ----------------------------------------- ------------------- --
: Summary of the scenarios described in Subsection \[ssec:illu\]. In all the cases we have assumed $\varphi_{10}=0^{\circ}$.[]{data-label="tab:Tab2"}
As in the case given by Eq. (\[eq:ObsCsCp0\]), the correlation between the observables $\mathcal{A}^{\mu\mu}_{\Delta \Gamma_s}$ and $\mathcal{S}_{\mu\mu}$ describes a circle with radius one. The overall phase $\varphi_P$ includes effects from the, in general, complex quantities $\mathcal{C}_{10}$ and $C_S$. This is particularly interesting if future measurements reveal $(\mathcal{A}_{\Delta\Gamma_s}^{\mu\mu})^2+(\mathcal{S}_{\mu\mu})^2$ compatible with the unit circle and if we have bounds available on the phase $\varphi_{10}$ from other processes. Then, results incompatible with Eq. (\[eq:ObsCsCp0\]) will indicate the potential presence of a scalar or pseudo-scalar contribution.
### $\Delta=180^\circ$ {#sec:Delta180}
In the case of $\Delta=180^\circ$, we obtain the following expressions for the $B^0_s\to\mu^+\mu^-$ observables: $$\label{r-Del180}
r|_{\Delta=180^\circ}=|{\cal C}_{10}|^2-2\left(\frac{1-|x|}{1+|x|}\right)\cos(\varphi_{10}-\varphi_S)|{\cal C}_{10}||S|
+2\left[\frac{1+|x|^2}{(1+|x|)^2}\right]|S|^2$$ $${\cal A}^{\mu\mu}_{\Delta\Gamma_s}|_{\Delta=180^\circ}=
\frac{(1+|x|)^2|{\cal C}_{10}|^2\cos2\varphi_{10}-2\left(1-|x|^2\right)
\cos(\varphi_{10}+\varphi_S)|{\cal C}_{10}||S| - 4 |x||S|^2\cos2\varphi_S}{(1+|x|)^2|{\cal C}_{10}|^2-2\left(1-|x|^2\right)
\cos(\varphi_{10}-\varphi_S)|{\cal C}_{10}||S| + 2 (1+|x|^2)|S|^2}
\label{eq:ADGx1-Del180}$$ $${\cal S}_{\mu\mu}|_{\Delta=180^\circ}=
\frac{(1+|x|)^2|{\cal C}_{10}|^2\sin2\varphi_{10}-2\left(1-|x|^2\right)
\sin(\varphi_{10}+\varphi_S)|{\cal C}_{10}||S| - 4 |x||S|^2\sin2\varphi_S}{(1+|x|)^2|{\cal C}_{10}|^2-2\left(1-|x|^2\right)
\cos(\varphi_{10}-\varphi_S)|{\cal C}_{10}||S| + 2 (1+|x|^2)|S|^2}
\label{eq:Smumux1-Del180}$$ $${\cal C}_{\mu\mu}|_{\Delta=180^\circ}=\frac{2|S|\left[(1+|x|)^2|{\cal C}_{10}|\cos(\varphi_{10}-\varphi_S)-
(1-|x|^2)|S|\right]}{(1+|x|)^2|{\cal C}_{10}|^2-2\left(1-|x|^2\right)
\cos(\varphi_{10}-\varphi_S)|{\cal C}_{10}||S| + 2 (1+|x|^2)|S|^2}.
\label{eq:Cmumux1-Del180}$$ These equations could be solved numerically to determine $|x|$, $|S|$ and $\varphi_S$, in analogy to the discussion of $\Delta=0^\circ$.
It is interesting to have a closer look at $x=-1$, i.e. $|x|=1$ and $\Delta=180^\circ$. In terms of the short-distance coefficients, this case corresponds to $$\label{eq:condition180}
C_S=-C'_S.$$ Using the relations in Eqs. (\[rel-PS\]) and (\[rel-PS-prime\]), we obtain furthermore $$C_P=C'_P,$$ implying $$P=\mathcal{C}_{10}.$$ Using Eqs. (\[R-expr\]) and (\[y-pm\]), we obtain $$\label{eq:S180}
|S|^2=\frac{\Bigl(1+y_s\Bigl)\overline{R}-\Bigl[1+y_s \cos(2\varphi_{10})\Bigl]|\mathcal{C}_{10}|^2}{1-
y_s \cos(2\varphi_{S})}.$$ Special care should be paid when using Eq. (\[eq:S180\]), since the expression on the right-hand side has to be greater than or equal to zero. This feature implies the following upper bound: $$|\mathcal{C}_{10}|\leq \sqrt{\left (\frac{1+y_s}{1-y_s}\right) \overline{R}},$$ where we have used that $1+ y_s \cos(2\varphi_{10})\geq1-y_s$, with $y_s$ given in Eq. (\[ys-exp\]). With the current experimental value of $\overline{R}$ in Eq. (\[eq:Rbar\]), the corresponding bound is given by $$|\mathcal{C}_{10}| \leq 0.98\pm0.09.$$
The different scenarios described in the previous subsections are presented in Table \[tab:Tab2\], where we show the connection between the standard parametrization used for the short distance contributions and the SMEFT one introduced in Sec. \[ssec:gen-frame\].
Experimental Aspects {#sec:Exp}
====================
Up to now we have not considered experimental uncertainties in the observables $\mathcal{A}^{\mu\mu}_{\Delta \Gamma_s}$, ${\cal S}_{\mu\mu}$ and $\mathcal{C}_{\mu\mu}$ when studying the different scenarios. Nevertheless, we would like to demonstrate the potential for the determination of the underlying parameters at future experiments. Since the asymmetries are not independent, due to the relation in Eq. (\[CP-rel\]), it is not possible to determine all four parameters $|S|$, $\varphi_S$, $|x|$ and $\Delta$. However, as discussed in Subsection \[ssec:extractionxDelta\], we expect to have a better picture of physics beyond the SM by the time the CP asymmetries of $B^0_s\to\mu^+\mu^-$ have been measured. Therefore, we consider some of the examples discussed in Subsection \[ssec:illu\], which correspond to specific values of $|x|$ or $\Delta$. We assume uncertainties for the observables, allowing us to extract the NP parameters through fits.
Unless specified otherwise, within this section we use a future measurement of $$\label{eq:rBarUp}
\overline{R} = 0.84 \pm 0.09,$$ where we have assumed a relative uncertainty of $10 \%$ for $\overline{\mathcal{B}}(B_s \to \mu^+\mu^-)$, which is achievable at the LHCb upgrade [@LHCbup], while keeping the current central value fixed. Notice that the relative uncertainty in our “measurement” for $\bar{R}$ in Eq. (\[eq:rBarUp\]) leads to a $2\sigma$ tension with the SM. Thus the statistical significance will not be high enough to claim for the discovery of NP effects. The major limiting factor of the precision is the ratio $f_d/f_s$ of the fragmentation functions of the $B^0_d$ and $B^0_s$ mesons [@FST], which is required for normalization purposes. To the best of our knowledge, no information about the expected precision of future measurements of ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$, ${\cal S}_{\mu\mu}$ and ${\cal C}_{\mu\mu}$ is available. The key question we want to address is the precision of the measurement of these observables that is required to establish in particular new (pseudo)-scalar contributions at the $5\,\sigma$ confidence level.
$x=0$ and $|x|\to\infty$ {#ssec:Expx0xinf}
------------------------
To begin with, we evaluate the impact of experimental errors for the observables in Example (a) of Subsection \[sec:x0xinfty\], corresponding to a scenario where $x = 0$. An absolute uncertainty of $\pm 0.2$ for the asymmetries leads to the following set of observables: $$\label{eq:obsx0fit}
{\cal A}_{\Delta\Gamma_s}^{\mu\mu} = 0.58 \pm 0.20,
\quad {\cal S}_{\mu\mu} = -0.80 \pm 0.20, \quad {\cal C}_{\mu\mu} = 0.16 \pm 0.20.$$ In such a situation, ${\cal S}_{\mu\mu}$ would indicate CP-violating NP effects at the $4 \sigma$ level, while ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$ and ${\cal C}_{\mu\mu}$ would deviate from the SM picture at the $2\sigma$ and $1 \sigma$ levels, respectively. Let us assume that the values above have been measured at a future experiment, and that there are strong reasons to consider models characterized by $x=0$. We will now illustrate through a $\chi^2$ fit how well we can reveal the underlying decay dynamics.
Let us first obtain the regions allowed for $|S|$ and $\varphi_S$ if we only include $\overline{R}$ and ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$ in the statistical analysis. Thus, using the expression in Eqs. (\[rx0-calc\]) and (\[ADGx0-calc\]) with the “data” in Eqs. (\[eq:rBarUp\]) and (\[eq:obsx0fit\]), we perform a $\chi^2$ fit to these two observables and obtain the blue contours in the left panel of Fig. \[fig:fitx0xinf\], which correspond to $1\sigma$ allowed regions. We indicate the input parameters used to determine our observables in Eq. (\[eq:obsx0fit\]) with the green dot. This plot allows us to establish a non-zero value for $|S|$ at the $3\sigma$ level. If we include also the “measurement” for ${\cal S}_{\mu\mu}$ indicated in Eq. (\[eq:obsx0fit\]), along with Eq. (\[Smumux0-calc\]), and repeat the $\chi^2$ fit, we can eliminate the dashed contour in the left panel and obtain the right plot of Fig. \[fig:fitx0xinf\].
As we have pointed out in Subsection \[sec:x0xinfty\], there is a symmetry between $x=0$ and $|x|\to\infty$, implying the same values of ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$ and ${\cal S}_{\mu\mu}$ for these two cases. Conversely, we could not distinguish $x=0$ and $|x|\to\infty$ at the phenomenological level having only measurements of these observables available. Indeed, repeating the $\chi^2$ fits assuming $|x|\to\infty$ for the same set of input observables leads to the red contours in Fig. \[fig:fitx0xinf\]. Although we use $x=0$ as our favoured model in this illustration, it would certainly be desirable to rule out the degenerate $|x|\to\infty$ scenario. As ${\cal C}_{\mu\mu}$ breaks the symmetry by an overall minus sign, we could actually exclude the $|x|\to\infty$ case through experimental information on the sign on this CP asymmetry. If we add ${\cal C}_{\mu\mu}$ to the analysis, a solution only arises in case of the $x=0$ scenario, thereby singling out the blue contour. We would then find $$|S| = 0.43_{-0.08}^{+0.07}, \qquad \varphi_S = (54_{-7}^{+6})^\circ,$$ where ${\cal C}_{\mu\mu}$ has a minor impact on the numerical values themselves, apart from excluding $|x|\to\infty$. In this scenario, the assumed experimental uncertainties in Eq. (\[eq:obsx0fit\]) would allow us to establish non-zero values of $|S|$ and $\varphi_S$ at the $5\sigma$ and $7\sigma$ levels, respectively, which would provide highly non-trivial insights into the underlying dynamics.
![Illustration of the determination of $|S|$ and $\varphi_S$ from the observables in Eq. (\[eq:obsx0fit\]) for a scenario with $x=0$, which is degenerate with $|x|\to\infty$. The contours correspond to the $1\sigma$ allowed regions obtained from $\chi^2$ fits. In the left panel, we show the result of the fit to only $\overline{R}$ and ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$, while in the right panel we have also included ${\cal S}_{\mu\mu}$. The blue contours were obtained by assuming $x=0$, whereas the red contours follow for $|x|\to\infty$. A measurement of the sign of ${\cal C}_{\mu\mu}$ would allow us to distinguish these cases, excluding the $|x|\to\infty$ scenario.[]{data-label="fig:fitx0xinf"}](fig13a.pdf "fig:"){width="45.00000%"} ![Illustration of the determination of $|S|$ and $\varphi_S$ from the observables in Eq. (\[eq:obsx0fit\]) for a scenario with $x=0$, which is degenerate with $|x|\to\infty$. The contours correspond to the $1\sigma$ allowed regions obtained from $\chi^2$ fits. In the left panel, we show the result of the fit to only $\overline{R}$ and ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$, while in the right panel we have also included ${\cal S}_{\mu\mu}$. The blue contours were obtained by assuming $x=0$, whereas the red contours follow for $|x|\to\infty$. A measurement of the sign of ${\cal C}_{\mu\mu}$ would allow us to distinguish these cases, excluding the $|x|\to\infty$ scenario.[]{data-label="fig:fitx0xinf"}](fig13b.pdf "fig:"){width="45.00000%"}
$\Delta=0^\circ$ {#delta0circ}
----------------
Let us now have a closer look at another interesting scenario: $\Delta=0^\circ$, where $C_S'$ and $C_S$ have the same CP-violating phases. The expressions in Eqs. (\[r-Del0\])–(\[eq:Cmumux1\]) form a system of three independent equations which allows us to determine $|S|$, $\varphi_S$ and $|x|$. Due to the highly non-linear structure of the mathematical expressions, we cannot provide analytical solutions in general. Instead we give an example of how to solve the system through a $\chi^2$ fit. We consider the input parameters $$\label{eq:inputDelta0fit1}
|x|=0.5, \qquad \varphi_S = 20^\circ, \qquad |{\cal C}_{10}| = 1, \qquad \varphi_{10} = 0^\circ,$$ allowing us to determine the following solution for $|S|$, which is consistent with the current central value for $\overline{R}$ shown in Eq. (\[eq:Rbar\]): $$\label{eq:inputDelta0fit2}
|S| = 0.55.$$ If we use the previous numerical values in Eqs. (\[eq:ADGx1\])–(\[eq:Cmumux1\]), our observables are $$\label{eq:obsDelta0fit}
{\cal A}_{\Delta\Gamma_s}^{\mu\mu} = -0.27 \pm 0.20, \quad
{\cal S}_{\mu\mu} = 0.46 \pm 0.20, \quad {\cal C}_{\mu\mu} = -0.85 \pm 0.20,$$ where we have considered the same absolute uncertainties as in Subsection \[ssec:Expx0xinf\]. Assuming that these observables have been measured correspondingly at a future experiment, ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$ would indicate NP at the $6\sigma$ level, while ${\cal S}_{\mu\mu}$ and ${\cal C}_{\mu\mu}$ would differ from the SM at the $2\sigma$ and $4\sigma$ levels, respectively. The latter observable would require a non-vanishing scalar contribution $S$. Performing a $\chi^2$ fit to these quantities, we can determine the underlying decay parameters $|x|$, $|S|$ and $\varphi_S$ simultaneously from the best fit point.
We start our statistical analysis by considering only $\overline{R}$, ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$ and ${\cal S}_{\mu\mu}$. In the left and right panels of Fig. \[fig:fitDelta0x05\], we show the corresponding $1\,\sigma$ confidence regions in the $\varphi_S$–$|S|$ and $\varphi_S$–$|x|$ planes, respectively. We obtain two solutions, given by the blue and red contours, as we expect based on the symmetry relations in Eq. (\[eq:Del-0-symm-r-ADG-smumu\]). Our input parameters are indicated by the green dot. Consequently, non-zero values of $|S|$ and $|x|$ at the $4\sigma$ and $6\sigma$ levels, respectively, could then be established.
![Illustration of the determination of $|S|$, $|x|$ and $\varphi_S$ in the scenario where we assume $\Delta=0^\circ$. The contours correspond to the $1\sigma$ allowed regions obtained by performing a $\chi^2$ fit to $\overline{R}$, ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$ and ${\cal S}_{\mu\mu}$ given in Eq. (\[eq:obsDelta0fit\]). We obtain two solutions, indicated in blue and red, as expected from the symmetry relations in Eq. (\[eq:Del-0-symm-r-ADG-smumu\]). The green dot marks the input parameters given in Eqs. (\[eq:inputDelta0fit1\]) and (\[eq:inputDelta0fit2\]).[]{data-label="fig:fitDelta0x05"}](fig14a.pdf "fig:"){width="45.00000%"} ![Illustration of the determination of $|S|$, $|x|$ and $\varphi_S$ in the scenario where we assume $\Delta=0^\circ$. The contours correspond to the $1\sigma$ allowed regions obtained by performing a $\chi^2$ fit to $\overline{R}$, ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$ and ${\cal S}_{\mu\mu}$ given in Eq. (\[eq:obsDelta0fit\]). We obtain two solutions, indicated in blue and red, as expected from the symmetry relations in Eq. (\[eq:Del-0-symm-r-ADG-smumu\]). The green dot marks the input parameters given in Eqs. (\[eq:inputDelta0fit1\]) and (\[eq:inputDelta0fit2\]).[]{data-label="fig:fitDelta0x05"}](fig14b.pdf "fig:"){width="45.00000%"}
![Illustration of the determination of $|S|$, $|x|$ and $\varphi_S$ in a scenario where we assume $\Delta=0^\circ$. The contours correspond to the $1\sigma$ allowed regions obtained by performing a $\chi^2$ fit to $\overline{R}$, ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$, ${\cal S}_{\mu\mu}$ and ${\cal C}_{\mu\mu}$ given in Eq. (\[eq:obsDelta0fit\]). The green dot marks the input parameters given in Eqs. (\[eq:inputDelta0fit1\]) and (\[eq:inputDelta0fit2\]).[]{data-label="fig:fitwithCmumuDelta0x05"}](fig15a.pdf "fig:"){width="45.00000%"} ![Illustration of the determination of $|S|$, $|x|$ and $\varphi_S$ in a scenario where we assume $\Delta=0^\circ$. The contours correspond to the $1\sigma$ allowed regions obtained by performing a $\chi^2$ fit to $\overline{R}$, ${\cal A}_{\Delta\Gamma_s}^{\mu\mu}$, ${\cal S}_{\mu\mu}$ and ${\cal C}_{\mu\mu}$ given in Eq. (\[eq:obsDelta0fit\]). The green dot marks the input parameters given in Eqs. (\[eq:inputDelta0fit1\]) and (\[eq:inputDelta0fit2\]).[]{data-label="fig:fitwithCmumuDelta0x05"}](fig15b.pdf "fig:"){width="45.00000%"}
If we include also ${\cal C}_{\mu\mu}$ in the analysis, we can eliminate the solution corresponding to the red contours, since ${\cal C}_{\mu\mu}$ breaks the symmetry relation in Eq. (\[sym-Del-0\]) by an overall minus sign. The resulting $1\sigma$ regions are shown in Fig. \[fig:fitwithCmumuDelta0x05\], corresponding to the results $$|S| = 0.55_{-0.10}^{+0.08}, \qquad \varphi_S=(20_{-5}^{+5})^\circ, \qquad |x| = 0.50_{-0.07}^{+0.07}.$$ As a matter of fact, non-zero values of these parameters could be pinned down at the $5\sigma$, $4\sigma$ and $7\sigma$ levels, respectively.
In general, the precision for the CP asymmetries required to determine $|S|$, $\varphi_S$ and $|x|$ with a given confidence level depends on the situation in parameter space. Moreover, we may end up with an ambiguity even after including ${\cal C}_{\mu\mu}$ in the $\chi^2$ fit. Nevertheless, this example nicely complements the one in Subsection \[ssec:Expx0xinf\] and shows the potential of the CP asymmetries to determine the (pseudo)-scalar contributions, and even to discriminate between the corresponding primed and unprimed Wilson coefficients.
Conclusions and Outlook {#sec:concl}
=======================
The rare decay $B^0_s\to\mu^+\mu^-$ has been in the focus of particle physics for decades, offering one of the theoretically cleanest probes for physics beyond the SM, in particular for new (pseudo)-scalar contributions, which are still largely unconstrained. Finally, this channel could be observed by the CMS and LHCb collaborations and is now an experimentally well established process, exhibiting a branching ratio encoded in $\overline{R}$ in the ballpark of the SM. The observable ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$, which is accessible thanks to the decay width difference $\Delta\Gamma_s$ and requires an untagged – but time-dependent – analysis, will play an important role to shed light on possible NP contributions to $B^0_s\to\mu^+\mu^-$. In general, these effects involve also CP-violating phases, which are usually neglected in theoretical analyses for simplicity.
In this paper, we have presented a comprehensive strategy for the future LHC upgrade(s), allowing us to reveal the presence of new sources of CP violation. The key role in this endeavour is played by the mixing-induced CP asymmetry ${\cal S}_{\mu\mu}$, which requires – in contrast to ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$ – also tagging information for the experimental analysis. Another observable, ${\cal C}_{\mu\mu}$, would become accessible if the helicity of the final-state muons could be determined; already sign information for this CP asymmetry would be very valuable information. These three observables do not depend on the decay constant $f_{B_s}$ and are not affected by theoretical uncertainties.
Interestingly, the interplay of $\overline{R}$ with ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$ and ${\cal S}_{\mu\mu}$ allows us to establish new (pseudo)-scalar contributions and new sources of CP violation. In general, we can only obtain constraints as we do not have sufficient independent observables to determine the short-distance coefficients $|S|$, $|P|$ and their phases $\varphi_S$, $\varphi_P$. To obtain further insights, additional information is required. This could either be obtained by assuming specific NP models, or in a model-independent way through relations between the short-distance coefficients $C_P^{(')}$, $C_S^{(')}$, which can be derived within the SMEFT approach. We have followed the latter avenue, discussing a variety of scenarios to illustrate how the corresponding parameters can be determined from the measured observables.
Since the pseudo-scalar coefficient $P$ involves ${\cal C}_{10}$, we need information on this quantity. By the time precise measurements of the observables ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$ and ${\cal S}_{\mu\mu}$ are available, we expect to have a detailed picture of ${\cal C}_{10}$, following from analyses of semileptonic rare $B\to K^{(*)}\mu^+\mu^-$ and $B_s\to \phi\mu^+\mu^-$ decays. Current anomalies in the data for the former and $B\to K^{(*)} e^+e^-$ decays indicate NP effects in ${\cal C}_{10}$, which we have also considered in our explorations. It will be important to utilize CP violation in the corresponding observables in the future.
To the best of our knowledge, experimental feasibility studies for the measurement of ${\cal S}_{\mu\mu}$ at the LHC upgrade(s) are not yet available. Performing fits to the observables for given future scenarios, we find that an absolute precision at the 0.2 level for ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$ and ${\cal S}_{\mu\mu}$ could have a dramatic impact on our search for new (pseudo)-scalar contributions in leptonic rare $B_s$ decays, allowing us to reveal the underlying dynamics. We urge the LHC collaborations to add studies of CP violation in rare $B^0_s\to\ell^+\ell^-$ decays to their physics agenda for the long-term future and super-high-precision era of $B$ physics.
Acknowledgements {#acknowledgements .unnumbered}
================
This research has been supported by the Netherlands Foundation for Fundamental Research of Matter (FOM) programme 156, “Higgs as Probe and Portal”, and by the National Organisation for Scientific Research (NWO). D.G.E. acknowledges the support through a fellowship by the Université Paris-Sud and the hospitality by Nikhef and the Vrije Universiteit Amsterdam during her visit. We would like to thank Marcel Merk for useful discussions.
Appendix {#sec:Obs .unnumbered}
========
In this appendix, we collect formulae which are useful for the analysis of $B_s^0 \to \mu^+ \mu^-$ within the SMEFT framework introduced in Subsection \[ssec:gen-frame\]. These expressions can be applied to any SMEFT scenario. In order to obtain the relevant observables in terms of the parameters $|S|$, $\varphi_S$, $|x|$ and $\Delta$, we write Eq. (\[conv-1\]) as $$|P|\cos\varphi_P=|{\cal C}_{10}|\cos\varphi_{10} - \frac{1}{w}
\left[\frac{\left(1-|x|^2\right)\cos\varphi_S-2|x|\sin\Delta\sin\varphi_S}{1-2|x|\cos\Delta+|x|^2}\right]|S|$$ $$|P|\sin\varphi_P=|{\cal C}_{10}|\sin\varphi_{10} - \frac{1}{w}
\left[\frac{\left(1-|x|^2\right)\sin\varphi_S+2|x|\sin\Delta\cos\varphi_S}{1-2|x|\cos\Delta+|x|^2}\right]|S|,$$ yielding $$\tan\varphi_P=\frac{|{\cal C}_{10}|\sin\varphi_{10}-\left[ \left(1-|x|^2\right)\sin\varphi_S+
2|x|\sin\Delta\cos\varphi_S \right]G}{|{\cal C}_{10}|\cos\varphi_{10}-
\left[ \left(1-|x|^2\right)\cos\varphi_S-2|x|\sin\Delta\sin\varphi_S \right]G}$$ with $$G\equiv\frac{|S|}{w \left(1-2|x|\cos\Delta+|x|^2 \right)}\,.$$ The scalar coefficient function is given as $$S \equiv |S|e^{i\varphi_S}= w \frac{M_{B_s}^2}{2m_\mu}\left(\frac{m_b}{m_b + m_s}\right)
\left( \frac{|C_{S}|}{C_{10}^{\rm SM}}\right)\left(1-|x|e^{i\Delta}\right) e^{i\tilde\varphi_S}
\label{eq:Sofx}$$ with $$\tan\varphi_S=\frac{\left(1-|x|\cos\Delta\right)\sin\tilde\varphi_S-|x|\sin\Delta\cos\tilde\varphi_S}{\left(1-|x|\cos\Delta\right)
\cos\tilde\varphi_S+|x|\sin\Delta\sin\tilde\varphi_S}.$$ As we noted in Eq. (\[C10-neg\]), $C_{10}^{\rm SM}$ is negative. We may also convert $\varphi_S$ into $\tilde\varphi_S$: $$\cos\tilde\varphi_S\propto |x|\cos(\varphi_S-\Delta)-\cos\varphi_S, \quad
\sin\tilde\varphi_S\propto |x|\sin(\varphi_S-\Delta)-\sin\varphi_S,$$ which yields $$\tan\tilde\varphi_S=\frac{(1-|x|\cos\Delta)\sin\varphi_S+|x|\sin\Delta\cos\varphi_S}{(1-|x|\cos\Delta)\cos\varphi_S
-|x|\sin\Delta\sin\varphi_S}.$$ Moreover, we have $$|C_{S}|=\frac{1}{w} \frac{2m_\mu}{M_{B_s}^2}\left(\frac{m_b + m_s}{m_b}\right)
\frac{|C_{10}^{\rm SM}|}{\sqrt{1-2|x|\cos\Delta+|x|^2}} |S|.$$
The observables in Eqs. (\[ADG-expr\]), and (\[S-expr\]) and (\[rr-def\]) require the quantities $$\hspace*{-2.0truecm}
|P|^2=|{\cal C}_{10}|^2-2\left[(1-|x|^2)\cos(\varphi_{10}-\varphi_S)+2|x|\sin\Delta\sin(\varphi_{10}-\varphi_S)\right]
|{\cal C}_{10}|G$$ $$+\left[ \left(1-|x|^2\right)^2 + \left( 2 |x| \sin\Delta\right)^2 \right]G^2,$$ $$\hspace*{-8.5truecm}|P|^2\cos2\varphi_P=|P|^2\left( \cos^2\varphi_P-\sin^2\varphi_P \right)$$ $$=|{\cal C}_{10}|^2\cos2\varphi_{10}
-2\left[\left(1-|x|^2\right)\cos(\varphi_{10}+\varphi_S)-2|x|\sin\Delta\sin(\varphi_{10}+\varphi_S) \right]|{\cal C}_{10}|G$$ $$+\left[\left\{\left(1-|x|^2\right)^2 - (2|x|\sin\Delta)^2\right\} \cos2\varphi_S - 4 |x| \left(1-|x|^2\right)\sin\Delta
\sin2\varphi_s \right]G^2,$$ $$\hspace*{-8.5truecm}|P|^2\sin2\varphi_P=2|P|\sin\varphi_P|P|\cos\varphi_P=$$ $$=|{\cal C}_{10}|^2\sin2\varphi_{10}
-2\left[\left(1-|x|^2\right)\sin(\varphi_{10}+\varphi_S)+2|x|\sin\Delta\cos(\varphi_{10}+\varphi_S) \right]|{\cal C}_{10}|G$$ $$+\left[\left\{\left(1-|x|^2\right)^2 - (2|x|\sin\Delta)^2\right\} \sin 2\varphi_S + 4 |x| \left(1-|x|^2\right)\sin\Delta
\cos 2\varphi_s \right]G^2,$$ while the CP asymmetry in Eq. (\[Cobs\]) involves $$|P||S|\cos(\varphi_P-\varphi_S)=|S|\left[|{\cal C}_{10}|\cos(\varphi_{10}-\varphi_S) - \left(\frac{1-|x|^2}{1-2|x|\cos\Delta
+|x|^2}\right)\frac{|S|}{w}\right].$$ In view of the complexity of the resulting general expressions, we refrain from listing them for the observables $r$, ${\cal A}^{\mu\mu}_{\Delta\Gamma_s}$, ${\cal S}_{\mu\mu}$ and ${\cal C}_{\mu\mu}$. However, we have given formulae for specific examples discussed in Subsection \[ssec:illu\].
[99]{} C. Bobeth, M. Gorbahn, T. Hermann, M. Misiak, E. Stamou and M. Steinhauser, Phys. Rev. Lett. [**112**]{} (2014) 101801 doi:10.1103/PhysRevLett.112.101801 \[arXiv:1311.0903 \[hep-ph\]\]. S. Aoki [*et al.*]{}, Eur. Phys. J. C [**77**]{} (2017) no.2, 112 doi:10.1140/epjc/s10052-016-4509-7 \[arXiv:1607.00299 \[hep-lat\]\]. G. Borissov, R. Fleischer and M. H. Schune, Ann. Rev. Nucl. Part. Sci. [**63**]{} (2013) 205 doi:10.1146/annurev-nucl-102912-144527 \[arXiv:1303.5575 \[hep-ph\]\]. V. Khachatryan [*et al.*]{} \[CMS and LHCb Collaborations\], Nature [**522**]{} (2015) 68 doi:10.1038/nature14474 \[arXiv:1411.4413 \[hep-ex\]\]. K. De Bruyn, R. Fleischer, R. Knegjens, P. Koppenburg, M. Merk, A. Pellegrino and N. Tuning, Phys. Rev. Lett. [**109**]{} (2012) 041801 doi:10.1103/PhysRevLett.109.041801 \[arXiv:1204.1737 \[hep-ph\]\]. A. J. Buras, R. Fleischer, J. Girrbach and R. Knegjens, JHEP [**1307**]{} (2013) 77 doi:10.1007/JHEP07(2013)077 \[arXiv:1303.3820 \[hep-ph\]\]. W. Altmannshofer, C. Niehoff and D. M. Straub, JHEP [**1705**]{} (2017) 076 doi:10.1007/JHEP05(2017)076 \[arXiv:1702.05498 \[hep-ph\]\]. R. Fleischer, R. Jaarsma and G. Tetlalmatzi-Xolocotzi, JHEP [**1705**]{} (2017) 156 doi:10.1007/JHEP05(2017)156 \[arXiv:1703.10160 \[hep-ph\]\]. R. Aaij [*et al.*]{} \[LHCb Collaboration\], Phys. Rev. Lett. [**118**]{} (2017) no.19, 191801 doi:10.1103/PhysRevLett.118.191801 \[arXiv:1703.05747 \[hep-ex\]\]. G. D’Ambrosio, G. F. Giudice, G. Isidori and A. Strumia, Nucl. Phys. B [**645**]{} (2002) 155 doi:10.1016/S0550-3213(02)00836-2 \[hep-ph/0207036\]. G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. [**68**]{} (1996) 1125 doi:10.1103/RevModPhys.68.1125 \[hep-ph/9512380\]. I. Dunietz, R. Fleischer and U. Nierste, Phys. Rev. D [**63**]{} (2001) 114015 doi:10.1103/PhysRevD.63.114015 \[hep-ph/0012219\]. K. De Bruyn, R. Fleischer, R. Knegjens, P. Koppenburg, M. Merk and N. Tuning, Phys. Rev. D [**86**]{} (2012) 014027 doi:10.1103/PhysRevD.86.014027 \[arXiv:1204.1735 \[hep-ph\]\]. Y. Amhis [*et al.*]{} \[Heavy Flavor Averaging Group\], arXiv:1612.07233 \[hep-ex\] and online update at <http://www.slac.stanford.edu/xorg/hfag/>. K. De Bruyn and R. Fleischer, JHEP [**1503**]{} (2015) 145 doi:10.1007/JHEP03(2015)145 \[arXiv:1412.6834 \[hep-ph\]\]. J. Charles [*et al.*]{}, Phys. Rev. D [**91**]{} (2015) no.7, 073007 doi:10.1103/PhysRevD.91.073007 \[arXiv:1501.05013 \[hep-ph\]\]; for updates, see <http://ckmfitter.in2p3.fr>. S. Chatrchyan [*et al.*]{} \[CMS Collaboration\], Phys. Rev. Lett. [**111**]{} (2013) 101804 doi:10.1103/PhysRevLett.111.101804 \[arXiv:1307.5025 \[hep-ex\]\]. C. Patrignani [*et al.*]{} \[Particle Data Group\], Chin. Phys. C [**40**]{} (2016) no.10, 100001. doi:10.1088/1674-1137/40/10/100001 M. Aaboud [*et al.*]{} \[ATLAS Collaboration\], Eur. Phys. J. C [**76**]{} (2016) no.9, 513 doi:10.1140/epjc/s10052-016-4338-8 \[arXiv:1604.04263 \[hep-ex\]\]. M. Beneke, C. Bobeth and R. Szafron, arXiv:1708.09152 \[hep-ph\]. C. S. Huang and W. Liao, Phys. Lett. B [**525**]{} (2002) 107 doi:10.1016/S0370-2693(01)01427-7 \[hep-ph/0011089\]. A. Dedes and A. Pilaftsis, Phys. Rev. D [**67**]{} (2003) 015012 doi:10.1103/PhysRevD.67.015012 \[hep-ph/0209306\]. P. H. Chankowski, J. Kalinowski, Z. Was and M. Worek, Nucl. Phys. B [**713**]{} (2005) 555 doi:10.1016/j.nuclphysb.2005.02.010 \[hep-ph/0412253\]. W. Buchmüller and D. Wyler, Nucl. Phys. B [**268**]{} (1986) 621. doi:10.1016/0550-3213(86)90262-2 B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, JHEP [**1010**]{} (2010) 085 doi:10.1007/JHEP10(2010)085 \[arXiv:1008.4884 \[hep-ph\]\]. R. Alonso, B. Grinstein and J. Martin Camalich, Phys. Rev. Lett. [**113**]{} (2014) 241802 doi:10.1103/PhysRevLett.113.241802 \[arXiv:1407.7044 \[hep-ph\]\]. F. Beaujean, C. Bobeth and S. Jahn, Eur. Phys. J. C [**75**]{} (2015) no.9, 456 doi:10.1140/epjc/s10052-015-3676-2 \[arXiv:1508.01526 \[hep-ph\]\]. W. Altmannshofer, C. Niehoff, P. Stangl and D. M. Straub, Eur. Phys. J. C [**77**]{} (2017) no.6, 377 doi:10.1140/epjc/s10052-017-4952-0 \[arXiv:1703.09189 \[hep-ph\]\]. W. Altmannshofer and D. M. Straub, Eur. Phys. J. C [**75**]{} (2015) no.8, 382 doi:10.1140/epjc/s10052-015-3602-7 \[arXiv:1411.3161 \[hep-ph\]\]. C. Bobeth, G. Hiller and D. van Dyk, JHEP [**1107**]{} (2011) 067 doi:10.1007/JHEP07(2011)067 \[arXiv:1105.0376 \[hep-ph\]\]. A. K. Alok, B. Bhattacharya, D. Kumar, J. Kumar, D. London and S. U. Sankar, Phys. Rev. D [**96**]{} (2017) no.1, 015034 doi:10.1103/PhysRevD.96.015034 \[arXiv:1703.09247 \[hep-ph\]\]. C. Bobeth, A. J. Buras, A. Celis and M. Jung, JHEP [**1704**]{} (2017) 079 doi:10.1007/JHEP04(2017)079 \[arXiv:1609.04783 \[hep-ph\]\]. R. Aaij [*et al.*]{} \[LHCb Collaboration\], Eur. Phys. J. C [**73**]{} (2013) 2373 doi:10.1140/epjc/s10052-013-2373-2 \[arXiv:1208.3355 \[hep-ex\]\].
R. Fleischer, N. Serra and N. Tuning, Phys. Rev. D [**82**]{} (2010) 034038 doi:10.1103/PhysRevD.82.034038 \[arXiv:1004.3982 \[hep-ph\]\].
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---
abstract: 'We review the recent theoretical developments towards understanding the Mott phases and quantum phase transition of extended Bose-Hubbard models on lattices in two spatial dimensions . We focuss on the description of these systems using the dual vortex picture and point out the crucial role played by the geometry of underlying lattices in determining the nature of both the Mott phases and the quantum phase transitions. We also briefly compare the results of dual vortex theory with quantum Monte Carlo results.'
author:
- 'K. Sengupta'
title: 'Mott phases and quantum phase transitions of extended Bose-Hubbard models in 2+1 dimensions'
---
Introduction
============
The superfluid to Mott insulators transitions of strongly correlated lattice bosons systems, described by extended Hubbard models, in two spatial dimensions have recently received a great deal of theoretical interest. One of the reasons for this renewed attention is the possibility of experimental realization of such models using cold atoms trapped in optical lattices [@bloch1; @kasevitch1]. However, such transitions are also of interest from a purely theoretical point of view, since they provide us with a test bed for exploring the recently developed theoretical paradigm of non-Landau-Ginzburg-Wilson (LGW) phase transitions [@senthil1]. In the particular context of lattice bosons in two spatial dimensions, a general framework for such non-LGW transitions has been developed and applied to the case of square lattice [@balents1]. The application of this framework to triangular [@burkov1], kagome [@sengupta1] lattices has also been carried out. These theoretical works has also supplemented by quantum Monte Carlo (QMC) studies which serves as a rigorous test for predicted theoretical results. Such studies, for extended Bose Hubbard models, has been carried out for square [@squaremonte], triangular [@melko1; @damle1], kagome [@isakov1; @damle2], and hexagonal lattices [@wessel1]. In this review, we present a brief overview of the theoretical ideas developed in the above-mentioned works focussing on role of the geometry of the underlying lattices in determining the nature of both the Mott phases and the quantum phase transitions. We also briefly compare the results of dual vortex theory with QMC results.
The typical paradigm of non-LGW transitions that has been proposed in the context of lattice bosons in Refs. is the following. It is proposed that for non-integer rational fillings $f=p/q$ of bosons per unit cell of the underlying lattice ($p$ and $q$ are integers), the theory of phase transition from the superfluid to the Mott insulator state is described in terms of the vortices which are non-local topological excitations of the superfluid phase, living on the dual lattice.[@fisher1; @tesanovic1] These vortices are not the order parameters of either superfluid or Mott insulating phases in the usual LGW sense. Thus the theory of the above mentioned phase transitions are not described in terms of the order parameters on either side of the transition which is in contrast with the usual LGW paradigm of phase transitions. Also, as first explicitly demonstrated in Refs. , although these vortices are excitations of a featureless superfluid phase, they exhibit a quantum order which depends on the filling fraction $f$. It is shown that the vortex fields describing the transition form multiplets transforming under projective symmetry group (projective representations of the space group of the underlying lattice and that this property of the vortices naturally and necessarily predicts broken translational symmetry of the Mott phase, where the vortices condense. Since this translational symmetry breaking is dependent on the symmetry group of the underlying lattice, geometry of the lattice naturally plays a key role in determining the competing ordered states of the Mott phase and in the theory of quantum phase transition between the Mott and the superfluid phases.
In what follows, we shall develop the above-mentioned ideas concentrating on square and Kagome lattices. The relevant results for the triangular lattice can be found in Ref.. To this end, we consider the extended Bose-Hubbard Hamiltonian $$\begin{aligned}
H_{\rm boson}&=& -t\sum_{\left<ij\right> }\left( b_i^{\dagger} b_j +
{\rm h.c.} \right) +\frac{U}{2} \sum_i n_i\left(n_i-1\right) \nonumber\\
&& + V \sum_{\left<ij\right>} n_i n_j -\mu \sum_i n_i.
\label{boson1}\end{aligned}$$ Here $t$ is the boson hopping amplitude between nearest neighbor sites, $U$ is the on-site interaction, $V$ denotes the strength of the nearest neighbor interaction between the bosons and $\mu$ is the chemical potential. We note at this point that a simple Holstein-Primakoff transformations maps this boson model (Eq.\[boson1\]), in the limit of hardcore bosons ($U \rightarrow
\infty$), to XXZ models with ferromagnetic $J_x$ and antiferromagnetic $J_z$ interaction in a longitudinal magnetic field $B_l$ $$\begin{aligned}
H_{\rm XXZ} &=& -J_x \sum_{\left<ij\right>} \left( S_i^x S_j^x +
S_i^y
S_j^y \right) + J_z \sum_{\left<ij\right>} S_i^z S_j^z \nonumber\\
&& -B_l \sum_i S^z_i \label{spinmodel1}\end{aligned}$$ where $J_x >0 $ and $J_z> 0$ are the strengths of transverse and longitudinal nearest neighbor interactions and $B_l$ is a longitudinal magnetic field. The parameters $J_x$, $J_z$ and $B_l$ of $H_{\rm XXZ}$ can be mapped to those of $H_{\rm boson}$: $J_{x}=
2t$, $J_z= V$, and $B_l = zV\left(f-\frac{1}{2}\right)$, where $z$ denotes the coordination number of the underlying lattice and $f$ is the average boson filling.
Competing Mott states and quantum phase transition {#Mott}
==================================================
The basic premise underlying the theories developed in Refs. is that the quantum phase transition of the extended Bose-Hubbard model occurs due to destabilization of the superfluid phases by proliferation of vortices which are topological excitations of the superfluid phase. To analyze such a transition, one therefore needs to obtain an effective action for the vortex excitation. Such an action can be obtained by performing a duality analysis of the Bose-Hubbard model [@dasgupta1; @balents1]. The final form of the dual action $S_d$ can be written as [@balents1] $$\begin{aligned}
S_d &=& \frac{1}{2e^2} \sum_b \left(\epsilon_{\mu \nu \lambda}
\Delta_{\nu} A_{b\lambda} - f\delta_{\mu \tau} \right)^2 \nonumber\\
&& - y_v \sum_b \left(\psi_{b +\mu} e^{2\pi i A_{b \mu}} \psi_b +
{\rm h.c.} \right) \nonumber\\
&& + \sum_b \left( r \left|\psi_b \right|^2 + u \left|\psi_b
\right|^4 \right) \Bigg]\end{aligned}$$ where $\psi_b$ are the vortex field living on the site $b$ of the dual lattice, $A_{b\mu}$ is the U(1) dual gauge field so that $\epsilon_{\tau \nu \lambda} \Delta_{\nu} A_{b \lambda}= n_i$ where $n_i$ is the physical boson density at site $i$, $\sum_p$ denotes sum over elementary plaquette of the dual lattice$, \Delta_{\mu}$ denotes lattice derivative along $\mu=x,y,\tau$, and $f$ is the average boson density. Note that the vortex action $S_d$ is not self-dual to the boson action obtained from $H_{\rm boson}$. Therefore we can not, in general, obtain a mapping between the parameters of the two actions, except for identifying $\epsilon_{\tau \nu \lambda} \Delta_{\nu} A_{b\lambda}$ as the physical boson density [@balents1]. We shall therefore classify the phases of this action based on symmetry consideration and within the saddle point approximation as done in Refs..
The transition from a superfluid ($\left<\psi_b \right> =0$) to a Mott insulating phase in $S_d$ can be obtained by tuning the parameter $r$. For $r>0$, we are in the superfluid phase. Note that the saddle point of the gauge fields $A_{b\mu}$ in action corresponds to $\epsilon_{\tau \nu \lambda} \Delta_{\nu} {\bar
A}_{b\lambda}= f$, so that the magnetic field seen by the vortices is pinned to the average boson filling $f$. Now as we approach the phase transition point $r=0$, the fluctuations about this saddle point ($\left<\psi_b \right> =0$, ${\bar A}_{by} = f x$) increase and ultimately destabilize the superfluid phase in favor of a Mott phase with $\left<\psi_b \right> \ne 0$. Clearly, in the above scenario, the most important fluctuations of the vortex field $\psi_b$ are the ones which has the lowest energy. This prompts us to detect the minima of the vortex spectrum by analyzing the kinetic energy term of the vortices $H_{\rm kinetic} = -y_v \sum_{b,\alpha }
\left(\psi_{b+\alpha} ^{\dagger} e^{2\pi i {\bar A}_{b}} \psi_{b} +
{\rm h.c} \right)$, where the sum over $\alpha$ is carried out over the sites of the dual lattice which are nearest neighbors to $b$. The analysis of $H_{\rm kinetic}$ therefore amounts to solving the Hofstadter problem on the dual lattice which has been carried out in Refs. , , and for square, triangular and Kagome lattices respectively.
Square lattice {#square}
--------------
The dual lattice in this case is also a square lattice with its sites being at the center of the plaquettes of the direct lattice. Thus here one needs to solve the Hofstadter problem on a square lattice with $f=p/q$ flux quanta per plaquette of the dual square lattice. It has been shown in Ref. that for $f=p/q$, there are $q$ minima within the magnetic Brillouin zone at the wave-vectors ${\bf Q}= (k_x a,k_y a) =(0,2\pi l p/q)$, where $l$ runs from $0$ to $q-1$, $k_{x,y}$ are the wavevectors of the magnetic Brillouin zone and $a$ is the lattice spacing. The eigenfunctions $\psi(k_x,k_y)$ corresponding to these minima can be easily constructed [@balents1; @senthil1].
Once the positions of these minima are located, the next task is to identify the low lying excitations around these minima which are going to play a leading role in destabilizing the superfluid phase. These excitations, represented by the bosonic fields $\varphi_l$, has definite transformation properties under the symmetry operations of the underlying dual lattice [@balents1]. For the case of square lattice, the relevant symmetry operations are translations along $x$ and $y$ directions by one lattice spacing ($T_x$ and $T_y$), reflections about $x$ and $y$ axes ($I_x$ and $I_y$) and rotation by $\pi/2$ ($R_{\pi/2}$). It has been shown in Ref. that the transformation properties of $\varphi_l$ under these transformations are the following: $T_x:
\,\varphi_{l} \rightarrow \varphi_{l-1}$, $T_y: \, \varphi_{l}
\rightarrow \varphi_l \omega^{l}$, $R_{\pi/2}: \, \varphi_l
\rightarrow \frac{1}{\sqrt{q}} \sum_{l'=0}^{q-1} \omega^{ l l'}
\varphi_{l'}$, $I_x: \, \varphi_l \rightarrow \varphi_{l}^{\ast}$, and $I_y: \, \varphi_l \rightarrow \varphi_{-l}^{\ast}$, where $\omega = \exp(-2\pi i f)$.
Finally, we need to construct an effective LGW action in terms of the low-energy fields $\varphi_l$ which is invariant under all the symmetry operations. For the sake of definiteness, we shall focuss on the case $q=2$ and $q=3$ from now on. First, let us consider the case $q=2$ for which, the effective action reads [@balents1; @senthil2], $$\begin{aligned}
S_v &=& \int d^2 r dt (L_2 + L_4 +L_8) \nonumber\\
L_2 &=& \left( \sum_{\ell = 0}^{1} \left[ |(\partial_\mu - i A_{\mu}
) \varphi_l |^2 + s |\varphi_l |^2 \right] + \frac{1}{2e^2} \left(
\epsilon_{\mu\nu\lambda}
\partial_\nu A_\lambda \right)^2 \right). \nonumber\\
L_4 &=& \frac{\gamma_{00}}{4} \left( |\varrho_0|^2 + |\varrho_1|^2
\right)^2 - \frac{\gamma_{01}}{4} \left( |\varrho_0|^2 - |
\varrho_1|^2 \right)^2 \nonumber\\
L_8 &=& \lambda (\varrho_0 \varrho_1^\ast )^4 + {\rm h.c}.
\label{q2lang}\end{aligned}$$ where $ \varrho_{0(1)}= \left(\varphi_0 \pm \varphi_1\right)/{\sqrt
2}$. Note that we have included a $8^{\rm th}$ order term $L_8$ in the effective action since this is the lowest order term, invariant under all the symmetry operations, which determines the relative phase of the vortex fields $\varrho_0$ and $\varrho_1$.
The Mott phases, which are stabilized for $s < 0$, can be obtained for $q=2$ by a straightforward analysis of $L_4$ and $L_8$. To concisely present the results for these phases, it is convenient to define a generalized density $\delta \rho ({\bf r}) =
\sum_{m,n=-1,1} \rho_{mn} e^{i(m r_x + n r_y)}$, where $\rho_{mn}=
S\left(|{\bf Q}_{mn}|\right ) \omega^{mn/2} \sum_{l = 0}^{1}
\varphi^{\ast}_l \varphi_{l+n} \omega^{l m}$ denotes the most general gauge-invariant bilinear combinations of the vortex fields $\varphi_l$ with appropriate transformation properties of square lattice space group [@balents1]. As shown in Ref., the values of $\delta \rho ({\bf r})$ on sites of the dual lattice (integer $r_x$ and $r_y$) can be considered a measure of the ring-exchange amplitude of bosons around the plaquette whereas those with ${\bf r}$ half-odd-integer co-ordinates represent sites of the direct lattice, and the values of $\delta \rho ({\bf r})$ on such sites measure the boson density on these sites. Finally, ${\bf r}$ values with $r_x$ integer and $r_y$ half-odd-integer correspond to vertical links of the square lattice (and vice versa for horizontal links), and the values of $\delta \rho ({\bf r})$ on the links is a measure of the mean boson kinetic energy; if the bosons represent a spin system, this is a measure of the spin exchange energy. We also note two fundamental points from the definition of $\rho_{mn}$. First, as long as $\langle\varphi_l \rangle \ne 0$ for at least one non-zero ${\bf
Q}_{mn}$, the Mott state is characterized by a non-trivial density wave order. Second, the relative phase of the boson fields $\varrho_l$ which plays a crucial role in determining the nature of the density wave order is fixed by the $8^{\rm th}$ order term $L_8$.
The Mott phases for $q=2$ is shown in Fig. \[fig1\]. We note that there are three possible phases. For $\gamma_{01}>0$, only one of the two vortex fields condenses and the resulting Mott state (state A in Fig. \[fig1\]) has a charge density wave order. For $\gamma_{01} <0$, both the vortex fields condense and this leads to two possible Mott states (states B and C in Fig. \[fig1\]). For these states, all sites of the direct lattice are equivalent. These therefore represent valence bond solid (VBS) states with columnar or plaquette VBS order parameters.
Next, we address the issue of quantum phase transition between the superfluid and the Mott state for $q=2$. Such a transition, within the premise of dual vortex theory discussed here, is described by $S_v$ and not, as would be expected by LGW paradigm, by an action written in terms of order parameters fields at either side of the transition. The phase transition can either be first or second order. If it turns out to be second order and if $v <0$, the relative phase of the vortex fields, which is pinned in the Mott phase by the dangerously irrelevant (in the renormalization group sense) $8^{\rm th}$ order term $L_8$, becomes a gapless mode at the transition [@balents1]. Thus such a transition has an emergent gapless mode at the critical point. It provides an example of a deconfined quantum critical point [@senthil1] and can be shown to accompanied by boson fractionalization [@balents1].
The Mott phases for $q=3$ can also be similarly obtained [@balents1]. Here the quartic term in the effective Lagrangian which determines the Mott states are given by $$\begin{aligned}
\mathcal{L}_4 &=& \frac{\gamma_{00}}{4} \left( |\varphi_0|^2 +
|\varphi_1|^2 + |\varphi_2|^2 \right)^2 \nonumber \\ &+&
\frac{\gamma_{01}}{2} \left( \varphi_0^\ast \varphi_1^\ast
\varphi_2^2 + \varphi_1^\ast \varphi_2^\ast \varphi_0^2 +
\varphi_2^\ast \varphi_0^\ast \varphi_ - \varphi_0^\ast
\varphi_1^2 + \mbox{c.c.} \right. \nonumber \\
&-& \left. 2 |\varphi_0|^2 |\varphi_1|^2 - 2 |\varphi_1|^2
|\varphi_2|^2- 2 |\varphi_2|^2 |\varphi_0|^2\right). \label{m6}\end{aligned}$$ Note that here the ordering of the Mott states are completely determined by ${\mathcal L}_4$ and retaining higher order terms are not essential. It turns out that there are two sixfold degenerate VBS states with columnar or diagonal VBS orders as shown in Fig.\[fig2\]. The superfluid-Mott insulator quantum phase transition is again described by a dual vortex action, but does not lead to deconfined quantum criticality as pointed out in Ref..
QMC studies on Bose-Hubbard models on square lattice with nearest neighbor interaction and near half-filling has been carried out in Ref. . These studies indicate a checkerboard Mott state and a strong first order transition between the superfluid and checkerboard Mott state. Analogous QMC studies on triangular lattice has also been carried out in Refs..
Kagome lattice
--------------
The dual lattice corresponding to Kagome is the well-known dice lattice which has three inequivalent sites denoted commonly by $A$, $B$ and $C$ [@vidal1]. To obtain the minima of the vortex spectrum, we therefore need to solve the Hofstadter problem on the dice lattice. Here we shall concentrate only for $f=1/2$ and $2/3$ (or equivalently $f=1/3$).
For $f=1/2$, it has been shown [@vidal1; @ye1] that the vortex spectrum on dice lattice do not have well-defined minima and collapse to three degenerate bands. Physically, the collapse of the vortex spectrum can be tied to localization of the vortex within the so-called Aharanov-Bohm cages as explicitly demonstrated in Ref.. An example of such a cage is shown in Fig.\[fig3\]. A vortex whose initial wavepacket is localized at the central (white) $A$ site can never propagate beyond the black sites which form the border of the cage. This can be understood in terms of destructive Aharanov-Bohm interference: The vortex has two paths $0\rightarrow1\rightarrow 3$ and $0\rightarrow 2\rightarrow 3$ to reach the rim site $3$ from the starting site $0$. The amplitudes from these paths destructively interfere for $f=1/2$ to cancel each other. Thus the vortex remain within the cage. Such dynamic localization of the vortex wavepackets, termed as Aharanov-Bohm caging, makes it energetically unfavorable to condense vortices. Thus superfluidity persists for arbitrarily small values of $t/V$. In the language of spins, this also explains the absence of $S_z$ ordering for XXZ model in a Kagome lattice for $B_l=0$ and $J_z \gg
J_x$. Such a persistence of superfluidity has also been confirmed by QMC studies [@isakov1; @damle2].
In contrast to $f=1/2$, the vortex spectrum has two well-defined minima for $f=2/3$ within the magnetic Brillouin zone located at $
(k_x a, \kappa) = (0, \pi/3)\, {\rm and} \,(2\pi/3, 2 \pi/3)$, where $\kappa = \sqrt{3}k_y a/2$. Thus the low energy properties of the vortex system can be characterized in terms of the two low-energy bosonic fields $\varphi_1$ and $\varphi_2$ similar to the case of square lattice at $f=1/2$. However, as shown in Ref., in contrast to the case of the square lattice, the space group of symmetry transformations of the dice lattice involves translations by lattice vectors $u= ()$ and $v=()$ ($T_u$ and $T_v$), rotation by $\pi/3$ ($R_{\pi/3}$), and reflections around $x$ and $y$ ($I_x$ and $I_y$). The transformation properties of the vortex fields under these symmetry operations are the following: $T_{u}:\, \varphi_{1} \rightarrow \varphi_{1}
\exp(-i\pi/3),\, \varphi_{2} \rightarrow \varphi_{2} \exp(i\pi/3)$, $T_{v}: \, \varphi_{1} \rightarrow \varphi_{1} \exp(i\pi/3),\,
\varphi_{2} \rightarrow \varphi_{2} \exp(-i\pi/3)$, $I_x:\,
\varphi_{1(2)} \rightarrow \varphi^{\ast}_{1(2)}$, $I_y:\,\varphi_{1(2)} \rightarrow \varphi^{\ast}_{2(1)}$, and $R_{\pi/3}: \varphi_{1(2)}\rightarrow \varphi_{2(1)}$ [@sengupta1].
The simplest Landau-Ginzburg theory for the vortex fields which respects all the symmetries is [@sengupta1] $$\begin{aligned}
L_v &=& L_v^{(2)}+ L_v^{(4)}+ L_v^{(6)} \\
L_v^{(2)} &=& \sum_{\alpha=1,2} \left[ \left|\left(\partial_{\mu} -
e A_{\mu}\right) \phi_{\alpha} \right|^2 +
r \left|\varphi_{\alpha}\right|^2 \right]\\
L_v^{(4)} &=& u\left(\left|\varphi_1\right|^4 +
\left|\varphi_2\right|^4 \right) + v \left|\varphi_1\right|^2
\left|\varphi_2\right|^2 \\
L_v^{(6)} &=& w \left[ \left(\varphi_1^{\ast} \varphi_2 \right)^3 +
{\rm h.c.} \right]\end{aligned}$$ Here we find that $L_v^{(6)}$ turns out to be the lowest-order term which breaks the $U(1)$ symmetry associated with the relative phase of the vortex fields.
A simple power counting shows that $L_v^{(6)}$ is marginal at tree level. Unfortunately, the relevance/irrelevance of such a term beyond the tree level, is not easily determined analytically [@sengupta1]. If it so turns out that $L_v^{(6)}$ is irrelevant, the situation here will be identical to that of bosons on square lattice at $f=1/2$. The relative phase of the vortices would emerge as a gapless low energy mode at the critical point. The quantum critical point would be deconfined and shall be accompanied by boson fractionalization [@balents1]. On the other hand, if $L_v^{(6)}$ is relevant, the relative phase degree of the bosons will always remain gapped and there will be no deconfinement at the quantum critical point. Here there are two possibilities depending on the sign of $u$ and $v$. If $u,v<0$ and $w>0$, the transition may become weakly first order whereas for $u>0$, it remains second order (but without any deconfinement). QMC studies seem to indicate a weak first order transition in this case [@isakov1]. However, it is possible that a second order quantum phase transition can still be possible at special points on the phase diagram and this issue remains to be settled [@isakov1; @damle2; @damle3].
The vortices condense for $r<0$ signifying the onset of Mott phases. It turns out that the Mott phase for $v>0$ do not lead to any density wave order [@sengupta1]. For $v <0$, there are two possible Mott phases as demonstrated in Ref.. One of these phases with $9$ by $9$ ordering pattern is shown in Fig. \[fig4\]. Here the bosons are localized in red, blue, green (closed circle) and black (open circle) sites whereas the green (open circle) and black (closed circle) sites are vacant leading to a filling of $f=2/3$. The net occupation of the hexagons takes values $0$, $4$ and $2$ as shown in Fig. \[fig4\]. Note that interchanging the occupations of all the black and green sites of this state (while leaving the red and the blue sites filled) has the effect of $2\leftrightarrow 4$ for the boson occupation of the hexagons labeled $2$ and $4$ in Fig. \[fig4\] while leaving those for hexagons labeled $0$ unchanged. The resultant state is degenerate (within mean-field theory) to the state shown in Fig. \[fig4\]. This allows us to conjecture that the effect of quantum fluctuations on the state is shown in Fig.\[fig4\] might lead to stabilization of a partial resonating state with a $3$ by $3$ $R-3-3$ ordering pattern [@sengupta1; @burkov2]. Such a state is indeed found to be the true ground state in exact diagonalization [@cabra1] and QMC studies [@isakov1]. Note that the mean-field analysis of vortex theory which neglects quantum fluctuations can not directly give this partially resonating state.
Discussion {#conclusion}
==========
In conclusion, we have briefly reviewed theoretical developments on duality analysis of Bose-Hubbard and equivalently XXZ models on two dimensional lattices. We have focussed mainly at predictions such an analysis for square and Kagome lattices at half and one-third filling. We have compared the predictions of such dual vortex theories with QMC results wherever possible. Finally, we would like to point out a couple of issues that still remain unsettled in this regard. It is yet unknown, whether it is possible to have a second order quantum phase transition in these models as predicted by the vortex theory. Direct QMC studies [@squaremonte; @damle1; @melko1; @isakov1; @damle2; @damle3] as well as several other studies [@kuklov1] seem to find at best a weak first order transition. However, it has been recently conjectured that at least for the Kagome lattice, a second order transition might occur at special particle-hole symmetric points in the phase diagrams [@damle3]. Since QMC studies, at least for large enough system sizes, can only reach close to this point, this issue is not settled yet. Second, it remains to be seen whether such duality analysis for boson theories with $U(1)$ symmetry can be extended to either models with Fermions and/or bosonic models with higher symmetry group.
The author thanks L. Balents, L. Bartosch, A. Burkov, S. Isakov, Y.B. Kim, R. Melko, S. Sachdev, and S. Wessel for collaborations on related earlier works, and K. Damle, T. Senthil, and A. Vishwanath for helpful discussions.
[99]{}
M. Greiner, O. Mandel, T.Esslinger, T.W. Ha¨nsch, and I. Bloch, Nature (London) [**415**]{}, 39 (2002).
C. Orzel, A.K. Tuchman, M.L. Fenselau, M. Yasuda, and M.A. Kasevich, Science [**291**]{}, 2386 (2001).
T. Senthil [*et al.*]{}, Science [**303**]{}, 1490 (2004); T. Senthil [*et al.*]{}, Phys. Rev. B, [**70**]{}, 144407 (2004).
L. Balents ${\it et\, al.}$, Phys. Rev. B [**71**]{}, 144508 (2005); [*ibid*]{} [**71**]{}, 144509 (2005); L. Balents ${\it
et\, al.}$, Prog. Theor. Phys. Supp., [bf 160]{}, 314 (2005).
A. Burkov and L. Balents, Phys. Rev. B [**72**]{}, 134502 (2005).
K. Sengupta, S. Isakov, and Y.B. Kim, Phys. Rev. B [**72**]{}, 134502 (2005).
A. Burkov and E. Demler, Phys. Rev. Lett. [**96**]{}, 180406 (2006).
G.G. Batrouni [*et al.*]{}, Phys. Rev. Lett. [**74**]{}, 2527 (1995); R.T. Scalettar [*et al.*]{}, Phys. Rev. B [**51**]{}, 8467 (1995); G.G. Batrouni and R.T. Scalettar, Phys. Rev. Lett. [**84**]{}, 1599 (2000); F. Herbert [it et al.]{}, Phys. Rev. B [**65**]{}, 014513 (2002); G. Schmid [*et al.*]{}, Phys. Rev. Lett. [**88**]{}, 167208 (2002); A. Kuklov, N. Prokof’ev, and B. Svistunov, Phys. Rev. Lett. 93, 230402 (2004).
R. G. Melko [*et al.*]{}, Phys. Rev. Lett. [**95**]{}, 127207 (2005).
D. Heidarian and K. Damle, Phys. Rev. Lett. [**95**]{}, 127206 (2005).
S. V. Isakov [*etal.*]{}, Phys. Rev. Lett. [**97**]{}, 147202 (2006).
K. Damle and T. Senthil, Phys. Rev. Lett. [**97**]{}, 067202 (2006).
S. Wessel, cond-mat/0701337 (unpublished).
M. P. A. Fisher and D. H. Lee, Phys. Rev. B [**39**]{}, 2756 (1989).
Z. Tesanovic, Phys. Rev. Lett. [**93**]{}, 217004 (2004); A. Melikyan and Z. Tesanovic, Phys. Rev. B [**71**]{}, 214511 (2005).
C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. [**47**]{}, 1556 (1981)
M. Wallin, E. Sorensen, A.P. Young and S.M. Girvin, Phys. Rev. B [**49**]{}, 12115 (1994).
C. Lannert, M.P.A Fisher, and T. Senthil, Phys. Rev. B [**63**]{}, 134510 (2001).
J. Vidal ${\it et\,al.}$, Phys. Rev. B [**64**]{}, 155306 (2001); M. Rizzi, V. Cataudella, R. Fazio, Phys. Rev. B [**73**]{}, 144511 (2006).
L. Jiang and J. Ye, cond-mat/0601083 (unpublished).
R. Moessner, S.L. Sondhi, and P. Chandra, Phys. Rev. Lett. [**84**]{}, 4457 (2000).
D.C. Cabra [*etal.*]{}, Phys. Rev. B [**71**]{}, 144420 (2005).
A. Sen, K. Damle, and T. Senthil, cond-mat/0701476.
A. Kuklov [*etal.*]{}, cond-mat/0602466 (unpublished); A. Kuklov, N. Prokof’ev, and B. Svistunov, cond-mat/0501052 (unpublished).
|
---
abstract: |
Stimulated by a scholium in Newton’s Principia we find some beautiful results in classical mechanics which can be interpreted in terms of the orbits in the field of a mass endowed with a gravomagnetic monopole. All the orbits lie on cones! When the cones are slit open and flattened the orbits are exactly the ellipses and hyperbolae that one would have obtained without the gravomagnetic monopole.
The beauty and simplicity of these results has led us to explore the similar problems in Atomic Physics when the nuclei have an added Dirac magnetic monopole. These problems have been explored by others and we sketch the derivations and give details of the predicted spectrum of monopolar hydrogen.
Finally we return to gravomagnetic monopoles in general relativity. We explain why NUT space has a non-spherical metric although NUT space itself is the spherical space-time of a mass with a gravomagnetic monopole. We demonstrate that all geodesics in NUT space lie on cones and use this result to study the gravitational lensing by bodies with gravomagnetic monopoles.
We remark that just as electromagnetism would have to be extended beyond Maxwell’s equations to allow for magnetic monopoles and their currents so general relativity would have to be extended to allow torsion for general distributions of gravomagnetic monopoles and their currents. Of course if monopoles were never discovered then it would be a triumph for both Maxwellian Electromagnetism and General Relativity as they stand!
address: |
*$^1$Institute of Astronomy, The Observatories, Cambridge. CB3 0HA\
$^2$Current address Physics Department, The Queen’s University,\
Belfast. BT7 1NN*
author:
- 'D Lynden-Bell$^{1,2}$ & M Nouri-Zonoz$^1$'
title: 'Classical Monopoles: Newton, NUT-space, gravomagnetic lensing and atomic spectra'
---
Introduction {#I}
============
One of us was asked to review Chandrasekhar’s (1995) book on Newton’s Principia (1687) for Notes and Records of the Royal Society (Lynden-Bell 1996). This led to reading passages of Cajori’s translation of Principia. In his first proposition Newton shows that motion under the influence of a central force will be in a plane and that equal areas will be swept by the radius vector in equal times. In his second proposition he shows that if a radius from a point $S$ to a body sweeps out equal areas in equal times then the force is central. There follows this scholium: “A body may be urged by a centripetal force compounded of several forces; in which case the meaning of the proposition is that the force which results out of all tends to the point $S$. But if any force acts continually in the direction of lines perpendicular to the [*described surface*]{}, this force will make the body to deviate from the plane of its motion; but it will neither augment nor diminish the area of the [*described surface*]{} and is therefore to be neglected in the composition of forces.”
What does this mean?
The words [*described surface*]{} have been translated from a Latin word that carries the extra connotation of a surface described by its edge. We shall take this to be the surface swept out by the radius vector to the body that is now describing the non-coplanar path. A force normal to this surface at the body must be perpendicular to ${\bf r}$ and ${\bf v}$ which are both within the surface, so Newton is considering extra forces of the form $Nm_0 {\bf r} \times {\bf v}$ where $N$ may depend on ${\bf r},{\bf v}, t$ etc. We write the equation of motion $$m_0 d^2 {\bf r} / dt^2 = -V'(r) {\hat {\bf r}} +
N \,{\bf L}, \eqno (1.1)$$ where $V(r)$ is the potential for the central force, ${\hat {\bf r}}$ is the unit radial vector, and $${\bf L} = m_0 {\bf r} \times {\bf v}\,. \eqno (1.2)$$ Taking the cross product ${\hat {\bf r}} \times (1.1)$ we have $$d {\bf L}/dt = N {\bf r} \times {\bf L} \eqno (1.3)$$ from which it follows either geometrically à la Newton or by dotting with ${\bf L}$ that $$|{\bf L}| = \ {\rm constant}. \eqno (1.4)$$ Now if $\varphi$ is the angle [*measured within the described surface*]{} between a fixed half-line ending at $S$ and the radius vector, $${\textstyle{1\over2}} r^2 {\dot \varphi} = {\textstyle{1\over2}}
|{\bf L}|/m_0 \eqno (1.5)$$ so equal areas are swept out in equal times just as Newton says. To see this angle more precisely it is perhaps worthwhile to work in axes which are continually tilting to keep up with the plane of the motion. In any axes rotating with angular velocity ${\bf \Omega} (t)$, the apparent acceleration $\ddot {\bf r}$ is related to the absolute acceleration $d^2 {\bf r}/dt^2$ by $$d^2 {\bf r}/dt^2 = {\ddot {\bf r}} + 2{\bf \Omega} \times {\dot {\bf
r}} + {\dot {\bf \Omega}} \times {\bf r} + {\bf \Omega} \times ({\bf
\Omega} \times {\bf r})\, . \eqno (1.6)$$ We shall apply this formula to axes which are always tilting about ${\hat {\bf r}}$ such that in these axes the motion appears as planar. Thus putting ${\bf \Omega} = \Omega {\hat {\bf r}}$ in Eq. (1.6) $$d^2 {\bf r}/dt^2 = {\ddot {\bf r}} + \Omega r^{-1} {\bf
L}/m_0\,. \eqno (1.7)$$ Inserting this into Eq. (1.1) and choosing $$\Omega = r N, \eqno (1.8)$$ we recover in these axes the equation we would have had in inertial axes had Newton’s extra force $\propto N$ been absent i.e., $$m_0 {\ddot {\bf r}} = -V' {\hat {\bf r}}\,. \eqno (1.9)$$ Thus relative to [*these moving axes*]{} ${\bf r} \times m_0 {\dot
{\bf r}} = {\bf L}$ is constant not only in magnitude but also in direction and $$|{\bf r} \times {\dot {\bf r}}| = r^2 {\dot \varphi} = L/m_0 \eqno (1.10)$$ where $\varphi$ is the angle at $S$ between some line fixed in the moving axes and the current radial line (this is of course equal to the earlier angle since this moving plane is ‘rolling’ on the [*described surface*]{} about the common radius vector).
We now return to the inertial axes in which the direction of ${\bf L}$ varies in accord with (1.3). Dotting Eq. (1.1) with ${\bf v} =
d{\bf r}/dt$ the $N$ term goes out so the energy equation is left unchanged and we have, remembering that $L^2$ is constant, $${m_0 \over 2} {\bf v}^2 + V = {m_0 \over 2} \left [{\dot r}^2 +
\left ({L \over m_0} \right )^2 r^{-2} \right ] + V = E\, . \eqno (1.11)$$ Here ${\dot r}$ is the same in fixed or rotating axes since this $r$ is scalar. Equation (1.11) demonstrates that the radial motion $r
(t)$ is precisely that which would have occurred had $N$ been zero. Furthermore (1.9) and (1.10) demonstrate that within the tilting axes, or \[using (1.5)\] within the described surface, the solution $r
(\varphi)$ is precisely the same function that we would have found for the truly planar motion that occurs with $N$ absent. Although this extension of Newton’s theorem is not in Principia it would surprise us if Newton had not seen and understood it. There is interesting historical research to be done here on Newton’s surviving manuscripts. We know from Whiteside that this scholium was not in the first draft of Newton’s De Motu Corporum written in Autumn 1684 but appears in its revision which is probably dated to the Spring of 1685.
Although (1.5) and (1.11) are sufficient for the solution of the motion within the described surface, we need to find that surface by solving (1.3) for a complete description of the motion. This is not particularly simple and to do it we need to prescribe how $N$ depends on ${\bf r}, {\bf v}, t$ etc. However $d{\bf L}/dt$ and $d{\hat {\bf
r}}/dt$ are always parallel since both are perpendicular to ${\bf r}$ and ${\bf L}$. This led us to consider under what circumstances they might be proportional. In particular $$d{\hat {\bf r}}/dt = -{\hat {\bf r}} \times ({\hat {\bf r}} \times
{\bf v})/r = -{\bf r} \times {\bf L}/(r^3 m_0) \eqno (1.12)$$ so in full generality we have from (1.3) $$d{\bf L}/dt = -(m_0 N r^3) d{\hat {\bf r}}/dt\, . \eqno (1.13)$$ This demonstrates that when $m_0 N r^3 = Q_* = \ {\rm constant}$ we have a beautifully simple solution to (1.13) to wit. $${\bf L} + Q_* {\hat {\bf r}} = {\bf j} = {\rm const} \eqno (1.14)$$ here ${\bf j}$ is the vector constant of integration; notice that $Q_*$ has the same dimensions as $L$. Since ${\bf L}$ and ${\hat {\bf
r}}$ are perpendicular we deduce, dotting with ${\hat {\bf r}}$ $${\bf j} \cdot {\hat {\bf r}} = Q_* \eqno (1.15)$$ which shows that the angle between ${\bf j}$ and ${\hat {\bf r}}$ is constant so ${\hat {\bf r}}$ moves on a cone whose axis is along ${\bf
j}$. Similarly dotting (1.14) with ${\bf L}$ we find $L^2 = {\bf j}
\cdot {\bf L}$ so likewise ${\bf L}$ moves on another cone with ${\bf
j}$ as its axis. If this cone has semi-angle $\chi$ then $L/|{\bf j}|
= \cos \chi$, but ${\hat {\bf r}}$ and ${\bf L}$ are orthogonal and by (1.14) they are coplanar with ${\bf j}$ so we may choose cf. (1.15) $$Q_*/|{\bf j}| = \sin \chi \eqno (1.16)$$ so the angle between ${\hat {\bf r}}$ and ${\bf j}$ is $\pi /2 - \chi$ as shown in Fig. 1. Notice from (1.16) that the angle of the cone is determined completely from $|{\bf L}|$ and the force constant $Q_*$. Orbits with larger $|{\bf j}|$ have smaller $\chi$ so the angular momentum then moves around a narrow cone and ${\hat {\bf r}}$ then moves around a very open one. For $|{\bf j}| \gg Q_*$ it is almost planar. Fig. 1 illustrates two circular orbits moving in opposite senses about the same axis. Notice that the one moving right-handedly about the upward pointing axis is displaced above the center sitting like a halo about it while that moving left handedly is displaced below the center like an Elizabethan ruff below the head. One might have supposed that for $j \gg Q_*$ these two circular orbits would approach the central plane but although the cone becomes much flatter and more open the displacement between the direct and retrograde orbits actually increases. For circular orbits at distance $a$ from $S$ we have, for a Newtonian potential, $L^2 = GMa m_0^2$ and the displacement is $$2{\hat {\bf j}} \cdot {\bf r} = a/ \sqrt{GMa
m_0^2 Q_*^{-2} +1} \rightarrow m_0^{-1} Q_* \sqrt{a/(GM)}.$$ We have been led to the case $m_0 N r^3 = Q_* = \ {\rm constant}$ for reasons of mathematical simplicity but this case is more than a mathematical curiosity because:
1\. Of all the forces of Newton’s $N$ type \[see Eq. (1.1)\] only those of the form $-{\bf v} \times {\hat {\bf r}} r^{-2} Q_*(\theta, \phi)$ derive from a Lagrangian. For a monopole $Q_*$ is constant.
2\. We may rewrite this force in the form $$N {\bf L} = -Q_* {\bf v} \times {\bf r}/r^3 = {m_0 \over c} {\bf v}
\times {\bf B}_g, \eqno (1.17)$$
where $${\bf B}_g = - Q {\hat {\bf r}}/r^2\;\ \ ;\, \ \ Q = Q_*
c/m_0. \eqno (1.18)$$ We have introduced the velocity of light $c$ to make the analogy with magnetic forces even more obvious. ${\bf B}_g$ is clearly the field of a magnetic monopole of strength $Q$ but since this sort of magnetism acts not on moving charges but rather on moving masses; it is a gravomagnetic field. Such fields are well known in general relativity see Landau & Lifshitz Theory of Fields (1966) §89 problem 1. They are position dependent Coriolis forces associated with what relativists less helpfully call the dragging of inertial frames. The field ${\bf B}_g$ as we have defined it has the same dimensions as ${\bf g}$ the acceleration due to gravity and $Q/G$ has the dimensions of mass. In electricity, like charges repel while in gravity, like masses attract. It is the same with like magnetic monopoles, they repel while the gravomagnetic monopoles of like sign attract one another, hence the negative sign in Eq. (1.18) is best left there rather than combined into a new definition of the pole strength $Q$. We may find the Lagrangian corresponding to the force (1.17) by analogy with the electrodynamic case. There we add a term $q {\bf
v} \cdot {\bf A}/c$ where $q$ is the charge and ${\bf A}$ is the vector potential. For any poloidal axi-symmetric magnetic field one may choose ${\bf A}$ to be of the form $A {\bf \nabla} \phi$ where $\phi$ is the azimuth around the axis. We require $$-Q {\hat {\bf r}}/r^2 = {\bf B}_g = \nabla \times (A {\bf \nabla}
\phi) = {\bf \nabla}A \times {\bf \nabla} \phi \eqno (1.19)$$ from which one readily finds $A = Q (1+\cos \theta)$ gives the right ${\bf B}_g$. Thus a Lagrangian for Eq. (1.1) is $${\cal L} = {\textstyle {1\over 2}} m_0v^2 - m_0 V(r) + Q_* (1 + \cos
\theta) {\bf v} \cdot {\bf \nabla} \phi. \eqno (1.20)$$ Although the dynamical system is spherically symmetrical the Lagrangian is not and can not be made so. The only spherically symmetrical vector fields are $f(r){\bf r}$. If ${\bf A}$ were of this form its curl would be zero and therefore could not be the field of a monopole. Of course we can choose any axis we like and measure $\theta$ and $\phi$ appropriately from it. The ${\bf A}$ field will then be quite different but it will give the same ${\bf B}_g$ field by construction. Thus the difference between any two such ${\bf A}$ fields will have zero curl showing that ${\bf
A}' = {\bf A} + {\bf \nabla} \chi$ i.e., a gauge transformation. The Lagrangian (1.20) is neither spherically symmetrical nor gauge invariant but it is a member of a whole class of equivalent Lagrangians with different axes which are related by gauge transformations. Whereas none of these is individually spherical the class of all of them is spherically symmetric. The moral is that it can be restrictive to impose symmetry on a single member of the class if the member is not gauge invariant.
So far everything holds for any spherical potential $V(r)$. We could for example choose it to be Henon’s (1959) isochrone potential $2aV_0/(a + \sqrt{r^2+a^2})$ or its better known limits the simple harmonic oscillator $a \gg r$ or the Newtonian potential $a \ll r$. For all isochrones the orbits can be solved using only trigonometric functions (see e.g., Lynden-Bell 1963, Evans [*et al*]{}. 1990). Here we shall stick to the Newtonian potential $V/m_0 = -GM/r$. We have already shown that the motion lies on a cone whose semi-angle is given by $\cos^{-1} (Q_*/|{\bf j}|)$; furthermore if we slit that cone along $\varphi = 0$ and flatten it, the orbit will be exactly what it would have been in the absence of $N$ i.e., a conic section. Of course when we slit and flatten the orbit’s cone a gap appears whose angle is $\gamma = 2 \pi \left [1-L/ \sqrt{L^2 + Q^2_*}\, \right ]$, see Fig. 2. An ellipse with focus at $S$ and apocentre at $\varphi = 0$ would get back to apocentre at $\varphi = 2 \pi$ but unfortunately the gap intervenes. On the cone we identify $\varphi = 0$ not with $\varphi =
2 \pi$ but rather with $\varphi = 2 \pi - \gamma$. Thus on the cone the ellipse will precess forwards by an angle $\gamma$ in each radial period, Fig. 3. This angle $\gamma$ is an angle like $\varphi$ measured at $S$ within the cone’s surface. It is perhaps more natural to measure angles $\eta$ around the axis of the cone; these angles are related through ${\dot \eta} = {\dot \varphi}/\cos
\chi = L/(m_0r^2 \cos \chi)$ so $\eta = \varphi \sec \chi = \varphi
|{\bf j}|/L$.
In these terms the precession per radial period is $$\Delta \eta = 2 \pi (|{\bf j}|/L-1). \eqno (1.21)$$ Newton in his proposition on revolving orbits showed that the addition of an inverse cube force led to an orbit of exactly the same shape but traced relative to axes that rotate at a rate proportional to $\dot \phi$ in the original orbit. It is natural to ask whether such an additional force can stop the precession around the cone of an orbit in the monopolar problem and so yield an orbit that closes on itself in fixed axes. Wonderfully a simple change in $V(r)$ does this not just for one orbit but for all orbits at once. We thus obtain a new superintegrable system in which all bound orbits close. By analogy with Hamilton’s derivation of his eccentricity vector (Hamilton 1847) we take the cross product of the equation of motion (1.1) with ${\bf
j} = {\bf L} + Q_* {\hat {\bf r}}$. On the right hand side two terms are zero and the remaining two are multiples of $d{\hat {\bf r}}/dt$ cf. (1.12) so we find $$m_0 {\bf j} \times d^2 {\bf r}/dt^2 = -(m_0 r^2 V' + Q_*^2 r^{-1})
d{\hat {\bf r}}/dt. \eqno (1.22)$$ This will integrate vectorially if the bracket is constant. Calling it $GM m_0^2$ we find the potential must be of the form $$V/m_0 = -GM r^{-1} + {1 \over 2} {Q^2_* \over m^2_0} r^{-2}. \eqno (1.23)$$ Evidently the required inverse cube repulsive force is proportional to the square of the monopole moment $Q$. Integrating (1.22) we have $$d{\bf r} /dt \times {\bf j} = GM m_0 ({\hat {\bf r}} + {\bf e})
\eqno (1.24)$$ where ${\bf e}$ is the vector constant of integration. Dotting (1.24) with ${\hat {\bf r}}$ we have $$\ell_*/r = ({\bf l} + {\bf e} \cdot {\hat {\bf r}}) \eqno (1.25)$$ where $\ell_* = {\bf L} \cdot {\bf j}/(GM m_0^2) = \ {\rm const}$. Equation (1.25) is the equation of a conic section of eccentricity ${\bf e}$ which defines the direction to pericentre. But we have not yet proved that the orbit lies in a plane so (1.25) actually defines a prolate spheroid, paraboloid or hyperboloid. Nick Manton, by analogy with his work on monopoles in Euclidean Taub Space (Gibbons & Manton 1986), showed us that the motion is in fact planar; for using (1.15) Eq. (1.25) becomes on multiplication by $Q_*r$ $$Q_* \ell_* = ({\bf j} + Q_* {\bf e}) \cdot {\bf r}$$ which demonstrates that the orbit lies on a plane whose normal is ${\bf j} + Q_* {\bf e}$. As ${\bf r}$ also lies on a cone this provides another proof that the motion lies along a conic section.
Notice that the vector integral ${\bf e}$ in (1.24) together with the integral ${\bf j}$ appears to provide six integrals of the motion. However they are not all independent because $- {\bf e}\cdot {\bf j} =
{\hat {\bf r}}\cdot {\bf j} = Q_*$. So they provide 5 independent integrals. Thus we have a new superintegrable dynamical system in which the bound orbits exactly close (cf. Evans 1990, 91).
It was the beauty and simplicity of these results for monopoles in classical mechanics that led us to believe that a similar simplicity might well be discernible both in quantum mechanics and in general relativity. We were not disappointed, both had already attracted attention. Hautot 1972 discusses the separation of variables in $r,
\theta, \phi$ coordinates. The vector integral ${\bf j}$ is preferable because the generality of motion on cones is then seen. For motion in special relativity ${\bf j}$ is still conserved provided ${\bf L}$ is interpreted as $m_0{\bf r} \times d{\bf r}/d\tau =
m_0{\bf r} \times {\bf v}/ \sqrt{1-v^2/c^2}$. Goddard and Olive (1978) in their excellent review of monopoles in gauge field theories quote Poincaré (1895) for this integral in the classical case of a pure electromagnetic monopole.
Dirac’s monopole and the spectra of monopolar atoms
===================================================
Gauge transformations, Schrödinger’s equation & Dirac’s quantised monopole
--------------------------------------------------------------------------
The Lagrangian for a particle of mass $m_0$ and charge $-e$ in an electromagnetic field is $${\cal L} = {\textstyle {1 \over 2}} m_0 {\dot {\bf r}}^2 - e {\dot
{\bf r}} \cdot {\bf A}/c + e \Phi (r)$$ where $\Phi$ is the electrostatic potential and ${\bf B} = \ {\rm
Curl} \ {\bf A}$ is the magnetic field. The momentum conjugate to [**r**]{} is $${\bf p} = \partial {\cal L}/ \partial {\dot r} = m_0 {\dot {\bf r}}
- e {\bf A}/c$$ which is not a gauge invariant quantity. However the particle’s momentum $m_0 {\dot {\bf r}} = {\bf p} + e {\bf A}/c$ is gauge invariant and therefore has greater physical significance. The Hamiltonian is given by $$H = {\bf p}\cdot {\bf v} - {\cal L} = (2m_o)^{-1} ({\bf p} + e {\bf
A}/c)^2 - e \Phi . \eqno (2.1)$$ For Schrödinger’s equation we replace ${\bf p}$ by $-i \hbar {\bf
\nabla}$ and solve $H\psi = E\psi$ for the wave function of a steady state. A given magnetic field ${\bf B}$ can be described by many different vector potentials ${\bf A}$ related by gauge transformations ${\bf A}' = {\bf A} + {\bf \nabla} \chi$. Each will give us a different Hamiltonian. Let us first see how the different wave functions corresponding to these are related. Define a new function $\psi '$ such that $$\psi = \, {\rm exp} [ie \chi/(\hbar c)] \psi ' \eqno (2.2)$$ then $(-i\hbar {\bf \nabla}+ e {\bf A}/c)\psi = \, {\rm exp}[ie
\chi/(\hbar c)](-i\hbar \nabla + e {\bf A}/c+e{\bf \nabla} \chi /c)
\psi '$ and the combination ${\bf A}' = {\bf A} + \nabla \chi$ has appeared. Applying the above operator twice we see that $$H \psi = \, {\rm exp} [ie \chi/(\hbar c)] H' \psi'$$ where $H'$ is $H$ with ${\bf A}$ replaced by ${\bf A}'$. It follows that Schrödinger’s equation $H \psi = E \psi$ implies $H' \psi' = E
\psi'$ so under gauge transformation $\psi$ transforms to $\psi'$ given by (2.2).
This is a perfectly good wave function whenever $\chi$ is single valued but following Aharonov and Böhm (1959) we now consider the wave function of a particle outside a small impenetrable cylinder $R=a$. If we take ${\bf \nabla} \chi = {\bf \nabla}(F\phi /2 \pi), R\geq a$ where $\phi$ is the azimuth this corresponds to the same magnetic field outside the cylinder but a different magnetic flux within it because $$\int{\bf B}' \cdot d {\bf S} = \oint {\bf A}' \cdot d
\mbox{\boldmath $\ell$} = \oint ({\bf A}+\nabla \chi) \cdot d
\mbox{\boldmath $\ell$} = \int {\bf B} \cdot d{\bf S}+F$$ which identifies the constant $F$ as the extra flux threading the cylinder. If we adopt our transformation of wave function for a gauge transformation we get the phase factor $${\rm exp} [-ie F\phi/(hc)] \eqno (2.3)$$ which is only single valued when the flux takes the special values $$F=N(hc/e) \eqno (2.4)$$ where $N$ is an integer (positive, negative or zero). (This $N$ is not the force coefficient of Section I). Thus while we get the correct wave function for those particular values of $F$, we need to solve the problem anew with the correct boundary condition that $\psi'$ must be periodic in $\phi$ whenever $F$ is not an integer multiple of the flux quantum $hc/e$. Indeed when it is not, there is interference between the two parts of a beam of electrons that pass on either side of such a cylinder just because their phases differ by $e \oint \Delta {\bf A} \cdot d \mbox{\boldmath
$\ell$}/(hc) = eF/(hc)$. It was just this phase shift that was observed in the experiments demonstrating the Aharanov Böhm effect of the magnetic flux even when the electron beams were untouched by the magnetic field. There is an intimate connection of this result with Dirac’s (1931) earlier quantum of magnetic monopole from which one flux unit (2.4) emanates.. This comes about because in the presence of a monopole $\oint {\bf A} \cdot d \mbox{\boldmath $\ell$}$ is itself multivalued.
Consider the integral $\oint {\bf A} \cdot d \mbox{\boldmath $\ell$}$ around a small loop; this is clearly the flux of ${\bf B}$ through the loop but such a flux is ambiguous in the presence of a monopole since it depends on whether the surface spanning the loop is chosen to pass above or below the monopole i.e., $S_1$ or $S_2$ in Fig. 4. The difference between these two estimates is just $\int_{S_1 - S_2}{\bf
B} \cdot d{\bf S} = 4 \pi Q$ by Gauss’s theorem. Inserting this $\Delta \int {\bf A} \cdot d\mbox{\boldmath $\ell$}$ in place of $F$ in (2.3) we see that the wave function will only have an unambiguous phase provided $$4 \pi Q=N(hc/e) \eqno (2.5)$$ i.e., provided that the monopole strength is quantized in Dirac units of ${1 \over 2} \hbar c/e \approx {137e \over 2}$.
The quantum of magnetic flux (2.4) is inversely proportional to the charge. Quanta of half this size are observed in the Josephson effect in superconductivity where the effect is due to paired electrons. There is some evidence for the larger unit (2.4) in ordinary conductors at low temperatures (Umbach [*et al*]{}. 1986).
Returning to Schrödinger’s equation (2.1) and using the vector potential \[cf. under (1.19)\] $${\bf A} = -Q(1+\cos \theta)\nabla \phi \eqno (2.6)$$ we have the correct magnetic field for a monopole of strength $Q$ but we notice that ${\bf A}$ is singular along the line $\theta = 0$ although it is regular along $\theta = \pi$. Near the singular line ${\bf A}
\rightarrow -2Q \nabla \phi$ which is the vector potential of a tube carrying a flux $4 \pi Q$ downwards, thus formula (2.6) represents the vector potential of a magnetic monopole fed its flux by the singular half line $\theta = 0$. This half line gives an unobservable Aharanov-Böhm effect provided $4\pi Q = Nhc/e$ that is provided the monopole is a multiple of the Dirac (1931, 48) unit. Extra interest in his monopole comes from his argument that it can also be read as a reason for charge quantization, because, if $Q$ is the least monopole, then $e$ must be a multiple of $\hbar c/Q$; thus in his picture, charge quantization and monopole quantization spring from the same source. It is of interest that ${\bf A}$ in (2.6) is single valued. It has avoided the multi-valuedness alluded to above by having the singular string at $\theta = 0$ down to the monopole. This plays the role of the cut in multivalued functions in the complex plane Wat & Yang (1976). An interesting historical remark is that Schrödinger in 1922 saw that quantum conditions in the old quantum theory led to $\Gamma \equiv {\textstyle {e \over c}}[\oint \Phi dt -{\bf A} \cdot
d{\bf x}] = nh$ while Weyl’s gauge theory led him to consider exp $(-\Gamma/ \gamma)$ with $\gamma$ as yet unspecified. He realized that the identification $\gamma = -i\hbar$ would lead naturally to such quantum numbers and after de Bröglie (1925), he built on this idea to invent his wave mechanics in 1926. (See Yang 1987).
Solution of Schrödinger’s Equation
----------------------------------
Written in spherical polar coordinates Schrödinger’s equation is $$\begin{aligned}
-{\hbar ^2\over2 m_0 r^2} \left\{ {\partial \over\partial r} \left(r^2
{\partial \psi \over\partial r} \right) + {\partial\over\partial\mu}
\left[(1-\mu^2) {\partial\psi\over\partial\mu} \right] + \right. \\
\mbox{}+ \left. {1\over 1-\mu^2}
\left[{\partial^2\psi\over\partial\phi^2} - iN (\mu+1)
{\partial\psi\over\partial\phi} - {\textstyle {1\over 4}}N^2 (\mu +
1)^2 \psi \right] \right \} - \end{aligned}$$$$- e\, \Phi\, \psi = E \psi
\eqno (2.7)$$ here $\mu$ has been written for $\cos \theta$ and $N$ is the number of Dirac monopoles on the nucleus. $\phi$ only occurs as $\partial / \partial\phi$ in the above equation so we may take one Fourier component with $\psi \propto \ {\rm exp}\, (im \phi)$ and $m$ an integer positive, negative or zero in order that $\psi$ be single valued. On multiplication by $-2 m_0 r^2/(\hbar ^2 \psi)$ we then find the separated equation $$\displaylines{{1 \over \psi} {\partial \over \partial \mu} \left[(1-
\mu^2) {\partial \psi \over \partial \mu} \right] - {[m-N {\textstyle
{1 \over 2}} (\mu +1)]^2 \over 1-\mu ^2} = -C = \cr \hfill{} = -{1
\over \psi} {\partial \over \partial r} \left(r^2 {\partial \psi \over
\partial r}\right) - {r^2 2m_0 \over
\hbar^2}(E+e\Phi). \hfill{(2.8)}\cr}$$ Writing $\psi=\psi_r(r)
\psi_\mu (\mu)$ the left hand side is a function of $\mu$ alone and the right hand side is a function of $r$ alone so both must be a constant which we call $-C$. The resultant equation for $\psi_\mu$ has regular singular points at $\mu=\pm 1$. The indicial equations for the series solutions about $\mu =\pm 1$ have regular solutions behaving as $(1-\mu)^{{\textstyle {1 \over 2}}|m-N|}$ and $(1+\mu)^{{\textstyle {1 \over 2}}|m|}$ respectively, so we remove those factors by writing $$\psi_\mu = (1-\mu)^{{\textstyle {1 \over
2}}|m-N|} (1+\mu)^{{\textstyle {1 \over 2}}|m|} F(\mu). \eqno
(2.9)$$After some algebra the equation for $F$ takes the form $$\displaylines{(1-\mu^2)F'' + [ (|m| + 1)(1-\mu) -
(|m-N|+1)(1+\mu)]F' +\cr +{\textstyle {1 \over 2}} [2C - m(m-N) -
|m||m-N|-|m-N|-|m|] F = 0.\cr \hfill {(2.10)}\cr}$$
We now write $z = {\textstyle {1\over 2}}(1+\mu)$, so $z(1-z)=(1-\mu^2)/4$ and $dz={\textstyle {1\over2}} d \mu$ which reduces the above equation into the standard form for the hypergeometric equation i.e., $$z(1-z)d^2 F/dz^2 + [c-(a+b+1)z] \, dF/dz - abF=0 \eqno (2.11)$$ where $$c = |m| + 1 \eqno (2.12)$$ $$a+b = |m| + |m-N| +1 \eqno (2.13)$$ and $-2ab$ is the final square bracket in Eq. (2.10).
The hypergeometric function finite at $\mu = -1, z=0$ diverges like $(1-\mu)^{c-a-b}$ at $\mu =1$, that is twice as fast as the first factor in (2.9) converges, so in order to get convergence the hypergeometric series must terminate. This occurs only if $a$ or $b$ is a negative integer or zero. w.l.g. taking $b=-K$ we find that $F$ reduces to a Jacobi polynomial $P^{\alpha \beta}_K (\mu)$ so that $\psi_\mu$ takes the form $$\psi_\mu = C_{kmn}(1-\mu)^{1/2 |m-N|}(1+\mu)^{1/2 |m|}
P^{|m-N|,|m|}_K \eqno (2.14)$$ here $\int^{+1}_{-1} \psi^2_\mu d\mu = 1$ and $C_{kmn}$ is the normalization $$\left[{(2K+|m-N|+|m|+1) K! (K+|m-N|+|m|)! \over
2^{|m-N|+|m|+1} (K+|m-N|)! (K+|m|)!} \right]^{{\textstyle{1\over2}}}.$$ The condition that $b=-K$ gives $$-2ab = 2K(|m| +|m-N| +1+K)$$ and hence (noticing that $K=0$ leaves (2.14) finite) we have $$\displaylines{C=K(K+1)+K(|m| +|m-N|) + \cr \hfill{}
+{\textstyle {1\over
2}}\left[m(m-N)+|m||m-N|+|m-N|+|m|\, \right]. \hfill{(2.15)}\cr}$$ If we write $j=K+ {\textstyle {1 \over 2}}(|m| +|m-N|)$, then we notice that $j$ is a positive half-integer and $j\geq {\textstyle {1
\over 2}}(|m| + |m-N|)$ $$C = j(j+1) - N^2/4. \eqno (2.16)$$ Thus $C$ and $j$ are only integers when $N$ is an [*even*]{} integer. When $N$ is odd $C$ and $j$ are an integer $\pm {\textstyle {1 \over
2}}$. For given $j$ and $N \geq 0, \ \ \ m- {\textstyle {N \over 2}}$ takes the $2 j+1$ values from $-j$ to $+j$ in steps of 1. For $N=1$ the ground state has $j={\textstyle {1\over 2}}$ and $C= {\textstyle
{1\over 2}}$ rather than the values $0$ familiar from the normal hydrogen atom. The $j={\textstyle {1\over 2}}$ states with $m=1$ and $m=0$ are degenerate, see Fig. 5.
With the value (2.16) for $C$ we now turn to the radial equation for $\psi_r$, (2.8). Here the treatment is very close to the classical case clearly laid out by Pauling and Wilson (1935). We take $\Phi =
Ze/r, Ze$ being the nuclear charge, and $E$ negative. We write $$\alpha^2 = -2m_0 E/\hbar ^2 \eqno (2.17)$$ $$\zeta = m_0 Z e^2 h^{-2} \alpha^{-1} \eqno (2.18)$$ and use a normalized radius $\tilde {r} = 2\alpha r$. As all the radii in the rest of this section are so normalized we shall forget the and take it as read. Eq. (2.8) now takes the form $${1 \over r^2} {d \over dr} \left (r^2 {d\psi_r \over dr}\right ) +
\left (-{1\over 4} - {C \over r^2} + {\zeta \over r} \right ) \psi_r =
0. \eqno (2.19)$$ The asymptotic form of this equation for large $r$ is $\psi_r'' =
\psi_r/4$ so $\psi_r \rightarrow \, {\rm exp} (r/2)$ or ${\rm exp} -
(r/2)$. Of these only the second is acceptable so we write $$\psi_r =
\, {\rm exp} (-r/2) r^s f(r) \eqno (2.20)$$ where $f(r)$ may be expanded in series about the origin in the form $$\sum^\infty_{p=0}\,
a_p\, r^p$$ and $s$ is chosen so that $a_0 \not=0$. The indicial equation found by substitution of the series (2.20) into (2.19) is $$[s(s+1) -C]a_0 =0$$ but by hypothesis $a_0 \not=0$ so using (2.16) $s$ is given by $$\left (s+{\textstyle {1 \over 2}}\right )^2 = {\textstyle {1 \over
4}}+s \left (s+1 \right)={\textstyle {1 \over 4}} + C = \left
(j+{\textstyle {1 \over 2}}\right )^2 - {\textstyle {1\over 4}} N^2.
\eqno (2.21)$$ The recurrence relation for general $p$ is then $$p(p+2s+1)a_p = (s+p-\zeta)a_{p-1}$$ and the asymptotic form for large $p$ is $a_p \rightarrow a_{p-1}/p$ which shows that $f \rightarrow e^r$. In that case $\psi _r$ would diverge at large $r$. This is unacceptable so the series must terminate. Thus there must be a positive integer $p=n'+1$ such that $$\zeta = p+s =n'+s+1 \eqno (2.22)$$ with $s$ given by (2.21). Returning to (2.18) and (2.17) this gives the eigenvalues for the energy in the form $$E = -{m_0 Z^2e^4 \over 2\hbar ^2} {1 \over (n'+s+1)^2} = -{m_0Z^2e^4
\over 2 \hbar ^2} {1 \over (n+ \Delta)^2} \eqno (2.23)$$ where $n=n'+J+1$ where $J$ takes values 0, 1, 2 $\ldots$ replaces the usual $\mbox{\boldmath $\ell$}$ and $$J = j - {\textstyle {1 \over 2}}|N| \geq 0$$ $N$ is the number of Dirac monopoles on the nucleus and $$\Delta = \sqrt{(J+{\textstyle {1\over2}}) (J+{\textstyle {1\over2}}
+ |N|)} - \left(J+{\textstyle {1\over2}} \right) \ \ \ \geq 0.$$ Notice that $\Delta$ depends on $J$ as well as $|N|$ and is only zero when $N = 0$. For large $J/|N|$, $$\Delta \rightarrow {\textstyle {1 \over 2}}|N|
\left[ 1- {1\over 4} {|N| \over J+{\textstyle {1 \over 2}}} + {1 \over
8} \left( {|N| \over J+ {\textstyle {1 \over 2}}} \right) ^2 - \cdots
\right],$$ while for the ground state $J = 0$ $$\Delta = {\textstyle {1 \over 2}} \left( \sqrt{2|N|+1} - 1 \right)$$ which becomes ${\textstyle {1 \over 2}} \left( \sqrt{3} -1 \right)$ for $N=1$. So $\Delta$ is [*not*]{} small. For a spinless electron the degeneracy of a state of given $J$ and $n$ is $2j+1 = 2J+1+|N|$ with $m-{\textstyle {1 \over 2}} |N|$ taking values from $-j$ to $+j$. Notice that the ground state $J=0,n=1$ is a doublet for $N = 1$ and has $j$ value ${\textstyle {1\over 2}}$ with $m = 0$ and $m=+1$ states even [*before*]{} we have allowed for further degeneracy due to electron spin. A single Dirac monopole thus gives some effects reminiscent of spin ${\textstyle {1\over 2}}$ particles (Goldhaber 1976).
The dependence of $\Delta$ upon $J$ lifts the degeneracy of the different $J$ states ($\ell$ states) that occurs in normal hydrogen. The energy levels are near to those for an atom with a true spinning electron laid out in Figs. 6, 7, 8 and Tables 1 & 2. The degeneracy would return if the extra repulsive potential ${\textstyle
{1\over 2}} Q^2/(m_0r^2c^2)$ were included. Then the $-{\textstyle {1
\over 4}}N^2$ in (2.21) would be cancelled so $s$ would become equal to $j$.
Selection Rules
---------------
The string to the monopole makes it look non-spherical but this is not truly the case as putting the string in any other direction can be achieved by a mere gauge transformation. Therefore without loss of generality we may evaluate transition moments by taking the displacement in the $z$ direction in which case we get $$\displaylines{ R_{ab} = \int \psi^*_a r \mu \psi_b d^3 r= \cr = 2\pi
\delta_{m_am_b} \int^\infty _0 \psi_{ra} \psi_{rb} r^3 dr
\int^{+1}_{-1} \psi_{\mu a} \mu \psi_{\mu b} d\mu. \cr}$$ The radial integral is that for normal hydrogen but the scales have changed since $s$ in (2.22) is no longer $\ell$ but is given instead by (2.21). We shall concentrate on the important change in selection rules given by the final integral.
Whereas for the Legendre Polynomials in normal hydrogen wave functions we have $$\mu P_\ell (\mu) = {\ell + 1 \over 2\ell +1} P_{\ell +1} + {\ell
\over 2\ell +1} P_{\ell -1}$$ so that $\int^{+1}_{-1} P_{\ell '} \mu P_\ell d\mu$ is only non-zero when $\ell' - \ell = \pm 1$, for the Jacobi polynomials in monopolar hydrogen $$\mu P_K^{\alpha \beta} = \left (a_1 P_{K+1}^{\alpha \beta} + a_2
P^{\alpha \beta}_K +a_4 P^{\alpha \beta}_{K-1} \right)/a_3$$ where $$\begin{aligned}
a_1 & = & 2(K+1)(K+\alpha + \beta +1)(2K +\alpha + \beta) \nonumber \\
a_2 & = & (2K + \alpha + \beta +1)(\alpha^2 - \beta^2) \nonumber \\
a_3 & = & (2K +\alpha +\beta)(2K + \alpha + \beta +1)(2K + \alpha + \beta
+2) \nonumber \\
a_4 & = & 2(K+\alpha) (K+\beta)(2K+\alpha +\beta +2) \nonumber\end{aligned}$$ so that $\int ^{+1}_{-1} (1-\mu)^\alpha (1+\mu)^\beta P^{\alpha
\beta}_{K'} \mu P_K^{\alpha \beta} d\mu$ will be non-zero when $K' - K
=\pm 1 \, {\rm or} \, 0$. \[The 0 term is only absent when $a_2 =0$ i.e., $\alpha \equiv |m-N| = \beta \equiv |m|$. This occurs for $N=0$ always, for $N=2$ when $m=1$, but never for $N=1$.\]
Thus there is a significant change in the selection rules for electric dipole transitions. Some might imagine that magnetic dipole transitions should be important but the magnetic monopole is on a heavy nucleus and barely responds to an oscillating magnetic field so it is still the electric dipole transitions of the electron that are important. $m$ is unchanged for a dipole along the z-axis so $\Delta
K=\pm 1$ or $0$ leads directly to $\Delta j$ and hence $\Delta J = \pm
1 \ {\rm or} \ 0$ for such transitions.
Even order of magnitude estimates show that the interaction of the electron spin’s magnetic moment with unit monopole gives not a delicate fine structure but significant changes in the eigenvalues! Thus to find the true eigenvalues the Dirac equation is a necessity! Before treating it we clear up some details. We took the form (2.2) for ${\bf A}$ corresponding to a monopole with a string along $\mu
=+1$. For $|N| \geq 2$ we could have taken two or more inwardly directed strings of flux. Are these different string configurations really different monopoles or do they all give the same eigenvalues? The effect of such a change is to add a unit flux string along the $z$ axis. It is simple to show that this is equivalent to adding one to $m$ everywhere that it occurs. Provided we do that also to $m$ in the definition of $j$ under (2.15) the final spectrum remains unchanged. What does change are the $K$ and $m$ values associated with a given $j$ value.
A second detail is the value of $m_0$ which for $N=0$ would be the reduced mass of the electron so for hydrogen it is $m_0 = m_em_p/(m_e+m_p)$.
Particle physicists expect a heavy mass for any monopole so any monopolar hydrogen will have a nucleus much heavier than the proton and $m_e$ should be substituted for $m_0$ in predicting spectra. A third detail for later reference is the energy spectrum of the relativistic Klein-Gordon equation. Here we follow Schiff’s treatment and obtain writing $\alpha_z = Ze^2/(\hbar c)$ $$E = m_0c^2 \left\{ \left[ 1+ {\alpha^2_z \over (n+\Delta_1)^2}
\right] ^{-1/2}-1 \right\} \eqno (2.25)$$ where $$\Delta_1 = \sqrt{(J+{\textstyle {1 \over 2}})(J+ {\textstyle
{1\over 2}}+|N|) - \alpha^2_z} - \left( J+{\textstyle {1 \over 2}}
\right). \eqno (2.26)$$
Angular Momentum
----------------
Returning to the classical conserved quantity cf. (1.14) we see the conserved quantity is not the particle’s angular momentum ${\bf L} =
{\bf r} \times m_0 {\bf v}$ but rather that supplemented by $eQc^{-1}
{\hat {\bf r}}$. The physics behind this supplement lies in the Poynting vector of the electromagnetic field which carries an angular momentum
$${}$$ $$\begin{aligned}
{1 \over 4\pi c} \int {\bf r}' \times \left( {\bf E} \times
{Q \over r'^2} {\hat {\bf r}}' \right) d^3r' = {Q \over 4 \pi c} \int \left(
{\bf E} \cdot {\bf \nabla} \right) {\hat {\bf r}}' d^3r' = \\ \mbox{}
= {-Q \over 4 \pi c} \int {\hat {\bf r}}' {\bf \nabla} \cdot {\bf E}
d^3 r' = + {eQ \over c} {\hat {\bf r}} = + {\textstyle {1\over 2}} N
\hbar {\hat {\bf r}}\end{aligned}$$ where ${\bf \nabla} \cdot {\bf E}
= - e4 \pi \delta^3 ({\bf r}' - {\bf r})$. The total angular momentum is thus ${\bf j} = {\bf L} + eQc^{-1} {\hat {\bf r}}$.
As we saw above (2.1) $m_0{\bf v} = {\bf p} + e {\bf A}/c$ in the presence of a magnetic field so the operator representing ${\bf j}$ is ${\bf r} \times \left( -i \hbar {\bf \nabla} + e{\bf A}/c \right) +
{\textstyle {1\over 2}} N \hbar {\hat {\bf r}}$. The commutators $\left[ -i\hbar \partial_j + e A_j/c, - i\hbar \partial_k + eA_k/c
\right] = -i \hbar e\, c^{-1} \varepsilon_{jkl} B_l = -i \hbar
eQc^{-1} \varepsilon_{jkl} x^l/r^3$ and $[-i \hbar \partial_j + eA_j/c,
x^k] = -i \hbar \partial_j^k$ enable one to derive the commutator $$[j_j,j_k] = i \hbar \varepsilon_{jkl}j_l$$ which demonstrates that ${\bf j}$ obeys the angular-momentum algebra of the rotation group. One may also demonstrate that ${\bf j}^2$ commutes with ${\bf j}$ and that $j_{\pm} = j_x \pm ij_y$ are the raising and the lowering operators for $j_z$. From which it follows by the usual argument that the eigenvalues of $j_z$ are $-j \hbar$ to $+j
\hbar$ and that the eigenvalues of ${\bf j}^2$ are $j(j+1) \hbar ^2$. But $|{\bf j}|^2 = |{\bf L}|^2 + {\textstyle {1 \over 4}} N^2 \hbar
^2$ so the eigenvalues of $|{\bf L}|^2$ are $\left[ j(j+1) -
{\textstyle {1 \over 4}} N^2 \right] \hbar ^2$. Now looking at our separation of variables expression (2.8) we see that the LHS is just $-\hbar ^{-2} |{\bf L}^2|$ by construction and hence $C = \left( j
(j+1) - {\textstyle {1\over 4}}N^2 \right)$ which agrees with (2.16) and identifies the $j$ defined there with the generalized angular momentum eigenvalue defined in this section. Note that for a single Dirac monopole and a non-spinning electron we showed (2.16) that $j$ took [*half odd integer values*]{}.
In the next section we look at the Dirac equation for a spinning electron. There the correct generalization is ${\bf j} = {\bf L} +
{\textstyle {1 \over 2}} N \hbar {\hat {\bf r}} + {\textstyle {1\over
2}} \hbar \mbox{\boldmath $\sigma$}$.
This new ${\bf j}$ obeys the angular momentum algebra of the rotation group but now its eigenvalues are $j(j+1) \hbar ^2$ with $j$ taking integer (or half integer) values $\geq {N+1 \over 2}$ depending on whether $N$ is odd or even.
Dirac Equation
--------------
The Dirac equation may be written in standard notation $$H\psi =
\left[ -c \mbox{\boldmath $\alpha$} \cdot ({\bf p} + e{\bf A}/c) -
\beta m_0c^2 + V \right] \psi = E\psi$$ with the newly defined ${\bf
j}$ we find ${d{\bf j} \over dt} = [{\bf j}, H] = 0$ so that each component of this generalized ${\bf j}$ commutes with the Hamiltonian always provided that ${\bf A}$ is a vector potential for the monopole. Following Schiff’s treatment (1955) we define $p_r = r^{-1} ({\bf r}
\cdot {\bf p} - i\hbar)$ and $\alpha r = r^{-1}(\mbox{\boldmath
$\alpha$} \cdot {\bf r})$ and $\hbar {\bf k} = \beta \sigma' \cdot
\left( {\bf r} \times \left( {\bf p} + {e {\bf A} \over c} \right) +
\hbar \right)$. No ${\bf A}$ term is needed in $p_r$ since ${\bf r}
\cdot {\bf A} = 0$ for our monopole.
The Hamiltonian is now written $$H = -c \alpha_r p_r -i {\hbar c \over r} \alpha_r \beta k - \beta
m_0c^2 + V$$ and as before $\alpha_r , \beta$ and $p_r$ all commute with ${\bf k}$. The eigenvalues of ${\bf k}$ follow by squaring the definition $$\hbar ^2 k^2 = (\mbox{\boldmath $ \sigma$}' \cdot {\bf L})^2 + 2
\hbar \mbox{\boldmath $\sigma$}' \cdot {\bf L} + h^2 = L^2 +
{\textstyle {1 \over 4}} \hbar ^2.$$ In the last section we showed that $L^2$ has eigenvalues $j (j+1) \hbar ^2 - {N^2 \over 4} \hbar ^2$ where $j$ was an integer ($N$ odd) or half odd integer ($N$ even) so $k^2$ has eigenvalues $\left( j+{\textstyle {1 \over 2}} \right)^2 -
{\textstyle {1 \over 4}} N^2$. Save for this change of $k$ the usual separation of the Dirac equation goes through unscathed and following Schiff one obtains the energy levels $$E = m_0c^2 \left\{ \left[ 1+ {\alpha^2_z \over (s + n')^2}
\right]^{-1/2} -1 \right\}$$ where $s = \left( k^2 - \alpha^2_z \right)^{1/2}$ and $\alpha_z =
Ze^2/(\hbar c)$.
$n'$ is the radial quantum number. Inserting our eigenvalues $k^2 =
\left(j + {\textstyle {1\over 2}} \right)^2 - {\textstyle {1\over 4}}
N^2$ with $j = J + {\textstyle {1 \over 2}} (|N|+1)$ and $J=0,1,2$, etc. we have $$E = m_0c^2 \left\{ \left[1 + {\alpha ^2_z \over (n + \Delta)^2}
\right]^{-1/2} -1 \right\}$$ where $n = n'+J+1$ and $$\Delta = \sqrt{(J+1)(J+1+|N|) - \alpha^2_z} - (J+1).$$ These energy levels were first derived by Hautot (1972, 73) generalizing Harish-Chandra’s (1968) separation of the Dirac equation[^1]. For the scattering by monopoles see also Goldhaber (1965), Kazama & Yang (1976) and Kazama [*et al*]{}. (1977). We have drawn the bound energy levels that result Figs. 6, 7 and 8 and derived the wavelengths of the lines of “Monopolar Hydrogen” with one or two Dirac monopoles attached to the nucleus. Tables 1 & 2. Schwinger (1966) has suggested that the unit monopole should have the strength of two Dirac monopoles. With colleagues he has also calculated the motions of charged monopoles, dyons, under their mutual attraction (Schwinger [*et al*]{}. 1976). While monopoles may seem esoteric it is worthwhile looking for lines of monopolar hydrogen in the spectra of exotic astronomical objects.
Gravomagnetic Monopoles in General Relativity, NUT Space
========================================================
NUT space the general spherically symmetric gravity field
---------------------------------------------------------
Zelmanov (1956) and Landau & Lifshitz (1966) in developing their very physical approach to general relativity consider stationary space-times and put the metric in the form $$ds^2 = e^{-2 \nu} (dx^0 - A_\alpha dx^\alpha)^2 - \gamma_{\alpha,
\beta} dx^\alpha dx^\beta \eqno (3.1)$$ where $\nu \geq 0,A_\alpha$ and $\gamma_{\alpha \beta}$ are independent of $x^0 = ct$. (Our $\nu$ is $-{\textstyle {1 \over 2}} \nu$ of Landau & Lifshitz).
However this form is not unique since a transformation of time zero $x'^0 = x^0 + \chi (x^\alpha)$ leads to $$ds^2 = e^{-2 \nu} (dx'^0 - A'_\alpha dx^\alpha)^2 - \gamma_{\alpha
\beta} dx^\alpha dx^\beta$$ where $A'_\alpha = A_\alpha + \nabla_\alpha \chi$ so under such a change ${\bf A}$ undergoes a gauge transformation. Landau & Lifshitz also show that $\gamma_{\alpha \beta}$ can be regarded as a metric of space i.e., the quotient space $V^4/L^1$ (Geroch 1970) – as opposed to space-time. They show that test bodies following geodesics of space-time depart from the geodesics of space as if acted on by gravitational forces which in our notation take the form $${\bf f} = {m_0 \over \sqrt{1-v^2/c^2}} \left[{\bf E}_g + {{\bf v}
\over c} \times e^{- \nu} {\bf B}_g \right] \eqno (3.2)$$ where the gravitational field $${\bf E}_g = c^2 {\bf \nabla}\nu \eqno
(3.3)$$ and $${\bf B}_g = c^2 \ {\rm Curl} \ {\bf A}. \eqno (3.4)$$ The conserved energy of the particle in motion is $$\varepsilon = m_0c^2e^{-\nu} \left( 1-v^2/c^2 \right)^{-1/2} \eqno (3.5)$$ $e^{-\nu} <1$ is the redshift factor by which energy is degraded.
Rewriting Landau & Lifshitz’s form of Einstein’s equations (§95 problem) we find $${\rm div} \ {\bf B}_g = 0 \eqno (3.6)$$ $${\rm Curl} \ {\bf E}_g = 0 \eqno (3.7)$$ $$\displaylines{{\rm div} \ {\bf E}_g = -c^{-2} \left[ 4\pi G {(\rho
c^2+3p) + {v^2 \over c^2} (\rho c^2-p) \over 1-v^2/c^2}\right. - \cr
\hfill{} \left. -{\textstyle {1 \over 2}} e^{-2 \nu} {\bf B}_g^2 -
{\bf E}_g^2 \Biggr] \right. \hfill{(3.8)} \cr}$$ where $\rho$ is the energy density in the rest frame of the fluid, $3p$ is the trace of its pressure tensor and $v$ its velocity defined locally by local time synchronized along the fluid’s motion. For non-relativistic velocities this equation reduces to Poisson’s equation with the primary term on the right being $4 \pi G\rho$. The remaining term has the form of a negative energy density contributed by the gravity fields. The next equation takes the form $${\rm Curl} \ (e^{-\nu} {\bf B}_g) = -c^{-3} [16 \pi G {\bf j}_g - 2c
{\bf E}_g \times e^{- \nu} {\bf B}_g]. \eqno (3.9)$$ Notice that $e^{- \nu} {\bf B}_g$ occurs also in that combination in the expression for the force. It is attractive to regard the final term as an energy current corresponding to a Poynting vector flux of gravitational field energy. ${\bf j}_g$ the matter energy current is given by $${\bf j}_g = {\rho c^2 + p \over 1 - v^2/c^2} \ {\bf v}.$$ The final Landau & Lifshitz equation for the 3 stress tensor is $$\displaylines{P^{\alpha \beta} - E_g^{\alpha; \beta} = \left(
T^{\alpha \beta} + {\textstyle {1 \over 2}} {\dot T} \gamma^{\alpha
\beta} \right) + \cr \hfill{}+e^{-2 \nu} (B^\alpha _g B^\beta _g -
B^2_g \gamma ^{\alpha \beta}) + E^\alpha _g E^\beta _g,
\hfill{(3.10)}\cr}$$ $P^{\alpha \beta}$ is the 3 dimensional Ricci Tensor constructed from the metric $\gamma^{\alpha \beta}$. Those familiar with the Maxwell stresses of magnetic and electric fields in say magnetohydrodynamics will find some interest in the field terms on the right. The matter terms may be rewritten as physical quantities for an isotropic fluid in motion $$T^{\alpha \beta} + {\textstyle {1 \over 2}} T \gamma^{\alpha \beta}
= {8 \pi G \over c^4} \left[ {(p + \rho c^2) v^\alpha v^\beta \over c^2-v^2}
+ {\textstyle {1 \over 2}} (\rho c^2 - p) \gamma^{\alpha \beta}
\right].$$ It should be stressed that all these equations hold good even when space-time is strongly curved. Unlike some treatments they are not restricted to nearly flat-space but it is assumed that the space-time is stationary.
To find the general spherically symmetric solution for empty space we take $dl^2 = \gamma _{\alpha \beta} dx^ \alpha dx^ \beta = e^{2
\lambda}dr^2 + r^2 d{\hat {\bf r}}^2$ where ${\hat {\bf r}}$ is the unit Cartesian vector $(\sin \theta \cos \phi$, $\sin \theta \sin
\phi, \ \cos \theta)$. Then $d {\hat {\bf r}}^2 = d \theta^2 + \sin
^2 \theta \, d\phi ^2$ but the advantage of the vector notation is that no axis for $\theta,\phi$ need be taken. In spherical symmetry ${\bf B}_g$ must be radial and divergenceless so Gauss’s theorem gives $|{\bf B}_g|r^2 = Q$ = const which is the field of a gravomagnetic monopole $$B_g^r = -Qe^{- \lambda}/r^2. \eqno (3.11)$$ Reinserting ${\bf E}_g = c^2 \nabla\nu$ into (3.8) we have $$R_{00} = -\nu'' + \nu'^2 - 2 \nu'/r + \lambda'\nu' + {\textstyle {1
\over 2}}e^{2(\lambda - \nu)} Q^2(cr)^{-4} = 0. \eqno (3.12)$$ To form $P^\alpha \! _\beta$ we need the 3 dimensional Christofel symbols $$\lambda ^\sigma _{\mu \nu} = {\textstyle {1 \over 2}} \gamma^{\sigma
\eta} (\gamma_{\eta \mu, \nu} + \gamma _{\eta \nu, \mu} - \gamma _{\mu
\nu, \eta}) \eqno (3.13)$$ which are $${}$$
$\begin{array}{lll} \lambda^\sigma_{\mu \sigma} = {\textstyle {1 \over
2 \gamma}} \, \gamma,_\mu & & \ \ \ \ \lambda^\sigma_{\phi \phi} =
-{\textstyle {1 \over 2}} \gamma^{\sigma \eta} \gamma_{\phi \phi,
\eta} \\ \lambda^\sigma _{rr} \, = \left\{{\ \ \ \ 0 \ \ \ \ \, \ \
\sigma \not= r \atop {\textstyle {1 \over 2}} \gamma^{rr}\gamma_{rr,r}
\ \sigma = r} \right. & & \ \ \ \ \lambda^\sigma_{\theta \theta} \, =
\left\{{0 \atop -{\textstyle {1 \over 2}} \gamma^{rr}\gamma_{\theta
\theta,r}} {\sigma \not= r \atop \sigma = r}\right. \\
\lambda^\sigma_{\phi \sigma} = 0 & & \ \ \ \ \lambda^\sigma_{r \tau}
\, = {\textstyle {1 \over 2}} \gamma^{\sigma \eta} \gamma_{\eta
\tau,r}\end{array}$ $$\eqno (3.14)$$
(3.6), (3.7) and (3.9) are identically satisfied. The surviving equations of (3.10) are $$R^{rr} = -\nu'' + \nu'^2 + \lambda'\nu - 2\lambda' /r = 0 \eqno (3.15)$$ and $$\displaylines{R^{\theta \theta} = R^{\phi \phi} = \lambda 'e^{-2
\lambda} - {e^{-2 \lambda} \over r} + {1 \over r} + {1 \over 2}e^{-2
\nu} Q^2c^{-4}r^{-3} + \cr \hfill{} + \nu'e^{-2 \lambda} = 0. \hfill
{(3.16)} \cr}$$ Equations (3.12), (3.15) and (3.16) must be solved for $\nu$ and $\lambda$. Eliminating $\nu''$ from (3.12) and (3.15) we find $$2(\lambda' - \nu') + {\textstyle {1 \over 2}} e^{2(\lambda - \nu)}
Q^2c^{-4}r^{-3} = 0 \eqno (3.17)$$ which integrates on division by $e^{2(\lambda - \nu)}$ giving $$e^{-2(\lambda - \nu)} = - q^2r^{-2} + C \eqno (3.18)$$ where $q = Q/2c^2$ which has the dimensions of a length. Multiplying (3.16) by $e^{2 \nu}$ and using (3.17) and (3.18) we have $$(C - q^2 r^{-2})(r^{-1} - 2\nu') = q^2r^{-3} + e^{+2 \nu}/r$$ dividing by $e^{2\nu}(C - q ^2 r^{-2})$ we obtain $$(e^{-2 \nu})' + {1 \over r} \left( {Cr^2 - 2 q^2 \over Cr^2 - q^2}
\right) e^{-2 \nu} - {r \over Cr^2 - q^2} = 0$$ which is linear in $e^{-2 \nu}$ and readily solved by integrating factor to give $$e^{-2 \nu} = {1 \over C} - {2q^2 \over Cr^2} + {2 {\overline C}
\over r^2} \sqrt{Cr^2 - q^2} \eqno (3.19)$$ where $C$ and $\overline C$ are both constants. It follows from (3.18) $$\gamma_{rr} = e^{2 \lambda} = (C -q^2 r^{-2})^{-1}e^{+2 \nu}. \eqno
(3.20)$$ To get $e^{-2 \nu}$ and $\gamma_{\alpha \beta}$ asymptotically of Schwarzschild form we need $C=1$ and $\overline C =
- \widetilde {m}$, the asymptotic mass $GM/c^2$. Thus we find $$g_{00} = e^{-2 \nu} = 1 - 2r^{-2} \left( q^2 + \widetilde
{m} \sqrt{r^2 - \ell ^2} \ \right) \eqno (3.21)$$ $$\gamma_{rr} = (1 - q^2 r^{-2}) e^{+2 \nu} \eqno (3.22)$$ which are the metric components of NUT space. Notice that when $Q =
2qc^2 = 0$ this reduces to Schwarzschild’s metric. The metric is completed by taking a vector potential $A_ \alpha$ for the gravomagnetic field ${\bf B}_g$; any one will do since they are connected by gauge transformation which merely changes the zero point of time. As we saw in Section I it is impossible to choose a spherically symmetric vector potential but this does not affect the spherical symmetry of the physics. A suitable vector potential is that given in (1.19) which gives us the metric $$\displaylines{ds^2 = e^{-2 \nu} \left( cdt-2q (1 + \cos \theta) d
\phi \right)^2 - \cr \hfill{}- (1 -q^2/r^2) e^{+2 \nu} dr^2 - r^2 d
{\hat {\bf r}}^2 \hfill{(3.23)} \cr}$$ where $e^{-2 \nu}$ is given by (3.21). This metric is more commonly written in terms of the radial variable $\tilde {r} = \sqrt{r^2 - q ^2}$ because the square roots disappear leaving an analytic expression, however we have preferred the variable that makes the surface area of the sphere $4 \pi r^2$ as in Schwarzschild space. Of course the metric (3.23) appears to have a preferred axis but this is illusory because we can switch it into any direction we like by a gauge transformation, see the discussion under (1.20). The horizon where $g_{00}$ changes sign is given by $\tilde
{r} = \widetilde {m} + \sqrt{q^2 + \widetilde {m} ^2}$ at which point $\gamma_{rr}$ changes sign also as in Schwarzschild space.
NUT-Space was discovered in Ehlers’ thesis (1957) and rediscovered by Newman, Tamburino & Unti (1963). It is closely related to Taub’s (1951) metric and their relationship has been beautifully illuminated by Misner & Taub (1969). The fact that NUT space has a gravomagnetic monopole was found by Demianski & Newman (1966), who also found a NUT version of Kerr space. See also Dowker & Roche (1967).
Orbits and Gravitational Lensing by NUT Space
---------------------------------------------
The geodesics of NUT space may be determined from $\delta \int ds=0$ using the metric in the form (3.1). When $ds^2 \not= 0$ we write $\tau$ for the proper time and when $ds^2 \not= 0$ we replace it by an affine parameter (which we also call $\tau$). Varying $\dot t = dt/d
\tau$ and using the fact that the metric is stationary we have $$e^{-2
\nu} (c \dot t - A_\alpha \dot x ^\alpha ) = \varepsilon = \ {\rm
constant}. \eqno (3.24)$$ Varying $x^\alpha$ we find $$\displaylines{\delta x^\alpha \left\{{d \over d\tau} \left[ e^{-2
\nu} (c \dot t - A_\beta \dot x ^\beta) A_\alpha + \gamma_{\alpha
\beta} \dot x ^\beta \right] \right. + \cr \hfill{}
+\left. {\textstyle {1 \over 2}} {\partial \over \partial x^\alpha}
\left[ e^ {-2 \nu} (c \dot t - A_\beta \dot x^ \beta)^2 -
\gamma_{\beta \gamma} \dot x ^\beta \dot x^\gamma \right]
\right\}. \hfill{(3.25)}\cr}$$ Using (3.24) and transferring the $\varepsilon d A_\alpha/d \tau = \varepsilon \dot x^\beta
\partial_\beta A_\alpha$ term into the second bracket we find the equation of motion in which ${\bf A}$ only occurs through $\partial_\alpha A_\beta - \partial _\beta A_\alpha = \eta_{\alpha
\beta \gamma} B^\gamma$ where $\eta_{\alpha \beta \gamma}$ is the antisymmetric tensor, $\sqrt{\gamma}$ times the alternating symbol. $$\displaylines{\delta x^\alpha \left[{d \over d\tau} (\gamma_{\alpha
\beta} \dot x ^\beta) + {\textstyle {1 \over 2}} {\partial \over
\partial x^\alpha} (e^ {-2 \nu}) \varepsilon ^2 e^{+4 \nu}\right. -
\cr \hfill - \left. {\textstyle {1 \over 2}} \gamma_{\beta \gamma,
\alpha} \dot x^ \beta \dot x^\gamma - \varepsilon \eta_{\alpha \beta
\gamma} B^ \gamma \dot x^ \beta \Bigr] \right. = 0 \hfill{(3.26)}
\cr}$$ We now write $\gamma_{\alpha \beta}$ in the form involving the unit Cartesian vector ${\hat {\bf r}}$, $$\gamma_{\alpha \beta} dx^\alpha dx^\beta = e^{2 \lambda} dr^2 + r^2
(d{\hat {\bf r}})^2.$$ $\delta {\hat {\bf r}}$ the variation of ${\hat {\bf r}}$ is an arbitrary small vector perpendicular to ${\hat {\bf r}}$. Thus making variations with $r$ fixed we deduce from (3.26) Using (3.11) for $B_g^r$ $$\delta {\hat {\bf r}} \cdot \left[ {d \over d \tau} \left( r^2 {d
{\hat {\bf r}} \over d \tau} \right) + \varepsilon {d {\hat {\bf r}}
\over d \tau} \times Q {\hat {\bf r}} \right] = 0$$ since $\delta {\hat {\bf r}}$ is an arbitrary vector perpendicular to ${\hat {\bf r}}$ we deduce that $${\hat {\bf r}} \times {d \over dt} \left( r^2 {d {\hat {\bf r}}
\over d \tau} \right) = {d {\bf L} \over d \tau} = - {d \over d \tau}
\left( \varepsilon Q {\hat {\bf r}} \right)$$ $${\bf L} + \varepsilon Q {\hat {\bf r}} = {\bf j} = \ {\rm const}.
\eqno (3.27)$$ Except for the factor $\varepsilon$ which reduces to $m_0c^2$ in the non-relativistic case we see that this is precisely the vector integral (1.14). Dotting Eq. (3.27) with ${\hat {\bf
r}}$ we find ${\bf j} \cdot {\hat {\bf r}} = \varepsilon Q$ showing that ${\hat {\bf r}}$ lies on a cone similarly ${\bf L} \cdot {\bf j}
= L^2 = {\bf j}^2 - \varepsilon ^2 Q^2 = \ {\rm const}$ so ${\bf L}$ moves around a cone. The radial equation of motion is redundant since we may use the energy and the equation $(ds/d \tau)^2 = U = 1 \ {\rm
or} \ 0$ instead. $U$ is 1 for time like geodesics and 0 for light-like ones.
This gives us $$\varepsilon ^2 e^{+2 \nu} - \dot r ^2 e^{2 \lambda} - L ^2 r^{-2} =
U. \eqno (3.28)$$ To see the geometry of the trajectory we introduce the curvilinear angle $\varphi$ of $\S$1 measured around the cone’s surface. Then $r^2 \dot \varphi = L$ so Eq. (3.1) can be integrated by quadrature $$\varphi - \varphi_0 = \int {L r^{-2} dr \over \sqrt{\varepsilon ^2
e^{-2 (\lambda - \nu)} - (U + L^2 r^{-2}) e^{-2 \lambda}}}. \eqno
(3.29)$$ In general this integral can not be performed explicitly for the $\lambda$ and $\nu$ of NUT space even after substitution in terms of $\tilde {r}$ to make it more analytic. We therefore turn to the $r^2 \gg q^2 + \widetilde {m} ^2$ limit well away from the event horizon. This is the important case in all gravitational lenses observed to date. In that limit the $q^2/r^2$ term in the effective potential is attractive and therefore of the wrong sign to give the non-precessing orbits of Section I. The precession around the cones is faster than in the classical Kepler + monopole problem by a factor 3/2. To the first order in $\widetilde {m}/b$ where $b$ is the impact parameter, we find a bending angle measured like $\varphi$ of $\Delta
\varphi = 4 \widetilde {m}/b$ just as for the Schwarzschild metric; however the difference is that this angle is measured around a cone not in a plane. Again to first order we can find the effect of the gravomagnetic field by integrating the momentum transfer along the unperturbed straight line path. This gives an out-of-plane bending of $4q/b$; a result that is confirmed by the full NUT space calculation. Nouri-Zonoz, M. & Lynden-Bell, D. (1997). Thus the major effect of the gravomagnetic monopole $Q$ is to twist the rays that pass it. While the bending angle is proportional to $b^{-1}$, the effect is exaggerated when looking down the line toward the NUT lens by the factor $D_L/b$, so the twist around the lens is $4qD_L/b^2$. Here $D_L$ is the distance from the observer to the lens. The same exaggeration factor occurs for the normal gravitational bending so for a source at infinity and an image at $(b, \theta)$ in the plane of the sky at the lens’s distance, the apparent position of the source is $$(b_s, \theta _s) = \left( b \left( 1 - {4
\widetilde {m} D_L \over b^2} + {8q^2D^2_L \over b^4} \right) , \theta
- {4qD_L \over b^2} \right).$$ This expression defines a map from image to source. From this map one can work out both the shear and the magnification of a NUT lens in the large impact parameter régime. The magnification of area and thus luminosity is $$db^2/db^2_s = [1-16b^{-4}D^2_L (m^2 + q^2)]^{-1}. \eqno (3.30)$$ A small circular source will be imaged into an ellipse of axial ratio $${b^2 +4 \ D_L (m + \sqrt{m^2 + q^2}) \over b^2 + 4 \ D_L (m-
\sqrt{m^2 + q^2})}$$ with the short axis of the ellipse inclined to the radius at the angle (see Fig. 9) $$\tan ^{-1} \left( {q \over m+ \sqrt{m^2 + q^2}}
\right).$$ This is $45^\circ$ for $q \gg m$ and $13^\circ$ for $Q = 2qc^2 =
mc^2$. This spiral conformation of the images about a NUT lens is very characteristic. It is not displayed in normal gravitational imaging and the gravomagnetic lens due to a rotating object seen pole on does not show it because the twist of the ray as it approaches such a lens is cancelled by the opposite twist as it recedes. Thus the discovery of a spiral shear field about a lens would indicate the presence of a gravomagnetic monopole. Such effects should be looked for by those studying gravitational lenses. The expectation must be small but the reward might be an amazing discovery.
Quantization of Gravomagnetic Monopoles and their Classical Physics
-------------------------------------------------------------------
By analogy with Dirac’s argument for the quantization of magnetic monopoles and charges, Dowker & Roche (1967), Dowker (1974), Hawking (1979) and Zee (1985) have suggested quantization of gravomagnetic monopoles and energy. Corresponding to Dirac’s $Q_me= {\textstyle {1
\over 2}} N \hbar c$ for magnetic $Q_m$, they have $Qm_0 = {\textstyle
{1 \over 2}} N \hbar c$ for gravomagnetic monopole $Q$. This implies that both $Q$ and mass $m_0$ are quantized in conjugate units $Q_1$ and $m_1$ obeying $Q_1 m_1 = {\textstyle {1 \over 2}} \hbar c$. Whereas such ideas are naturally attractive they do not naturally lead to a self-consistent relativistic theory. For instance, looking at the Klein-Gordon equation in NUT space and separating variables with $\psi \propto e^{i(m \phi + \omega t)}$, one finds an eigenvalue equation for $\omega$. The Dirac monopole quantization condition, $Q(\hbar \omega /c^2) = {\textstyle {1 \over 2}} N \hbar c$ with $N$ an integer, shows us that the only possible eigenvalues $\omega$ are integer multiples of ${\textstyle {1 \over 2}} c^3/Q$ and the corresponding energy $\hbar \omega$ is the total energy of the ‘orbit’ including rest mass. However this condition conflicts with the energies of the bound states[^2] which are not integer multiples of any unit. Mueller & Perry (1986). Thus if such ideas are viable at all a more radical change in basic theory is needed. In $+++-$ NUT space it does not appear to be possible to build a consistent quantum theory like Dirac’s magnetic monopole theory. This is what Ross (1983) concluded and is related to Misner’s (1963) finding that NUT space contains closed timelike lines, with time being periodic every $8 \pi q/c$. For a discussion of the energy levels in $++++$ Taub-NUT space, the reader is referred to the papers by Gibbons & Manton (1986). This space was shown to be relevant to the interactions of monopoles by Atiyah & Hitchin (1985). §If magnetic monopoles exist, Maxwell’s equations must be changed to include ${\rm div} \ {\bf B} = 4 \pi \rho_ m$, ${\rm Curl} \ {\bf E} + {1 \over c} {\partial {\bf B} \over \partial t} = 4
\pi {\bf j}_m$, where $\rho_m$ is the monopole density and ${\bf j} m$ is the monopole current density. Such modified Maxwell equations do not come with a vector potential. It is natural to ask how general relativity must be modified to allow for gravomagnetic monopole densities and currents. While this is not so obvious we conjecture the generalization will be to spaces with unsymmetric affine connections which have non-zero torsion. It would be interesting to demonstrate this conjecture as it could introduce a greater degree of physical understanding of those spaces.
Observability
=============
Following Kibble’s (1980) suggestion that magnetic monopoles would be a natural consequence of the Big Bang, they have long been sought.
We have concentrated on the spectra of monopolar atoms and the lensing properties of gravomagnetic monopoles since these are ways in which, at least in principle, monopoles might be discovered observationally. Spectroscopically one may argue that the best place to look is in the spectra of supernovae, quasars or active galactic nuclei where the basic Ly $\alpha$ lines of Table I or II might be seen as very weak absorption lines in very high resolution spectra. Quasars have the advantage that these lines will be shifted into the visible. We have looked at IUE ultra-violet spectra of Supernova 1987A and seen no lines at the wavelengths 2774.62 or 2733.78. More supernovae and stacked high resolution spectra of quasars should be pursued. Although in regions of observed magnetic fields the limits obtained spectroscopically will fall far short of the Parker (1970) bound. While the nature of the dark matter that constitutes most mass in the universe remains unknown, such esoteric possibilities are worth pursuit.
Searches on Earth have produced one unrepeatable event and a monopolar observatory under the Grand Sasso that has so far found no monopoles in Cosmic Rays. There has been a speculative suggestion, Kephart & Weiler (1996), that the leveling up on the numbers of cosmic rays at the highest energies might be due to monopoles but there is no confirmation of that idea. To date the best limit on the numbers of monopoles in interstellar space comes from the Parker (1970-71) bound. This arises from the idea that too many magnetic monopoles would ‘short out’ the galactic magnetic fields that are observed. A good general discussion of such limits may be found in the book of Kolb & Turner (1991). For more recent work on monopoles in field theory see reviews by Olive (1996, 97), and the papers by Sen (1994) and by Seiberg & Witten (1994). More details of the fundamental work on monopoles in field theory by ‘t Hooft (1974) and by Polyakov (1974) can be found in the review by Goddard & Olive.
We thank P. Goddard, D.G. Walmsley and D. Crothers and G.W. Gibbons for discussions. M. Mathioudakis brought us the IUE spectrum of SN1987a and F. McKenna looked for coincidences with spectral lines in RR Telescopii and pointed out that there is a HeII line at $\lambda$2733.28A which almost coincides with the predicted $\lambda$2733.78 line of monopolar hydrogen. We thank the Director of the Armagh Observatory where this work began and the Physics Department of The Queen’s University, Belfast, for their hospitality.
Aharanov, Y. and D. Böhm, 1959, Phys. Rev. [**115**]{}, 485.
Atiyah, M.F. and N.J. Hitchin, 1985, Phys. Lett. A. [**107**]{}, 21.
Atiyah, M.F. and N.J. Hitchin, 1985, Philos. Trans. R. Soc. London A [**315**]{}, 459.
Cajori, F., 1934, [*Newton’s Principia, Motte’s Translation Revised*]{} (Univ. of California Press, Berkeley).
Chandrasekhar, S., 1995, [*Newton’s Principia for the Common Reader*]{} (Oxford).
de Bröglie, L., 1925, Ann. d. Phys. [**3**]{}, 22.
Demianski, M. and E.T. Newman, 1966, Bull. de L’Acad. Polon des Sciences Series Math. Astr. Phys. XIV, 653.
Dirac, P.A.M., 1931, Proc. R. Soc. [**133**]{}, 60.
Dirac, P.A.M., 1948, Phys. Rev. [**74**]{}, 817.
Dowker, J.S., 1974, Gen. Rel. Grav. [**5**]{}, 603.
Dowker, J.S. and J.A. Roche, 1967, Proc. Phys. Soc. [**92**]{}, 1.
Ehlers, J., 1957, Dissertation Univ. Hambourg, Hambourg.
Evans, N.W., 1990, Phys. Rev. A [**41**]{}, 5666.
Evans, N.W., 1991, J. Math. Phys. [**32**]{}, 3369.
Evans, N.W., P.T. de Zeeuw and D. Lynden-Bell, 1990, Mon. Not. R. Astron. Soc. [**244**]{}, 111.
Geroch, R., 1971, J. Math. Phys. [**12**]{}, 918.
Gibbons, G.W. and N.S. Manton, 1986, Nucl. Phys. B. [**274**]{}, 183.
Goddard, P. and D.I. Olive, 1978, Rep. Prog. Phys. [**41**]{}, 1357.
Goldhaber, A.S., 1965, Phys. Rev. B. [**140**]{}, 1407.
Goldhaber, A.S., 1976, Phys. Rev. Lett. [**36**]{}, 1122.
Hamilton, W.R., 1847, in [*Proc. R. Irish Acad. III*]{} Appendix, pp. 36.
Harish-Chandra, H., 1948, Phys. Rev. [**74**]{}, 883.
Hautot, A., 1972, J. Math. Phys. [**13**]{}, 710.
Hautot, A., 1973, J. Math. Phys. [**14**]{}, 201.
Hawking, S.W., 1979, [*General Relativity an Einstein Survey*]{}, edited by S.W. Hawking and W. Israel (Cambridge University Press), p. 746.
Henon, M., 1959, Annals d’Astrophysique [**22**]{}, 126 and [**23**]{}, 474.
Hoang, L.V., L.X. Hai, L.I. Komarov and J.S. Romanova, 1992, J. Phys. A. [**25**]{}, 6461.
Kazama, Y. and C.N. Yang, 1976, Phys. Rev. D. [**15**]{}, 2300.
Kazama, Y., C.N. Yang and A.S. Goldhaber, 1977, Phys. Rev. D. [**15**]{}, 2287.
Kephart, T.J. and T.W. Weiler, 1996, Astroparticle Physics [**4**]{}, 271.
Kibble, T.W.B., 1980, Phys. Rept. [**67**]{}, 183.
Kolb, E.W. and M.S. Turner, 1991, [*The Early Universe*]{} (Addison Wesley, New York).
Landau, L.D. and E.M. Lifshitz, 1966, [*Classical Theory of Fields §89 problem 1, §95 problem*]{}.
Lynden-Bell, D., 1963, Observatory [**83**]{}, 23.
Lynden-Bell, D., 1996, Notes & Records of the Royal Society, London [**50**]{}, 253.
Misner, C.W., 1963, J. Math. Phys. [**4**]{}, 924.
Misner, C.W. and A.H. Taub, 1969, Sov. Phys. JETP [**28**]{}, 122.
Mueller, M. and M.J. Perry, 1986, Class Quantum Grav. [**3**]{}, 65.
Newman, E.T., L. Tamburino and T. Unti, 1963, J. Math. Phys. [**4**]{}, 915.
Newton, I., 1686, [*Philosophiae Naturalis Principia Mathematicia*]{} (Royal Society, London).
Nouri-Zonoz, M. and D. Lynden-Bell, unpublished.
Olive, D.I., 1996, Nucl. Phys. B. Proc. Supp. [**45**]{}, 88.
Olive, D.I., 1997, [*Roy. Soc. Dirac Commemoration Symposium*]{}.
Parker, E.N., 1970, Astrophys. J. [**160**]{}, 383.
Parker, E.N., 1971, Astrophys. J. [**163**]{}, 225 & [**166**]{}, 295.
Pauling, L. and E.B. Wilson, 1935, [*Introduction to Quantum Mechanics*]{} (McGraw-Hill, New York), Ch. V.
Poincaré, H., 1896, C. R. Acad. Sci. Paris [**123**]{}, 530.
Polyakov, A.M., 1974, JETP Lett. [**20**]{}, 194.
Ross, D.K., 1983, J. Math. Phys. [**24**]{}, 1814.
Schiff, L.I., 1955, [*Quantum Mechanics, 2nd Edition*]{} (McGraw-Hill, New York), §44, pp. 331.
Schrödinger, E., 1922, Z. Phys. [**12**]{}, 13.
Schrödinger, E., 1926, Ann. d. Phys. [**79**]{}, 361, 489, 734.
Schwinger, J., 1966, Phys. Rev. [**144**]{}, 1087.
Schwinger, J., K.A. Milton, W.-Y. Tsai, L.L. De Raad and D.C. Clark, 1976, Ann. of Phys. [**101**]{}, 451.
Seiberg, N. and E. Witten, 1994, Nucl. Phys. B. [**431**]{}, 484.
Sen, A., 1994, Phys. Lett. B. [**329**]{}, 217.
Taub, A.H., 1951, Ann. Math. [**53**]{}, 472.
t’Hooft, G., 1974, Nucl. Phys. B. [**79**]{}, 276.
Umbach, C.P., C. Van Haesendonck, R.B. Laibowitz, S. Washburn and R.A. Webb, 1986, Phys. Rev. Lett. [**56**]{}, 386.
Villalba, V.M., 1994, Phys. Lett. [**193**]{}, 218.
Villalba, V.M., 1995, J. Math. Phys. [**36**]{}, 3332.
Wat, S. and C.N. Yang, 1976, Nucl. Phys. B. [**107**]{}, 365.
Weyl, H., 1918, [*Berlin Sitzungsberichte, 30th May*]{}, p. 465.
Whiteside, D.T., 1674-1691, [*Mathematical Papers of Isaac Newton VI 1674-1691*]{} (Cambridge University Press).
Yang, C.N., 1987, in [*Schrödinger Centenary*]{}, edited by C.W. Kilminster (Cambridge University Press).
Zee, A., 1985, Phys. Rev. Lett. [**55**]{}, 2379.
Zelmanov, A.L., 1956, Astron. Zh. USSR, 227.
[^1]: For the further generalization to the problem with an additional Böhm-Aharonov string, see Villalba (1994, 95) & Hoang [*et al*]{}. (1992).
[^2]: To get definite bound states one must impose a potential barrier so that the black hole is not reached.
|
---
author:
- 'Zhengjun Cao$^{1}$, Lihua Liu$^{2}$'
title: On the Weakness of Fully Homomorphic Encryption
---
> **Abstract**. Fully homomorphic encryption (FHE) allows anyone to perform computations on encrypted data, despite not having the secret decryption key. Since the Gentry’s work in 2009, the primitive has interested many researchers. In this paper, we stress that any computations performed on encrypted data are constrained to the encrypted domain (finite fields or rings). This restriction makes the primitive useless for most computations involving common arithmetic expressions and relational expressions. It is only applicable to the computations related to modular arithmetic. We want to reaffirm that cryptography uses modular arithmetic a lot in order to obscure and dissipate the redundancies in a plaintext message, not to perform any numerical calculations. We think it might be an overstated claim that FHE is of great importance to client-server computing or cloud computing.
>
> **Keywords.** fully homomorphic encryption; common arithmetic; modular arithmetic; encrypted domain; client-server computing
Introduction
============
Homomorphic encryption introduced by Rivest, Adleman and Dertouzos [@RAD78] in 1978, is a useful cryptographic primitive because it can translate an operation on the ciphertexts into an operation on the corresponding plaintexts. The property is useful for some applications, such as e-voting, watermarking and secret sharing schemes. For example, if an additively homomorphic encryption is used in an e-voting scheme, one can obtain an encryption of the sum of all ballots from their encryption. Consequently, it becomes possible that a single decryption will reveal the result of the election. That is, it is unnecessary to decrypt all ciphertexts one by one.
A fully homomorphic encryption (FHE) is defined as a scheme which allows anyone to perform arbitrarily computations on encrypted data, despite not having the secret decryption key. In 2009, Gentry [@G09] proposed a FHE scheme over ideal lattices, which is capable of evaluating some functions in the encrypted domain. Since then, the primitive has interested many researchers.
Related works
-------------
Homomorphic encryption schemes supporting either addition or multiplication operations (but not both) had been intensively studied, e.g., Goldwasser-Micali encryption [@GM82], ElGamal encryption [@E84], and Paillier encryption [@P99]. The Gentry encryption [@G09] is a fully homomorphic encryption scheme, which makes it possible to evaluate some functions in the encrypted domain. After that, some new FHE schemes appeared.
At Eurocrypt’10, Gentry, Halevi and Vaikuntanathan [@GHV10] proposed a FHE scheme based on the Learning With Error (LWE) problem. In 2010, van Dijk, et al. [@DGHV10] constructed a simple FHE scheme using only elementary modular arithmetic. At Crypto’11, a FHE scheme working over integers with shorter public keys and a FHE scheme based on ring-LWE were presented by Coron et al. [@C11], Brakerski and Vaikuntanathan [@BV11], separately. At FOCS’11, a FHE scheme based on standard LWE by Brakerski and Vaikuntanathan [@BV12; @BV14], and a FHE scheme using depth-3 arithmetic circuits by Gentry and Halevi [@GH11], have interested many audiences. In 2012, Brakerski, Gentry and Vaikuntanathan [@BGV12] designed a leveled FHE scheme without bootstrapping. At Eurocrypt’13, Cheon, et al. [@C13] investigated the problem of batching FHE schemes over integers. In 2013, Brakerski, Gentry and Halevi [@BGH13] discussed the problem of packing ciphertexts in LWE-based homomorphic encryption.
In 2015, Castagnos and Laguillaumie [@CL15] proposed a linearly homomorphic encryption scheme whose security relies on the hardness of the decisional Diffie-Hellman problem. Recently, Cheon and Kim [@CK15] introduced a hybrid homomorphic encryption which combines public-key encryption and somewhat homomorphic encryption in order to reduce the storage requirements for some applications.
FHE makes it possible to enable secure storage and computation on the cloud. However, current homomorphic encryption schemes are still inefficient. For example, key generation in Gentry’s FHE scheme takes from 2.5 seconds to 2.2 hours [@GHE11]. A recent implementation required 36 hours for a homomorphic evaluation of AES [@GHS12]. One of the most remarkable things about these implementations is that the computations did not involve common arithmetic expressions and relational expressions.
Our contributions
-----------------
In this paper, we want to stress that any computations performed on encrypted data are constrained to the encrypted domain (finite fields or rings). This restriction makes the primitive useless for most computations involving common arithmetic expressions, logical expressions and relational expressions. It is only applicable to the computations related to modular arithmetic. Some researchers have neglected the differences between common arithmetic and modular arithmetic, and falsely claimed that FHE enables arbitrary computations on encrypted data. We here reaffirm that cryptography uses modular arithmetic a lot in order to obscure and dissipate the redundancies in a plaintext message, not to perform any numerical calculations.
We revisit the Dijk-Gentry-Halevi-Vaikuntanathan FHE scheme [@DGHV10] and Nuida-Kurosawa FHE scheme [@NK15] under the client-server computing model. The former encrypts bit by bit. The latter works over the encrypted domain $\mathbb{Z}_Q$, where $Q$ is a prime. We find that in the Dijk-Gentry-Halevi-Vaikuntanathan scheme the server can not decide the carries by the encrypted data, and in the Nuida-Kurosawa scheme it is impossible to find an invertible transformation $\mathcal{T}$ from the real number set $\mathbb{R}$ to the field $\mathbb{Z}_Q$. Therefore, in both schemes the server can not return right values to the client even though the server is asked to help to evaluate the simple function $f(x, y)=x+y$.
In view of the limitations mentioned above, we believe it might be a false claim that FHE is of great importance to cloud computing. To the best of our knowledge, it is the first time to concretely discuss FHE schemes under the client-server computing model.
The real goal of using modular arithmetic in cryptography
=========================================================
Any calculation needs an describing expression, which consists of variables, constants and operators. There are three kinds of expressions: arithmetic expressions, logical expressions and relational expressions. Arithmetic operators include addition $(+)$, substraction $(-)$, multiplication ($*$), division $(/)$, integer-division $(\setminus)$, modulus (Mod), and so on.
Like common arithmetic, modular arithmetic is commutative, associative, and distributive. Suppose that $a, b$ are in the decrypted domain $\mathbb{Z}_p$ where $p$ is a prime, $E(\cdot)$ is a fully homomorphic encryption algorithm, and $D(\cdot)$ is the corresponding decryption algorithm. Then the following properties are obvious. $$D(E(a)+E(b))=D(E(a+b))=a+b \mod p$$ $$D(E(a)\cdot E(b))=D(E(ab))=ab \mod p.$$ Generally, $$a+b \not=(a+b \mod p), \qquad ab\not= (ab \mod p)$$ $$a<b \not\Longrightarrow E(a)<E(b), \qquad E(a)<E(b) \not\Longrightarrow a<b$$
We here want to stress that cryptography uses modular arithmetic a lot, because it can obscure the relationship between the plaintext and the ciphertext, and dissipate the redundancy of the plaintext by spreading it out over the ciphertext. It is well known that confusion and diffusion are the two basic techniques for obscuring the redundancies in a plaintext message. They could frustrate attempts to study the ciphertext looking for redundancies and statistical patterns. Practically speaking, the real goal of using modular arithmetic in cryptography is to *obscure and dissipate the redundancies in a plaintext message, not to perform any numerical calculations.*
To see this, we will have a close look at two typical FHE schemes proposed by van Dijk et al. [@DGHV10], Nuida and Kurosawa [@NK15]. The former encrypts bit by bit. The encrypted domain for the latter is $\mathbb{Z}_Q$, where $Q$ is a prime.
Analysis of Dijk-Gentry-Halevi-Vaikuntanathan FHE scheme under the client-server computing model
================================================================================================
Description of Dijk-Gentry-Halevi-Vaikuntanathan scheme
-------------------------------------------------------
At Eurocrypt 2010, van Dijk et al. [@DGHV10] constructed an FHE scheme. For convenience, we here only describe the symmetric version of the Dijk-Gentry-Halevi-Vaikuntanathan FHE scheme as follows.
**KeyGen**($\lambda$): For a security parameter $\lambda$, pick an odd number $p \in[2^{\lambda-1}, 2^{\lambda})$ and set it as the secret key.
**Encrypt**($p, m$): Given a bit $m\in\{0, 1\}$, compute the ciphertext as $$c = pq + 2r + m$$ where the integers $q, r$ are chosen at random in some other prescribed intervals, such that $2r$ is smaller than $p/2$ in absolute value.
**Decrypt**($p, c$): $m=(c \mod p) \mod 2$.
**Additively homomorphic property (under the modulus)**: If $c_1 = pq_1 + 2r_1 + m_1$ and $c_2 = pq_2 + 2r_2 + m_2$, then $m_1+m_2=(c_1+c_2\mod p)\mod 2$.
**Multiplicatively homomorphic property (under the modulus)**: If $c_1 = pq_1 + 2r_1 + m_1$ and $c_2 = pq_2 + 2r_2 + m_2$, then $m_1\cdot m_2=(c_1 \cdot c_2\mod p)\mod 2$.
Notice that these homomorphic properties hold only on the condition that computations are constrained by the prescribed modulus $p, 2$. This restriction makes the scheme impossible to deal with any numerical calculations without knowing the modulus.
An example for Dijk-Gentry-Halevi-Vaikuntanathan scheme
-------------------------------------------------------
Suppose that one client sets $p=7919$ as his secret key. He has two numbers $a=5$, $b=3$, and wants a server to help him to compute $c=a+b$. Now, he encrypts two numbers $a$ and $b$ as follows (see Table 1).
$a=5$
------- --------------------------------- --------------------------------- ---------------------------------
$7919\times 1325+2\times 57+1 $ $7919\times 3168+2\times 49+0 $ $7919\times 5247+2\times 63+1 $
$b=3$
$7919\times 5538+2\times 85+1 $ $7919\times 6214+2\times 74+1 $
Table 1: Ciphertexts of 5 and 3 w.r.t. the secret key $7919$
The client sends two ciphertexts $$\underbrace{\fbox{10492790},\fbox{25087490},\fbox{41551120}}_x \
\mbox{and}\ \underbrace{\fbox{43855593},\fbox{49208815}}_y$$ to a server and asks the server to compute the function $$f(x, y)=x+y.$$ Hence, the server may return the values $$\fbox{10492790},\fbox{68943083},\fbox{90759935}$$ to the client. Thus, the client decrypts the returned values as follows $$(10492790 \mod p) \mod 2= 1,$$ $$(68943083 \mod p) \mod 2= 1,$$ $$(90759935 \mod p) \mod 2= 0,$$ and obtains the number $(110)_2=6$, not the right number $8$. See the following Table 2 for the process.
Client Server
-------------------------------------------------------------------------------- --------------------------- ----------------------------------------------------------------------------------
Input: $p=7919$, $f(x, y)=x+y$
$a=5$, $b=3$
Encryption: $3 \rightarrow \underbrace{\fbox{43855593},\fbox{49208815}}_y $,
$5\rightarrow \underbrace{\fbox{10492790},\fbox{25087490},\fbox{41551120}}_x$.
$\xlongrightarrow{x, y}$
$\xlongleftarrow{\hat c}$ $ f(x, y)=\underbrace{\fbox{10492790},\fbox{68943083},\fbox{90759935}}_{\hat c}$
Decryption: $\hat c \rightarrow \underbrace{\fbox{1},\fbox{1},\fbox{0}}_c$
Table 2: An example for the Dijk-Gentry-Halevi-Vaikuntanathan FHE scheme
What is the problem with this process? The returned values miss all carries because *the server can not decide the carries by the encrypted data.*
**Remark 1**. One might argue that the client himself can construct a Boolean circuit which contains the carries and send the circuit to the server. The argument is unreasonable because the client is assumed to be of weak computational capability. If the client can construct such a Boolean circuit, then he can directly evaluate the circuit, instead of asking a server to help him to evaluate it.
Analysis of Nuida-Kurosawa FHE scheme under the client-server computing model
=============================================================================
In the Dijk-Gentry-Halevi-Vaikuntanathan FHE scheme, the message space is $\mathbb{Z}_2$. The scheme is very inefficient because it has to generate 256 or more bits in order to mask one bit. At Eurocrypt 2015, Nuida and Kurosawa [@NK15] extended the scheme to the message space $\mathbb{Z}_Q$ where $Q$ is any prime. We here only describe the symmetric version of Nuida-Kurosawa FHE scheme as follows.
Description of Nuida-Kurosawa FHE scheme
----------------------------------------
**KeyGen**($\lambda$): For a security parameter $\lambda$, pick an odd number $p \in[2^{\lambda-1}, 2^{\lambda})$ and a prime $Q$. Set $p$ as the secret key ($Q$ is published).
**Encrypt**($p, m$): Given a message $m\in \mathbb{Z}_Q$, compute the ciphertext as $$c = pq + Qr + m$$ where the integers $q, r$ are chosen at random in some other prescribed intervals, such that $Qr$ is smaller than $p/2$ in absolute value.
**Decrypt**($p, c$): $m=(c \mod p) \mod Q$.
**Additively homomorphic property (under the modulus)**: If $c_1 = pq_1 + Qr_1 + m_1$ and $c_2 = pq_2 + Qr_2 + m_2$, then $m_1+m_2=(c_1+c_2\mod p)\mod Q$.
**Multiplicatively homomorphic property (under the modulus)**: If $c_1 = pq_1 + Qr_1 + m_1$ and $c_2 = pq_2 + Qr_2 + m_2$, then $m_1\cdot m_2=(c_1 \cdot c_2\mod p)\mod Q$.
An example for Nuida-Kurosawa FHE scheme
----------------------------------------
Suppose that one client sets $p=22801763489$ as his secret key and sets $Q=15485863$. He has two numbers $a=0.1$, $b=2.3$, and wants a server to help him to compute $c=a+b$.
First, he has to transform $a=0.1$, $b=2.3$ into integers $\bar a, \bar b$ such that $\bar a, \bar b\in \mathbb{Z}_Q$. Denote the transformation by $\mathcal{T}$. Second, he encrypts $\bar a, \bar b$ and obtains the corresponding ciphertexts $\hat a, \hat b$. Third, he sends $\hat a, \hat b$ to a server. The server then takes $\hat a, \hat b$ as the inputs of the function $f(x, y)=x+y$. Finally, the server returns $\hat c=f(\hat a, \hat b)$ to the client. See the following Table 3 for the process.
-------------------------------------------------------------------------------- ------------------------------------- ---------------------------------------------------
Client Server
Input: $p=22801763489$, $Q=15485863$; $f(x, y)=x+y$
$a=0.1$, $b=2.3$
Transformation $\mathcal{T}$: $ a\rightarrow \bar a$, $ b\rightarrow \bar b$.
such that $\bar a, \bar b\in \mathbb{Z}_Q$.
Encryption: $\bar a\rightarrow\hat a$, $\bar b\rightarrow \hat b$. $\xlongrightarrow{\hat a, \hat b} $
$\xlongleftarrow{\hat c}$ Computation $ f(\hat a,\hat b)\rightarrow \hat c$
Decryption: $\hat c \rightarrow \bar c$
Inverse Transformation $\mathcal{T}^{-1}$: $\bar c \rightarrow c. $
-------------------------------------------------------------------------------- ------------------------------------- ---------------------------------------------------
Table 3: An example for the Nuida-Kurosawa FHE scheme
What is the problem with this process? *It is impossible to find an invertible transformation $\mathcal{T}$ from the real number set $\mathbb{R}$ to the field $\mathbb{Z}_Q$.*
Note that most encryption algorithms must run over some finite field or ring. One has to transform all inputting characters into integers in the field or ring. That means an invertible encoding algorithm is necessary for any encryption scheme.
This condition is easily satisfied if all inputting characters are indeed viewed as characters. But when some inputting characters are viewed as real numbers and they are used for some arithmetic computations, it is impossible to find such an invertible encoding algorithm that maps any real number to an integer in a prescribed field or ring.
character ASCII code character ASCII code
----------- ------------ ----------- ------------
0 48 6 54
1 49 7 55
2 50 8 56
3 51 9 57
4 52 $\cdot$ 250
5 53
We here describe a possible encryption-decryption process for the real numbers $0.1$ and $2.3$. The ASCII coding method will map $0.1, 2.3$ to two integers in the field $\mathbb{Z}_{15485863}$.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$a=0.1\xlongrightarrow{\mbox{ASCII}} \fbox{48}\fbox{250}\fbox{49}\xlongrightarrow{\mathcal{T}}\bar a=48\times 256^2+250\times 256+49=3209777\xlongrightarrow{q=3215964,r=13} $
$\hat a= 73329650721664392\xlongrightarrow{\mod p, \mod Q} \bar a=3209777
\xlongrightarrow{\mathcal{T}^{-1}}\fbox{48}\fbox{250}\fbox{49}\xlongrightarrow{\mbox{ASCII}} 0.1 $
$b=2.3\xlongrightarrow{\mbox{ASCII}} \fbox{50}\fbox{250}\fbox{51}\xlongrightarrow{\mathcal{T}}\bar b=50\times 256^2+250\times 256+51=3340851\xlongrightarrow{q=6490231,r=9} $
$\hat b= 147988712393689577\xlongrightarrow{\mod p, \mod Q}\bar b = 3340851
\xlongrightarrow{\mathcal{T}^{-1}}\fbox{50}\fbox{250}\fbox{51}\xlongrightarrow{\mbox{ASCII}} 2.3 $
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
If a server performs the operator of addition on the encrypted data, $\hat a, \hat b $, then it gives $$\hat c= \hat a+\hat b = 73329650721664392+147988712393689577= 221318363115353969.$$ The server returns the value to the client. The client will obtain $$\bar c= (221318363115353969 \mod p) \mod Q=6550628.$$ Notice that $$6550628=99\times 256^2+244\times 256+100\xlongrightarrow{\mathcal{T}^{-1}} \fbox{99}\fbox{244}\fbox{100}.$$ It does not correspond to the wanted number $2.4$ when ASCII coding method is used.
FHE is not applicable to client-server computing
================================================
Cloud computing refers to the practice of transferring computer services such as computation or data storage to other redundant offsite locations available on the Internet, which allows application software to be operated using internet-enabled devices. It benefits one from the existing technologies and paradigms, even though he is short of deep knowledge about or expertise with them. The cloud aims to cut costs, and helps the users focus on their core business instead of being impeded by IT obstacles. Usually, cloud computing adopts the client-server business model.
What computations do you want to outsource privately? Backup your phone’s contacts directory to the cloud? Ask the cloud to solve a mathematic problem in your homework? Do a private web search? $\cdots$. It seems obvious that the daily computational tasks are rarely constrained by some prescribed modulus. Moreover, the client-server computing model can not deal with relational expressions which are defined over plain data, not over encrypted data. This is because $$a<b \not\Longrightarrow E(a)<E(b), \qquad E(a)<E(b) \not\Longrightarrow a<b.$$ In view of this weakness of FHE and the flaws of two typical schemes mentioned above, we think, FHE is not applicable to cloud computing.
**Remark 2**. The problem that what computations are worth delegating privately by individuals and companies to untrusted devices or servers remains untouched. We think the cloud computing community has not yet found a good for-profit model convincing individuals to pay for this or that computational service.
Conclusion
==========
We reaffirm the role of modular arithmetic in modern cryptography and show that FHE is not applicable to cloud computing because any FHE scheme does work over some encrypted domains. When two decrypted number are added, one cannot decide the carries without knowing the secret decryption key. Moreover, there is no an invertible transformation from the real number set to the encrypted domain which makes it impossible to tackle numerical calculations. We think the primitive of FHE might be of little importance to client-server computing.
[4]{}
Z. Brakerski, C. Gentry, and S. Halevi: Packed ciphertexts in LWE-based homomorphic encryption. In: Proc. PKC’13, LNCS, vol. 7778, pp. 1-13, Springer-Verlag, 2013.
Z. Brakerski, C. Gentry, and V. Vaikuntanathan: (Leveled) fully homomorphic encryption without bootstrapping. In: Proc. 3rd Innovations in Theoretical Computer Science Conf., pp. 309-325, 2012.
Z. Brakerski and V. Vaikuntanathan, Fully homomorphic encryption from ring-LWE and security for key dependent messages. In: Proc. CRYPTO’11, LNCS, vol. 6841, pp. 505-524, Springer-Verlag, 2011.
Z. Brakerski and V. Vaikuntanathan: Efficient fully homomorphic encryption from (standard) LWE. In: Proc. IEEE 52nd Annu. Symp. on Foundations of Computer Science, pp. 97-106, 2011.
Z. Brakerski and V. Vaikuntanathan: Efficient Fully Homomorphic Encryption from (Standard) LWE. SIAM J. Comput. 43(2), 831-871, 2014.
J. Coron, et al.: Fully homomorphic encryption over the integers with shorter public keys. In: CRYPTO’11, LNCS, vol. 6841, pp. 487-504, Springer-Verlag, 2011.
J. Cheon, et al.: Batch fully homomorphic encryption over the integers. In: Proc. EUROCRYPT’13, LNCS, vol. 7881, pp. 315-335, Springer-Verlag, 2013.
J. H. Cheon and J. Kim: A Hybrid Scheme of Public-Key Encryption and Somewhat Homomorphic Encryption. IEEE Transactions on Information Forensics and Security, vol. 10, No. 5, 1052-1063, 2015.
G. Castagnos, F. Laguillaumie: Linearly Homomorphic Encryption from DDH. In: Proc. CT-RSA 2015, LNCS, vol. 9048, pp. 487-505, Springer-Verlag, 2015.
M. van Dijk, C. Gentry, S. Halevi, and V. Vaikuntanathan: Fully homomorphic encryption over the integers. In Proc. EUROCRYPT’10, LNCS, vol. 6110, pp. 24-43, Springer-Verlag, 2010.
T. ElGamal: A public key cryptosystem and a signature scheme based on discrete logarithms. In: Proc. CRYPTO’84, pp. 10-18, Springer-Verlag, 1984.
C. Gentry: Fully homomorphic encryption using ideal lattices. In: Proc. STOC’09, ACM, pp. 169-178, 2009.
C. Gentry and S. Halevi: Fully Homomorphic Encryption without Squashing Using Depth-3 Arithmetic Circuits. In: Proc. 52nd Annual IEEE Symposium on Foundations of Computer Science, pp. 107-116, 2011.
C. Gentry and S. Halevi: Implementing Gentry’s fully-homomorphic encryption scheme. In: Proc. EUROCRYPT’11, LNCS, vol. 6632, pp. 129-148, Springer-Verlag, 2011.
C. Gentry, S. Halevi, and N. Smart: Homomorphic evaluation of the AES circuit, IACR Cryptology ePrint Archive, vol. 2012, paper 99, 2012.
C. Gentry, S. Halevi, and V. Vaikuntanathan: A simple BGN-type cryptosystem from LWE. In: Proc. EUROCRYPT’10, LNCS, vol. 6110, pp. 506-522, Springer-Verlag, 2010.
S. Goldwasser and S. Micali: Probabilistic encryption and how to play mental poker keeping secret all partial information. In: Proc. STOC’82, ACM, pp. 365-377, 1982.
K. Nuida and K. Kurosawa: (Batch) Fully Homomorphic Encryption over Integers for Non-Binary Message Spaces. In: Proc. EUROCRYPT’15, LNCS, vol. 9056, pp. 537-555, 2015.
P. Paillier: Public-key cryptosystems based on composite degree residuosity classes. In: Proc. EUROCRYPT’99, pp. 223-238, Springer-Verlag, 1999.
R. Rivest, L. Adleman, and M. Dertouzos: On data banks and privacy homomorphisms. In: Foundations of Secure Computation, R. Demillo, et al. (Eds.), New York: Academic, pp. 169-180, 1978.
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---
abstract: 'Propagators of unstable particles are considered in the spectral representation which naturally follows from the concept of continuous mass. The spectral functions are found with the help of the most general formal and physical assumptions. Dressed propagators of unstable scalar, vector, and spinor fields are derived in an analytical way for a variant of parameter space. The structure of the propagators is in a correspondence with the complex mass scheme.'
author:
- 'V. I. Kuksa'
title: Complex mass definition and the concept of continuous mass
---
Introduction
============
Two standard definitions of the mass and width of unstable particles (UP), which are usually considered in the literature, have essentially different nature. The on-mass-shell (OMS) scheme defines the mass $M$ and width $\Gamma$ of a bosonic UP by the renormalization conditions [@1]: $$\label{1.1}
M^2=M^2_0 +\Re \Pi(M^2),\,\,\,M\Gamma=\frac{-\Im \Pi(M^2)}{1-\Re
\Pi^{'}_s(M^2)}\,.$$ In the pole scheme (PS) the definitions of the mass and width are based on the complex-valued position of the propagator pole $s_R-M^2_0-\Pi(s_R)=0$. For instance, in the case of a vector UP a possible definition is as follows [@1]: $$\label{1.2}
s_R=M^2_{\rho}-iM_{\rho}\Gamma_{\rho},\,\mbox{where}\,\,s_R=M^2_0+\Pi(s_R).$$ It should be noted that this definition of the pole mass $M_{\rho}$ and width $\Gamma_{\rho}$ (also called $m_2, \Gamma_2$ or $\tilde{M}_Z, \tilde{\Gamma}_Z$) is not unique. There has been considerable discussion concerning definition of the vector-boson mass [@2; @3; @4; @5; @6; @7; @8]. It was shown that OMS scheme contains spurious higher-order gauge-dependent terms while PS provides gauge invariant definition. Later on the PS definition was considered in detail [@9; @10; @11] and gauge-invariant treatment was developed in the frame of the complex-mass scheme (CMS) [@12; @13].
An alternative, semi-phenomenological, scheme based on the hypothesis of continuous (smeared) mass of UP was considered in [@14; @15] (and references therein). In this approach, the physical values of the mass and width are related to the parameters of continuous mass distribution. In the model the propagators of UP have spectral-representation form, so that the standard problems transform to constructing the spectral function which describes the mass distribution. The effective theory of UP was developed on the base of this approach in [@16], where the method of factorization of widths and cross-sections is presented.
In this work, we show that the concept of continuous mass along with some natural assumptions (which are considered in the next section) lead to the pole scheme with complex mass definition. This result follows directly from the propagators in spectral representation which, in turn, follows from the quantum-field realization of the mass-smearing concept [@14; @15].
The paper is organized as follows. In section \[sec:2\] we present the principal elements of the approach under consideration and analyze the general structure of the propagators in the model. The structure of the bosonic and fermionic propagators is considered in sections \[sec:3\] and \[sec:4\] respectively. Some conclusions concerning the physical status of the results are made in section \[sec:5\].
\[sec:2\]General structure of the propagator of an unstable particle
====================================================================
Traditional way to construct the dressed propagator of UP is the Dyson summation, which introduces the width and redefines the mass of UP. This procedure runs into some problems widely discussed in the literature. Let us consider one of the formal sources of the difficulties which follow from this summation. In the simplest case of a scalar UP this procedure looks like $$\begin{aligned}
\label{2.1}
D_{(1)}(q)={}&D_0(q)\sum_{k=0}^{\infty}(-i\Pi_{(1)}(q)D_0(q))^k\notag\\
={}&\frac{D_0(q)}{1+i\Pi_{(1)}(q)D_0(q)}=\frac{i}{q^2-M^2_0-\Pi_{(1)}(q)}\,,\end{aligned}$$ where $D_0(q)=i(q^2-M^2_0+i\epsilon)^{-1}$ is the propagator of a scalar free field and $\Pi_{(1)}(q)$ is the one-particle–irreducible self-energy. The dressed propagator (\[2.1\]) is formally incorrect in the near-resonance range, because of the finite radius of convergence of the series $$\label{2.2}
\frac{1}{1-z}=\sum_{k=0}^{\infty}z^k=1+z+z^2+...,\qquad |z|<1.$$ From Eqns. (\[2.1\]) and (\[2.2\]) it follows that the variable $z=\Pi_{(1)}(q)/(q^2-M^2_0)$ should be correctly redefined before summation and one can not use the procedure in the region $z>1$ after the redefinition. Thus, the dressed propagator in the form (\[2.1\]) contains an assumption of infinite series, that is, UP is described as a non-perturbative object. This, evidently, leads to difficulties in the scheme of sequential fixed-order calculations, which exhibit themselves in the violation of gauge invariance. As was pointed out in the previous section, such problems have been under considerable discussion for many years.
An alternative approach is based on the spectral representation of the propagator of UP. It has a long history [@17; @18; @19; @20; @21] and treats UP as a non-perturbative state or effective field (asymptotic free field [@20; @21]). Here, we consider this approach in the framework of the model of UP with continuous (smeared) mass [@14; @15]. In the works [@14; @15], the effective field function of UP is a continuous superposition of ordinary ones with a universal weight function $\omega(m)$ which describes the distribution of the mass parameter $m$. The value $m$ was interpreted as a random mass with continuous distribution from some threshold to infinity (physical range definition). Here, we consider some modification of this approach which leads to significant changes in the effective theory of UP. We consider a non-universal weight function $\omega(q,m)$, that is the function which depends on the momentum $q$ of the state. The second modification is the extension of the allowed values of the mass parameter $m^2$ from $[m^2_0,+\infty)$ to $(-\infty,+\infty)$. So, the parameter $m^2$ loses its previous simple interpretation. It acquires, however, the status of the momentum squared $q^2$ in the whole Minkowski space. Further we show that this modifications lead to PS definition.
The field function of UP in the momentum representation is defined in exact analogy with the definition in [@14] except for the above mentioned modification: $$\label{2.3}
\phi(q)=\int_{-\infty}^{+\infty}\phi(q,m^2)\,\omega(q,m^2)\,dm^2.$$ The canonical commutation relations are not modified—they contain, as in Ref. [@14], an additional delta-function $\delta(m^2-m'{}^{2})$. Using the definition (\[2.3\]) and the commutation relations from [@14; @15], one get the propagator of a scalar unstable field in spectral representation. Starting from the standard definition of the Green’s function, by straightforward calculations we get the Lehmann-like spectral representation of the propagator [@15; @16]: $$\label{2.4}
D(q)=i\int dx\, exp\,(-iqx)\,\langle
0|\hat{T}\phi(x)\phi(0)|0\rangle
=i\int_{-\infty}^{+\infty}\,\frac{\rho(q,m^2)\,dm^2}{q^2-m^2+i\epsilon},$$ where $\rho(q,m^2)=|\omega(q,m^2)|^2$. For the case $m^2<0$, that is $m=i|m|$, we leave the physical range of the continuous (smearing) mass. It should be noted, that the expression (\[2.4\]) is applicable for any values of $q^2$ and $m^2$ without restrictions. Moreover, it is not directly connected with the expansion of the Dyson type and perturbative constructions. So, the correct definition of the function $\rho(q,m^2)$ makes it possible to escape some problems which arise in a traditional approach.
Now, let us consider possible physical consequences of the presence of negative mass parameter $m^2<0$ in the spectral representation (\[2.3\]) and (\[2.4\]). Negative component of the spectrum leads to the states with imaginary mass parameters which usually interpreted as tachyon states. The problem of the existence of tachyons is under considerable discussion in the last decades. The main attention is paid to the principal problems, such as a violation of causality, tachyon vacuum, and radiation instability. There is no unique and consistent quantum field theory of tachyons, and various approaches are suggested to overcome the above mentioned difficulties. We note, that the first problem relates to UP as an observable object. Further we show that such object in the framework of our effective model is described by the positive mass square $M^2(q)$ and width $\Gamma(q)$. However, quantum field description of UP as the continuous superposition with a tachyon component encounters the problem of instability. In the third section, we estimate the tachyon fraction for the case of fundamental UP. It is found to be rather small; for instance, in the case of $Z$ boson it is approximately a percent. The analysis of an expression for the tachyon fraction leads to an interesting conclusion: tachyon instability is intrinsic property of UP; it can be interpreted as the cause of unstable particle decay.
The principal problem of the approach under consideration is to define the spectral function $\rho(q,m^2)$ which is the main characteristic of UP. Some phenomenological definitions were considered in [@14]. Here, we consider the construction of this function with an account of the most general formal and physical arguments. First of all, we choose two-parametric distributions, where the parameters are directly connected with the mean values of mass and width of UP. These parameters are some functions of the four-momentum, that is $M(q)$ and $\Gamma(q)$ are $q$-dependent characteristics of UP. The probability density $\rho(q,m^2)$ has to be normalized for any $q$. If $q\longrightarrow q_0$, where $q_0$ is the threshold value of the momentum, the value $\Gamma(q)\longrightarrow \Gamma(q_0)=0$. So, the particle become stable, its mass go to a fixed value $M(q_0)=M_0$ and the function $\rho(q,m^2)$ behaves as a delta-function $\delta (m-M_0)$. There are three well-known approximations of the delta-function which are continuously differentiable and normalized. These are as follows [@22]: $$\label{2.5}
(a)\,\,\,\frac{\alpha}{\pi(1+\alpha^2
x^2)};\,\,\,(b)\,\,\,\frac{\alpha}{\sqrt{\pi}}\exp(-\alpha^2
x^2);\,\,\,(c)\,\,\frac{\alpha}{\pi}\frac{\sin \alpha x}{\alpha
x}\qquad (\alpha\to\infty).$$ It is difficult to find the physical interpretation for the case of the oscillating distribution $(c)$ in Eq. (\[2.5\]). As was discussed in [@14], the exponential distribution (b) can be motivated when the smearing of mass is caused by stochastic mechanism of UP interaction with vacuum. It was noted that fast decreasing of the exponent does not describe some deeply virtual processes. One can see that the distribution (a) is the well-known Lorentzian distribution, where $\alpha \sim (\Gamma(q))^{-1}$ and $x=m^2-M^2(q)$ or $x=m-M(q)$. It is easy to check that this function is normalized to unity. So, the function (a) in Eq. (\[2.5\]) satisfies above mentioned formal and physical requirements and will be used in derivation of the propagators of UP.
\[sec:3\]Propagators of bosonic unstable particles
==================================================
In this section, we consider the structure of the propagators of scalar and vector UP. From Eq. (\[1.1\]) and dimension requirement we have to choose the parametrization $x=m^2-M^2(q)$ and $\alpha=(q\Gamma(q))^{-1}$ or $\alpha=(\Gamma(q))^{-2}$. Then, the probability density (\[2.5\]a) takes the form $$\label{3.1}
\rho(q,m^2)=\frac{1}{\pi}\frac{\alpha^{-1}}{x^2+\alpha^{-2}}=
\frac{1}{\pi}\frac{q\Gamma(q)}{[m^2-M^2(q)]^2+q^2\Gamma^2(q)},$$ where $M(q)$ and $\Gamma(q)$ are $q$-dependent parameters of the distribution. According to Eqs. (\[2.4\]) and (\[3.1\]) the propagator of a scalar UP can be written as $$\label{3.2}
D(q)=\frac{1}{\pi}\int_{-\infty}^{+\infty}\frac{P(q)\,dm^2}
{(q^2-m^2+i\epsilon)\{[m^2-M^2(q)]^2+P^2(q)\}},$$ where $P(q)=q\Gamma(q)$ and we omit $i$ for the simplicity of consideration. The integral in (\[3.2\]) can be calculated with the help of the integration rule $$\label{3.3}
\int_{-\infty}^{+\infty}\frac{f(x)\,dx}{x\pm i\epsilon}=\mp i\pi
f(0)+ \mathcal{P}\int_{-\infty}^{+\infty}\frac{f(x)}{x}dx,$$ which follows from the Sokhotski-Plemelj formula. In Eq. (\[3.3\]) $\mathcal{P}\int$ stand for the principal part of the integral. As a result we have: $$\begin{aligned}
\label{3.4}
\Im D(q)={}&-\pi\rho(q,q^2)=\frac{-P(q)}{[q^2-M^2(q)]^2+P^2(q)};\notag\\
\Re
D(q)={}&\mathcal{P}\int_{-\infty}^{+\infty}\frac{\rho(q,m^2)\,dm^2}
{q^2-m^2}=\frac{q^2-M^2(q)}{[q^2-M^2(q)]^2+P^2(q)}.\end{aligned}$$ It is easy to check that Eqs. (\[3.4\]) result in the following expression for the dressed propagator of a scalar UP: $$\label{3.5}
D(q)=\frac{1}{q^2-M^2(q)+iP(q)},$$ where $P(q)=q\Gamma(q)$. The same result can be got in a more simple way with the help of contour integration. Let us rewrite the function $D(q)$ as follows: $$\label{3.6}
D(q)=-\int_{-\infty}^{+\infty}\frac{\rho(q,m^2)\,dm^2}{m^2-(q^2+i\epsilon)}=-\frac{1}{\pi}
\int_{-\infty}^{+\infty}\frac{P(q)\,dm^2}{(m^2-z_0)(m^2-z_+)(m^2-z_-)},$$ where $z_0=q^2+i\epsilon$ and $z_{\pm}=M^2(q)\pm iP(q)$. Analytical continuation of the integrand function $f(z)$, where $m^2\to z$, has three poles $z_0,z_+,z_-$ in the complex plane. It decreases as $1/z^2$ for $z\to\infty$, that is, it satisfies the condition $|f(z)|<M/|z|^{1+\delta}$ for $|z|>R_0$, where $M$ and $\delta$ are positive numbers and $R_0\to\infty$. So, we can rearrange $D(q)$ as follows: $$\label{3.7}
D(q)=\mp\frac{P(q)}{\pi}\oint_{C_{\pm}}\frac{dz}{(z-z_0)(z-z_+)(z-z_-)}
=2\pi i \sum_{k=z_0,z_+,z_-} \operatorname{Res}(f(z),z_k),$$ where $\operatorname{Res}(f(z),z_k)$ is the residue at the pole $z_k$ and $C_{\pm}$ is a contour in the upper ($C_+$) or lower ($C_-$) half of the complex $z$-plane. The simplest way to perform the integration is to go along the contour $C_-$ which contains only one pole $z_-$: $$\begin{aligned}
\label{3.8}
D(q)={}&\frac{P(q)}{\pi}\oint_{C_-}\frac{dz}{(z-z_-)}\frac{1}{(z-z_+)(z-z_0)}\notag\\
={}&\frac{2i P(q)}{(z_--z_+)(z_--z_0)}=\frac{1}{q^2-M^2(q)+iP(q)}.\end{aligned}$$ In Eqs. (\[3.8\]) we have used the equality $z_--z_+=-2iP(q)$. One can check that the same result follows from the integration along the contour $C_+$.
Thus, UP can be described at two different hierarchical levels—“fundamental” level by the spectral representation (\[3.2\]) and phenomenological one by the effective theory after integrating out unobservable mass parameter $m^2$. In the framework of this theory, UP is described by the observed physical values $M^2(q)$ and $\Gamma(q)$, which can always be defined as a positive quantity. So, at this phenomenological level UP has no explicit tachyonic content that could lead to the problems noted in the Introduction. At the first level, however, the approach under consideration deals with explicitly tachyonic component in the field function of UP. As it was noted earlier, it can lead to an instability of the system. Let us estimate the tachyonic fraction, that is the probability of the states with $m^2<0$. From the Eq.(\[3.1\]) it follows that this probability $P_t(M(q),\Gamma(q))$ under the condition $M(q)/\Gamma(q)\ll 1$ is as follows: $$\label{3.8a}
P_t(M(q),\Gamma(q))=\int_{-\infty}^0 \rho(m^2; M^2(q),
\Gamma(q))\,dm^2 \approx \frac{\Gamma(q)}{\pi M(q)}.$$ Thus, for the majority of fundamental particles this value is very small. However, for the case of vector boson ($Z$ and $W$) and $t$-quark the tachyon fraction is appreciable ($~10^{-2}$). From the Eq.(\[3.8a\]) it follows that this fraction is defined by a value of $\Gamma(q)/M(q)$, that is the value which quantifies the effects of instability or finite-width effects (FWE) in the processes with UP participation. This makes it possible to suggest the direct connection between the tachyon component and instability of the particle. The decay of UP can be caused, for instance, by the instability of the tachyon vacuum.
To define the structure of the vector propagator, we assume that the spectral function $\rho(q,m^2)$ is the same as for a scalar UP. Using the standard vector propagator for a free vector particle with a fixed mass, we get: $$\label{3.9}
D_{\mu\nu}(q)=\frac{1}{\pi}\int_{-\infty}^{+\infty}\frac{-g_{\mu\nu}+q_{\mu}q_{\nu}/(m^2-i\epsilon)}
{q^2-m^2+i\epsilon}\,\frac{P(q)\,dm^2}{[m^2-M^2(q)]^2+P^2(q)}.$$ In the propagator term $q_{\mu}q_{\nu}/(m^2-i\epsilon)$ we use the same rule of going around pole as in the denominator $q^2-(m^2-i\epsilon)$. The integral in Eq. (\[3.9\]) can be evaluated with the help of the formula (\[3.3\]), however, it is easier to do it using the method of contour integration. The integration along the lower contour $C_-$ gives: $$\begin{aligned}
\label{3.10}
D_{\mu\nu}(q)={}&-\frac{P(q)}{\pi}\oint_{C_-}\frac{(g_{\mu\nu}-q_{\mu}q_{\nu}/(z-i\epsilon))\,dz}
{(z-z_-)(z-z_+)(z-z_0)}\notag\\={}&-2iP(q)\frac{g_{\mu\nu}-q_{\mu}q_{\nu}/(z_-)}{(z_--z_+)(z_--z_0)}=
\frac{-g_{\mu\nu}+q_{\mu}q_{\nu}/(M^2(q)-iP(q))}{q^2-M^2(q)+iP(q)}\,.\end{aligned}$$ One can check that the integration along the upper contour $C_+$ or with the help of the formula (\[3.3\]) leads to the same result.
We can see that both the scalar and vector propagators of UP can be represented in the form with universal complex mass squared: $$\label{3.11}
D(q)=\frac{1}{q^2-M^2_P(q)};\qquad
D_{\mu\nu}(q)=\frac{-g_{\mu\nu}+q_{\mu}q_{\nu}/M^2_P(q)}{q^2-M^2_P(q)}\,,$$ where $M^2_P(q)=M^2(q)-iP(q)$. Note that the dressed propagator of a bosonic UP can be get from the “free" propagator by the substitution $m^2-i\epsilon \longrightarrow
M^2(q)-iP(q)$. The inverse substitution, $P(q)\to i0,
M^2(q)\to M^2_0$, leads to the “free" propagator, if $q\to q_0$.
The $q$-dependent parameters $M(q)$ and $\Gamma(q)$ of the propagator pole $s_R(q)=M^2_P(q)=M^2(q)-iq\Gamma(q)$ are a generalization of the traditional pole definition of the mass and width of UP, which is based on the complex-valued position of the propagator pole $s_R-M^2_0-\Pi(s_R)=0$ [@1]. It should be noted that such pole mass and width are defined in the non-physical region of four-momentum squared $q^2=s_R$. In our analysis, propagator pole is $q$-dependent, $s_R(q)=M^2(q)-iq\Gamma(q)$, so we have $q$-dependent (running) pole definition of the mass and width. At fixed point $q^2=M^2_{\rho}$ we get an ordinary definition $s_R(M_{\rho})=M^2(M_{\rho})-M_{\rho}\Gamma(M_{\rho})$. The correctness and gauge invariance of such definition are strongly stipulated by the definition of the functions $M(q)$ and $\Gamma(q)$.
The traditional pole definition of the mass and width follows from the poles in the $S$-matrix. In practical calculations we deal with an expansion of the $S$-matrix, which is defined by the interaction Lagrangian. For instance, the process $e^+e^-\to Z\to f\bar{f}$ is described at the tree level by the second order term and the pole in the $S$-matrix in this approximation is defined by the pole of the boson propagator. So, this pole depends on the scheme of inclusion of the mass and width terms into the dressed propagator of $Z$-boson (see the discussion in Refs. [@23; @24]). We consider $q^2$ dependent mass and width as the most general case. But, this scheme is not obligatory for the consideration and we can include fixed mass and width terms which is valid in the vicinity of the resonance. As a result, we get a unique pole of the $S$-matrix and a pole definition of the fixed mass and width.
\[sec:4\]Propagator of a spinor unstable particle
=================================================
The propagator of a free fermion can be represented in two equivalent forms: $$\label{4.1}
\hat{D}(q)=\frac{1}{\hat{q}-m+i\epsilon}=\frac{\hat{q}+m-i\epsilon}{q^2-(m-i\epsilon)^2}.$$ According to the heuristic rule for constructing the dressed propagator, we have to make the substitution $m\to M(q)$ and $i\epsilon\to
i\Gamma(q)$, where the dimension $[\epsilon]=[m]$ was taken into account. Then, the dressed propagator of the spinor UP takes the form $$\label{4.2}
\hat{D}(q)=\frac{\hat{q}+M(q)-i\Gamma(q)}{q^2-(M(q)-i\Gamma(q))^2}=\frac{\hat{q}+M_P(q)}{q^2-M^2_P(q)},$$ where $M_P(q)=M(q)-i\Gamma(q)$ is $q$-dependent pole-type complex mass. Now, we show that the expression (\[4.2\]) can be derived in a more systematic way with the help of the spectral representation: $$\label{4.3}
\hat{D}(q)=\int\frac{\hat{q}+m}{q^2-(m-i\epsilon)^2}\,\rho(q,m)\,dm\,,$$ where the integration range is not defined yet. The spectral function $\rho(q,m)$ for fermions differs from the bosonic one, because of another parametrization $M(q)=M_0+\Re\Sigma(q)$ and $\Gamma(q)=\Im\Sigma(q)$. So, we have to take $x=m-M(q)$ and $\alpha=\Gamma(q)$ in the general expression for $\rho(q,m)$ in Eq. (\[2.5\]a). As a result, the spectral function for the case of the spinor UP is as follows: $$\label{4.4}
\rho(q,m)=\frac{1}{\pi}\,\,\frac{\Gamma(q)}{[m-M(q)]^2+\Gamma^2(q)}
=\frac{1}{\pi}\frac{\Gamma(q)}{(m-M_-(q))(m-M_+(q))},$$ where $M_{\pm}(q)=M(q)\pm \Gamma(q)$. The main difference between boson and spinor cases is a presence of the linear term $m$ instead the quadratic one $m^2$, which is defined at the whole real axis $m^2 \in (-\infty,+\infty)$. Here, we restrict the problem by the formal definition and consider a straightforward relation between the bosonic parameter range and spinor one. Thus, we have two intervals $(\pm i\infty , i0; 0 , \infty)$ for the value $m$. In the method of contour integration the signs $\pm$ correspond to integration along the contours $C_{\pm}$, which enclose the first or fourth quadrants of the complex plane. Then, from Eqs. (\[4.3\]) and (\[4.4\]) it follows: $$\label{4.5}
\hat{D}_{\pm}(q)=\pm\frac{\Gamma(q)}{\pi}\int_{C_{\pm}}\frac{(\hat{q}+z)\,dz}
{(z^2-z^2_0)(z-z_-)(z-z_+)}\,,$$ where $z^2_0=q^2+i\epsilon$, $z_{\pm}=M_{\pm}(q)$ and $C_{\pm}$ are the above described contours. By simple and straightforward calculations one can see that the result (\[4.2\]) follows from the integration along the contour $C_-$, while the integration along the $C_+$ leads to non-physical result. This is likely caused by the presence of the branch point $z^2_0$ in the first quadrant. From Eq. (\[4.5\]) it follows: $$\begin{aligned}
\label{4.6}
\hat{D}_{-}(q)=&-\frac{\Gamma(q)}{\pi}\int_{C_{-}}\frac{dz}{z-z_-}\,\frac{\hat{q}+z}
{(z^2-z^2_0)(z-z_+)}\notag\\=&-2i\Gamma(q)\frac{\hat{q}+z_-}{(z^2_--z^2_0)(z_--z_+)}
=\frac{\hat{q}+M_P(q)}{q^2-M^2_P(q)}.\end{aligned}$$ The integration along the contour $C_+$ leads to a more complicated expression. Note, that while the spinor $q$-dependent complex mass $M_P(q)=M(q)-i\Gamma(q)$ differs from the bosonic one, it has, however, the same pole-type structure. Then, the pole definition of the mass and width of the spinor UP is $M_P(M_{\rho})=M(M_{\rho})-i\Gamma(M_{\rho})$.
\[sec:5\]Conclusion
===================
The definitions of the mass and width of UP, as a rule, are closely connected with the construction of the dressed propagators. There are two main definitions, which follows from the on-shell re-normalization and pole structure of the propagators (see Introduction). We considered the general structure of the propagators of UP in the phenomenological approach based on the spectral representation. The spectral function was defined with account of the some formal and physical suggestions which naturally arise in the model of UP with continuous mass.
In this work, we analyzed a specific case—the spectral function depends on $q$-dependent parameters and random mass variable $m^2$ defined in the interval $(-\infty,+\infty)$. So, the variable $m$ can be imaginary, which is beyond the physical domain in the model of UP with continuous (smeared) mass [@14]. It was shown that this leads to the $q$-dependent complex mass which gives the pole definition of the mass and width of UP. We also suggest that the imaginary component of mass spectrum, which corresponds to tachyonic states, causes the instability of particle.
We have proved a simple heuristic rule for constructing the dressed propagators, which is based on the definition of $q$-dependent mass and width of UP. Physical $q$-dependence of the UP width makes it possible to introduce naturally the stable particle limit near threshold momentum and the narrow width approximation $\Gamma(q^2)\to 0$ when $q^2\to q^2_0$. Note, however, that the physical nature of the random mass parameter $m^2$ is vague at $m^2<0$.
[99]{} A. R. Bohm, N. L. Harshman, [*Nucl. Phys. B*]{} [**581**]{}, 91 (2000). S. Willenbrock and G. Valencia, [*Phys. Lett. B*]{} [**259**]{}, 373 (1991). R. G. Stuart, [*Phys. Lett. B*]{} [**262**]{}, 113 (1991). G. Lopez Castro, J. L. M. Lucio, J. Pestieau, [*Mod. Phys. Lett . A*]{} [**6**]{}, 3679 (1991). A. Sirlin, [*Phys. Lett. B*]{} [**267**]{}, 240 (1991). A. Leike, T. Riemann, J. Rose, [*Phys. Lett. B*]{} [**273**]{}, 513 (1991). M. Nowakowski, A. Pilaftsis, [*Z. Phys. C*]{} [**60**]{}, 121 (1993). H. Veltman, [*Z. Phys. C*]{} [**62**]{}, 35 (1994). A. Aeppli, G. J. van Oldenborgh, D. Wyler, [*Nucl. Phys. B*]{} [**428**]{}, 126 (1994). M. Faisst, J. H. Kuhn, O. Veretin, [*Phys. Lett. B*]{} [**589**]{}, 35 (2004). B. A. Kniehl and A. Sirlin, [*Phys. Rev. D*]{} [**77**]{}, 116012 (2008). A. Denner, S. Dittmaier, M. Roth, D. Wackeroth, [*Nucl. Phys. B*]{} [**560**]{}, 33 (1999). A. Denner, S. Dittmaier, M. Roth, L. H. Wieders, [*Nucl. Phys. B*]{} [**724**]{}, 247 (2005). V. I. Kuksa, [*Int. J. Mod. Phys. A*]{} [**24**]{}, 1185 (2009). V. I. Kuksa, [*Phys. Part. and Nucl.*]{} [**45**]{}, 568 (2014). V. I. Kuksa, N. I. Volchanskiy, [*Cent. Eur. J. Phys.*]{} [**11**]{}, 182 (2013). P. T. Mathews and A. Salam, [*Phys. Rev.*]{} [**112**]{}, 283 (1958). J. Schwinger, [*Ann. Phys. (New York)*]{} [**9**]{}, 169 (1960). R. Jacob and R. G. Sachs, [*Phys. Rev.*]{} [**121**]{}, 350 (1961). O. W. Greenberg, [*Ann. Phys. (New York)*]{} [**16**]{}, 158 (1961). A. L. Lich, [*Ann. Phys. (New York)*]{} [**34**]{}, 161 (1965). G. A. Korn, T. M. Korn, [*Mathematical handbook.*]{} (McGraw-Hill Book Company, New York, 1968). M. H. Seymour, [*Phys. Lett. B*]{} [**354**]{}, 409 (1995). S. D. Dittmaier and M. Roth, [*Nucl. Phys. B*]{} [**642**]{}, (2002).
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abstract: 'We present a fluid dynamics video showing the adhesion of a drop to a superhydrophobic surface. We use environmental scanning electron microscopy to observe depinning events at the microscale. As the drop moves along the surface, the advancing portion of the contact line simply lies down onto the upcoming roughness features, contributing negligibly to adhesion. After measuring the local receding contact angle of capillary bridges formed on a micropillar array, we find that these depinning events follow the Gibbs depinning criterion. We further extend this technique to two-scale hierarchical structures to reveal a self-similar depinning mechanism in which the adhesion of the entire drop depends only on the pinning at the very smallest level of roughness hierarchy. With this self-similar depinning mechanism we develop a model to predict the adhesion of drops to superhydrophobic surfaces that explains both the low adhesion on sparsely structured surfaces and the surprisingly high adhesion on surfaces whose features are densely spaced or tortuously shaped.'
author:
- |
Adam Paxson, Kripa K. Varanasi\
Massachusetts Institute of Technology, Cambridge
title: 'Sticking around: an up-close look at droplet adhesion'
---
Brief Explanation of Video Submission
=====================================
This video depicts the imaging technique used to visualize droplet pinning and depinning on discrete textured superhydrophobic surfaces, and some examples of motion of drops across different superhydrophobic textures. A Zeiss EVO-55 ESEM was used for imaging. A custom fixture was fabricated to mount the samples vertically in the ESEM and provide cooling with a Deben Coolstage Mk II Peltier cooling stage. A 10 ml water drop was affixed to the end of a copper wire that allowed the drop to be held against a vertically oriented surface, and also provided a cooling flux to minimize evaporation. By rotating the motorized stage of the ESEM, the droplet was swept across the surface at a velocity of 2 mm/s while observing the depinning events of the contact line at micron length scales. The chamber of the ESEM was maintained at a pressure of 1,000 Pa.
References
==========
A. T. Paxson, K. K. Varanasi, “Self-similarity of contact line depinning from textured surfaces,” Nature Communications, 4, 1492, 2013.
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abstract: |
We consider a problem of considerable practical interest: the recovery of a data matrix from a sampling of its entries. Suppose that we observe $m$ entries selected uniformly at random from a matrix $M$. Can we complete the matrix and recover the entries that we have not seen?
We show that one can perfectly recover most low-rank matrices from what appears to be an incomplete set of entries. We prove that if the number $m$ of sampled entries obeys $$m \ge C \, n^{1.2} r \log n$$ for some positive numerical constant $C$, then with very high probability, most $n \times n$ matrices of rank $r$ can be perfectly recovered by solving a simple convex optimization program. This program finds the matrix with minimum nuclear norm that fits the data. The condition above assumes that the rank is not too large. However, if one replaces the 1.2 exponent with 1.25, then the result holds for all values of the rank. Similar results hold for arbitrary rectangular matrices as well. Our results are connected with the recent literature on compressed sensing, and show that objects other than signals and images can be perfectly reconstructed from very limited information.
author:
- |
Emmanuel J. Candès$^{\dagger}$ and Benjamin Recht$^{\sharp}$\
\
$\dagger$ Applied and Computational Mathematics, Caltech, Pasadena, CA 91125\
\
$\sharp$ Center for the Mathematics of Information, Caltech, Pasadena, CA 91125
bibliography:
- 'MatrixCompletionFinal3.bib'
date: May 2008
title: Exact Matrix Completion via Convex Optimization
---
[**Keywords.**]{} Matrix completion, low-rank matrices, convex optimization, duality in optimization, nuclear norm minimization, random matrices, noncommutative Khintchine inequality, decoupling, compressed sensing.
Introduction {#sec:intro}
============
In many practical problems of interest, one would like to recover a matrix from a sampling of its entries. As a motivating example, consider the task of inferring answers in a partially filled out survey. That is, suppose that questions are being asked to a collection of individuals. Then we can form a matrix where the rows index each individual and the columns index the questions. We collect data to fill out this table but unfortunately, many questions are left unanswered. Is it possible to make an educated guess about what the missing answers should be? How can one make such a guess? Formally, we may view this problem as follows. We are interested in recovering a data matrix ${\bm{M}}$ with $n_1$ rows and $n_2$ columns but only get to observe a number $m$ of its entries which is comparably much smaller than $n_1 n_2$, the total number of entries. Can one recover the matrix ${\bm{M}}$ from $m$ of its entries? In general, everyone would agree that this is impossible without some additional information.
In many instances, however, the matrix we wish to recover is known to be structured in the sense that it is low-rank or approximately low-rank. (We recall for completeness that a matrix with $n_1$ rows and $n_2$ columns has rank $r$ if its rows or columns span an $r$-dimensional space.) Below are two examples of practical scenarios where one would like to be able to recover a low-rank matrix from a sampling of its entries.
- [*The Netflix problem.*]{} In the area of recommender systems, users submit ratings on a subset of entries in a database, and the vendor provides recommendations based on the user’s preferences [@Rennie05; @SrebroThesis]. Because users only rate a few items, one would like to infer their preference for unrated items.
A special instance of this problem is the now famous Netflix problem [@NetflixPrize]. Users (rows of the data matrix) are given the opportunity to rate movies (columns of the data matrix) but users typically rate only very few movies so that there are very few scattered observed entries of this data matrix. Yet one would like to complete this matrix so that the vendor (here Netflix) might recommend titles that any particular user is likely to be willing to order. In this case, the data matrix of all user-ratings may be approximately low-rank because it is commonly believed that only a few factors contribute to an individual’s tastes or preferences.
- [*Triangulation from incomplete data.*]{} Suppose we are given partial information about the distances between objects and would like to reconstruct the low-dimensional geometry describing their locations. For example, we may have a network of low-power wirelessly networked sensors scattered randomly across a region. Suppose each sensor only has the ability to construct distance estimates based on signal strength readings from its nearest fellow sensors. From these noisy distance estimates, we can form a partially observed distance matrix. We can then estimate the true distance matrix whose rank will be equal to two if the sensors are located in a plane or three if they are located in three dimensional space [@Linial95; @So07]. In this case, we only need to observe a few distances per node to have enough information to reconstruct the positions of the objects.
These examples are of course far from exhaustive and there are many other problems which fall in this general category. For instance, we may have some very limited information about a covariance matrix of interest. Yet, this covariance matrix may be low-rank or approximately low-rank because the variables only depend upon a comparably smaller number of factors.
Impediments and solutions
-------------------------
Suppose for simplicity that we wish to recover a square $n \times n$ matrix ${\bm{M}}$ of rank $r$.[^1] Such a matrix ${\bm{M}}$ can be represented by $n^2$ numbers, but it only has $(2 n - r ) r$ degrees of freedom. This fact can be revealed by counting parameters in the singular value decomposition (the number of degrees of freedom associated with the description of the singular values and of the left and right singular vectors). When the rank is small, this is considerably smaller than $n^2$. For instance, when ${\bm{M}}$ encodes a 10-dimensional phenomenon, then the number of degrees of freedom is about $20\, n$ offering a reduction in dimensionality by a factor about equal to $n/20$. When $n$ is large (e.g. in the thousands or millions), the data matrix carries much less information than its ambient dimension suggests. The problem is now whether it is possible to recover this matrix from a sampling of its entries without having to probe all the $n^2$ entries, or more generally collect $n^2$ or more measurements about ${\bm{M}}$.
### Which matrices?
In general, one cannot hope to be able to recover a low-rank matrix from a sample of its entries. Consider the rank-1 matrix ${\bm{M}}$ equal to $$\label{eq:problem1}
{\bm{M}} = {\bm{e}}_1 {\bm{e}}_n^* = \begin{bmatrix}
0 & 0 & \cdots & 0 & 1 \\
0 & 0 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
0 & 0 & \cdots & 0 & 0
\end{bmatrix},$$ where here and throughout, ${\bm{e}}_i$ is the $i$th canonical basis vector in Euclidean space (the vector with all entries equal to 0 but the $i$th equal to 1). This matrix has a 1 in the top-right corner and all the other entries are 0. Clearly this matrix cannot be recovered from a sampling of its entries unless we pretty much see all the entries. The reason is that for most sampling sets, we would only get to see zeros so that we would have no way of guessing that the matrix is not zero. For instance, if we were to see 90% of the entries selected at random, then 10% of the time we would only get to see zeroes.
It is therefore impossible to recover [*all*]{} low-rank matrices from a set of sampled entries but can one recover [*most*]{} of them? To investigate this issue, we introduce a simple model of low-rank matrices. Consider the singular value decomposition (SVD) of a matrix ${\bm{M}}$ $$\label{eq:svd}
{\bm{M}} = \sum_{k = 1}^r \sigma_k {\bm{u}}_k {\bm{v}}_k^*,$$ where the ${\bm{u}}_k$’s and ${\bm{v}}_k$’s are the left and right singular vectors, and the $\sigma_k$’s are the singular values (the roots of the eigenvalues of ${\bm{M}}^*{\bm{M}}$). Then we could think of a [*generic*]{} low-rank matrix as follows: the family $\{{\bm{u}}_k\}_{1 \le k \le r}$ is selected uniformly at random among all families of $r$ orthonormal vectors, and similarly for the the family $\{{\bm{v}}_k\}_{1 \le k \le r}$. The two families may or may not be independent of each other. We make no assumptions about the singular values $\sigma_k$. In the sequel, we will refer to this model as the [*random orthogonal model*]{}. This model is convenient in the sense that it is both very concrete and simple, and useful in the sense that it will help us fix the main ideas. In the sequel, however, we will consider far more general models. The question for now is whether or not one can recover such a generic matrix from a sampling of its entries.
### Which sampling sets?
Clearly, one cannot hope to reconstruct any low-rank matrix ${\bm{M}}$—even of rank $1$—if the sampling set avoids any column or row of ${\bm{M}}$. Suppose that ${\bm{M}}$ is of rank 1 and of the form ${\bm{x}} {\bm{y}}^*$, ${\bm{x}}, {\bm{y}} \in {\mathbb{R}}^n$ so that the $(i,j)$th entry is given by $$M_{ij} = x_i y_j.$$ Then if we do not have samples from the first row for example, one could never guess the value of the first component $x_1$, by any method whatsoever; no information about $x_1$ is observed. There is of course nothing special about the first row and this argument extends to any row or column. To have any hope of recovering an unknown matrix, one needs at least one observation per row and one observation per column.
We have just seen that if the sampling is adversarial, e.g. one observes all of the entries of ${\bm{M}}$ but those in the first row, then one would not even be able to recover matrices of rank $1$. But what happens for most sampling sets? Can one recover a low-rank matrix from almost all sampling sets of cardinality $m$? Formally, suppose that the set $\Omega$ of locations corresponding to the observed entries ($(i,j) \in \Omega$ if $M_{ij}$ is observed) is a set of cardinality $m$ sampled uniformly at random. Then can one recover a generic low-rank matrix $M$, perhaps with very large probability, from the knowledge of the value of its entries in the set $\Omega$?
### Which algorithm?
If the number of measurements is sufficiently large, and if the entries are sufficiently uniformly distributed as above, one might hope that there is only [*one*]{} low-rank matrix with these entries. If this were true, one would want to recover the data matrix by solving the optimization problem $$\label{eq:l0}
\begin{array}{ll}
\textrm{minimize} & \quad {\operatorname{rank}}({\bm{X}})\\
\textrm{subject to} & \quad X_{ij} = M_{ij} \quad (i,j) \in \Omega,
\end{array}$$ where ${\bm{X}}$ is the decision variable and ${\operatorname{rank}}({\bm{X}})$ is equal to the rank of the matrix ${\bm{X}}$. The program is a common sense approach which simply seeks the simplest explanation fitting the observed data. If there were only one low-rank object fitting the data, this would recover ${\bm{M}}$. This is unfortunately of little practical use because this optimization problem is not only NP-hard, but all known algorithms which provide exact solutions require time doubly exponential in the dimension $n$ of the matrix in both theory and practice [@Grigoriev84].
If a matrix has rank $r$, then it has exactly $r$ nonzero singular values so that the rank function in is simply the number of nonvanishing singular values. In this paper, we consider an alternative which minimizes the sum of the singular values over the constraint set. This sum is called the *nuclear norm*, $$\label{eq:nuclear}
\|{\bm{X}}\|_* = \sum_{k = 1}^n \sigma_k({\bm{X}})$$ where, here and below, $\sigma_k({\bm{X}})$ denotes the $k$th largest singular value of ${\bm{X}}$. The heuristic optimization is then given by $$\label{eq:sdp}
\begin{array}{ll}
\textrm{minimize} & \quad \|{\bm{X}}\|_*\\
\textrm{subject to} & \quad X_{ij} = M_{ij} \quad (i,j) \in \Omega.
\end{array}$$ Whereas the rank function counts the number of nonvanishing singular values, the nuclear norm sums their amplitude and in some sense, is to the rank functional what the convex $\ell_1$ norm is to the counting $\ell_0$ norm in the area of sparse signal recovery. The main point here is that the nuclear norm is a convex function and, as we will discuss in Section \[sec:background\] can be optimized efficiently via semidefinite programming.
### A first typical result
Our first result shows that, perhaps unexpectedly, this heuristic optimization recovers a generic ${\bm{M}}$ when the number of randomly sampled entries is large enough. We will prove the following:
\[teo:main\] Let ${\bm{M}}$ be an $n_1 \times n_2$ matrix of rank $r$ sampled from the random orthogonal model, and put $n = \max(n_1, n_2)$. Suppose we observe $m$ entries of ${\bm{M}}$ with locations sampled uniformly at random. Then there are numerical constants $C$ and $c$ such that if $$\label{eq:main1} m \ge C \, n^{5/4} r \log n\, ,$$ the minimizer to the problem is unique and equal to ${\bm{M}}$ with probability at least $1 - c n^{-3}$; that is to say, the semidefinite program $\eqref{eq:sdp}$ recovers all the entries of ${\bm{M}}$ with no error. In addition, if $r \leq n^{1/5}$, then the recovery is exact with probability at least $1 - c n^{-3}$ provided that $$\label{eq:main2} m \ge C \, n^{6/5}r \log n\,.$$
The theorem states that a surprisingly small number of entries are sufficient to complete a generic low-rank matrix. For small values of the rank, e.g. when $r = O(1)$ or $r = O(\log n)$, one only needs to see on the order of $n^{6/5}$ entries (ignoring logarithmic factors) which is considerably smaller than $n^2$—the total number of entries of a squared matrix. The real feat, however, is that the recovery algorithm is tractable and very concrete. Hence the contribution is twofold:
- Under the hypotheses of Theorem \[teo:main\], there is a unique low-rank matrix which is consistent with the observed entries.
- Further, this matrix can be recovered by the convex optimization . In other words, for most problems, the nuclear norm relaxation is [*formally equivalent*]{} to the combinatorially hard rank minimization problem .
Theorem \[teo:main\] is in fact a special instance of a far more general theorem that covers a much larger set of matrices ${\bm{M}}$. We describe this general class of matrices and precise recovery conditions in the next section.
Main results {#sec:main}
------------
As seen in our first example , it is impossible to recover a matrix which is equal to zero in nearly all of its entries unless we see all the entries of the matrix. To recover a low-rank matrix, this matrix cannot be in the null space of the sampling operator giving the values of a subset of the entries. Now it is easy to see that if the singular vectors of a matrix ${\bm{M}}$ are highly concentrated, then ${\bm{M}}$ could very well be in the null-space of the sampling operator. For instance consider the rank-2 symmetric matrix ${\bm{M}}$ given by $${\bm{M}} = \sum_{k = 1}^2 \sigma_k {\bm{u}}_k {\bm{u}}_k^*, \quad
\begin{array}{ll}
{\bm{u}}_1 & = ({\bm{e}}_1 + {\bm{e}}_2)/\sqrt{2},\\
{\bm{u}}_2 & = ({\bm{e}}_1 - {\bm{e}}_2)/\sqrt{2},
\end{array}$$ where the singular values are arbitrary. Then this matrix vanishes everywhere except in the top-left $2 \times 2$ corner and one would basically need to see all the entries of ${\bm{M}}$ to be able to recover this matrix exactly by any method whatsoever. There is an endless list of examples of this sort. Hence, we arrive at the notion that, somehow, the singular vectors need to be sufficiently spread—that is, uncorrelated with the standard basis—in order to minimize the number of observations needed to recover a low-rank matrix.[^2] This motivates the following definition.
\[def:coherence\] Let $U$ be a subspace of $\mathbb{R}^n$ of dimension $r$ and ${\bm{P}}_U$ be the orthogonal projection onto $U$. Then the *coherence* of $U$ (vis-à-vis the standard basis $({\bm{e}}_i)$) is defined to be $$\label{eq:coherence} \mu(U) \equiv \frac{n}{r} \max_{1 \le i \le n}
\|{\bm{P}}_U
{\bm{e}}_i\|^2.$$
Note that for any subspace, the smallest $\mu(U)$ can be is $1$, achieved, for example, if $U$ is spanned by vectors whose entries all have magnitude $1/\sqrt{n}$. The largest possible value for $\mu(U)$ is $n/r$ which would correspond to any subspace that contains a standard basis element. We shall be primarily interested in subspace with low coherence as matrices whose column and row spaces have low coherence cannot really be in the null space of the sampling operator. For instance, we will see that the random subspaces discussed above have nearly minimal coherence.
To state our main result, we introduce two assumptions about an $n_1\times n_2$ matrix ${\bm{M}}$ whose SVD is given by ${\bm{M}} =
\sum_{1 \le k \le r} \sigma_k {\bm{u}}_k {\bm{v}}_k^*$ and with column and row spaces denoted by $U$ and $V$ respectively.
[A0]{}
: The coherences obey $\max(\mu(U), \mu(V)) \le \mu_0$ for some positive $\mu_0$.
[A1]{}
: The $n_1 \times n_2$ matrix $\sum_{1 \le k \le r}
{\bm{u}}_k {\bm{v}}_k^*$ has a maximum entry bounded by $\mu_1
\sqrt{r/(n_1 n_2)}$ in absolute value for some positive $\mu_1$.
The $\mu$’s above may depend on $r$ and $n_1, n_2$. Moreover, note that [**A1**]{} always holds with $\mu_1 = \mu_0 \, \sqrt{r}$ since the $(i,j)$th entry of the matrix $\sum_{1\le k \le r} {\bm{u}}_k
{\bm{v}}_k^*$ is given by $\sum_{1\le k \le r} u_{ik} v_{jk}$ and by the Cauchy-Schwarz inequality, $$\left|\sum_{1\le k \le r} u_{ik} v_{jk}\right| \le \sqrt{\sum_{1\le k
\le r} |u_{ik}|^2}\, \sqrt{\sum_{1\le k \le r} |v_{jk}|^2} \le
\frac{\mu_0 r}{\sqrt{n_1 n_2}}.$$ Hence, for sufficiently small ranks, $\mu_1$ is comparable to $\mu_0$. As we will see in Section \[sec:whichlow\], for larger ranks, both subspaces selected from the uniform distribution and spaces constructed as the span of singular vectors with bounded entries are not only incoherent with the standard basis, but also obey [**A1**]{} with high probability for values of $\mu_1$ at most logarithmic in $n_1$ and/or $n_2$. Below we will assume that $\mu_1$ is greater than or equal to 1.
We are in the position to state our main result: if a matrix has row and column spaces that are incoherent with the standard basis, then nuclear norm minimization can recover this matrix from a random sampling of a small number of entries.
\[teo:main2\] Let ${\bm{M}}$ be an $n_1 \times n_2$ matrix of rank $r$ obeying [**A0**]{} and [**A1**]{} and put $n = \max(n_1,n_2)$. Suppose we observe $m$ entries of ${\bm{M}}$ with locations sampled uniformly at random. Then there exist constants $C$, $c$ such that if $$\label{eq:main3}
m \ge \, C
\max(\mu_1^2, \mu_0^{1/2} \mu_1, \mu_0 n^{1/4}) \, n r (\beta \log n)$$ for some $\beta>2$, then the minimizer to the problem is unique and equal to ${\bm{M}}$ with probability at least $1 - c
n^{-\beta}$. For $r \le \mu_0^{-1} n^{1/5}$ this estimate can be improved to $$\label{eq:main4} m \ge C \, \mu_0
\, n^{6/5} r (\beta \log n)$$ with the same probability of success.
Theorem \[teo:main2\] asserts that if the coherence is low, few samples are required to recover ${\bm{M}}$. For example, if $\mu_0 =
O(1)$ and the rank is not too large, then the recovery is exact with large probability provided that $$\label{eq:incoherent-teo} m \ge C \, n^{6/5} r \log n\,.$$ We give two illustrative examples of matrices with incoherent column and row spaces. This list is by no means exhaustive.
1. The first example is the random orthogonal model. For values of the rank $r$ greater than $\log n$, $\mu(U)$ and $\mu(V)$ are $O(1)$, $\mu_1 = O(\log n)$ both with very large probability. Hence, the recovery is exact provided that $m$ obeys or . Specializing Theorem \[teo:main2\] to these values of the parameters gives Theorem \[teo:main\]. Hence, Theorem \[teo:main\] is a special case of our general recovery result.
2. The second example is more general and, in a nutshell, simply requires that the components of the singular vectors of ${\bm{M}}$ are small. Assume that the ${\bm{u}}_j$ and ${\bm{v}}_j$’s obey $$\label{eq:muB}
\max_{ij} |{\langle}{\bm{e}}_i, {\bm{u}}_j{\rangle}|^2 \le \mu_B/n, \quad \max_{ij} |{\langle}{\bm{e}}_i, {\bm{v}}_j{\rangle}|^2
\le \mu_B/n,$$ for some value of $\mu_B=O(1)$. Then the maximum coherence is at most $\mu_B$ since $\mu(U) \le \mu_B$ and $\mu(V) \le \mu_B$. Further, we will see in Section \[sec:whichlow\] that ${\bf A1}$ holds most of the time with $\mu_1 = O(\sqrt{\log n})$. Thus, for matrices with singular vectors obeying , the recovery is exact provided that $m$ obeys for values of the rank not exceeding $\mu_B^{-1} n^{1/5}$.
Extensions {#sec:extensions}
----------
Our main result (Theorem \[teo:main2\]) extends to a variety of other low-rank matrix completion problems beyond the sampling of entries. Indeed, suppose we have two orthonormal bases ${\bm{f}}_1,\ldots, {\bm{f}}_{n}$ and ${\bm{g}}_1,\ldots, {\bm{g}}_{n}$ of ${\mathbb{R}}^n$, and that we are interested in solving the rank minimization problem $$\label{eq:rank-min-prob-general}
\begin{array}{ll}
\textrm{minimize} & \quad \textrm{rank}({\bm{X}})\\
\textrm{subject to} & \quad {\bm{f}}_i^* {\bm{X}} {\bm{g}}_j =
{\bm{f}}_i^* {\bm{M}} {\bm{g}}_j, \quad (i,j) \in \Omega,
\end{array}\,.$$ This comes up in a number of applications. As a motivating example, there has been a great deal of interest in the machine learning community in developing specialized algorithms for the *multiclass* and *multitask* learning problems (see, e.g., [@Abernethy06; @Argyriou07; @Amit07]). In multiclass learning, the goal is to build multiple classifiers with the same training data to distinguish between more than two categories. For example, in face recognition, one might want to classify whether an image patch corresponds to an eye, nose, or mouth. In multitask learning, we have a large set of data, but have a variety of different classification tasks, and, for each task, only partial subsets of the data are relevant. For instance, in activity recognition, we may have acquired sets of observations of multiple subjects and want to determine if each observed person is walking or running. However, a different classifier is to be learned for each individual, and it is not clear how having access to the full collection of observations can improve classification performance. Multitask learning aims precisely to take advantage of the access to the full database to improve performance on the individual tasks.
In the abstract formulation of this problem for linear classifiers, we have $K$ classes to distinguish and are given training examples ${\bm{f}}_1, \ldots, {\bm{f}}_n$. For each example, we are given partial labeling information about which classes it belongs or does not belong to. That is, for each example ${\bm{f}}_j$ and class $k$, we may either be told that ${\bm{f}}_j$ belongs to class $k$, be told ${\bm{f}}_j$ does not belong to class $k$, or provided no information about the membership of ${\bm{f}}_j$ to class $k$. For each class $1
\le k \le K$, we would like to produce a linear function ${\bm{w}}_k$ such that ${\bm{w}}_k^* {\bm{f}}_i
> 0$ if ${\bm{f}}_i$ belongs to class $k$ and ${\bm{w}}_k^* {\bm{f}}_i
< 0$ otherwise. Formally, we can search for the vector ${\bm{w}}_k$ that satisfies the equality constraints ${\bm{w}}_k^* {\bm{f}}_i =
y_{ik}$ where $y_{ik}=1$ if we are told that ${\bm{f}}_i$ belongs to class $k$, $y_{ik}=-1$ if we are told that ${\bm{f}}_i$ does not belong to class $k$, and $y_{ik}$ unconstrained if we are not provided information. A common hypothesis in the multitask setting is that the ${\bm{w}}_k$ corresponding to each of the classes together span a very low dimensional subspace with dimension significantly smaller than $K$ [@Abernethy06; @Argyriou07; @Amit07]. That is, the basic assumption is that $${\bm{W}} = [{\bm{w}}_1, \ldots, {\bm{w}}_K]$$ is low-rank. Hence, the multiclass learning problem can be cast as with observations of the form ${\bm{f}}_i^* {\bm{W}} {\bm{e}}_j$.
To see that our theorem provides conditions under which can be solved via nuclear norm minimization, note that there exist unitary transformations ${\bm{F}}$ and ${\bm{G}}$ such that ${\bm{e}}_j = {\bm{F}} {\bm{f}}_j$ and ${\bm{e}}_j = {\bm{G}} {\bm{g}}_j$ for each $j = 1,\ldots, n$. Hence, $${\bm{f}}_i^* {\bm{X}} {\bm{g}}_j = {\bm{e}}_i^* ({\bm{F}} {\bm{X}} {\bm{G}}^*) {\bm{e}}_j.$$ Then if the conditions of Theorem \[teo:main2\] hold for the matrix ${\bm{F}} {\bm{X}} {\bm{G}}^*$, it is immediate that nuclear norm minimization finds the unique optimal solution of when we are provided a large enough random collection of the inner products ${\bm{f}}_i^* {\bm{M}}
{\bm{g}}_j$. In other words, all that is needed is that the column and row spaces of ${\bm{M}}$ be respectively incoherent with the basis $({\bm{f}}_i)$ and $({\bm{g}}_i)$.
From this perspective, we additionally remark that our results likely extend to the case where one observes a small number of arbitrary linear functionals of a hidden matrix ${\bm{M}}$. Set $N= n^2$ and ${\bm{A}}_1,\ldots, {\bm{A}}_{N}$ be an orthonormal basis for the linear space of $n \times n$ matrices with the usual inner product ${\langle}{\bm{X}}, {\bm{Y}}{\rangle}= {\operatorname{trace}}({\bm{X}}^* {\bm{Y}})$. Then we expect our results should also apply to the rank minimization problem $$\label{eq:generalrank}
\begin{array}{ll}
\textrm{minimize} & \quad \textrm{rank}({\bm{X}})\\
\textrm{subject to} & \quad {\langle}{\bm{A}}_k, {\bm{X}}{\rangle}=
{\langle}{\bm{A}}_k, {\bm{M}}{\rangle}\quad k \in \Omega,
\end{array}$$ where $\Omega \subset\{1,\ldots, N\}$ is selected uniformly at random. In fact, is when the orthobasis is the canonical basis $({\bm{e}}_i {\bm{e}}_j^*)_{1 \le i, j \le
n}$. Here, those low-rank matrices which have small inner product with all the basis elements ${\bm{A}}_k$ may be recoverable by nuclear norm minimization. To avoid unnecessary confusion and notational clutter, we leave this general low-rank recovery problem for future work.
Connections, alternatives and prior art {#sec:background}
---------------------------------------
Nuclear norm minimization is a recent heuristic introduced by Fazel in [@FazelThesis], and is an extension of the trace heuristic often used by the control community, see e.g. [@Beck98; @Mesbahi97]. Indeed, when the matrix variable is symmetric and positive semidefinite, the nuclear norm of ${\bm{X}}$ is the sum of the (nonnegative) eigenvalues and thus equal to the trace of ${\bm{X}}$. Hence, for positive semidefinite unknowns, would simply minimize the trace over the constraint set: $$\begin{array}{ll}
\textrm{minimize} & \quad {\operatorname{trace}}({\bm{X}})\\
\textrm{subject to} & \quad X_{ij} = M_{ij} \quad (i,j) \in \Omega\\
& \quad {\bm{X}} \succeq 0
\end{array}.$$ This is a semidefinite program. Even for the general matrix ${\bm{M}}$ which may not be positive definite or even symmetric, the nuclear norm heuristic can be formulated in terms of semidefinite programming as, for instance, the program is equivalent to $$\begin{array}{ll}
\textrm{minimize} & \quad {\operatorname{trace}}({\bm{W}}_1) + {\operatorname{trace}}({\bm{W}}_2)\\
\textrm{subject to} & \quad X_{ij} = M_{ij} \quad (i,j) \in \Omega\\
& \quad \begin{bmatrix} {\bm{W}}_1 & {\bm{X}}\\
{\bm{X}}^* & {\bm{W}}_2
\end{bmatrix} \succeq 0
\end{array}$$ with optimization variables ${\bm{X}}$, ${\bm{W}}_1$ and ${\bm{W}}_2$, (see, e.g., [@FazelThesis; @Vandenberghe96]). There are many efficient algorithms and high-quality software available for solving these types of problems.
Our work is inspired by results in the emerging field of [*compressive sampling*]{} or [*compressed sensing*]{}, a new paradigm for acquiring information about objects of interest from what appears to be a highly incomplete set of measurements [@CRT06; @OptimalRecovery; @DonohoCS]. In practice, this means for example that high-resolution imaging is possible with fewer sensors, or that one can speed up signal acquisition time in biomedical applications by orders of magnitude, simply by taking far fewer specially coded samples. Mathematically speaking, we wish to reconstruct a signal ${\bm{x}}\in {\mathbb{R}}^n$ from a small number measurements ${\bm{y}} = \Phi {\bm{x}}$, ${\bm{y}} \in {\mathbb{R}}^m$, and $m$ is much smaller than $n$; i.e. we have far fewer equations than unknowns. In general, one cannot hope to reconstruct ${\bm{x}}$ but assume now that the object we wish to recover is known to be structured in the sense that it is sparse (or approximately sparse). This means that the unknown object depends upon a smaller number of unknown parameters. Then it has been shown that $\ell_1$ minimization allows recovery of sparse signals from remarkably few measurements: supposing $\Phi$ is chosen randomly from a suitable distribution, then with very high probability, all sparse signals with about $k$ nonzero entries can be recovered from on the order of $k \log n$ measurements. For instance, if ${\bm{x}}$ is $k$-sparse in the Fourier domain, i.e. ${\bm{x}}$ is a superposition of $k$ sinusoids, then it can be perfectly recovered with high probability—by $\ell_1$ minimization—from the knowledge of about $k \log n$ of its entries sampled uniformly at random [@CRT06].
From this viewpoint, the results in this paper greatly extend the theory of compressed sensing by showing that other types of interesting objects or structures, beyond sparse signals and images, can be recovered from a limited set of measurements. Moreover, the techniques for proving our main results build upon ideas from the compressed sensing literature together with probabilistic tools such as the powerful techniques of Bourgain and of Rudelson for bounding norms of operators between Banach spaces.
Our notion of incoherence generalizes the concept of the same name in compressive sampling. Notably, in [@CR07], the authors introduce the notion of the incoherence of a unitary transformation. Letting ${\bm{U}}$ be an $n \times n$ unitary matrix, the *coherence* of ${\bm{U}}$ is given by $$\mu({\bm{U}}) = n\max_{j,k} |U_{jk}|^2.$$ This quantity ranges in values from $1$ for a unitary transformation whose entries all have the same magnitude to $n$ for the identity matrix. Using this notion, [@CR07] showed that with high probability, a $k$-sparse signal could be recovered via linear programming from the observation of the inner product of the signal with $m = \Omega(\mu({\bm{U}}) k\, \log n)$ randomly selected columns of the matrix ${\bm{U}}$. This result provided a generalization of the celebrated results about partial Fourier observations described in [@CRT06], a special case where $\mu({\bm{U}})=1$. This paper generalizes the notion of incoherence to problems beyond the setting of sparse signal recovery.
In [@Recht07], the authors studied the nuclear norm heuristic applied to a related problem where partial information about a matrix ${\bm{M}}$ is available from $m$ equations of the form $$\label{eq:linearfunctional}
{\langle}{\bm{A}}^{(k)}, M{\rangle}= \sum_{ij} A^{(k)}_{ij} M_{ij} = b_k, \qquad k = 1, \ldots, m,$$ where for each $k$, $\{A^{(k)}_{ij}\}_{ij}$ is an i.i.d. sequence of Gaussian or Bernoulli random variables and the sequences $\{{\bm{A}}^{(k)}\}$ are also independent from each other (the sequences $\{{\bm{A}}^{(k)}\}$ and $\{b_k\}$ are available to the analyst). Building on the concept of [*restricted isometry*]{} introduced in [@Candes05] in the context of sparse signal recovery, [@Recht07] establishes the first sufficient conditions for which the nuclear norm heuristic returns the minimum rank element in the constraint set. They prove that the heuristic succeeds with large probability whenever the number $m$ of available measurements is greater than a constant times $2 n r \log n$ for $n \times n$ matrices. Although this is an interesting result, a serious impediment to this approach is that one needs to essentially measure random projections of the unknown data matrix—a situation which unfortunately does not commonly arise in practice. Further, the measurements in give some information about [*all*]{} the entries of ${\bm{M}}$ whereas in our problem, information about most of the entries is simply not available. In particular, the results and techniques introduced in [@Recht07] do not begin to address the matrix completion problem of interest to us in this paper. As a consequence, our methods are completely different; for example, they do not rely on any notions of restricted isometry. Instead, as we discuss below, we prove the existence of a Lagrange multiplier for the optimization (\[eq:sdp\]) that certifies the unique optimal solution is precisely the matrix that we wish to recover.
Finally, we would like to briefly discuss the possibility of other recovery algorithms when the sampling happens to be chosen in a very special fashion. For example, suppose that ${\bm{M}}$ is generic and that we precisely observe every entry in the first $r$ rows and columns of the matrix. Write ${\bm{M}}$ in block form as $${\bm{M}} = \left[\begin{array}{cc} {\bm{M}}_{11} & {\bm{M}}_{12}\\ {\bm{M}}_{21} & {\bm{M}}_{22}
\end{array}\right]$$ with ${\bm{M}}_{11}$ an $r\times r$ matrix. In the special case that ${\bm{M}}_{11}$ is invertible and ${\bm{M}}$ has rank $r$, then it is easy to verify that ${\bm{M}}_{22} =
{\bm{M}}_{21}{\bm{M}}_{11}^{-1}{\bm{M}}_{12}$. One can prove this identity by forming the SVD of ${\bm{M}}$, for example. That is, if ${\bm{M}}$ is generic, and the upper $r\times r$ block is invertible, and we observe *every* entry in the first $r$ rows and columns, we can recover ${\bm{M}}$. This result immediately generalizes to the case where one observes precisely $r$ rows and $r$ columns and the $r \times r$ matrix at the intersection of the observed rows and columns is invertible. However, this scheme has many practical drawbacks that stand in the way of a generalization to a completion algorithm from a general set of entries. First, if we miss *any* entry in these rows or columns, we cannot recover ${\bm{M}}$, nor can we leverage any information provided by entries of ${\bm{M}}_{22}$. Second, if the matrix has rank less than $r$, and we observe $r$ rows and columns, a combinatorial search to find the collection that has an invertible square sub-block is required. Moreover, because of the matrix inversion, the algorithm is rather fragile to noise in the entries.
Notations and organization of the paper {#sec:notations}
---------------------------------------
The paper is organized as follows. We first argue in Section \[sec:whichlow\] that the random orthogonal model and, more generally, matrices with incoherent column and row spaces obey the assumptions of the general Theorem \[teo:main2\]. To prove Theorem \[teo:main2\], we first establish sufficient conditions which guarantee that the true low-rank matrix ${\bm{M}}$ is the unique solution to in Section \[sec:duality\]. One of these conditions is the existence of a dual vector obeying two crucial properties. Section \[sec:architecture\] constructs such a dual vector and provides the overall architecture of the proof which shows that, indeed, this vector obeys the desired properties provided that the number of measurements is sufficiently large. Surprisingly, as explored in Section \[sec:coupon\], the existence of a dual vector certifying that ${\bm{M}}$ is unique is related to some problems in random graph theory including “the coupon collector’s problem.” Following this discussion, we prove our main result via several intermediate results which are all proven in Section \[sec:proofs\]. Section \[sec:numerical\] introduces numerical experiments showing that matrix completion based on nuclear norm minimization works well in practice. Section \[sec:discussion\] closes the paper with a short summary of our findings, a discussion of important extensions and improvements. In particular, we will discuss possible ways of improving the 1.2 exponent in so that it gets closer to 1. Finally, the Appendix provides proofs of auxiliary lemmas supporting our main argument.
Before continuing, we provide here a brief summary of the notations used throughout the paper. Matrices are bold capital, vectors are bold lowercase and scalars or entries are not bold. For instance, ${\bm{X}}$ is a matrix and $X_{ij}$ its $(i,j)$th entry. Likewise ${\bm{x}}$ is a vector and $x_i$ its $i$th component. When we have a collection of vectors ${\bm{u}}_k\in {\mathbb{R}}^n$ for $1\leq k \leq d$, we will denote by $u_{ik}$ the $i$th component of the vector ${\bm{u}}_k$ and $[{\bm{u}}_1,\ldots,{\bm{u}}_d]$ will denote the $n\times d$ matrix whose $k$th column is ${\bm{u}}_k$.
A variety of norms on matrices will be discussed. The spectral norm of a matrix is denoted by $\|{\bm{X}}\|$. The Euclidean inner product between two matrices is ${\langle}{\bm{X}}, {\bm{Y}}{\rangle}= {\operatorname{trace}}({\bm{X}}^*
{\bm{Y}})$, and the corresponding Euclidean norm, called the Frobenius or Hilbert-Schmidt norm, is denoted $\|{\bm{X}}\|_F$. That is, $\|{\bm{X}}\|_F={\langle}{\bm{X}},{\bm{X}}{\rangle}^{1/2}$. The nuclear norm of a matrix ${\bm{X}}$ is $\|{\bm{X}}\|_*$. The maximum entry of ${\bm{X}}$ (in absolute value) is denoted by $\|{\bm{X}}\|_\infty \equiv \max_{ij}
|X_{ij}|$. For vectors, we will only consider the usual Euclidean $\ell_2$ norm which we simply write as $\|{\bm{x}}\|$.
Further, we will also manipulate linear transformation which acts on matrices and will use caligraphic letters for these operators as in ${\cal A}({\bm{X}})$. In particular, the identity operator will be denoted by ${\mathcal{I}}$. The only norm we will consider for these operators is their spectral norm (the top singular value) denoted by $\|{\cal A}\| = \sup_{{\bm{X}} :
\|{\bm{X}}\|_F \le 1} \, \|{\cal A}({\bm{X}})\|_F$.
Finally, we adopt the convention that $C$ denotes a numerical constant independent of the matrix dimensions, rank, and number of measurements, whose value may change from line to line. Certain special constants with precise numerical values will be ornamented with subscripts (e.g., $C_R$). Any exceptions to this notational scheme will be noted in the text.
Which matrices are incoherent? {#sec:whichlow}
==============================
In this section we restrict our attention to square $n \times n$ matrices, but the extension to rectangular $n_1\times n_2$ matrices immediately follows by setting $n=\max(n_1,n_2)$.
Incoherent bases span incoherent subspaces
------------------------------------------
Almost all $n\times n$ matrices ${\bm{M}}$ with singular vectors $\{{\bm{u}}_k\}_{1 \le k \le r}$ and $\{{\bm{v}}_k\}_{1 \le k \le r}$ obeying the size property also satisfy the assumptions [**A0**]{} and [**A1**]{} with $\mu_0 = \mu_B$, $\mu_1 = C
\mu_B \sqrt{\log n}$ for some positive constant $C$. As mentioned above, [**A0**]{} holds automatically, but, observe that [**A1**]{} would not hold with a small value of $\mu_1$ if two rows of the matrices $[{\bm{u}}_1, \ldots, {\bm{u}}_r]$ and $[{\bm{v}}_1, \ldots,
{\bm{v}}_r]$ are identical with all entries of magnitude $\sqrt{\mu_B/n}$ since it is not hard to see that in this case $$\|\sum_k {\bm{u}}_k {\bm{v}}_k^*\|_\infty = \mu_B \, r/n.$$ Certainly, this example is constructed in a very special way, and should occur infrequently. We now show that it is generically unlikely.
Consider the matrix $$\label{eq:example-inc-svd}
\sum_{k=1}^r \epsilon_k {\bm{u}}_k {\bm{v}}_k^*,$$ where $\{\epsilon_k\}_{1 \le k \le r}$ is an arbitrary sign sequence. For almost all choices of sign sequences, [**A1**]{} is satisfied with $\mu_1 = O(\mu_B \sqrt{\log n})$. Indeed, if one selects the signs uniformly at random, then for each $\beta > 0$, $$\label{eq:incoherent1}
{\operatorname{\mathbb{P}}}(\| \sum_{k=1}^r \epsilon_k {\bm{u}}_k {\bm{v}}_k\|_\infty \ge \mu_B \, \sqrt{8\beta r \log n}/n) \le (2n^2) \, n^{-\beta}.$$ This is of interest because suppose the low-rank matrix we wish to recover is of the form $$\label{eq:example-inc-svd2}
{\bm{M}} = \sum_{k=1}^r \lambda_k {\bm{u}}_k {\bm{v}}_k^*$$ with scalars $\lambda_k$. Since the vectors $\{{\bm{u}}_k\}$ and $\{{\bm{v}}_k\}$ are orthogonal, the singular values of ${\bm{M}}$ are given by $|\lambda_k|$ and the singular vectors are given by ${\textrm{sgn}}(\lambda_k) {\bm{u}}_k$ and ${\bm{v}}_k$ for $k = 1,\ldots, r$. Hence, in this model [**A1**]{} concerns the maximum entry of the matrix given by with $\epsilon_k={\textrm{sgn}}(\lambda_k)$. That is to say, for most sign patterns, the matrix of interest obeys an appropriate size condition. We emphasize here that the only thing that we assumed about the ${\bm{u}}_k$’s and ${\bm{v}}_k$’s was that they had small entries. In particular, they could be equal to each other as would be the case for a symmetric matrix.
The claim is a simple application of Hoeffding’s inequality. The $(i,j)$th entry of is given by $$Z_{ij} = \sum_{1 \le k \le r} \epsilon_k u_{ik} v_{jk},$$ and is a sum of $r$ zero-mean independent random variables, each bounded by $\mu_B/n$. Therefore, $${\operatorname{\mathbb{P}}}(|Z_{ij}| \geq \lambda \mu_B\sqrt{r}/n) \leq 2e^{-\lambda^2/8}.$$ Setting $\lambda$ proportional to $\sqrt{\log n}$ and applying the union bound gives the claim.
To summarize, we say that ${\bm{M}}$ is sampled from the [ *incoherent basis model*]{} if it is of the form $$\label{eq:incoherentmodel}
{\bm{M}} = \sum_{k=1}^r \epsilon_k \sigma_k {\bm{u}}_k {\bm{v}}_k^*;$$ $\{\epsilon_k\}_{1 \le k \le r}$ is a random sign sequence, and $\{{\bm{u}}_k\}_{1 \le k \le r}$ and $\{{\bm{v}}_k\}_{1 \le k \le r}$ have maximum entries of size at most $\sqrt{\mu_B/n}$.
\[teo:incoherentbases\] There exist numerical constants $c$ and $C$ such that for any $\beta > 0$, matrices from the incoherent basis model obey the assumption [**A1**]{} with $\mu_1 \leq C\mu_B
\sqrt{(\beta + 2) \log n}$ with probability at least $1-c n^{-\beta}$.
Random subspaces span incoherent subspaces
------------------------------------------
In this section, we prove that the random orthogonal model obeys the two assumptions [**A0**]{} and [**A1**]{} (with appropriate values for the $\mu$’s) with large probability.
\[teo:gaussian\] Set $\bar r = \max(r,\log n)$. Then there exist constants $C$ and $c$ such that the random orthogonal model obeys:[^3]
1. $\max_i \|{\bm{P}}_U {\bm{e}}_i\|^2 \le C \, \bar r/n$,
2. $\|\sum_{1 \le k \le r} {\bm{u}}_k {\bm{v}}_k^*\|_\infty \le C
\, \log n \, \sqrt{\bar r}/n$.
with probability $1- c n^{-3} \log n$.
We note that an argument similar to the following proof would give that if $C$ of the form $K \beta$ where $K$ is a fixed numerical constant, we can achieve a probability at least $1 - c n^{-\beta}$ provided that $n$ is sufficiently large. To establish these facts, we make use of the standard result below [@LaurentMassart].
Let $Y_d$ be distributed as a chi-squared random variable with $d$ degrees of freedom. Then for each $t > 0$ $$\label{eq:chi2}
{\operatorname{\mathbb{P}}}(Y_d - d \ge t \, \sqrt{2d} + t^2)
\le e^{-t^2/2} \quad
\text{ and } \quad {\operatorname{\mathbb{P}}}(Y_d - d \le -t \,
\sqrt{2d}) \le e^{-t^2/2}.$$
We will use as follows: for each $\epsilon \in (0,1)$ we have $$\label{eq:chi2b}
{\operatorname{\mathbb{P}}}(Y_d \ge d \, (1-\epsilon)^{-1}) \le e^{-\epsilon^2 d/4} \quad
\text{ and } \quad {\operatorname{\mathbb{P}}}(Y_d \le d \, (1-\epsilon)) \le e^{-\epsilon^2 d/4}.$$
We begin with the second assertion of Lemma \[teo:gaussian\] since it will imply the first as well. Observe that it follows from $$\label{eq:PU}
\|{\bm{P}}_U {\bm{e}}_i\|^2 = \sum_{1 \le k \le r} u_{ik}^2,$$ that $Z_r \equiv \|{\bm{P}}_U {\bm{e}}_i\|^2$ ($a$ is fixed) is the squared Euclidean length of the first $r$ components of a unit vector uniformly distributed on the unit sphere in $n$ dimensions. Now suppose that $x_1, x_2, \ldots, x_n$ are i.i.d. $N(0,1)$. Then the distribution of a unit vector uniformly distributed on the sphere is that of ${\bm{x}}/\|{\bm{x}}\|$ and, therefore, the law of $Z_r$ is that of $Y_r/Y_n$, where $Y_r = \sum_{k \le r} x_k^2$. Fix $\epsilon > 0$ and consider the event $A_{n,\epsilon} = \{Y_n/n \ge
1-\epsilon\}$. For each $\lambda > 0$, it follows from that $$\begin{aligned}
{\operatorname{\mathbb{P}}}(Z_r - r/n \ge \lambda \sqrt{2r}/n) & = {\operatorname{\mathbb{P}}}(Y_r \ge
[r+\lambda\sqrt{2r}] Y_n/n)\\
& \le {\operatorname{\mathbb{P}}}(Y_r \ge
[r+\lambda\sqrt{2r}] Y_n/n \text{ and } A_{n,\epsilon}) + {\operatorname{\mathbb{P}}}(A_{n,\epsilon}^c)\\
& \le {\operatorname{\mathbb{P}}}(Y_r \ge [r+\lambda\sqrt{2r}][1-\epsilon]) + e^{-\epsilon^2 n/4}\\
& = {\operatorname{\mathbb{P}}}(Y_r - r \ge \lambda\sqrt{2r}[1- \epsilon - \epsilon
\sqrt{r/2\lambda^2}]) + e^{-\epsilon^2 n/4}.\end{aligned}$$ Now pick $\epsilon = 4 (n^{-1} \log n)^{1/2}$, $\lambda = 8\sqrt{2\log
n}$ and assume that $n$ is sufficiently large so that $$\epsilon(1+ \sqrt{r/2\lambda^2}) \le 1/2.$$ Then $${\operatorname{\mathbb{P}}}(Z_r - r/n \ge \lambda \sqrt{2r}/n) \le {\operatorname{\mathbb{P}}}(Y_r - r \ge (\lambda/2)
\sqrt{2r}) + n^{-4}.$$ Assume now that $r \ge 4\log n$ (which means that $\lambda \le 4
\sqrt{2r}$). Then it follows from that $${\operatorname{\mathbb{P}}}(Y_r - r \ge (\lambda/2)
\sqrt{2r}) \le {\operatorname{\mathbb{P}}}(Y_r - r \ge (\lambda/4)
\sqrt{2r} + (\lambda/4)^2) \le e^{-\lambda^2/32} = n^{-4}.$$ Hence $${\operatorname{\mathbb{P}}}(Z_r - r/n \ge 16 \sqrt{r \log n}/n) \le 2n^{-4}$$ and, therefore, $$\label{eq:conclusion1} {\operatorname{\mathbb{P}}}(\max_i \|{\bm{P}}_U {\bm{e}}_i\|^2 - r/n \ge
16 \sqrt{r \log n}/n) \le 2 n^{-3}$$ by the union bound. Note that establishes the first claim of the lemma (even for $r < 4 \log n$ since in this case $Z_r \le Z_{\lceil 4\log n \rceil}$).
It remains to establish the second claim. Notice that by symmetry, ${\bm{E}} = \sum_{1 \le k \le r} {\bm{u}}_k {\bm{v}}_k^*$ has the same distribution as $${\bm{F}} = \sum_{k = 1}^r \epsilon_k {\bm{u}}_k {\bm{v}}_k^*,$$ where $\{\epsilon_k\}$ is an independent Rademacher sequence. It then follows from Hoeffding’s inequality that conditional on $\{{\bm{u}}_k\}$ and $\{{\bm{v}}_k\}$ we have $${\operatorname{\mathbb{P}}}(|F_{ij}| > t) \le 2 e^{-t^2/2\sigma_{ij}^2}, \quad \sigma_{ij}^2
= \sum_{1 \le k \le r} u_{ik}^2 v_{ik}^2.$$ Our previous results indicate that $\max_{ij} |v_{ij}|^2 \le (10\log
n)/n$ with large probability and thus $$\sigma^2_{ij} \le 10 \, \frac{\log n}{n} \, \|{\bm{P}}_U
{\bm{e}}_i\|^2.$$ Set $\bar r = \max(r,\log n)$. Since $\|{\bm{P}}_U {\bm{e}}_i\|^2 \le
C \bar r/n$ with large probability, we have $$\sigma^2_{ij} \le C (\log n) \, \bar r/n^2$$ with large probability. Hence the marginal distribution of $F_{ij}$ obeys $${\operatorname{\mathbb{P}}}(|F_{ij}| > \lambda \sqrt{\bar r}/n) \le 2 e^{-\gamma
\lambda^2/\log n} + {\operatorname{\mathbb{P}}}(\sigma_{ij}^2 \ge C (\log n) {\bar r}/n^2).$$ for some numerical constant $\gamma$. Picking $\lambda = \gamma' \log
n$ where $\gamma'$ is a sufficiently large numerical constant gives $$\|{\bm{F}}\|_\infty \le C\, (\log n) \, \sqrt{\bar r}/n$$ with large probability. Since ${\bm{E}}$ and ${\bm{F}}$ have the same distribution, the second claim follows.
The claim about the size of $\max_{ij} |v_{ij}|^2$ is straightforward since our techniques show that for each $\lambda > 0$ $${\operatorname{\mathbb{P}}}(Z_1 \ge \lambda (\log n)/n) \le\ {\operatorname{\mathbb{P}}}(Y_1 \ge \lambda (1-\epsilon) \log
n) + e^{-\epsilon^2 n/4}.$$ Moreover, $${\operatorname{\mathbb{P}}}(Y_1 \ge \lambda (1-\epsilon) \log n) = {\operatorname{\mathbb{P}}}(|x_1| \ge
\sqrt{\lambda(1-\epsilon) \log n}) \le 2e^{-\frac{1}{2}\lambda
(1-\epsilon) \log
n}.$$ If $n$ is sufficiently large so that $\epsilon \le 1/5$, this gives ${\operatorname{\mathbb{P}}}(Z_1 \ge 10 (\log n)/n) \le 3n^{-4}$ and, therefore, $${\operatorname{\mathbb{P}}}(\max_{ij} |v_{ij}|^2 \ge 10 (\log n)/n) \le 12 n^{-3} \log n$$ since the maximum is taken over at most $4 n \log n$ pairs.
Duality {#sec:duality}
=======
Let ${\cal R}_\Omega:{\mathbb{R}}^{n_1\times n_2}\rightarrow {\mathbb{R}}^{|\Omega|}$ be the sampling operator which extracts the observed entries, ${\cal
R}_\Omega({\bm{X}}) = (X_{ij})_{ij \in
\Omega}$, so that the constraint in becomes ${\cal
R}_\Omega({\bm{X}}) = {\cal R}_\Omega({\bm{M}})$. Standard convex optimization theory asserts that ${\bm{X}}$ is solution to if there exists a dual vector (or Lagrange multiplier) $\lambda\in{\mathbb{R}}^{|\Omega|}$ such that ${\cal R}_\Omega^* \, \lambda$ is a [*subgradient*]{} of the nuclear norm at the point ${\bm{X}}$, which we denote by $$\label{eq:kkt}
{\cal R}_\Omega^* \, \lambda \in \partial \|{\bm{X}}\|_*$$ (see, e.g. [@BertsekasConvexBook]). Recall the definition of a subgradient of a convex function $f : {\mathbb{R}}^{n_1
\times n_2} {\rightarrow}{\mathbb{R}}$. We say that ${\bm{Y}}$ is a subgradient of $f$ at ${\bm{X}}_0$, denoted ${\bm{Y}} \in \partial f({\bm{X}}_0)$, if $$\label{eq:subgradient}
f({\bm{X}}) \ge f({\bm{X}}_0) + {\langle}{\bm{Y}}, {\bm{X}} - {\bm{X}}_0{\rangle}$$ for all ${\bm{X}}$.
Suppose ${\bm{X}}_0 \in {\mathbb{R}}^{n_1 \times n_2}$ has rank $r$ with a singular value decomposition given by $$\label{eq:near-svd}
{\bm{X}}_0 = \sum_{1 \le k \le r} \sigma_k \, {\bm{u}}_k {\bm{v}}_k^*,$$ With these notations, ${\bm{Y}}$ is a subgradient of the nuclear norm at ${\bm{X}}_0$ if and only if it is of the form $$\label{eq:sub-decomp}
{\bm{Y}} = \sum_{1 \le k \le r} {\bm{u}}_k {\bm{v}}_k^* + {\bm{W}},$$ where ${\bm{W}}$ obeys the following two properties:
- the column space of ${\bm{W}}$ is orthogonal to $U \equiv
{\operatorname{span}{({\bm{u}}_1, \ldots, {\bm{u}}_r)}}$, and the row space of ${\bm{W}}$ is orthogonal to $V \equiv {\operatorname{span}{({\bm{v}}_1, \ldots, {\bm{v}}_r)}}$;
- the spectral norm of ${\bm{W}}$ is less than or equal to 1.
(see, e.g., [@Lewis03; @Watson92]). To express these properties concisely, it is convenient to introduce the orthogonal decomposition ${\mathbb{R}}^{n_1 \times n_2} = T \oplus T^\perp$ where $T$ is the linear space spanned by elements of the form ${\bm{u}}_k {\bm{x}}^*$ and ${\bm{y}}
{\bm{v}}_k^*$, $1 \le k \le r$, where ${\bm{x}}$ and ${\bm{y}}$ are arbitrary, and $T^\perp$ is its orthogonal complement. Note that $\textrm{dim}(T) = r(n_1+n_2-r)$, precisely the number of degrees of freedom in the set of $n_1 \times n_2$ matrices of rank $r$. $T^\perp$ is the subspace of matrices spanned by the family $({\bm{x}}{\bm{y}}^*)$, where ${\bm{x}}$ (respectively ${\bm{y}}$) is any vector orthogonal to $U$ (respectively $V$).
The orthogonal projection ${{\cal P}_T}$ onto $T$ is given by $$\label{eq:PT}
{{\cal P}_T}({\bm{X}}) = {\bm{P}}_U {\bm{X}} + {\bm{X}} {\bm{P}}_V - {\bm{P}}_U {\bm{X}} {\bm{P}}_V,$$ where ${\bm{P}}_{U}$ and ${\bm{P}}_V$ are the orthogonal projections onto $U$ and $V$. Note here that while ${\bm{P}}_{U}$ and ${\bm{P}}_V$ are matrices, ${{\cal P}_T}$ is a linear operator mapping matrices to matrices. We also have $${{\cal P}_{T^\perp}}({\bm{X}}) = ({\mathcal{I}}- {{\cal P}_T})({\bm{X}}) = ({\bm{I}}_{n_1} -
{\bm{P}}_{U}) {\bm{X}} ({\bm{I}}_{n_2} - {\bm{P}}_{V})$$ where ${\bm{I}}_d$ denotes the $d\times d$ identity matrix. With these notations, ${\bm{Y}} \in
\partial \|{\bm{X}}_0\|_*$ if
- ${\cal P}_T({\bm{Y}}) = \sum_{1 \le k \le r} {\bm{u}}_k {\bm{v}}_k^*$,
- and $\|{\cal P}_{T^\perp} {\bm{Y}}\| \le 1$.
Now that we have characterized the subgradient of the nuclear norm, the lemma below gives sufficient conditions for the uniqueness of the minimizer to .
\[teo:unicity\] Consider a matrix ${\bm{X}}_0 = \sum_{k = 1}^r \sigma_k
\, {\bm{u}}_k {\bm{v}}_k^*$ of rank $r$ which is feasible for the problem , and suppose that the following two conditions hold:
1. there exists a dual point $\lambda$ such that ${\bm{Y}} = {\cal
R}^*_\Omega \lambda$ obeys $$\label{eq:key-decomp} {\cal P}_T({\bm{Y}}) = \sum_{k = 1}^r {\bm{u}}_k
{\bm{v}}_k^*,
\qquad \|{\cal P}_{T^\perp}({\bm{Y}})\| < 1;$$
2. the sampling operator ${\cal R}_\Omega$ restricted to elements in $T$ is injective.
Then ${\bm{X}}_0$ is the unique minimizer.
Before proving this result, we would like to emphasize that this lemma provides a clear strategy for proving our main result, namely, Theorem \[teo:main2\]. Letting ${\bm{M}} = \sum_{k = 1}^r \sigma_k
\, {\bm{u}}_k {\bm{v}}_k^*$, ${\bm{M}}$ is the unique solution to if the injectivity condition holds and if one can find a dual point $\lambda$ such that ${\bm{Y}} = {\cal R}^*_\Omega
\lambda$ obeys .
The proof of Lemma \[teo:unicity\] uses a standard fact which states that the nuclear norm and the spectral norm are dual to one another.
\[teo:dual\] For each pair ${\bm{W}}$ and ${\bm{H}}$, we have $${\langle}{\bm{W}}, {\bm{H}}{\rangle}\le \|{\bm{W}}\| \, \|{\bm{H}}\|_{*}.$$ In addition, for each ${\bm{H}}$, there is a ${\bm{W}}$ obeying $\|{\bm{W}}\| = 1$ which achieves the equality.
A variety of proofs are available for this Lemma, and an elementary argument is sketched in [@Recht07]. We now turn to the proof of Lemma \[teo:unicity\].
Consider any perturbation ${\bm{X}}_0 + {\bm{H}}$ where ${\cal
R}_\Omega({\bm{H}}) = 0$. Then for any ${\bm{W}}^0$ obeying (i)–(ii), $\sum_{k = 1}^r {\bm{u}}_k {\bm{v}}_k^* + {\bm{W}}^0$ is a subgradient of the nuclear norm at $X_0$ and, therefore, $${{\left\lVert{{\bm{X}}_0+{\bm{H}}}\right\rVert}}_* \ge {{\left\lVert{{\bm{X}}_0}\right\rVert}}_* + {\langle}\sum_{k=1}^r
{\bm{u}}_k {\bm{v}}_k^* + {\bm{W}}^0,{\bm{H}}{\rangle}.$$ Letting ${\bm{W}} = {{\cal P}_{T^\perp}}({\bm{Y}})$, we may write $\sum_{k=1}^r
{\bm{u}}_k {\bm{v}}_k^* = {\cal R}^*_\Omega \lambda - {\bm{W}}$. Since $\|{\bm{W}}\| < 1$ and ${\cal R}_\Omega({\bm{H}}) = 0$, it then follows that $${{\left\lVert{{\bm{X}}_0+{\bm{H}}}\right\rVert}}_* \ge {{\left\lVert{{\bm{X}}_0}\right\rVert}}_* + {\langle}{\bm{W}}^0 -
{\bm{W}},{\bm{H}}{\rangle}.$$ Now by construction $${\langle}{\bm{W}}^0 - {\bm{W}},{\bm{H}}{\rangle}= {\langle}{\cal P}_{T^\perp} ({\bm{W}}^0 -
{\bm{W}}), {\bm{H}}{\rangle}= {\langle}{\bm{W}}^0 - {\bm{W}}, {\cal
P}_{T^\perp} ({\bm{H}}){\rangle}.$$
We use Lemma \[teo:dual\] and set ${\bm{W}}^0 = {{\cal P}_{T^\perp}}({\bm{Z}})$ where ${\bm{Z}}$ is any matrix obeying $\|{\bm{Z}}\| \le 1$ and ${\langle}{\bm{Z}},{{\cal P}_{T^\perp}}({\bm{H}}){\rangle}= \|{{\cal P}_{T^\perp}}({\bm{H}})\|_*$. Then ${\bm{W}}^0 \in
T^\perp$, $\| {\bm{W}}^0\| \le 1$, and $${\langle}{\bm{W}}^0 - {\bm{W}},{\bm{H}}{\rangle}\ge (1 - \|{\bm{W}}\|) \,
\|{{\cal P}_{T^\perp}}({\bm{H}})\|_*,$$ which by assumption is strictly positive unless ${{\cal P}_{T^\perp}}({\bm{H}}) = 0$. In other words, $\|{\bm{X}}_0+{\bm{H}}\|_*
> \|{\bm{X}}_0\|_*$ unless ${{\cal P}_{T^\perp}}({\bm{H}}) = 0$. Assume then that ${{\cal P}_{T^\perp}}({\bm{H}}) = 0$ or equivalently that ${\bm{H}} \in T$. Then ${\cal R}_\Omega({\bm{H}}) = 0$ implies that ${\bm{H}} = 0$ by the injectivity assumption. In conclusion, $\|{\bm{X}}_0 + {\bm{H}}\|_*
> \|{\bm{X}}\|_*$ unless ${\bm{H}} = 0$.
Architecture of the proof {#sec:architecture}
=========================
Our strategy to prove that ${\bm{M}} = \sum_{1 \le k \le r} \sigma_k
{\bm{u}}_k {\bm{v}}_k^*$ is the unique minimizer to is to construct a matrix ${\bm{Y}}$ which vanishes on $\Omega^c$ and obeys the conditions of Lemma \[teo:unicity\] (and show the injectivity of the sampling operator restricted to matrices in $T$ along the way). Set ${{\cal P}_{\Omega}}$ to be the orthogonal projector onto the indices in $\Omega$ so that the $(i,j)$th component of ${{\cal P}_{\Omega}}({\bm{X}})$ is equal to $X_{ij}$ if $(i,j) \in \Omega$ and zero otherwise. Our candidate ${\bm{Y}}$ will be the solution to $$\label{eq:frob2}
\begin{array}{ll}
\textrm{minimize} & \quad \|{\bm{X}}\|_F \\
\textrm{subject to} & \quad ({\cal P}_T {\cal P}_\Omega)({\bm{X}}) = \sum_{k = 1}^r {\bm{u}}_k {\bm{v}}_k^*.
\end{array}$$ The matrix ${\bm{Y}}$ vanishes on $\Omega^c$ as otherwise it would not be an optimal solution since ${{\cal P}_{\Omega}}({\bm{Y}})$ would obey the constraint and have a smaller Frobenius norm. Hence ${\bm{Y}} = {{\cal P}_{\Omega}}({\bm{Y}})$ and ${{\cal P}_T}({\bm{Y}}) = \sum_{k = 1}^r {\bm{u}}_k {\bm{v}}_k^*$. Since the Pythagoras formula gives $$\begin{aligned}
\|{\bm{Y}}\|^2_{F} = \|{{\cal P}_T}({\bm{Y}})\|_F^2 +
\|{{\cal P}_{T^\perp}}({\bm{Y}})\|_F^2 & = \|\sum_{k = 1}^r {\bm{u}}_k {\bm{v}}_k^*\|_F^2 + \|{{\cal P}_{T^\perp}}({\bm{Y}})\|_F^2\\
& = r + \|{{\cal P}_{T^\perp}}({\bm{Y}})\|_F^2,\end{aligned}$$ minimizing the Frobenius norm of ${\bm{X}}$ amounts to minimizing the Frobenius norm of ${{\cal P}_{T^\perp}}({\bm{X}})$ under the constraint ${{\cal P}_T}({\bm{X}})
= \sum_{k = 1}^r {\bm{u}}_k {\bm{v}}_k^*$. Our motivation is twofold. First, the solution to the least-squares problem has a closed form that is amenable to analysis. Second, by forcing ${{\cal P}_{T^\perp}}({\bm{Y}})$ to be small in the Frobenius norm, we hope that it will be small in the spectral norm as well, and establishing that $\|{{\cal P}_{T^\perp}}({\bm{Y}})\| < 1$ would prove that ${\bm{M}}$ is the unique solution to .
To compute the solution to , we introduce the operator ${\mathcal{A}_{\Omega T}}$ defined by $${\mathcal{A}_{\Omega T}}({\bm{M}}) = {{\cal P}_{\Omega}}{{\cal P}_T}({\bm{M}}).$$ Then, if ${\mathcal{A}_{\Omega T}}^* {\mathcal{A}_{\Omega T}}= {{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T}$ has full rank when restricted to $T$, the minimizer to is given by $$\label{eq:Lambda1}
{\bm{Y}} = {\mathcal{A}_{\Omega T}}({\mathcal{A}_{\Omega T}}^* {\mathcal{A}_{\Omega T}})^{-1}({\bm{E}}),
\qquad {\bm{E}} \equiv \sum_{k = 1}^r {\bm{u}}_k {\bm{v}}_k^*.$$ We clarify the meaning of to avoid any confusion. $({\mathcal{A}_{\Omega T}}^* {\mathcal{A}_{\Omega T}})^{-1}({\bm{E}})$ is meant to be that element ${\bm{F}}$ in $T$ obeying $({\mathcal{A}_{\Omega T}}^* {\mathcal{A}_{\Omega T}})({\bm{F}}) = {\bm{E}}$.
To summarize the aims of our proof strategy,
- We must first show that ${\mathcal{A}_{\Omega T}}^* {\mathcal{A}_{\Omega T}}= {{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T}$ is a one-to-one linear mapping from $T$ onto itself. In this case, ${\mathcal{A}_{\Omega T}}= {{\cal P}_{\Omega}}{{\cal P}_T}$—as a mapping from $T$ to ${\mathbb{R}}^{n_1 \times
n_2}$—is injective. This is the second sufficient condition of Lemma \[teo:unicity\]. Moreover, our ansatz for ${\bm{Y}}$ given by is well-defined.
- Having established that ${\bm{Y}}$ is well-defined, we will show that $$\|{\cal P}_{T^\perp}({\bm{Y}})\| < 1,$$ thus proving the first sufficient condition.
The Bernoulli model {#sec:bernoulli}
-------------------
Instead of showing that the theorem holds when $\Omega$ is a set of size $m$ sampled uniformly at random, we prove the theorem for a subset $\Omega'$ sampled according to the [*Bernoulli model*]{}. Here and below, $\{\delta_{ij}\}_{1\le i \le n_1, 1\le j \le n_2}$ is a sequence of independent identically distributed $0/1$ Bernoulli random variables with $$\label{eq:delta}
{\operatorname{\mathbb{P}}}(\delta_{ij} = 1) = p \equiv \frac{m}{n_1n_2},$$ and define $$\label{eq:Omega}
\Omega' = \{ (i,j) : \delta_{ij} = 1\}.$$ Note that ${\operatorname{\mathbb{E}}}|\Omega'| = m$, so that the average cardinality of $\Omega'$ is that of $\Omega$. Then following the same reasoning as the argument developed in Section II.C of [@CRT06] shows that the probability of ‘failure’ under the uniform model is bounded by 2 times the probability of failure under the Bernoulli model; the failure event is the event on which the solution to is not exact. Hence, we can restrict our attention to the Bernoulli model and from now on, we will assume that $\Omega$ is given by . This is advantageous because the Bernoulli model admits a simpler analysis than uniform sampling thanks to the independence between the $\delta_{ij}$’s.
The injectivity property {#sec:injectivity}
------------------------
We study the injectivity of ${\mathcal{A}_{\Omega T}}$, which also shows that ${\bm{Y}}$ is well-defined. To prove this, we will show that the linear operator $p^{-1} {{\cal P}_T}({{\cal P}_{\Omega}}- p {\mathcal{I}}) {{\cal P}_T}$ has small operator norm, which we recall is $\sup_{\|{\bm{X}}\|_F \le 1} \, p^{-1}\|{{\cal P}_T}({{\cal P}_{\Omega}}- p {\mathcal{I}})
{{\cal P}_T}({\bm{X}})\|_F$.
\[teo:rudelson\] Suppose $\Omega$ is sampled according to the Bernoulli model – and put $n =
\max(n_1,n_2)$. Suppose that the coherences obey $\max(\mu(U),\mu(V)) \le \mu_0$. Then, there is a numerical constants $C_R$ such that for all $\beta>1$, $$\label{eq:near-isometry}
p^{-1} \, \|{{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T}- p {{\cal P}_T}\| \le C_R \,
\sqrt{\frac{\mu_0 \, nr (\beta \log n)}{m}}$$ with probability at least $1-3n^{-\beta}$ provided that $C_R \,
\sqrt{\frac{\mu_0 \, nr (\beta \log n)}{m}} < 1$.
Decompose any matrix ${\bm{X}}$ as ${\bm{X}} = \sum_{ab} {\langle}{\bm{X}},
{{\bm{e}}_{a}{\bm{e}}_{b}^*}{\rangle}{{\bm{e}}_{a}{\bm{e}}_{b}^*}$ so that $${{\cal P}_T}({\bm{X}}) = \sum_{ab} {\langle}{{\cal P}_T}({\bm{X}}), {{\bm{e}}_{a}{\bm{e}}_{b}^*}{\rangle}{{\bm{e}}_{a}{\bm{e}}_{b}^*}= \sum_{ab}
{\langle}{\bm{X}}, {{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*}){\rangle}{{\bm{e}}_{a}{\bm{e}}_{b}^*}.$$ Hence, ${{\cal P}_{\Omega}}{{\cal P}_T}({\bm{X}}) = \sum_{ab} \delta_{ab} \, {\langle}{\bm{X}},
{{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*}){\rangle}\, {{\bm{e}}_{a}{\bm{e}}_{b}^*}$ which gives $$({{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T})({\bm{X}}) = \sum_{ab} \delta_{ab} \, {\langle}{\bm{X}},
{{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*}){\rangle}\,{{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*}).$$ In other words, $${{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T}= \sum_{ab} \delta_{ab} \, {{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*}) \otimes {{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*}).$$ It follows from the definition of ${{\cal P}_T}$ that $$\label{eq:PTeab}
{{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*}) = ({\bm{P}}_U {\bm{e}}_a) {\bm{e}}_b^* + {\bm{e}}_a ({\bm{P}}_V {\bm{e}}_b)^* - ({\bm{P}}_U {\bm{e}}_a)({\bm{P}}_V {\bm{e}}_b)^*.$$ This gives $$\label{eq:PTeabF2} \|{{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*})\|_F^2 = {\langle}{{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*}), {{\bm{e}}_{a}{\bm{e}}_{b}^*}{\rangle}= \|{\bm{P}}_U
{\bm{e}}_a\|^2 + \|{\bm{P}}_V
{\bm{e}}_b\|^2 - \|{\bm{P}}_U {\bm{e}}_a\|^2 \, \|{\bm{P}}_V {\bm{e}}_b\|^2$$ and since $\|{\bm{P}}_U {\bm{e}}_a\|^2 \le \mu(U)r/n_1$ and $\|{\bm{P}}_V
{\bm{e}}_b\|^2 \le \mu(U)r/n_2$, $$\|{{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*})\|_F^2 \le 2\mu_0 r/\min(n_1,n_2).
\label{eq:PTeabF}$$ Now the fact that the operator ${{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T}$ does not deviate from its expected value $${\operatorname{\mathbb{E}}}({{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T}) = {{\cal P}_T}({\operatorname{\mathbb{E}}}{{\cal P}_{\Omega}}) {{\cal P}_T}= {{\cal P}_T}(p {\mathcal{I}}) {{\cal P}_T}= p{{\cal P}_T}$$ in the spectral norm is related to Rudelson’s selection theorem [@rudelson]. The first part of the theorem below may be found in [@CR07] for example, see also [@RV_JACM] for a very similar statement.
[@CR07] \[teo:rudelson2\] Let $\{\delta_{ab}\}$ be independent 0/1 Bernoulli variables with ${\operatorname{\mathbb{P}}}(\delta_{ab} = 1) = p = \frac{m}{n_1 n_2}$ and put $n = \max(n_1,
n_2)$. Suppose that $\|{{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*})\|^2_F \le 2\mu_0 r/n$. Set $$Z \equiv p^{-1} \|\sum_{ab}
(\delta_{ab} -p)\, {{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*}) \otimes {{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*})\| =
p^{-1} \|{{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T}- p {{\cal P}_T}\|.$$
1. There exists a constant $C'_R$ such that $$\label{eq:ERudel}
{\operatorname{\mathbb{E}}}Z \le C'_R \, \sqrt{\frac{\mu_0 \, nr \, \log n}{m}}$$ provided that the right-hand side is smaller than 1.
2. Suppose ${\operatorname{\mathbb{E}}}Z \le 1$. Then for each $\lambda > 0$, we have $$\label{eq:largeRudel}
{\operatorname{\mathbb{P}}}\left(|Z - {\operatorname{\mathbb{E}}}Z| > \lambda \, \sqrt{\frac{\mu_0 \, nr \,\log n}{m}}\right)
\le 3 \exp\left(-\gamma'_0 \, \min \left\{\lambda^2 \log n, \lambda \sqrt{\frac{m \log n}{\mu_0\, nr}}\right\}\right)$$ for some positive constant $\gamma'_0$.
As mentioned above, the first part, namely, is an application of an established result which states that if $\{y_i\}$ is a family of vectors in ${\mathbb{R}}^d$ and $\{\delta_i\}$ is a 0/1 Bernoulli sequence with ${\operatorname{\mathbb{P}}}(\delta_i = 1) = p$, then $$p^{-1} \|\sum_i (\delta_i -p) y_i \otimes y_i\| \le C\,
\sqrt{\frac{\log d}{p}} \max_i \|y_i\|$$ for some $C > 0$ provided that the right-hand side is less than 1. The proof may be found in the cited literature, e.g. in [@CR07]. Hence, the first part follows from applying this result to vectors of the form ${{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*})$ and using the available bound on $\|{{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*})\|_F$. The second part follows from Talagrand’s concentration inequality and may be found in the Appendix.
Set $\lambda = \sqrt{\beta/\gamma'_0}$ and assume that $m >
(\beta/\gamma'_0)\mu_0 \, nr \log n$. Then the left-hand side of is bounded by $3n^{-\beta}$ and thus, we established that $$Z \le C'_R \sqrt{\frac{\mu_0 \, nr \, \log n}{m}} +
\frac{1}{\sqrt{\gamma'_0}} \sqrt{\frac{\mu_0 \, nr \, \beta \log
n}{m}}$$ with probability at least $1 - 3n^{-\beta}$. Setting $C_R = C'_R +
1/\sqrt{\gamma'_0}$ finishes the proof.
Take $m$ large enough so that $C_R \, \sqrt{\mu_0 \, (nr/m) \log n}
\le 1/2$. Then it follows from that $$\label{eq:near-isometry2} \frac{p}{2} \|{{\cal P}_T}({\bm{X}})\|_F \le \|({{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T})({\bm{X}})\|_F \le \frac{3p}{2} \|{{\cal P}_T}({\bm{X}})\|_F$$ for all ${\bm{X}}$ with large probability. In particular, the operator ${\mathcal{A}_{\Omega T}}^* {\mathcal{A}_{\Omega T}}= {{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T}$ mapping $T$ onto itself is well-conditioned and hence invertible. An immediate consequence is the following:
\[teo:POPT\] Assume that $C_R \, \sqrt{\mu_0 nr (\log n)/m} \le 1/2$. With the same probability as in Theorem \[teo:rudelson\], we have $$\label{eq:POPT} \|{{\cal P}_{\Omega}}{{\cal P}_T}({\bm{X}})\|_F \le \sqrt{3p/2}
\|{{\cal P}_T}({\bm{X}})\|_F.$$
We have $\|{{\cal P}_{\Omega}}{{\cal P}_T}({\bm{X}})\|_F^2 = {\langle}{\bm{X}}, ({{\cal P}_{\Omega}}{{\cal P}_T})^* ({{\cal P}_{\Omega}}{{\cal P}_T})
{\bm{X}}{\rangle}= {\langle}{\bm{X}}, ({{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T}) {\bm{X}}{\rangle}$ and thus $$\|{{\cal P}_{\Omega}}{{\cal P}_T}({\bm{X}})\|_F^2 = {\langle}{{\cal P}_T}{\bm{X}}, ({{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T}) {\bm{X}}{\rangle}\le \|{{\cal P}_T}({\bm{X}})\|_F \, \|({{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T})({\bm{X}})\|_F,$$ where the inequality is due to Cauchy-Schwarz. The conclusion follows from .
The size property {#sec:size}
-----------------
In this section, we explain how we will show that $\|{{\cal P}_{T^\perp}}({\bm{Y}})\|
< 1$. This result will follow from five lemmas that we will prove in Section \[sec:proofs\]. Introduce $$\mathcal{H} \equiv {{\cal P}_T}- p^{-1} {{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T},$$ which obeys $\|\mathcal{H}({\bm{X}})\|_F \le C_R \, \sqrt{\mu_0
(nr/m) \, \beta \log n} \|{{\cal P}_T}({\bm{X}})\|_F$ with large probability because of Theorem \[teo:rudelson\]. For any matrix ${\bm{X}}\in
T$, $({{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T})^{-1}({\bm{X}})$ can be expressed in terms of the power series $$({{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T})^{-1}({\bm{X}}) = p^{-1}({\bm{X}} + \mathcal{H}({\bm{X}})
+ \mathcal{H}^2({\bm{X}}) + \ldots)$$ for $\mathcal{H}$ is a contraction when $m$ is sufficiently large. Since ${\bm{Y}} = {{\cal P}_{\Omega}}{{\cal P}_T}({{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T})^{-1}(\sum_{1\le k \le r}
{\bm{u}}_k {\bm{v}}_k^*)$, ${{\cal P}_{T^\perp}}({\bm{Y}})$ may be decomposed as $$\label{eq:size} {{\cal P}_{T^\perp}}({\bm{Y}}) = p^{-1} (\mathcal{P}_{T^\perp} {{\cal P}_{\Omega}}{{\cal P}_T})
({\bm{E}} + \mathcal{H}({\bm{E}}) + {\cal
H}^2({\bm{E}}) + \ldots), \qquad {\bm{E}} = \sum_{1 \le k \le r} {\bm{u}}_k
{\bm{v}}_k^*.$$ To bound the norm of the left-hand side, it is of course sufficient to bound the norm of the summands in the right-hand side. Taking the following five lemmas together establishes Theorem \[teo:main2\].
\[teo:order0\] Fix $\beta \ge 2$ and $\lambda \ge 1$. There is a numerical constant $C_0$ such that if $m \ge \lambda \, \mu_1^2 \, nr \beta
\log n$, then $$\label{eq:order0} p^{-1} \, \|({{\cal P}_{T^\perp}}{{\cal P}_{\Omega}}{{\cal P}_T}) {\bm{E}}\| \le
C_0 \, \lambda^{-1/2}.$$ with probability at least $1-n^{-\beta}$.
\[teo:order1\] Fix $\beta \ge 2$ and $\lambda \ge 1$. There are numerical constants $C_1$ and $c_1$ such that if $m \ge \lambda \, \mu_1
\max(\sqrt{\mu_0}, \mu_1) \, nr \beta \log n$, then $$\label{eq:order1} p^{-1} \, \|({{\cal P}_{T^\perp}}{{\cal P}_{\Omega}}{{\cal P}_T}) \mathcal{H}({\bm{E}})\|
\le C_1 \, \lambda^{-1}$$ with probability at least $1-c_1 n^{-\beta}$.
\[teo:order2\] Fix $\beta \ge 2$ and $\lambda \ge 1$. There are numerical constants $C_2$ and $c_2$ such that if $m \ge \lambda \, \mu_0^{4/3} \,
nr^{4/3} \beta \log n$, then $$\label{eq:order2} p^{-1} \, \|({{\cal P}_{T^\perp}}{{\cal P}_{\Omega}}{{\cal P}_T}) \mathcal{H}^2({\bm{E}})\|
\le C_2 \, \lambda^{-3/2}$$ with probability at least $1-c_2 n^{-\beta}$.
\[teo:order3\] Fix $\beta \ge 2$ and $\lambda \ge 1$. There are numerical constants $C_3$ and $c_3$ such that if $m \ge \lambda \mu_0^2 \, nr^2 \beta
\log n$, then $$\label{eq:order3} p^{-1} \, \|({{\cal P}_{T^\perp}}{{\cal P}_{\Omega}}{{\cal P}_T})
\mathcal{H}^3({\bm{E}})\| \le C_3 \, \lambda^{-1/2}$$ with probability at least $1-c_3 n^{-\beta}$.
\[teo:highorder\] Under the assumptions of Theorem \[teo:rudelson\], there is a numerical constant $C_{k_0}$ such that if $m \ge
(2C_R)^2 \mu_0 nr \beta \log n$, then $$\label{eq:highorder} p^{-1} \, \|({{\cal P}_{T^\perp}}{{\cal P}_{\Omega}}{{\cal P}_T}) \sum_{k \ge k_0}
\mathcal{H}^{k}({\bm{E}})\| \le C_{k_0} \, \left(\frac{n^2 r}{m}\right)^{1/2} \, \left(\frac{\mu_0 nr \beta \log n}{m}\right)^{k_0/2}$$ with probability at least $1-n^{-\beta}$.
Let us now show how we may combine these lemmas to prove our main results. Under all of the assumptions of Theorem \[teo:main2\], consider the four Lemmas \[teo:order0\], \[teo:order1\], \[teo:order2\] and \[teo:highorder\], the latter applied with $k_0
= 3$. Together they imply that there are numerical constants $c$ and $C$ such that $\|{{\cal P}_{T^\perp}}({\bm{Y}})\| < 1$ with probability at least $1-c n^{-\beta}$ provided that the number of samples obeys $$\label{eq:bound1}
m \ge C \, \, \max(\mu_1^2, \mu_0^{1/2} \mu_1, \mu_0^{4/3} r^{1/3}, \mu_0 n^{1/4}) \, nr \beta \log n$$ for some constant $C$. The four expressions in the maximum come from Lemmas \[teo:order0\], \[teo:order1\], \[teo:order2\] and \[teo:highorder\] in this order. Now the bound is only interesting in the range when $\mu_0 n^{1/4} r$ is smaller than a constant times $n$ as otherwise the right-hand side is greater than $n^2$ (this would say that one would see all the entries in which case our claim is trivial). When $\mu_0 r \le n^{3/4}$, $(\mu_0 r)^{4/3}
\le \mu_0 n^{5/4} r$ and thus the recovery is exact provided that $m$ obeys .
For the case concerning small values of the rank, we consider all five lemmas and apply Lemma \[teo:highorder\], the latter applied with $k_0 = 4$. Together they imply that $\|{{\cal P}_{T^\perp}}({\bm{Y}})\| < 1$ with probability at least $1-c n^{-\beta}$ provided that the number of samples obeys $$\label{eq:bound2}
m \ge C \max(\mu_0^2 r, \mu_0 n^{1/5}) \, nr \beta \log n$$ for some constant $C$. The two expressions in the maximum come from Lemmas \[teo:order3\] and \[teo:highorder\] in this order. The reason for this simplified formulation is that the terms $\mu_1^2$, $\mu_0^{1/2} \mu_1$ and $\mu_0^{4/3} r^{1/3}$ which come from Lemmas \[teo:order0\], \[teo:order1\] and \[teo:order2\] are bounded above by $\mu_0^2 r$ since $\mu_1 \le \mu_0 \sqrt{r}$. When $\mu_0 r \le n^{1/5}$, the recovery is exact provided that $m$ obeys .
Connections with Random Graph Theory {#sec:coupon}
====================================
The injectivity property and the coupon collector’s problem
-----------------------------------------------------------
We argued in the Introduction that to have any hope of recovering an unknown matrix of rank 1 by any method whatsoever, one needs at least one observation per row and one observation per column. Sample $m$ entries uniformly at random. Viewing the row indices as bins, assign the $k$th sampled entry to the bin corresponding to its row index. Then to have any hope of recovering our matrix, all the bins need to be occupied. Quantifying how many samples are required to fill all of the bins is the famous *coupon collector’s problem*.
Coupon collection is also connected to the injectivity of the sampling operator ${{\cal P}_{\Omega}}$ restricted to elements in $T$. Suppose we sample the entries of a rank 1 matrix equal to ${\bm{x}} {\bm{y}}^*$ with left and right singular vectors ${\bm{u}} = {\bm{x}}/\|{\bm{x}}\|$ and ${\bm{v}} = {\bm{y}}/\|{\bm{y}}\|$ respectively and have not seen anything in the $i$th row. Then we claim that ${{\cal P}_{\Omega}}$ (restricted to $T$) has a nontrivial null space and thus ${{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T}$ is not invertible. Indeed, consider the matrix ${\bm{e}}_i {\bm{v}}^*$. This matrix is in $T$ and $${{\cal P}_{\Omega}}({\bm{e}}_i {\bm{v}}^*) = 0$$ since ${\bm{e}}_i {\bm{v}}^*$ vanishes outside of the $i$th row. The same applies to the columns as well. If we have not seen anything in column $j$, then the rank-1 matrix ${\bm{u}} {\bm{e}}_j^* \in T$ and ${{\cal P}_{\Omega}}({\bm{u}}
{\bm{e}}_j^*) = 0$. In conclusion, the invertibility of ${{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T}$ implies a complete collection.
When the entries are sampled uniformly at random, it is well known that one needs on the order of $n \log n$ samples to sample all the rows. What is interesting is that Theorem \[teo:rudelson\] implies that ${{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T}$ is invertible—a stronger property—when the number of samples is also on the order of $n \log n$. A particular implication of this discussion is that the logarithmic factors in Theorem \[teo:rudelson\] are unavoidable.
The injectivity property and the connectivity problem
-----------------------------------------------------
To recover a matrix of rank 1, one needs much more than at least one observation per row and column. Let $R$ be the set of row indices, $1 \le i \le n$, and $C$ be the set of column indices, $1 \le
j \le n$, and consider the bipartite graph connecting vertices $i \in
R$ to vertices $j \in C$ if and only if $(i,j) \in \Omega$, i.e. the $(i,j)$th entry is observed. We claim that if this graph is not fully connected, then one cannot hope to recover a matrix of rank 1.
To see this, we let $I$ be the set of row indices and $J$ be the set of column indices in any connected component. We will assume that $I$ and $J$ are nonempty as otherwise, one is in the previously discussed situation where some rows or columns are not sampled. Consider a rank 1 matrix equal to ${\bm{x}} {\bm{y}}^*$ as before with singular vectors ${\bm{u}} = {\bm{x}}/\|{\bm{x}}\|$ and ${\bm{v}} =
{\bm{y}}/\|{\bm{y}}\|$. Then all the information about the values of the $x_i$’s with $i \in I$ and of the $y_j$’s with $j \in J$ are given by the sampled entries connecting $I$ to $J$ since all the other observed entries connect vertices in $I^c$ to those in $J^c$. Now even if one observes all the entries $x_i y_j$ with $i \in I$ and $j \in J$, then at least the signs of $x_i$, $i \in I$, and of $y_j$, $j \in J$, would remain undetermined. Indeed, if the values $(x_i)_{i \in I}$, $(y_j)_{j \in J}$ are consistent with the observed entries, so are the values $(-x_i)_{i \in I}$, $(-y_j)_{j \in J}$. However, since the same analysis holds for the sets $I^c$ and $J^c$, there are at least two matrices consistent with the observed entries and exact matrix completion is impossible.
The connectivity of the graph is also related to the injectivity of the sampling operator ${{\cal P}_{\Omega}}$ restricted to elements in $T$. If the graph is not fully connected, then we claim that ${{\cal P}_{\Omega}}$ (restricted to $T$) has a nontrivial null space and thus ${{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T}$ is not invertible. Indeed, consider the matrix $${\bm{M}} = {\bm{a}} {\bm{v}}^* + {\bm{u}} {\bm{b}}^*,$$ where $a_i = -u_i$ if $i \in I$ and $a_i = u_i$ otherwise, and $b_j =
v_j$ if $j \in J$ and $b_j = - v_j$ otherwise. Then this matrix is in $T$ and obeys $$M_{ij} = 0$$ if $(i,j) \in I \times J$ or $(i,j) \in I^c \times J^c$. Note that on the complement, i.e. $(i,j) \in I \times J^c$ or $(i,j) \in I^c
\times J$, one has $M_{ij} = 2 u_i v_j$ and one can show that ${\bm{M}} \neq 0$ unless ${\bm{u}}{\bm{v}}^* = 0$. Since $\Omega$ is included in the union of $I \times J$ and $I^c \times J^c$, we have that ${{\cal P}_{\Omega}}({\bm{M}}) = 0$. In conclusion, the invertibility of ${{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T}$ implies a fully connected graph.
When the entries are sampled uniformly at random, it is well known that one needs on the order of $n \log n$ samples to obtain a fully connected graph with large probability (see, e.g., [@BollobasGraphsBook]). Remarkably, Theorem \[teo:rudelson\] implies that ${{\cal P}_T}{{\cal P}_{\Omega}}{{\cal P}_T}$ is invertible—a stronger property—when the number of samples is also on the order of $n \log n$.
Proofs of the Critical Lemmas {#sec:proofs}
=============================
In this section, we prove the five lemmas of Section \[sec:size\]. Before we begin, however, we develop a simple estimate which we will use throughout. For each pair $(a,b)$ and $(a',b')$, it follows from the expression of ${{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*})$ that $$\label{eq:PTeabp}
{\langle}{{\cal P}_T}({{\bm{e}}_{a'} {\bm{e}}_{b'}^*}), {{\bm{e}}_{a}{\bm{e}}_{b}^*}{\rangle}= {\langle}{\bm{e}}_a, {\bm{P}}_U {\bm{e}}_{a'}{\rangle}\, 1_{\{b = b'\}} +
{\langle}{\bm{e}}_b, {\bm{P}}_V {\bm{e}}_{b'}{\rangle}\, 1_{\{a = a'\}} - {\langle}{\bm{e}}_a, {\bm{P}}_U {\bm{e}}_{a'}{\rangle}{\langle}{\bm{e}}_b, {\bm{P}}_V {\bm{e}}_{b'}{\rangle}.$$ Fix $\mu_0$ obeying $\mu(U) \le \mu_0$ and $\mu(V) \le \mu_0$ and note that $$|{\langle}{\bm{e}}_a, {\bm{P}}_U {\bm{e}}_{a'}{\rangle}| = |{\langle}{\bm{P}}_U {\bm{e}}_a,
{\bm{P}}_U {\bm{e}}_{a'}{\rangle}| \le \|{\bm{P}}_U {\bm{e}}_a\| \, \|{\bm{P}}_U
{\bm{e}}_{a'}\| \le \mu_0 r/n_1$$ and similarly for ${\langle}{\bm{e}}_b, {\bm{P}}_V {\bm{e}}_{b'}{\rangle}$. Suppose that $b = b'$ and $a \neq a'$, then $$|{\langle}{{\cal P}_T}({{\bm{e}}_{a'} {\bm{e}}_{b'}^*}), {{\bm{e}}_{a}{\bm{e}}_{b}^*}{\rangle}| = |{\langle}{\bm{e}}_a, {\bm{P}}_U {\bm{e}}_{a'}{\rangle}| (1
- \|{\bm{P}}_V {\bm{e}}_b\|^2) \le \mu_0 r/n_1.$$ We have a similar bound when $a = a'$ and $b \neq b'$ whereas when $a
\neq a'$ and $b \neq b'$, $$|{\langle}{{\cal P}_T}({{\bm{e}}_{a'} {\bm{e}}_{b'}^*}), {{\bm{e}}_{a}{\bm{e}}_{b}^*}{\rangle}| \le (\mu_0 r)^2/(n_1 n_2).$$ In short, it follows from this analysis (and from for the case where $(a,b) = (a',b')$) that $$\label{eq:useful}
\max_{ab, a'b'} |{\langle}{{\cal P}_T}({{\bm{e}}_{a'} {\bm{e}}_{b'}^*}), {{\bm{e}}_{a}{\bm{e}}_{b}^*}{\rangle}| \le 2\mu_0 r/\min(n_1, n_2).$$ A consequence of is the estimate: $$\begin{aligned}
\nonumber \sum_{a'b'} |{\langle}{{\cal P}_T}({{\bm{e}}_{a'} {\bm{e}}_{b'}^*}), {{\bm{e}}_{a}{\bm{e}}_{b}^*}{\rangle}|^2 & = \sum_{a'b'}
|{\langle}{{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*}), {{\bm{e}}_{a'} {\bm{e}}_{b'}^*}{\rangle}|^2 \\
\label{eq:useful2}
& = \|{{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*})\|_F^2 \le 2\mu_0 r/\min(n_1,n_2),\end{aligned}$$ which we will apply several times. A related estimate is this: $$\label{eq:Cauchy}
\max_a \sum_{b} |E_{ab}|^2 \le \mu_0r/\min(n_1,n_2),$$ and the same is true by exchanging the role of $a$ and $b$. To see this, write $$\sum_{b} |E_{ab}|^2 = \|{\bm{e}}_a^* {\bm{E}}\|^2 = \|\sum_{j \le r}
{\bm{v}}_j {\langle}{\bm{u}}_j, e_a{\rangle}\|^2 = \sum_{j\le r} |{\langle}{\bm{u}}_j,
e_a{\rangle}|^2 = \|{\bm{P}}_U {\bm{e}}_a\|^2,$$ and the conclusion follows from the coherence property.
We will prove the lemmas in the case where $n_1 = n_2 = n$ for simplicity, i.e. in the case of square matrices of dimension $n$. The general case is treated in exactly the same way. In fact, the argument only makes use of the bounds , (and sometimes ), and the general case is obtained by replacing $n$ with $\min(n_1,n_2)$.
Each of the following subsections computes the operator norm of some random variable. In each section, we denote ${\bm{S}}$ as the quantity whose norm we wish to analyze. We will also frequently use the notation ${\bm{H}}$ for some auxiliary matrix variable whose norm we will need to bound. Hence, we will reuse the same notation many times rather than introducing a dozens new names—just like in computer programming where one uses the same variable name in distinct routines.
Proof of Lemma \[teo:order0\] {#sec:order0}
-----------------------------
In this section, we develop a bound on $$\begin{aligned}
p^{-1} \|{{\cal P}_{T^\perp}}{{\cal P}_{\Omega}}{{\cal P}_T}({\bm{E}})\| & = p^{-1} \|{{\cal P}_{T^\perp}}({{\cal P}_{\Omega}}- p{\mathcal{I}})
{{\cal P}_T}({\bm{E}})\|\\
& \le p^{-1} \|({{\cal P}_{\Omega}}- p{\mathcal{I}})({\bm{E}})\|,\end{aligned}$$ where the equality follows from ${{\cal P}_{T^\perp}}{{\cal P}_T}= 0$, and the inequality from ${{\cal P}_T}({\bm{E}}) = {\bm{E}}$ together with $\|{{\cal P}_{T^\perp}}({\bm{X}})\| \le
\|{\bm{X}}\|$ which is valid for any matrix ${\bm{X}}$. Set $$\label{eq:S} {\bm{S}} \equiv p^{-1} ({{\cal P}_{\Omega}}- p{\mathcal{I}}) ({\bm{E}}) = p^{-1}
\sum_{ab} (\delta_{ab} - p) E_{ab} {{\bm{e}}_{a}{\bm{e}}_{b}^*}.$$ We think of ${\bm{S}}$ as a random variable since it depends on the random $\delta_{ab}$’s, and note that ${\operatorname{\mathbb{E}}}{\bm{S}} = 0$.
The proof of Lemma \[teo:order0\] operates by developing an estimate on the size of $({\operatorname{\mathbb{E}}}\|{\bm{S}}\|^q)^{1/q}$ for some $q \ge
1$ and by applying Markov inequality to bound the tail of the random variable $\|{\bm{S}}\|$. To do this, we shall use a symmetrization argument and the noncommutative Khintchine inequality. Since the function $f({\bm{S}}) = \|{\bm{S}}\|^q$ is convex, Jensen’s inequality gives that $${\operatorname{\mathbb{E}}}\|{\bm{S}}\|^q \le {\operatorname{\mathbb{E}}}\|{\bm{S}} - {\bm{S}}'\|^q,$$ where ${\bm{S}}' = p^{-1} \sum_{ab} (\delta'_{ab} - p) E_{ab} {{\bm{e}}_{a}{\bm{e}}_{b}^*}$ is an independent copy of ${\bm{S}}$. Since $(\delta_{ab} -
\delta'_{ab})$ is symmetric, ${\bm{S}} - {\bm{S}}'$ has the same distribution as $$p^{-1} \, \sum_{ab} \epsilon_{ab} (\delta_{ab} - \delta'_{ab})
E_{ab} {{\bm{e}}_{a}{\bm{e}}_{b}^*}\equiv {\bm{S}}_\epsilon - {\bm{S}}_\epsilon',$$ where $\{\epsilon_{ab}\}$ is an independent Rademacher sequence and ${\bm{S}}_\epsilon = p^{-1} \sum_{ab} \epsilon_{ab} \delta_{ab} E_{ab}
{\bm{e}}_a {\bm{e}}_b^*$. Further, the triangle inequality gives $$({\operatorname{\mathbb{E}}}\|{\bm{S}}_\epsilon-{\bm{S}}_\epsilon'\|^q)^{1/q} \le ({\operatorname{\mathbb{E}}}\|{\bm{S}}_\epsilon\|^q)^{1/q} + ({\operatorname{\mathbb{E}}}\|{\bm{S}}_\epsilon'\|^q)^{1/q} =
2 ({\operatorname{\mathbb{E}}}\|{\bm{S}}_\epsilon\|^q)^{1/q}$$ since ${\bm{S}}_\epsilon$ and ${\bm{S}}_\epsilon'$ have the same distribution and, therefore, $$({\operatorname{\mathbb{E}}}\|{\bm{S}}\|^q)^{1/q} \le 2p^{-1} \left({\operatorname{\mathbb{E}}}_{\delta} {\operatorname{\mathbb{E}}}_\epsilon
\|\sum_{ab}
\epsilon_{ab} \delta_{ab} \, E_{ab} {\bm{e}}_a {\bm{e}}_b^*\|^q\right)^{1/q}.$$
We are now in position to apply the noncommutative Khintchine inequality which bounds the Schatten norm of a Rademacher series. For $q\geq 1$, the *Schatten q-norm* of a matrix is denoted by $$\|{\bm{X}}\|_{S_q} = \left(\sum_{i=1}^n \sigma_i({\bm{X}})^q
\right)^{1/q}\,.$$ Note that the nuclear norm is equal to the Schatten 1-norm and the Frobenius norm is equal to the Schatten 2-norm. The following theorem was originally proven by Lust-Picquard [@LustPicquard86], and was later sharpened by Buchholz [@Buchholz01].
Let $({\bm{X}}_i)_{1 \le
i \le r}$ be a finite sequence of matrices of the same dimension and let $\{\epsilon_i\}$ be a Rademacher sequence. For each $q \ge 2$ $$\left[{\operatorname{\mathbb{E}}}_\epsilon {{\left\lVert{\sum_i \epsilon_i {\bm{X}}_i}\right\rVert}}_{S_q}^q
\right]^{1/q} \le C_K \, \sqrt{q} \, \max\left[{{\left\lVert{\left(\sum_{i}
{\bm{X}}_i^*
{\bm{X}}_i\right)^{1/2}}\right\rVert}}_{S_q}, {{\left\lVert{\left(\sum_{i} {\bm{X}}_i
{\bm{X}}_i^*\right)^{1/2}}\right\rVert}}_{S_q}\right],$$ where $C_K = 2^{-1/4} \sqrt{\pi/e}$.
For reference, if ${\bm{X}}$ is an $n \times n$ matrix and $q \ge
\log n$, we have $$\|{\bm{X}}\| \le \|{\bm{X}}\|_{S_q} \le e \|{\bm{X}}\|,$$ so that the Schatten $q$-norm is within a multiplicative constant from the operator norm. Observe now that with $q' \ge q$ $$\left({\operatorname{\mathbb{E}}}_{\delta} {\operatorname{\mathbb{E}}}_\epsilon \|{\bm{S}}_\epsilon\|^q\right)^{1/q}
\le \left({\operatorname{\mathbb{E}}}_{\delta} {\operatorname{\mathbb{E}}}_\epsilon
\|{\bm{S}}_\epsilon\|_{S_{q'}}^q\right)^{1/q} \le \left({\operatorname{\mathbb{E}}}_{\delta}
{\operatorname{\mathbb{E}}}_\epsilon \|{\bm{S}}_\epsilon\|_{S_{q'}}^{q'}\right)^{1/q'}.$$ We apply the noncommutative Khintchine inequality with $q' \ge \log
n$, and after a little algebra, obtain $$\left({\operatorname{\mathbb{E}}}_{\delta} {\operatorname{\mathbb{E}}}_\epsilon
\|{\bm{S}}_\epsilon\|_{S_{q'}}^{q'}\right)^{1/q'} \le C_K \, \frac{e \,
\sqrt{q'}}{p} \, \left( {\operatorname{\mathbb{E}}}_\delta \max \left[\|\sum_{ab}
\delta_{ab} E^2_{ab} {\bm{e}}_a {\bm{e}}_a^*\|^{q'/2}, \|\sum_{ab} \delta_{ab}
E^2_{ab} {\bm{e}}_b {\bm{e}}_b^*\|^{q'/2}\right]\right)^{1/q'}.$$ The two terms in the right-hand side are essentially the same and if we can bound any one of them, the same technique will apply to the other. We consider the first and since $\sum_{ab}
\delta_{ab} E^2_{ab} {\bm{e}}_a {\bm{e}}_a^*$ is a diagonal matrix, $$\|\sum_{ab} \delta_{ab} E^2_{ab} {\bm{e}}_a {\bm{e}}_a^*\| = \max_{a}
\sum_{b} \delta_{ab} E^2_{ab}.$$ The following lemma bounds the $q$th moment of this quantity.
\[teo:E-max-indep2\] Suppose that $q$ is an integer obeying $1
\le q \le np$ and assume $np \ge 2\log n$. Then $$\label{eq:E-max-indep2}
{\operatorname{\mathbb{E}}}_\delta \left(\max_{a} \sum_{b}
\delta_{ab} E^2_{ab}\right)^q \le 2 \, \left(2np \, \|{\bm{E}}\|^2_\infty \right)^q.$$
The proof of this lemma is in the Appendix. The same estimate applies to ${\operatorname{\mathbb{E}}}\left(\max_{b} \sum_{a} \delta_{ab} E^2_{ab}\right)^q$ and thus for each $q \ge 1$ $${\operatorname{\mathbb{E}}}_\delta \max \left[\|\sum_{ab} \delta_{ab} E^2_{ab} {\bm{e}}_a
{\bm{e}}_a^*\|^{q}, \|\sum_{ab} \delta_{ab} E^2_{ab} {\bm{e}}_b
{\bm{e}}_b^*\|^{q}\right] \le 4\, \left(2np \, \|{\bm{E}}\|^2_\infty
\right)^q.$$ (In the rectangular case, the same estimate holds with $n = \max(n_1,
n_2)$.)
Take $q = \beta \log n$ for some $\beta \ge 1$, and set $q' = q$. Then since $\|{\bm{E}}\|_\infty \le \mu_1 \sqrt{r}/n$, we established that $$\left({\operatorname{\mathbb{E}}}\|{\bm{S}}\|^q\right)^{1/q} \le C \, \frac{1}{p} \,
\sqrt{\beta \log n} \,\sqrt{np} \, \|{\bm{E}}\|_\infty = C \, \mu_1
\, \sqrt{\frac{n r \, \beta \log n}{m}} \equiv K_0.$$ Then by Markov’s inequality, for each $t > 0$, $${\operatorname{\mathbb{P}}}(\|{\bm{S}}\| > t K_0) \le t^{-q},$$ and for $t = e$, we conclude that $${\operatorname{\mathbb{P}}}\left(\|{\bm{S}}\| > C e \, \mu_1 \, \sqrt{\frac{nr\, \beta \log
n}{m}}\right) \le n^{-\beta}$$ with the proviso that $m \ge \max(\beta, 2) \, n \log n$ so that Lemma \[teo:E-max-indep2\] holds.
We have not made any assumption in this section about the matrix ${\bm{E}}$ (except that we have a bound on the maximum entry) and, therefore, have proved the theorem below, which shall be used many times in the sequel.
\[teo:PO\] Let ${\bm{X}}$ be a fixed $n\times n$ matrix. There is a constant $C_0$ such that for each $\beta > 2$ $$\label{eq:PObound} p^{-1} \|({{\cal P}_{\Omega}}- p{\mathcal{I}})({\bm{X}})\|
\le C_0\, \left(\frac{\beta n \log n}{p}\right)^{1/2} \,
\|{\bm{X}}\|_\infty$$ with probability at least $1 - n^{-\beta}$ provided that $np \ge \beta \log n$.
Note that this is the same $C_0$ described in Lemma \[teo:order0\].
Proof of Lemma \[teo:order1\] {#sec:order1}
-----------------------------
We now need to bound the spectral norm of ${{\cal P}_{T^\perp}}{{\cal P}_{\Omega}}{{\cal P}_T}\,
\mathcal{H}({\bm{E}})$ and will use some of the ideas developed in the previous section. Just as before, $$p^{-1} \|{{\cal P}_{T^\perp}}{{\cal P}_{\Omega}}{{\cal P}_T}\, \mathcal{H}({\bm{E}})\| \le p^{-1} \|({{\cal P}_{\Omega}}-
p{\mathcal{I}}) \, \mathcal{H}({\bm{E}})\|,$$ and put $${\bm{S}} \equiv p^{-1} ({{\cal P}_{\Omega}}- p{\mathcal{I}}) \, \mathcal{H}(E) = p^{-2}
\sum_{ab,
a'b'} \xi_{ab} \xi_{a'b'} \, E_{a'b'} {\langle}{{\cal P}_T}{{\bm{e}}_{a'} {\bm{e}}_{b'}^*},
{{\bm{e}}_{a}{\bm{e}}_{b}^*}{\rangle}{{\bm{e}}_{a}{\bm{e}}_{b}^*},$$ where here and below, $\xi_{ab} \equiv \delta_{ab} - p$. Decompose ${\bm{S}}$ as $$\label{eq:diagoff} {\bm{S}} \,\,\, = \,\,\, p^{-2} \sum_{(a,b) = (a',b')} \,\,\, +
\,\,\, p^{-2} \sum_{(a,b) \, \neq \, (a',b')} \,\,\, \equiv \,\,\,
{\bm{S}}_0 + {\bm{S}}_1.$$ We bound the spectral norm of the diagonal and off-diagonal contributions separately.
We begin with ${\bm{S}}_0$ and decompose $(\xi_{ab})^2$ as $$\xi_{ab}^2 = ({\delta_{ab}}- p)^2 = (1-2p)({\delta_{ab}}- p) + p(1-p) =
(1-2p)\xi_{ab} + p(1-p),$$ which allows us to express ${\bm{S}}_0$ as $$\label{eq:S0} {\bm{S}}_0 = \frac{1-2p}{p} \sum_{ab} {\xi_{ab}}\, H_{ab}
{{\bm{e}}_{a}{\bm{e}}_{b}^*}+ (1-p) \sum_{ab} H_{ab} {{\bm{e}}_{a}{\bm{e}}_{b}^*}, \quad H_{ab} \equiv p^{-1} \,
E_{ab} {\langle}{{\cal P}_T}{{\bm{e}}_{a}{\bm{e}}_{b}^*}, {{\bm{e}}_{a}{\bm{e}}_{b}^*}{\rangle}.$$ Theorem \[teo:PO\] bounds the spectral norm of the first term of the right-hand side and we have $$p^{-1} \|\sum_{ab} {\xi_{ab}}\, H_{ab} {{\bm{e}}_{a}{\bm{e}}_{b}^*}\| \le C_0 \, \sqrt{\frac{n^3
\beta \log n}{m}} \, \|{\bm{H}}\|_\infty$$ with probability at least $1 - n^{-\beta}$. Now since $\|{\bm{E}}\|_\infty \le \mu_1\sqrt{r}/n$ and $|{\langle}{{\cal P}_T}{{\bm{e}}_{a}{\bm{e}}_{b}^*}, {{\bm{e}}_{a}{\bm{e}}_{b}^*}{\rangle}| \le
2\mu_0 r/n$ by , $\|{\bm{H}}\|_\infty \le \mu_0\mu_1(2r/np)
\, \sqrt{r}/n$, and $$p^{-1} \| \sum_{ab} {\xi_{ab}}\, H_{ab} {{\bm{e}}_{a}{\bm{e}}_{b}^*}\| \le C \mu_0 \mu_1 \,
\frac{nr}{m} \, \sqrt{\frac{ n r \beta \log n}{m}}$$ with the same probability. The second term of the right-hand side in is deterministic and we develop an argument that we will reuse several times. We record a useful lemma.
\[teo:lemma0p\] Let ${\bm{X}}$ be a fixed matrix and set ${\bm{Z}} \equiv \sum_{ab}
X_{ab} {\langle}{{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*}), {{\bm{e}}_{a}{\bm{e}}_{b}^*}{\rangle}{{\bm{e}}_{a}{\bm{e}}_{b}^*}$. Then $$\|{\bm{Z}}\| \le \frac{2\mu_0 r}{n} \|{\bm{X}}\|.$$
Let ${\bm{\Lambda}}_U$ and ${\bm{\Lambda}}_V$ be the diagonal matrices with entries $\|{\bm{P}}_U e_a\|^2$ and $\|{\bm{P}}_V e_b\|^2$ respectively, $$\label{eq:Lambda}
{\bm{\Lambda}}_U = \textrm{diag}(\|{\bm{P}}_U e_a\|^2), \quad
{\bm{\Lambda}}_V = \textrm{diag}(\|{\bm{P}}_V e_b\|^2).$$ To bound the spectral norm of ${\bm{Z}}$, observe that it follows from that $$\label{eq:LambdaH}
{\bm{Z}} = {\bm{\Lambda}}_U {\bm{X}} + {\bm{X}} {\bm{\Lambda}}_V -
{\bm{\Lambda}}_U {\bm{X}} {\bm{\Lambda}}_V = {\bm{\Lambda}}_U {\bm{X}} ({\bm{I}} - {\bm{\Lambda}}_V) + {\bm{X}} {\bm{\Lambda}}_V.$$ Hence, since $\|{\bm{\Lambda}}_U\|$ and $\|{\bm{\Lambda}}_V\|$ are bounded by $\min(\mu_0 r/n,1)$ and $\|{\bm{I}} - {\bm{\Lambda_V}}\| \le
1$, we have $$\|{\bm{Z}}\| \le \|{\bm{\Lambda}}_U\| \|{\bm{X}}\| \|{\bm{I}} -
{\bm{\Lambda}}_V\| + \|{\bm{X}}\| \|{\bm{\Lambda}}_V\| \le (2\mu_0 r/n)
\|{\bm{X}}\|.$$
Clearly, this lemma and $\|{\bm{E}}\| = 1$ give that ${\bm{H}}$ defined in obeys $\|{\bm{H}}\| \le 2\mu_0 r/np$. In summary, $$\|{\bm{S}}_0\| \le C \frac{nr}{m}\left(\mu_0 \mu_1 \sqrt{\frac{\beta nr\log
n}{m}} + \mu_0\right)$$ for some $C > 0$ with the same probability as in Lemma \[teo:order0\].
It remains to bound the off-diagonal term. To this end, we use a useful decoupling lemma:
[@delaPena2] \[teo:decoupling\] Let $\{\eta_i\}_{1\le i \le n}$ be a sequence of independent random variables, and $\{x_{ij}\}_{i \neq j}$ be elements taken from a Banach space. Then $$\label{eq:decoupling}
{\operatorname{\mathbb{P}}}(\|\sum_{i \neq j} \eta_i \eta_j x_{ij}\| \ge t) \le C_D {\operatorname{\mathbb{P}}}(\|\sum_{i \neq j} \eta_i \eta'_j x_{ij}\| > t/C_D),$$ where $\{\eta'_i\}$ is an independent copy of $\{\eta_i\}$.
This lemma asserts that it is sufficient to estimate ${\operatorname{\mathbb{P}}}(\|{\bm{S}}_1'\| \ge t)$ where ${\bm{S}}'_1$ is given by $$\label{eq:Sp1}
{\bm{S}}'_1 \equiv p^{-2} \sum_{ab \neq
a'b'} {\xi_{ab}}{\xi'_{a'b'}}\, E_{a'b'} {\langle}{{\cal P}_T}{{\bm{e}}_{a'} {\bm{e}}_{b'}^*},
{{\bm{e}}_{a}{\bm{e}}_{b}^*}{\rangle}{{\bm{e}}_{a}{\bm{e}}_{b}^*}$$ in which $\{\xi'_{ab}\}$ is an independent copy of $\{{\xi_{ab}}\}$. We write ${\bm{S}}'_1$ as $$\label{eq:Sp1b} {\bm{S}}'_1 = p^{-1} \sum_{ab} {\xi_{ab}}\, H_{ab} {{\bm{e}}_{a}{\bm{e}}_{b}^*},
\qquad H_{ab} \equiv p^{-1} \sum_{a'b': (a',b') \neq (a,b)} {\xi'_{a'b'}}\,
E_{a'b'} {\langle}{{\cal P}_T}{{\bm{e}}_{a'} {\bm{e}}_{b'}^*}, {{\bm{e}}_{a}{\bm{e}}_{b}^*}{\rangle}.$$ To bound the tail of $\|{\bm{S}}'_1\|$, observe that $${\operatorname{\mathbb{P}}}(\|{\bm{S}}'_1\| \ge t) \le {\operatorname{\mathbb{P}}}(\|{\bm{S}}'_1\| \ge t \,\, | \,\,
\|{\bm{H}}\|_{\infty} \le K) + {\operatorname{\mathbb{P}}}(\|{\bm{H}}\|_{\infty} > K).$$ By independence, the first term of the right-hand side is bounded by Theorem \[teo:PO\]. On the event $\{\|{\bm{H}}\|_\infty \le K\}$, we have $$p^{-1} \|\sum_{ab} {\xi_{ab}}\, H_{ab} {{\bm{e}}_{a}{\bm{e}}_{b}^*}\| \le C \, \sqrt{\frac{n^3
\beta \log n}{m}} \, K.$$ with probability at least $1 - n^{-\beta}$. To bound $\|{\bm{H}}\|_\infty$, we use Bernstein’s inequality.
\[teo:bernstein\] Let ${\bm{X}}$ be a fixed matrix and define $\mathcal{Q}({\bm{X}})$ as the matrix whose $(a,b)$th entry is $$[\mathcal{Q}({\bm{X}})]_{ab} = p^{-1} \sum_{a'b' : (a',b') \neq
(a,b)} (\delta_{a'b'} - p) \, X_{a'b'} {\langle}{{\cal P}_T}{{\bm{e}}_{a'} {\bm{e}}_{b'}^*}, {{\bm{e}}_{a}{\bm{e}}_{b}^*}{\rangle},$$ where $\{\delta_{ab}\}$ is an independent Bernoulli sequence obeying ${\operatorname{\mathbb{P}}}(\delta_{ab} = 1) = p$. Then $$\label{eq:Hinf}
{\operatorname{\mathbb{P}}}\left(\|\mathcal{Q}({\bm{X}})\|_\infty > \lambda \, \sqrt{\frac{\mu_0 r}{np}} \, \|{\bm{X}}\|_\infty\right) \le
2 n^2\, \exp\left(-\frac{\lambda^2}{2 + \frac{2}{3} \sqrt{\frac{\mu_0 r}{np}} \lambda}\right).$$ With $\lambda = \sqrt{3\beta \log n}$, the right-hand side is bounded by $2n^{2-\beta}$ provided that $np \ge \frac{4\beta}{3} \mu_0 r\log
n$. In particular, for $\lambda = \sqrt{6\beta \log n}$ with $\beta >
2$, the bound is less than $2n^{-\beta}$ provided that $np \ge
\frac{8\beta}{3} \mu_0 r\log n$.
The inequality is an application of Bernstein’s inequality, which states that for a sum of uniformly bounded independent zero-mean random variables obeying $|Y_k| \le c$, $$\label{eq:bern2}
{\operatorname{\mathbb{P}}}\left(|\sum_{k = 1}^n Y_k| > t\right) \le 2e^{-t^2/(2\sigma^2 + 2ct/3)},$$ where $\sigma^2$ is the sum of the variances, $\sigma^2 \equiv \sum_{k
= 1}^n \textrm{Var}(Y_k)$. We have $$\begin{aligned}
\textrm{Var}([\mathcal{Q}({\bm{X}})]_{ab}) & = \frac{1-p}{p} \sum_{a'b': (a',b')
\neq (a,b)} |X_{a'b'}|^2 |{\langle}{{\cal P}_T}{{\bm{e}}_{a'} {\bm{e}}_{b'}^*}, {{\bm{e}}_{a}{\bm{e}}_{b}^*}{\rangle}|^2\\
& \le \frac{1-p}{p} \, \|{\bm{X}}\|_\infty^2 \sum_{a'b' : (a',b') \neq (a,b)}
|{\langle}{{\cal P}_T}{{\bm{e}}_{a}{\bm{e}}_{b}^*}, {{\bm{e}}_{a'} {\bm{e}}_{b'}^*}{\rangle}|^2 \le \frac{1-p}{p} \, \|{\bm{X}}\|_\infty^2 \, 2\mu_0 r/n\end{aligned}$$ by . Also, $$p^{-1} \, \left|(\delta_{a'b'} - p) X_{a'b'} {\langle}{{\cal P}_T}{{\bm{e}}_{a'} {\bm{e}}_{b'}^*}, {{\bm{e}}_{a}{\bm{e}}_{b}^*}{\rangle}\right| \le p^{-1} \, \|{\bm{X}}\|_\infty \, 2\mu_0 r/n$$ and hence, for each $t > 0$, gives $$\label{eq:bern3}
{\operatorname{\mathbb{P}}}(|[\mathcal{Q}({\bm{X}})]_{ab}| > t) \le 2 \exp\left(-\frac{t^2}{2\frac{\mu_0 r}{np}
\|{\bm{X}}\|_\infty^2 + \frac{2}{3} \frac{\mu_0 r}{np} \|{\bm{X}}\|_\infty t}\right).$$ Putting $t = \lambda \sqrt{\mu_0 r/np} \|{\bm{X}}\|_\infty$ for some $\lambda > 0$ and applying the union bound gives .
Since $\|{\bm{E}}\|_\infty \le \mu_1 \sqrt{r}/n$ it follows that ${\bm{H}}=\mathcal{Q}({\bm{E}})$ introduced in obeys $$\|{\bm{H}}\|_\infty \le C \, \frac{\mu_1 \sqrt{r}}{n} \,
\sqrt{\frac{\mu_0 nr \beta \log n}{m}}$$ with probability at least $1 - 2n^{-\beta}$ for each $\beta > 2$ and, therefore, $$\|{\bm{S}}'_1\| \le C \, \sqrt{\mu_0} \mu_1 \frac{nr \beta\log n}{m}$$ with probability at least $1 - 3n^{-\beta}$. In conclusion, we have $$\label{eq:important2}
p^{-1} \|({{\cal P}_{\Omega}}- p{\mathcal{I}}) \, \mathcal{H}({\bm{E}})\| \le C
\frac{nr}{m}\left(\sqrt{\mu_0} \mu_1 \left(\sqrt{\frac{\mu_0 nr\beta \log n}{m}} + \beta \log n\right) + \mu_0 \right)$$ with probability at least $1 - (1+3C_D)n^{-\beta}$. A simple algebraic manipulation concludes the proof of Lemma \[teo:order1\]. Note that we have not made any assumption about the matrix ${\bm{E}}$ and, therefore, established the following:
\[teo:PO2\] Let ${\bm{X}}$ be a fixed $n\times n$ matrix. There is a constant $C'_0$ such that $$\label{eq:PObound2} p^{-2} \|\sum_{(a,b) \neq (a',b')} {\xi_{ab}}{\xi_{a'b'}}X_{ab} {\langle}{{\cal P}_T}({{\bm{e}}_{a'} {\bm{e}}_{b'}^*}), {{\bm{e}}_{a}{\bm{e}}_{b}^*}{\rangle}{{\bm{e}}_{a}{\bm{e}}_{b}^*}\| \le C'_0 \frac{\sqrt{\mu_0 r} \, \beta \log n}{p} \|{\bm{X}}\|_\infty$$ with probability at least $1 - O(n^{-\beta})$ for all $\beta > 2$ provided that $np \ge 3 \mu_0 r \beta \log n$.
Proof of Lemma \[teo:order2\] {#sec:order2}
-----------------------------
To prove Lemma \[teo:order2\], we need to bound the spectral norm of $p^{-1} \, ({{\cal P}_{\Omega}}- p{\mathcal{I}}) \, \mathcal{H}^2({\bm{E}})$, a matrix given by $$p^{-3} \sum_{a_1 b_1, a_2 b_2, a_3 b_3} \xi_{a_1 b_1} \xi_{a_2 b_2}
\xi _{a_3 b_3} E_{a_3 b_3} {\langle}{{\cal P}_T}{{\bm{e}}_{a_3}{\bm{e}}_{b_3}^*}, {{\bm{e}}_{a_2}{\bm{e}}_{b_2}^*}{\rangle}{\langle}{{\cal P}_T}{{\bm{e}}_{a_2}{\bm{e}}_{b_2}^*},
{{\bm{e}}_{a_1}{\bm{e}}_{b_1}^*}{\rangle}{{\bm{e}}_{a_1}{\bm{e}}_{b_1}^*},$$ where $\xi_{ab} = \delta_{ab} - p$ as before. It is convenient to introduce notations to compress this expression. Set $\omega =
(a,b)$ (and $\omega_i = (a_i, b_i)$ for $i = 1, 2, 3$), ${\bm{F}}_\omega = {{\bm{e}}_{a}{\bm{e}}_{b}^*}$, and $P_{\omega' \omega} = {\langle}{{\cal P}_T}{{\bm{e}}_{a'} {\bm{e}}_{b'}^*},
{{\bm{e}}_{a}{\bm{e}}_{b}^*}{\rangle}$ so that $$p^{-1} \, ({{\cal P}_{\Omega}}- p{\mathcal{I}}) \, \mathcal{H}^2({\bm{E}}) = p^{-3}
\sum_{\omega_1, \omega_2, \omega_3} \xi_{\omega_1} \xi_{\omega_2}
\xi _{\omega_3} \, E_{\omega_3} P_{\omega_3 \omega_2} P_{\omega_2
\omega_1} {\bm{F}}_{\omega_1}.$$ Partition the sum depending on whether some of the $\omega_i$’s are the same or not $$\label{eq:equiv}
\frac{1}{p}({{\cal P}_{\Omega}}- p{\mathcal{I}})\mathcal{H}^2({\bm{E}}) = \frac{1}{p^3}
\left[\sum_{\omega_1 = \omega_2 = \omega_3} +
\sum_{\omega_1 \neq \omega_2 = \omega_3} + \sum_{\omega_1 = \omega_3
\neq \omega_2} + \sum_{\omega_1 = \omega_2 \neq \omega_3}
+ \sum_{\omega_1 \neq \omega_2 \neq \omega_3}\right].$$ The meaning should be clear; for instance, the sum $\sum_{\omega_1
\neq \omega_2 = \omega_3}$ is the sum over the $\omega$’s such that $\omega_2 = \omega_3$ and $\omega_1 \neq \omega_2$. Similarly, $\sum_{\omega_1 \neq \omega_2 \neq \omega_3}$ is the sum over the $\omega$’s such that they are all distinct. The idea is now to use a decoupling argument to bound each sum in the right-hand side of (except for the first which does not need to be decoupled) and show that all terms are appropriately small in the spectral norm.
We begin with the first term which is equal to $$\label{eq:onetwothree}
\frac{1}{p^{3}} \, \sum_\omega (\xi_{\omega})^3 \, E_{\omega} P_{\omega
\omega}^2 {\bm{F}}_{\omega} = \frac{1-3p+3p^2}{p^3} \, \sum_\omega
\xi_{\omega} \, E_{\omega} P_{\omega \omega}^2 {\bm{F}}_{\omega} +
\frac{1-3p+2p^2}{p^2} \sum_\omega E_{\omega} P_{\omega \omega}^2
{\bm{F}}_{\omega},$$ where we have used the identity $$(\xi_\omega)^3 = (1-3p+3p^2)\xi_\omega + p(1-3p+2p^2).$$ Set $H_{\omega} = E_\omega (p^{-1} P_{\omega \omega})^2$. For the first term in the right-hand side of , we need to control $\| \sum_\omega \xi_{\omega} \, H_\omega
{\bm{F}}_{\omega}\|$. This is easily bounded by Theorem \[teo:PO\]. Indeed, it follows from $$|H_{\omega}| \le \left(\frac{2\mu_0 r}{np}\right)^2 \,
\|{\bm{E}}\|_\infty$$ that for each $\beta > 0$, $$p^{-1} \| \sum_\omega \xi_{\omega} \, H_{\omega} \, {\bm{F}}_{\omega}\|
\le C \, \left(\frac{\mu_0 nr}{m}\right)^2 \, \mu_1 \sqrt{\frac{ nr
\beta \log n}{m}} = C \, \mu_0^2 \mu_1 \sqrt{\beta \log n} \,
\left(\frac{nr}{m}\right)^{5/2}$$ with probably at least $1 - n^{-\beta}$. For the second term in the right-hand side of , we apply Lemma \[teo:lemma0p\] which gives $$\|\sum_{\omega} E_\omega P_{\omega \omega}^2 {\bm{F}}_\omega\| \le (2\mu_0 r/n)^2$$ so that $\|{\bm{H}}\| \le (2\mu_0r/np)^2$. In conclusion, the first term in has a spectral norm which is bounded by $$C \, \left(\frac{nr}{m}\right)^2 \, \left(\mu_0^2 \mu_1 \left(\frac{nr
\beta \log n}{m}\right)^{1/2} + \mu_0^2\right)$$ with probability at least $1 - n^{-\beta}$.
We now turn our attention to the second term which can be written as $$\begin{gathered}
p^{-3} \sum_{\omega_1 \neq \omega_2}
\xi_{\omega_1} (\xi_{\omega_2})^2 \, E_{\omega_2} P_{\omega_2
\omega_2} P_{\omega_2 \omega_1} {\bm{F}}_{\omega_1} =
\frac{1-2p}{p^3} \sum_{\omega_1 \neq \omega_2} \xi_{\omega_1}
\xi_{\omega_2} \, E_{\omega_2} P_{\omega_2 \omega_2} P_{\omega_2
\omega_1} {\bm{F}}_{\omega_1} \\
+ \frac{1-p}{p^2} \sum_{\omega_1 \neq
\omega_2} \xi_{\omega_1} \, E_{\omega_2}
P_{\omega_2 \omega_2} P_{\omega_2 \omega_1} {\bm{F}}_{\omega_1}.\end{gathered}$$ Put ${\bm{S}}_1$ for the first term; bounding $\|{\bm{S}}_1\|$ is a simple application of Lemma \[teo:PO2\] with $X_\omega = p^{-1}
E_\omega P_{\omega \omega}$, which gives $$\|{\bm{S}}_1\| \le C\, \mu_0^{3/2} \mu_1 \, (\beta \log n) \,
\left(\frac{nr}{m}\right)^{2}$$ since $\|{\bm{E}}\|_\infty \le \mu_1 \sqrt{r}/n$. For the second term, we need to bound the spectral norm of ${\bm{S}}_2$ where $${\bm{S}}_2 \equiv p^{-1} \sum_{\omega_1} \xi_{\omega_1}
H_{\omega_1} {\bm{F}}_{\omega_1}, \qquad H_{\omega_1} = p^{-1}
\sum_{\omega_2 : \omega_2 \neq \omega_1} E_{\omega_2}
P_{\omega_2 \omega_2} P_{\omega_2 \omega_1}.$$ Note that ${\bm{H}}$ is deterministic. The lemma below provides an estimate about $\|{\bm{H}}\|_\infty$.
\[teo:Hp\] The matrix ${\bm{H}}$ obeys $$\label{eq:Hp}
\|{\bm{H}}\|_\infty \le
\frac{\mu_0 r}{np}\left(3\|{\bm{E}}\|_\infty + 2\frac{\mu_0 r}{n}\right).$$
We begin by rewriting ${\bm{H}}$ as $$pH_\omega = \sum_{\omega'} E_{\omega'} P_{\omega' \omega'}
P_{\omega'
\omega} - E_\omega P^2_{\omega \omega}.$$ Clearly, $|E_\omega P^2_{\omega \omega}| \le (\mu_0 r/n)^2
\|{\bm{E}}\|_\infty$ so that it suffices to bound the first term, which is the $\omega$th entry of the matrix $$\sum_{\omega, \omega'} E_{\omega'} P_{\omega' \omega'} P_{\omega'
\omega} {\bm{F}}_\omega = {{\cal P}_T}({\bm{\Lambda}}_U {\bm{E}} + {\bm{E}}
{\bm{\Lambda}}_V - {\bm{\Lambda}}_U {\bm{E}} {\bm{\Lambda}}_V).$$ Now it is immediate to see that ${\bm{\Lambda}}_U {\bm{E}} \in T$ and likewise for ${\bm{E}} {\bm{\Lambda}}_V$. Hence, $$\begin{aligned}
\| {{\cal P}_T}({\bm{\Lambda}}_U {\bm{E}} + {\bm{E}} {\bm{\Lambda}}_V -
{\bm{\Lambda}}_U {\bm{E}} {\bm{\Lambda}}_V)\|_\infty & \le \|{\bm{\Lambda}}_U
{\bm{E}}\|_\infty + \|{\bm{E}} {\bm{\Lambda}}_V\|_\infty + \|{{\cal P}_T}({\bm{\Lambda}}_U
{\bm{E}} {\bm{\Lambda}}_V)\|_\infty \\ & \le 2 \|{\bm{E}}\|_\infty
\mu_0r/n + \|{{\cal P}_T}({\bm{\Lambda}}_U {\bm{E}} {\bm{\Lambda}}_V)\|_\infty.\end{aligned}$$ We finally use the crude estimate $$\|{{\cal P}_T}({\bm{\Lambda}}_U {\bm{E}} {\bm{\Lambda}}_V)\|_\infty \le
\|{{\cal P}_T}({\bm{\Lambda}}_U {\bm{E}} {\bm{\Lambda}}_V)\| \le 2
\|{\bm{\Lambda}}_U {\bm{E}} {\bm{\Lambda}}_V\| \le 2(\mu_0 r/n)^2$$ to complete the proof of the lemma.
As a consequence of this lemma, Theorem \[teo:PO\] gives $$\|{\bm{S}}_2\| \le C \, \sqrt{\beta \log n} \,
\left(\frac{nr}{m}\right)^{3/2}(\mu_0 \mu_1 + \mu_0^2 \sqrt{r})$$ with probability at least $1 - n^{-\beta}$. In conclusion, the second term in has spectral norm bounded by $$C \, \sqrt{\beta \log n} \left(\frac{nr}{m}\right)^{3/2}\left(\mu_0
\mu_1 \sqrt{\frac{\mu_0 nr \beta \log n}{m}} + \mu_0 \mu_1 + \mu_0^2
\sqrt{r}\right)$$ with probability at least $1 - O(n^{-\beta})$.
We now examine the third term which can be written as $$\begin{gathered}
p^{-3} \sum_{\omega_1 \neq \omega_2} (\xi_{\omega_1})^2
\xi_{\omega_2} \, E_{\omega_1} P_{\omega_1 \omega_2} P_{\omega_2
\omega_1} {\bm{F}}_{\omega_1} = \frac{1-2p}{p^3} \sum_{\omega_1 \neq
\omega_2} \xi_{\omega_1}
\xi_{\omega_2} \, E_{\omega_1} P^2_{\omega_2 \omega_1} {\bm{F}}_{\omega_1} \\
+ \frac{1-p}{p^2} \sum_{\omega_1 \neq \omega_2} \xi_{\omega_2} \,
E_{\omega_1} P^2_{\omega_2 \omega_1} {\bm{F}}_{\omega_1}.\end{gathered}$$ We use the decoupling argument once more so that for the first term of the right-hand side, it suffices to estimate the tail of the norm of $${\bm{S}}_1 \equiv p^{-1} \sum_{\omega_1} \xi^{(1)}_{\omega_1}
E_{\omega_1} H_{\omega_1} {\bm{F}}_{\omega_1}, \qquad H_{\omega_1}
\equiv p^{-2} \sum_{\omega_2
: \omega_2 \neq \omega_1} \xi^{(2)}_{\omega_2} \, P^2_{\omega_2
\omega_1},$$ where $\{\xi^{(1)}_\omega\}$ and $\{\xi^{(2)}_\omega\}$ are independent copies of $\{\xi_\omega\}$. It follows from Bernstein’s inequality and the estimates $$|P_{\omega_2 \omega_1}| \le 2\mu_0 r/n$$ and $$\sum_{\omega_2 : \omega_2 \neq \omega_1} |P_{\omega_2 \omega_1}|^4 \le
\max_{\omega_2 : \omega_2 \neq \omega_1} |P_{\omega_2 \omega_1}|^2 \,
\sum_{\omega_2 : \omega_2 \neq \omega_1} |P_{\omega_2 \omega_1}|^2 \le
\left(\frac{2\mu_0 r}{n}\right)^2 \,\frac{2\mu_0 r}{n}$$ that for each $\lambda > 0$,[^4] $${\operatorname{\mathbb{P}}}\left(|H_{\omega_1}| > \lambda \left(\frac{2\mu_0
r}{np}\right)^{3/2}\right) \le 2 \exp\left(-\frac{\lambda^2}{2
+ \frac{2}{3} \lambda \left(\frac{2\mu_0
r}{np}\right)^{1/2}}\right).$$ It is now not hard to see that this inequality implies that $$P\left(\|{\bm{H}}\|_\infty > \sqrt{8\beta \log n} \,
\left(\frac{2 \mu_0 nr}{m}\right)^{3/2} \right) \le 2 \,
n^{-2\beta + 2}$$ provided that $m \ge \frac{16}{9} \mu_0 nr \, \beta\log n$. As a consequence, for each $\beta > 2$, Theorem \[teo:PO\] gives $$\|{\bm{S}}_1\| \le C \, \mu_0^{3/2} \mu_1 \, \beta \log n \left(\frac{nr}{m}\right)^{2}$$ with probability at least $1-3n^{-\beta}$. The other term is equal to $(1-p)$ times $\sum_{\omega_1} E_{\omega_1} H_{\omega_1}
{\bm{F}}_{\omega_1}$, and $$\begin{aligned}
\|\sum_{\omega_1} E_{\omega_1} H_{\omega_1} {\bm{F}}_{\omega_1}\| &
\le \|\sum_{\omega_1} E_{\omega_1} H_{\omega_1}
{\bm{F}}_{\omega_1}\|_F \\ & \le \|{\bm{H}}\|_\infty \|{\bm{E}}\|_F \le C \,
{\sqrt{\beta \log n}} \left(\frac{\mu_0 nr}{m}\right)^{3/2} \, \sqrt{r}.\end{aligned}$$ In conclusion, the third term in has spectral norm bounded by $$C \, \mu_0\, \sqrt{\beta \log n} \left(\frac{nr}{m}\right)^{3/2}
\left(\mu_1 \sqrt{\frac{\mu_0 n r \beta \log n}{m}} +
\sqrt{\mu_0 r}\right)$$ with probability at least $1-O(n^{-\beta})$.
We proceed to the fourth term which can be written as $$\begin{gathered}
p^{-3} \sum_{\omega_1 \neq \omega_3}
(\xi_{\omega_1})^2 \xi_{\omega_3} \, E_{\omega_3} P_{\omega_3
\omega_1} P_{\omega_1 \omega_1} {\bm{F}}_{\omega_1} =
\frac{1-2p}{p^3} \sum_{\omega_1 \neq \omega_3} \xi_{\omega_1}
\xi_{\omega_3} \, E_{\omega_3} P_{\omega_3 \omega_1} P_{\omega_1 \omega_1} {\bm{F}}_{\omega_1} \\
+ \frac{1-p}{p^2} \sum_{\omega_1 \neq \omega_3} \xi_{\omega_3} \,
E_{\omega_3} P_{\omega_3 \omega_1} P_{\omega_1 \omega_1}
{\bm{F}}_{\omega_1}.\end{gathered}$$ Let ${\bm{S}}_1$ be the first term and set $H_{\omega_1} = p^{-2}
\sum_{\omega_1 \neq \omega_3} \xi_{\omega_1} \xi_{\omega_3} \,
E_{\omega_3} P_{\omega_3 \omega_1} {\bm{F}}_{\omega_1}$. Then Lemma \[teo:lemma0p\] gives $$\|{\bm{S}}_1\| \le \frac{2\mu_0 r}{np} \|{\bm{H}}\| \le C \, \mu_0^{3/2}
\mu_1\, (\beta \log n) \, \left(\frac{nr}{m}\right)^{2}$$ where the last inequality is given by Lemma \[teo:PO2\]. For the other term—call it ${\bm{S}}_2$—set $H_{\omega_1} = p^{-1}
\sum_{\omega_3 : \omega_3 \neq \omega_1} \xi_{\omega_3} \,
E_{\omega_3} P_{\omega_3 \omega_1}$. Then Lemma \[teo:lemma0p\] gives $$\|{\bm{S}}_2\| \le \frac{2\mu_0 r}{np} \|{\bm{H}}\|.$$ Notice that $H_{\omega_1} = p^{-1} \sum_{\omega_3} \xi_{\omega_3} \,
E_{\omega_3} P_{\omega_3 \omega_1} - p^{-1} \xi_{\omega_1}
E_{\omega_1} P_{\omega_1 \omega_1}$ so that with $G_{\omega_1} =
E_{\omega_1} P_{\omega_1 \omega_1}$ $${\bm{H}} = p^{-1} [{{\cal P}_T}({{\cal P}_{\Omega}}- p{\mathcal{I}})({\bm{E}}) - ({{\cal P}_{\Omega}}-
p{\mathcal{I}})({\bm{G}})].$$ Now for any matrix ${\bm{X}}$, $\|{{\cal P}_T}({\bm{X}})\| = \|{\bm{X}} -
{{\cal P}_{T^\perp}}({\bm{X}})\| \le 2 \|{\bm{X}}\|$ and, therefore, $$\|{\bm{H}}\| \le 2p^{-1} \|({{\cal P}_{\Omega}}- p{\mathcal{I}})({\bm{E}})\| + p^{-1} \|({{\cal P}_{\Omega}}-
p{\mathcal{I}})({\bm{G}})\|.$$ As a consequence and since $\|{\bm{G}}\|_\infty \le
\|{\bm{E}}\|_\infty$, Theorem \[teo:PO\] gives for each $\beta > 2$, $$\|{\bm{H}}\| \le C \mu_1 \sqrt{\frac{nr \beta \log n}{m}}$$ with probability at least $1-n^{-\beta}$. In conclusion, the fourth term in has spectral norm bounded by $$C \, \mu_0 \mu_1 \sqrt{\beta \log n} \,
\left(\frac{nr}{m}\right)^{3/2} \left(\sqrt{\frac{\mu_0 nr \beta \log
n}{m}} + 1\right)$$ with probability at least $1-O(n^{-\beta})$.
We finally examine the last term $$p^{-3} \sum_{\omega_1 \neq \omega_2 \neq \omega_3} \xi_{\omega_1}
\xi_{\omega_2} \xi _{\omega_3} \, E_{\omega_3} P_{\omega_3
\omega_2} P_{\omega_2 \omega_1} {\bm{F}}_{\omega_1}.$$ Now just as one has a decoupling inequality for pairs of variables, we have a decoupling inequality for triples as well and we thus simply need to bound the tail of $${\bm{S}}_1 \equiv p^{-3} \sum_{\omega_1 \neq \omega_2 \neq \omega_3}
\xi^{(1)}_{\omega_1} \xi^{(2)}_{\omega_2} \xi^{(3)}_{\omega_3} \,
E_{\omega_3} P_{\omega_3 \omega_2} P_{\omega_2 \omega_1}
{\bm{F}}_{\omega_1}$$ in which the sequences $\{\xi_\omega^{(1)}\}$, $\{\xi_\omega^{(2)}\}$ and $\{\xi_\omega^{(3)}\}$ are independent copies of $\{\xi_\omega\}$. We refer to [@delaPena2] for details. We now argue as in Section \[sec:order1\] and write ${\bm{S}}_1$ as $${\bm{S}}_1 = p^{-1} \sum_{\omega_1} \xi^{(1)}_{\omega_1} H_{\omega_1}
{\bm{F}}_{\omega_1},$$ where $$\label{eq:forlater} H_{\omega_1} \equiv p^{-1} \sum_{\omega_2 :
\omega_2 \neq \omega_1} \xi^{(2)}_{\omega_2} \, G_{\omega_2} \,
P_{\omega_2 \omega_1}, \qquad G_{\omega_2} \equiv p^{-1}
\sum_{\omega_3 : \omega_3 \neq \omega_1,
\omega_3 \neq \omega_2} \xi^{(3)}_{\omega_3} \,
E_{\omega_3} \, P_{\omega_3 \omega_2}.$$ By Lemma \[teo:bernstein\], we have for each $\beta > 2$ $$\|{\bm{G}}\|_\infty \le C \, \sqrt{\frac{\mu_0 nr \beta \log n}{m}} \,
\|{\bm{E}}\|_\infty$$ with large probability and the same argument then gives $$\|{\bm{H}}\|_\infty \le C\, \sqrt{\frac{\mu_0 nr \beta \log n}{m}} \,
\|{\bm{G}}\|_\infty \le C \, \frac{\mu_0 nr \beta \log n}{m} \, \|{\bm{E}}\|_\infty$$ with probability at least $1 - 4n^{-\beta}$. As a consequence, Theorem \[teo:PO\] gives $$\| {\bm{S}} \| \le C \, \mu_0 \mu_1 \, \left(\frac{nr \beta \log
n}{m}\right)^{3/2}$$ with probability at least $1 - O(n^{-\beta})$.
To summarize the calculations of this section and using the fact that $\mu_0 \ge 1$ and $\mu_1 \le \mu_0 \sqrt{r}$, we have established that if $m \ge \mu_0 \, nr (\beta \log n)$, $$\begin{gathered}
p^{-1} \|({{\cal P}_{\Omega}}-
p{\mathcal{I}}) \, \mathcal{H}^2({\bm{E}})\| \le C \left(\frac{nr}{m}\right)^2\left(\mu_0^2 \mu_1 \sqrt{\frac{nr\beta \log n}{m}} + \mu_0^2\right)\\
+ C \sqrt{\beta \log n}
\left(\frac{nr}{m}\right)^{3/2} \mu_0^2 \sqrt{r}
+ C \left(\frac{nr\beta \log n}{m}\right)^{3/2} \mu_0 \mu_1\end{gathered}$$ with probability at least $1 - O(n^{-\beta})$. One can check that if $m = \lambda \, \mu_0^{4/3} n r^{4/3} \beta \log n$ for a fixed $\beta
\ge 2$ and $\lambda \ge 1$, then there is a constant $C$ such that $$\|p^{-1} \, ({{\cal P}_{\Omega}}- p{\mathcal{I}}) \, \mathcal{H}^2({\bm{E}})\| \le C
\lambda^{-3/2}$$ with probability at least $1 - O(n^{-\beta})$. This is the content of Lemma \[teo:order2\].
Proof of Lemma \[teo:order3\] {#sec:improved}
-----------------------------
Clearly, one could continue on the same path and estimate the spectral norm of $p^{-1} ({{\cal P}_{\Omega}}- p{\mathcal{I}}) \, \mathcal{H}^3({\bm{E}})$ by the same technique as in the previous sections. That is to say, we would write $$p^{-1} ({{\cal P}_{\Omega}}- p{\mathcal{I}}) \, \mathcal{H}^3({\bm{E}}) = p^{-4}
\sum_{\omega_1,
\omega_2, \omega_3, \omega_4} \, \left[ \prod_{i = 1}^4 \xi_{\omega_i} \right] \,
E_{\omega_4} \, \left[\prod_{i = 1}^3 P_{\omega_{i+1}
\omega_i}\right] \, {\bm{F}}_{\omega_1}$$ with the same notations as before, and partition the sum depending on whether some of the $\omega_i$’s are the same or not. Then we would use the decoupling argument to bound each term in the sum. Although this is a clear possibility, one would need to consider 18 cases and the calculations would become a little laborious. In this section, we propose to bound the term $p^{-1} ({{\cal P}_{\Omega}}- p{\mathcal{I}}) \,
\mathcal{H}^3({\bm{E}})$ with a different argument which has two main advantages: first, it is much shorter and second, it uses much of what we have already established. The downside is that it is not as sharp.
The starting point is to note that $$p^{-1} ({{\cal P}_{\Omega}}- p{\mathcal{I}}) \, \mathcal{H}^3({\bm{E}}) = p^{-1} (\Xi \circ
{\cal
H}^3({\bm{E}})),$$ where $\Xi$ is the matrix with i.i.d. entries equal to $\xi_{ab} =
\delta_{ab} - p$ and $\circ$ denotes the Hadamard product (componentwise multiplication). To bound the spectral norm of this Hadamard product, we apply an inequality due to Ando, Horn, and Johnson [@Ando87]. An elementary proof can be found in §5.6 of [@HJ2].
[@HJ2] \[teo:HJ\] Let ${\bm{A}}$ and ${\bm{B}}$ be two $n_1 \times n_2$ matrices. Then $$\label{eq:HJ} \|{\bm{A}} \circ {\bm{B}}\| \le \|{\bm{A}}\| \,
\nu({\bm{B}}),$$ where $\nu$ is the function $$\nu({\bm{B}}) = \inf \{c({\bm{X}}) c({\bm{Y}}) : {\bm{X}}{\bm{Y}}^* =
{\bm{B}}\},$$ and $c({\bm{X}})$ is the maximum Euclidean norm of the rows $$c({\bm{X}})^2 = \max_{1 \le i \le n} \sum_{j} X_{ij}^2.$$
To apply , we first notice that one can estimate the norm of $\Xi$ via Theorem \[teo:PO\]. Indeed, let ${\bm{Z}} = {\bf
1} {\bf 1}^*$ be the matrix with all entries equal to one. Then $p^{-1} \Xi = p^{-1} ({{\cal P}_{\Omega}}- p{\mathcal{I}})({\bm{Z}})$ and thus $$\label{eq:Xi}
p^{-1} \|\Xi\| \le C\, \left(\frac{n^3\beta \log n}{m}\right)^{1/2}$$ with probability at least $1-n^{-\beta}$. One could obtain a similar result by appealing to the recent literature on random matrix theory and on concentration of measure. Potentially this could allow to derive an upper bound without the logarithmic term but we will not consider these refinements here. (It is interesting to note in passing, however, that the two page proof of Theorem \[teo:PO\] gives a large deviation result about the largest singular value of a matrix with i.i.d. entries which is sharp up to a multiplicative factor proportional to at most $\sqrt{\log n}$.)
Second, we bound the second factor in via the following estimate:
\[teo:HE\] There are numerical constants $C$ and $c$ so that for each $\beta > 2$, $\mathcal{H}^3({\bm{E}})$ obeys $$\label{eq:HE} \nu(\mathcal{H}^3({\bm{E}})) \le C \mu_0 r/n$$ with probability at least $1-O(n^{-\beta})$ provided that $m \ge c
\, \mu_0^{4/3} \, nr^{5/3} (\beta \log n)$.
The two inequalities and give $$p^{-1} \|\Xi \circ \mathcal{H}^3({\bm{E}})\| \le C \,
\sqrt{\frac{\mu_0^2 \, nr^2 \, \beta \log
n}{m}},$$ with large probability. Hence, when $m$ is substantially larger than a constant times $\mu_0^2 n r^{2} (\beta \log n)$, we have that the spectral norm of $p^{-1}({{\cal P}_{\Omega}}- p{\mathcal{I}}) \, \mathcal{H}^3({\bm{E}})$ is much less than $1$. This is the content of Lemma \[teo:order3\].
The remainder of this section proves Lemma \[teo:HE\]. Set ${\bm{S}}
\equiv \mathcal{H}^3({\bm{E}})$ for short. Because ${\bm{S}}$ is in $T$, ${\bm{S}} = {{\cal P}_T}({\bm{S}}) = {\bm{P}}_U {\bm{S}} + {\bm{S}} {\bm{P}}_V -
{\bm{P}}_U {\bm{S}} {\bm{P}}_V$. Writing ${\bm{P}}_U = \sum_{j = 1}^r
{\bm{u}}_j {\bm{u}}_j^*$ and similarly for ${\bm{P}}_V$ gives $${\bm{S}} = \sum_{j = 1}^r {\bm{u}}_j ({\bm{u}}_j^* {\bm{S}}) + \sum_{j =
1}^r ((I - {\bm{P}}_{U}) {\bm{S}} {\bm{v}}_j) {\bm{v}}_j^*.$$ For each $1 \le j \le r$, let ${\bm{\alpha}}_j \equiv {\bm{S}} {\bm{v}}_j$ and ${\bm{\beta}}_j^* \equiv {\bm{u}}_j^* {\bm{S}}$. Then the decomposition $${\bm{S}} = \sum_{j = 1}^r {\bm{u}}_j {\bm{\beta}}_j^* + \sum_{j = 1}^r
({\bm{P}}_{U^\perp} {\bm{\alpha}}_j) {\bm{v}}_j^*,$$ where ${\bm{P}}_{U^\perp} = I - {\bm{P}}_U$, provides a factorization of the form $${\bm{S}} = {\bm{X}}{\bm{Y}}^*, \quad \begin{cases}
{\bm{X}} = [{\bm{u}}_1, \ldots, {\bm{u}}_r, {\bm{P}}_{U^\perp} {\bm{\alpha}}_1, \ldots, {\bm{P}}_{U^\perp} {\bm{\alpha}}_r],\\
{\bm{Y}} = [{\bm{v}}_1, \ldots, {\bm{v}}_r, {\bm{\beta}}_1, \ldots, {\bm{\beta}}_r].
\end{cases}$$ It follows from our assumption that $$c^2([{\bm{u}}_1, \ldots, {\bm{u}}_r]) = \max_{1 \le i \le n} \sum_{1
\le
j \le r} u_{ij}^2 = \max_{1 \le i \le n} \|{\bm{P}}_U {\bm{e}}_i\|^2 \le \mu_0 r/n,$$ and similarly for $[{\bm{v}}_1, \ldots, {\bm{v}}_r]$. Hence, to prove Lemma \[teo:HE\], it suffices to prove that the maximum row norm obeys $c([{\bm{\beta}}_1, \ldots, {\bm{\beta}}_r]) \le C \sqrt{\mu_0 r/n}$ for some constant $C > 0$, and similarly for the matrix $[{\bm{P}}_{U^\perp} {\bm{\alpha}}_1, \ldots, {\bm{P}}_{U^\perp}
{\bm{\alpha}}_r]$.
\[teo:toshow\] There is a numerical constant $C$ such that for each $\beta > 2$, $$\label{eq:toshow}
c([{\bm{\alpha}}_1, \ldots, {\bm{\alpha}}_r]) \le C \sqrt{\mu_0 r/n}$$ with probability at least $1-O(n^{-\beta})$ provided that $m$ obeys the condition of Lemma \[teo:HE\].
A similar estimate for $[{\bm{\beta}}_1, \ldots, {\bm{\beta}}_r]$ is obtained in the same way by exchanging the roles of ${\bm{u}}$ and ${\bm{v}}$. Moreover, a minor modification of the argument gives $$\label{eq:toshow1}
c([{\bm{P}}_{U^\perp}{\bm{\alpha}}_1, \ldots, {\bm{P}}_{U^\perp} {\bm{\alpha}}_r]) \le C \sqrt{\mu_0 r/n}$$ as well, and we will omit the details. In short, the estimate implies Lemma \[teo:HE\].
To prove , we use the notations of the previous section and write $$\begin{aligned}
{\bm{\alpha}}_j & = p^{-3} \sum_{a_1 b_1, a_2 b_2, a_3 b_3} \xi_{a_1
b_1} \xi_{a_2 b_2} \xi _{a_3 b_3} E_{a_3 b_3} {\langle}{{\cal P}_T}{{\bm{e}}_{a_3}{\bm{e}}_{b_3}^*},
{{\bm{e}}_{a_2}{\bm{e}}_{b_2}^*}{\rangle}{\langle}{{\cal P}_T}{{\bm{e}}_{a_2}{\bm{e}}_{b_2}^*}, {{\bm{e}}_{a_1}{\bm{e}}_{b_1}^*}{\rangle}{{\cal P}_T}({{\bm{e}}_{a_1}{\bm{e}}_{b_1}^*}) {\bm{v}}_j\\
& = p^{-3} \sum_{\omega_1, \omega_2, \omega_3} \xi_{\omega_1}
\xi_{\omega_2} \xi _{\omega_3} \, E_{\omega_3} P_{\omega_3 \omega_2} P_{\omega_2 \omega_1} {{\cal P}_T}({\bm{F_{\omega_1}}}) {\bm{v}}_j\\
& = p^{-3} \sum_{\omega_1, \omega_2, \omega_3} \xi_{\omega_1}
\xi_{\omega_2} \xi _{\omega_3} \, E_{\omega_3} P_{\omega_3
\omega_2} P_{\omega_2 \omega_1} ({\bm{F_{\omega_1}}} {\bm{v}}_j)\end{aligned}$$ since for any matrix ${\bm{X}}$, ${{\cal P}_T}({\bm{X}}) {\bm{v}}_j = {\bm{X}}
{\bm{v}}_j$ for each $1 \le j \le r$. We then follow the same steps as in Section \[sec:order2\] and partition the sum depending on whether some of the $\omega_i$’s are the same or not $$\label{eq:equiv1}
{\bm{\alpha}}_{j} = p^{-3} \left[\sum_{\omega_1 = \omega_2 = \omega_3} ~ + ~
\sum_{\omega_1 \neq \omega_2 = \omega_3} ~ + ~ \sum_{\omega_1 = \omega_3
\neq \omega_2} ~ + ~ \sum_{\omega_1 = \omega_2 \neq \omega_3}
~ + ~ \sum_{\omega_1 \neq \omega_2 \neq \omega_3}\right].$$ The idea is this: to establish , it is sufficient to show that if ${\bm{\gamma}}_{j}$ is any of the five terms above, it obeys $$\label{eq:fifth}
\sqrt{\sum_{1 \le j \le r} |\gamma_{ij}|^2} \le C \sqrt{\mu_0 r/n}$$ ($\gamma_{ij}$ is the $i$th component of ${\bm{\gamma}}_{j}$ as usual) with large probability. The strategy for getting such estimates is to use decoupling whenever applicable.
Just as Theorem \[teo:PO\] proved useful to bound the norm of $p^{-1} ({{\cal P}_{\Omega}}-p{\mathcal{I}}) \mathcal{H}^2({\bm{E}})$ in Section \[sec:order2\], the lemma below will help bounding the magnitudes of the components of ${\bm{\alpha}}_{j}$.
\[teo:sij\] Define ${\bm{S}} \equiv p^{-1} \sum_{ij} \sum_\omega \xi_\omega
H_\omega {\langle}{\bm{e}}_i, {\bm{F}}_\omega {\bm{v}}_j{\rangle}{\bm{e}}_i {\bm{e}}_j^*$. Then for each $\lambda > 0$ $$\label{eq:sij}
{\operatorname{\mathbb{P}}}(\|{\bm{S}}\|_\infty \ge \sqrt{\mu_0/n}) \le 2n^2
\exp\left(-\frac{1}{\frac{2n}{\mu_0 p} \|H\|_\infty^2 + \frac{2}{3p}
\sqrt{r} \|H\|_\infty}\right).$$
The proof is an application of Bernstein’s inequality . Note that ${\langle}{\bm{e}}_i, {\bm{F}}_\omega
{\bm{v}}_j{\rangle}= 1_{\{a = i\}} v_{bj}$ and hence $$\textrm{Var}(S_{ij}) \le p^{-1} \|{\bm{H}}\|_\infty^2 \sum_{\omega}
|{\langle}{\bm{e}}_i, {\bm{F}}_\omega {\bm{v}}_j{\rangle}|^2 = p^{-1}
\|{\bm{H}}\|_\infty^2$$ since $\sum_\omega |{\langle}{\bm{e}}_i, {\bm{F}}_\omega {\bm{v}}_j{\rangle}|^2 = 1$, and $|p^{-1} H_\omega {\langle}{\bm{e}}_i, {\bm{F}}_\omega {\bm{v}}_j{\rangle}| \le
p^{-1} \, \|{\bm{H}}\|_\infty \, \sqrt{\mu_0 r/n}$ since $|{\langle}{\bm{e}}_i, {\bm{F}}_\omega {\bm{v}}_j{\rangle}| \le |v_{bj}|$ and $$|v_{bj}| \le \|{\bm{P}}_V {\bm{e}}_b\| \le \sqrt{\mu_0 r/n}.$$
Each term in is given by the corresponding term in after formally substituting ${\bm{F}}_\omega$ with ${\bm{F}}_\omega {\bm{v}}_j$. We begin with the first term whose $i$th component is equal to $$\label{eq:term1}
\gamma_{ij} \equiv p^{-3}(1-3p+3p^2) \sum_\omega \xi_{\omega} \, E_{\omega}
P_{\omega \omega}^2 {{{\langle}{\bm{e}}_i, {\bm{F}}_\omega {\bm{v}}_j{\rangle}}}+ p^{-2}(1-3p+2p^2) \sum_\omega E_{\omega}
P_{\omega \omega}^2 {{{\langle}{\bm{e}}_i, {\bm{F}}_\omega {\bm{v}}_j{\rangle}}}.$$ Ignoring the constant factor $(1-3p+3p^2)$ which is bounded by 1, we write the first of these two terms as $$({\bm{S}}_0)_{ij} \equiv p^{-1} \sum_\omega \xi_\omega \, H_\omega {{{\langle}{\bm{e}}_i, {\bm{F}}_\omega {\bm{v}}_j{\rangle}}}, \qquad
H_\omega = E_\omega \, (p^{-1} P_{\omega \omega})^2.$$ Since $\|{\bm{H}}\|_\infty \le (\mu_0 nr/m)^2 \, \mu_1 \sqrt{r}/n$, it follows from Lemma that $${\operatorname{\mathbb{P}}}\left(\|{\bm{S}}_0\|_\infty \ge
\sqrt{\mu_0/n}\right) \le 2n^2 \,
{\mathrm{e}}^{-1/D}, \quad D \le C\left(\mu_0^3 \mu_1^2 \left(\frac{nr}{m}\right)^5 + \mu_0^{2} \mu_1 \left(\frac{nr}{m}\right)^3\right)$$ for some numerical $C > 0$. Since $\mu_1 \le \mu_0 \sqrt{r}$, we have that when $m \ge \lambda \mu_0 \, nr^{6/5} (\beta \log n)$ for some numerical constant $\lambda > 0$, $\|{\bm{S}}_0\|_\infty \ge
\sqrt{\mu_0/n}$ with probability at most $2n^2 e^{-(\beta \log n)^3}$; this probability is inversely proportional to a superpolynomial in $n$. For the second term, the matrix with entries $E_{\omega}
P_{\omega \omega}^2$ is given by $${\bm{\Lambda}}^2_U {\bm{E}} + {\bm{E}} {\bm{\Lambda}}^2_V +
2{\bm{\Lambda}}_U {\bm{E}} {\bm{\Lambda}}_V + {\bm{\Lambda}}^2_U {\bm{E}} {\bm{\Lambda}}^2_V
- 2{\bm{\Lambda}}^2_U {\bm{E}} {\bm{\Lambda}}_V - 2{\bm{\Lambda}}_U
{\bm{E}} {\bm{\Lambda}}^2_V$$ and thus $$\sum_\omega E_{\omega} P_{\omega \omega}^2 {{{\langle}{\bm{e}}_i, {\bm{F}}_\omega {\bm{v}}_j{\rangle}}}= {\langle}{\bm{e}}_i,
({\bm{\Lambda}}^2_U {\bm{E}} + {\bm{E}} {\bm{\Lambda}}^2_V +
2{\bm{\Lambda}}_U {\bm{E}} {\bm{\Lambda}}_V + {\bm{\Lambda}}^2_U {\bm{E}} {\bm{\Lambda}}^2_V
- 2{\bm{\Lambda}}^2_U {\bm{E}} {\bm{\Lambda}}_V - 2{\bm{\Lambda}}_U
{\bm{E}} {\bm{\Lambda}}^2_V) {\bm{v}}_j{\rangle}.$$ This is a sum of six terms and we will show how to bound the first three; the last three are dealt in exactly the same way and obey better estimates. For the first, we have $${\langle}{\bm{e}}_i, {\bm{\Lambda}}^2_U {\bm{E}} {\bm{v}}_j {\rangle}= {\langle}{\bm{\Lambda}}^2_U {\bm{e}}_i, {\bm{E}} {\bm{v}}_j {\rangle}= \|{\bm{P}}_U
{\bm{e}}_i\|^4 {\langle}{\bm{e}}_i, {\bm{u}}_j{\rangle}$$ Hence $$p^{-2} \sqrt{\sum_{1 \le j \le r} |{\langle}{\bm{e}}_i, {\bm{\Lambda}}^2_U
{\bm{E}} {\bm{v}}_j {\rangle}|^2} = p^{-2} \|{\bm{P}}_U {\bm{e}}_i\|^4
\sqrt{\sum_{1 \le j \le r} | {\langle}{\bm{e}}_i, {\bm{u}}_j{\rangle}|^2} = p^{-2}
\|{\bm{P}}_U {\bm{e}}_i\|^5 \le \left(\frac{\mu_0 r}{np}\right)^2
\sqrt{\frac{\mu_0 r}{n}}.$$ In other words, when $m \ge \mu_0 nr$, the right hand-side is bounded by $\sqrt{{\mu_0 r}/{n}}$ as desired. For the second term, we have $${\langle}{\bm{e}}_i, {\bm{E}} {\bm{\Lambda}}^2_V {\bm{v}}_j {\rangle}= \sum_{b}
\|{\bm{P}}_V {\bm{e}}_b\|^4 v_{bj} {\langle}{\bm{e}}_i, {\bm{E}} {\bm{e}}_b{\rangle}=
\sum_{b} \|{\bm{P}}_V {\bm{e}}_b\|^4 v_{bj} E_{ib}.$$ Hence it follows from the Cauchy-Schwarz inequality and that $$p^{-2} |{\langle}{\bm{e}}_i, {\bm{E}} {\bm{\Lambda}}^2_V {\bm{v}}_j {\rangle}| \le
\left(\frac{\mu_0 r}{np}\right)^2 \sqrt{\frac{\mu_0 r}{n}}.$$ In other words, when $m \ge \mu_0 nr^{5/4}$, $$\label{eq:lambda6}
p^{-2} \sqrt{\sum_{1 \le j \le r} |{\langle}{\bm{e}}_i, {\bm{E}} {\bm{\Lambda}}^2_V {\bm{v}}_j {\rangle}|^2} \le \sqrt{\frac{\mu_0 r}{n}}$$ as desired. For the third term, we have $${\langle}{\bm{e}}_i, {\bm{\Lambda}}_U {\bm{E}} {\bm{\Lambda}}_V {\bm{v}}_j {\rangle}=
\|{\bm{P}}_U {\bm{e}}_i\|^2 \sum_{b} \|{\bm{P}}_V {\bm{e}}_b\|^2 v_{bj}
E_{ib}.$$ The Cauchy-Schwarz inequality gives $$2p^{-2}|{\langle}{\bm{e}}_i, {\bm{\Lambda}}_U {\bm{E}} {\bm{\Lambda}}_V {\bm{v}}_j
{\rangle}| \le 2\left(\frac{\mu_0 r}{np}\right)^2 \sqrt{\frac{\mu_0 r}{n}}$$ just as before. In other words, when $m \ge \mu_0 nr^{5/4}$, $2p^{-2}\sqrt{\sum_{ 1\le j \le r} |{\langle}{\bm{e}}_i, {\bm{\Lambda}}_U
{\bm{E}} {\bm{\Lambda}}_V {\bm{v}}_j {\rangle}|^2}$ is bounded by $2\sqrt{\mu_0 r/n}$. The other terms obey as well when $m \ge \mu_0 nr^{5/4}$. In conclusion, the first term in obeys with probability at least $1 - O(n^{-\beta})$ provided that $m \ge \mu_0
n r^{5/4} (\beta \log n)$.
We now turn our attention to the second term which can be written as $$\begin{gathered}
\gamma_{ij} \equiv p^{-3} ({1-2p}) \sum_{\omega_1 \neq \omega_2}
\xi_{\omega_1} \xi_{\omega_2} \, E_{\omega_2} P_{\omega_2 \omega_2}
P_{\omega_2
\omega_1} {{{\langle}{\bm{e}}_i, {\bm{F}}_{\omega_1} {\bm{v}}_j{\rangle}}}\\
+ p^{-2}({1-p}) \sum_{\omega_1 \neq \omega_2} \xi_{\omega_1} \,
E_{\omega_2} P_{\omega_2 \omega_2} P_{\omega_2 \omega_1} {{{\langle}{\bm{e}}_i, {\bm{F}}_{\omega_1} {\bm{v}}_j{\rangle}}}.\end{gathered}$$ We decouple the first term so that it suffices to bound $$({\bm{S}}_0)_{ij} \equiv p^{-1} \sum_{\omega_1} \xi^{(1)}_{\omega_1} H_{\omega_1}
{{{\langle}{\bm{e}}_i, {\bm{F}}_{\omega_1} {\bm{v}}_j{\rangle}}}, \qquad H_{\omega_1} \equiv p^{-2} \sum_{\omega_2 : \omega_2
\neq \omega_1} \xi^{(2)}_{\omega_2} \, E_{\omega_2} P_{\omega_2
\omega_2} P_{\omega_2 \omega_1},$$ where the sequences $\{\xi_\omega^{(1)}\}$ and $\{\xi_\omega^{(2)}\}$ are independent. The method from Section \[sec:order1\] shows that $$\|{\bm{H}}\|_\infty \le C \, \sqrt\frac{\mu_0 nr \beta \log n}{m} \,
\sup_\omega |E_\omega (p^{-1} P_{\omega \omega})| \le C\, \sqrt{\beta
\log n} \, \left(\frac{\mu_0 nr}{m}\right)^{3/2} \,
\|{\bm{E}}\|_\infty$$ with probability at least $1-2n^{-\beta}$ for each $\beta >
2$. Therefore, Lemma \[teo:sij\] gives $$\label{eq:threetimesa}
{\operatorname{\mathbb{P}}}\left(\|{\bm{S}}_0\|_\infty \ge \sqrt{\mu_0/n}\right) \le 2n^2
e^{-1/D},$$ where $D$ obeys $$\label{eq:threetimesb}
D \le C \left(\mu_0^2 \mu_1^2 (\beta \log n)
\left(\frac{nr}{m}\right)^4 + \mu_0^{3/2} \mu_1 \sqrt{\beta \log
n}\left(\frac{nr}{m}\right)^{5/2}\right).$$ for some positive constant $C$. Hence, when $m \ge \lambda \mu_0 \,
nr^{5/4} (\beta \log n)$ for some sufficiently large numerical constant $\lambda > 0$, we have that $\|{\bm{S}}_0\|_\infty \ge
\sqrt{\mu_0/n}$ with probability at most $2n^2 e^{-(\beta
\log n)^2}$. This is inversely proportional to a superpolynomial in $n$. We write the second term as $$({\bm{S}}_1)_{ij} \equiv p^{-1} \sum_{\omega_1 \neq \omega_2} \xi_{\omega_1}
H_{\omega_1} {{{\langle}{\bm{e}}_i, {\bm{F}}_{\omega_1} {\bm{v}}_j{\rangle}}}, \qquad H_{\omega_1} = p^{-1} \sum_{\omega_2 :
\omega_2 \neq \omega_1} E_{\omega_2} P_{\omega_2 \omega_2}
P_{\omega_2 \omega_1}.$$ We know from Section \[sec:order2\] that ${\bm{H}}$ obeys $\|{\bm{H}}\|_\infty \le C \, \mu_0^2\, r^2/m$ since $\mu_1 \le \mu_0
\sqrt{r}$ so that Lemma \[teo:sij\] gives $${\operatorname{\mathbb{P}}}\left(\|{\bm{S}}_1\|_\infty \ge \sqrt{\mu_0/n}\right) \le
2n^2 e^{-1/D}, \quad D \le C \left(\mu_0^3 \frac{n^3 r^4}{m^3} +
\mu_0^2 \frac{n^2 r^{5/2}}{m^2}\right)$$ for some $C > 0$. Hence, when $m \ge \lambda \mu_0 \, nr^{4/3} (\beta
\log n)$ for some numerical constant $\lambda > 0$, we have that $\|{\bm{S}}_1\|_\infty \ge \sqrt{\mu_0/n}$ with probability at most $2n^2 e^{-(\beta \log n)^2}$. This is inversely proportional to a superpolynomial in $n$. In conclusion and taking into account the decoupling constants in , the second term in obeys with probability at least $1-O(n^{-\beta})$ provided that $m$ is sufficiently large as above.
We now examine the third term which can be written as $$p^{-3}({1-2p}) \sum_{\omega_1 \neq \omega_2} \xi_{\omega_1}
\xi_{\omega_2} \, E_{\omega_1} P^2_{\omega_2 \omega_1} {{{\langle}{\bm{e}}_i, {\bm{F}}_{\omega_1} {\bm{v}}_j{\rangle}}}+
p^{-2}({1-p}) \sum_{\omega_1 \neq \omega_2} \xi_{\omega_2} \,
E_{\omega_1} P^2_{\omega_2 \omega_1} {{{\langle}{\bm{e}}_i, {\bm{F}}_{\omega_1} {\bm{v}}_j{\rangle}}}.$$ For the first term of the right-hand side, it suffices to estimate the tail of $$({\bm{S}}_0)_{ij} \equiv p^{-1} \sum_{\omega_1} \xi^{(1)}_{\omega_1} E_{\omega_1}
H_{\omega_1} {{{\langle}{\bm{e}}_i, {\bm{F}}_{\omega_1} {\bm{v}}_j{\rangle}}}, \qquad H_{\omega_1} \equiv p^{-2} \sum_{\omega_2
: \omega_2 \neq \omega_1} \xi^{(2)}_{\omega_2} \, P^2_{\omega_2
\omega_1},$$ where $\{\xi^{(1)}_\omega\}$ and $\{\xi^{(2)}_\omega\}$ are independent. We know from Section \[sec:order2\] that $\|{\bm{H}}\|_\infty$ obeys $\|{\bm{H}}\|_\infty \le C \, \sqrt{\beta
\log n} \, (\mu_0 nr/m)^{3/2}$ with probability at least $1-2n^{-\beta}$ for each $\beta > 2$. Thus, Lemma shows that ${\bm{S}}_0$ obeys – just as before. The other term is equal to $(1-p)$ times $\sum_{\omega_1} E_{\omega_1}
H_{\omega_1} {{{\langle}{\bm{e}}_i, {\bm{F}}_{\omega_1} {\bm{v}}_j{\rangle}}}$, and by the Cauchy-Schwarz inequality and $$\left|\sum_{\omega_1} E_{\omega_1} H_{\omega_1} {{{\langle}{\bm{e}}_i, {\bm{F}}_{\omega_1} {\bm{v}}_j{\rangle}}}\right| \le
\|{\bm{H}}\|_\infty \, \|{\bm{e}}_i^* {\bm{E}}\| \, \left(\sum_{b}
v_{bj}^2\right)^{1/2} \le C \sqrt{\frac{\mu_0}{n}} \, \sqrt{\beta
\log n} \, \left(\frac{\mu_0
nr^{4/3}}{m}\right)^{3/2}$$ on the event where $\|{\bm{H}}\|_\infty \le C \, \sqrt{\beta \log n} \,
(\mu_0 nr/m)^{3/2}$. Hence, when $m \ge \lambda \mu_0 \, nr^{4/3} \,
(\beta \log n)$ for some numerical constant $\lambda > 0$, we have that $|\sum_{\omega_1} E_{\omega_1} H_{\omega_1} {{{\langle}{\bm{e}}_i, {\bm{F}}_{\omega_1} {\bm{v}}_j{\rangle}}}| \le
\sqrt{\mu_0/n}$ on this event. In conclusion, the third term in obeys with probability at least $1-O(n^{-\beta})$ provided that $m$ is sufficiently large as above.
We proceed to the fourth term which can be written as $$p^{-3}({1-2p}) \sum_{\omega_1 \neq \omega_3} \xi_{\omega_1}
\xi_{\omega_3} \, E_{\omega_3} P_{\omega_3 \omega_1} P_{\omega_1
\omega_1} {{{\langle}{\bm{e}}_i, {\bm{F}}_{\omega_1} {\bm{v}}_j{\rangle}}}+ p^{-2}({1-p}) \sum_{\omega_1 \neq \omega_3}
\xi_{\omega_3} \, E_{\omega_3} P_{\omega_3 \omega_1} P_{\omega_1
\omega_1} {{{\langle}{\bm{e}}_i, {\bm{F}}_{\omega_1} {\bm{v}}_j{\rangle}}}.$$ We use the decoupling trick for the first term and bound the tail of $$({\bm{S}}_0)_{ij} \equiv p^{-1} \sum_{\omega_1} \xi^{(1)}_{\omega_1} H_{\omega_1}
(p^{-1} P_{\omega_1 \omega_1}) \, {{{\langle}{\bm{e}}_i, {\bm{F}}_{\omega_1} {\bm{v}}_j{\rangle}}}, \qquad H_{\omega_1} \equiv
p^{-1} \sum_{\omega_3 : \omega_3 \neq \omega_1} \xi^{(3)}_{\omega_3}
\, E_{\omega_3} P_{\omega_3 \omega_1},$$ where $\{\xi^{(1)}_\omega\}$ and $\{\xi^{(3)}_\omega\}$ are independent. We know from Section \[sec:order1\] that $$\|{\bm{H}}\|_\infty \le C \, \sqrt{\frac{\mu_0 nr\beta \log n}{m}} \,
\|{\bm{E}}\|_\infty$$ with probability at least $1-2n^{-\beta}$ for each $\beta >
2$. Therefore, Lemma \[teo:sij\] shows that ${\bm{S}}_0$ obeys – just as before. The other term is equal to $(1-p)$ times $\sum_{\omega_1} H_{\omega_1}
(p^{-1} P_{\omega_1 \omega_1}) \, {{{\langle}{\bm{e}}_i, {\bm{F}}_{\omega_1} {\bm{v}}_j{\rangle}}}$, and the Cauchy-Schwarz inequality gives $$\left|\sum_{\omega_1} H_{\omega_1} (p^{-1} P_{\omega_1 \omega_1}) \,
{{{\langle}{\bm{e}}_i, {\bm{F}}_{\omega_1} {\bm{v}}_j{\rangle}}}\right| \le \sqrt{n} \, \|{\bm{H}}\|_\infty \, \frac{\mu_0 nr}{m}
\le
C\, \frac{\mu_1 \sqrt{r \beta \log n}}{\sqrt{n}} \,
\left(\frac{\mu_0 nr}{m}\right)^{3/2}$$ on the event $\|{\bm{H}}\|_\infty \le C \, \sqrt{\mu_0 nr (\beta \log
n)/m} \, \|{\bm{E}}\|_\infty$. Because $\mu_1 \le \mu_0 \sqrt{r}$, we have that whenever $m \ge \lambda \, \mu_0^{4/3} nr^{5/3} \, (\beta
\log n)$ for some numerical constant $\lambda > 0$, $p^{-1}
|\sum_{\omega_1} H_{\omega_1} P_{\omega_1 \omega_1} {{{\langle}{\bm{e}}_i, {\bm{F}}_{\omega_1} {\bm{v}}_j{\rangle}}}| \le
\sqrt{\mu_0/n}$ just as before. In conclusion, the fourth term in obeys with probability at least $1-O(n^{-\beta})$ provided that $m$ is sufficiently large as above.
We finally examine the last term $$p^{-3} \sum_{\omega_1 \neq \omega_2 \neq \omega_3} \xi_{\omega_1}
\xi_{\omega_2} \xi _{\omega_3} \, E_{\omega_3} P_{\omega_3
\omega_2} P_{\omega_2 \omega_1} {{{\langle}{\bm{e}}_i, {\bm{F}}_{\omega_1} {\bm{v}}_j{\rangle}}}.$$ Just as before, we need to bound the tail of $$({\bm{S}}_0)_{ij} \equiv p^{-1} \sum_{\omega_1, \omega_2, \omega_3}
\xi^{(1)}_{\omega_1} H_{\omega_1} {{{\langle}{\bm{e}}_i, {\bm{F}}_{\omega_1} {\bm{v}}_j{\rangle}}},$$ where ${\bm{H}}$ is given by . We know from Section \[sec:order2\] that ${\bm{H}}$ obeys $$\|{\bm{H}}\|_\infty \le C \,
(\beta \log n)\, \frac{\mu_0 nr}{m} \, \mu_1 \frac{\sqrt{r}}{n}$$ with probability at least $1-4n^{-\beta}$ for each $\beta > 2$. Therefore, Lemma \[teo:sij\] gives $${\operatorname{\mathbb{P}}}\left(\|{\bm{S}}_0\|_\infty \ge \frac{1}{5} \sqrt{\mu_0/n}\right) \le 2n^2
e^{-1/D}, \quad D \le C \left(\mu_0 \mu_1^2 (\beta \log n)^2
\left(\frac{n r}{m}\right)^3 + \mu_0 \mu_1 (\beta \log n)
\left(\frac{n r}{m}\right)^2\right)$$ for some $C > 0$. Hence, when $m \ge \lambda \mu_0 \, nr^{4/3} (\beta
\log n)$ for some numerical constant $\lambda > 0$, we have that $\|{\bm{S}}_0\|_\infty \ge \frac{1}{5} \sqrt{\mu_0/n}$ with probability at most $2n^2 e^{-(\beta \log n)}$. In conclusion, the fifth term in obeys with probability at least $1-O(n^{-\beta})$ provided that $m$ is sufficiently large as above.
To summarize the calculations of this section, if $m = \lambda \,
\mu_0^{4/3} nr^{5/3} \, (\beta \log n)$ where $\beta \ge 2$ is fixed and $\lambda$ is some sufficiently large numerical constant, then $$\sum_{1\le j \le r} |\alpha_{ij}|^2 \le \mu_0 r/n$$ with probability at least $1 - O(n^{-\beta})$. This concludes the proof.
Proof of Lemma \[teo:highorder\]
--------------------------------
It remains to study the spectral norm of $p^{-1} ({{\cal P}_{T^\perp}}{{\cal P}_{\Omega}}{{\cal P}_T})\sum_{k
\ge k_0} \mathcal{H}^k({\bm{E}})$ for some positive integer $k_0$, which we bound by the Frobenius norm $$\begin{aligned}
p^{-1} \|({{\cal P}_{T^\perp}}{{\cal P}_{\Omega}}{{\cal P}_T})\sum_{k \ge k_0} \mathcal{H}^k({\bm{E}})\| &
\le p^{-1} \|({{\cal P}_{\Omega}}{{\cal P}_T}) \sum_{k \ge
k_0} \mathcal{H}^k({\bm{E}})\|_F \\
& \le \sqrt{3/2p} \, \|\sum_{k \ge k_0} \mathcal{H}^k({\bm{E}})\|_F,\end{aligned}$$ where the inequality follows from Corollary \[teo:POPT\]. To bound the Frobenius of the series, write $$\begin{aligned}
\|\sum_{k \ge k_0} \mathcal{H}^k({\bm{E}})\|_F & \le \|\mathcal{H}\|^{k_0} \|{\bm{E}}\|_F +
\|\mathcal{H}\|^{k_0+1} \|{\bm{E}}\|_F + \ldots \\
& \le \frac{ \|\mathcal{H}\|^{k_0}}{1- \|\mathcal{H}\|} \, \|{\bm{E}}\|_F.\end{aligned}$$ Theorem \[teo:rudelson\] gives an upper bound on $\|\mathcal{H}\|$ since $\|\mathcal{H}\| \le C_R \, \sqrt{\mu_0 nr \beta \log n/m}<1/2$ on an event with probability at least $1 - 3n^{-\beta}$. Since $\|{\bm{E}}\|_F = \sqrt{r}$, we conclude that $$p^{-1} \|({{\cal P}_{\Omega}}{{\cal P}_T}) \sum_{k \ge k_0} \mathcal{H}^k({\bm{E}})\|_F \le C
\, \frac{1}{\sqrt{p}} \, \left(\frac{\mu_0 nr \beta \log
n}{m}\right)^{k_0/2} \, \sqrt{r} = C \, \left(\frac{n^2
r}{m}\right)^{1/2} \, \left(\frac{\mu_0 nr \beta \log
n}{m}\right)^{k_0/2}$$ with large probability. This is the content of Lemma \[teo:highorder\].
Numerical Experiments {#sec:numerical}
=====================
To demonstrate the practical applicability of the nuclear norm heuristic for recovering low-rank matrices from their entries, we conducted a series of numerical experiments for a variety of the matrix sizes $n$, ranks $r$, and numbers of entries $m$. For each $(n,m,r)$ triple, we repeated the following procedure $50$ times. We generated ${\bm{M}}$, an $n\times n$ matrix of rank $r$, by sampling two $n\times r$ factors ${\bm{M}}_L$ and ${\bm{M}}_R$ with i.i.d. Gaussian entries and setting ${\bm{M}}={\bm{M}}_L{\bm{M}}_R^*$. We sampled a subset $\Omega$ of $m$ entries uniformly at random. Then the nuclear norm minimization $$\begin{array}{ll}
\textrm{minimize} & \quad \|{\bm{X}}\|_*\\
\textrm{subject to} & \quad X_{ij} = M_{ij}, \quad (i,j) \in
\Omega
\end{array}$$ was solved using the SDP solver SDPT3 [@SDPT3]. We declared ${\bm{M}}$ to be recovered if the solution returned by the SDP, ${\bm{X_{\mathrm{opt}}}}$, satisfied $\|{\bm{X_{\mathrm{opt}}}}-{\bm{M}}\|_F/\|{\bm{M}}\|_F<10^{-3}$. Figure \[fig:phase-trans-full\] shows the results of these experiments for $n=40$ and $50$. The $x$-axis corresponds to the fraction of the entries of the matrix that are revealed to the SDP solver. The $y$-axis corresponds to the ratio between the dimension of the set of rank $r$ matrices, $d_r=r(2n-r)$, and the number of measurements $m$. Note that both of these axes range from zero to one as a value greater than one on the $x$-axis corresponds to an overdetermined linear system where the semidefinite program always succeeds, and a value of greater than one on the $y$-axis corresponds to a situation where there is always an infinite number of matrices with rank $r$ with the given entries. The color of each cell in the figures reflects the empirical recovery rate of the $50$ runs (scaled between $0$ and $1$). White denotes perfect recovery in all experiments, and black denotes failure for all experiments. Interestingly, the experiments reveal very similar plots for different $n$, suggesting that our asymptotic conditions for recovery may be rather conservative.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![ For each $(n,m,r)$ triple, we repeated the following procedure $50$ times. A matrix ${\bm{M}}$ of rank $r$ and a subset of $m$ entries were selected at random. Then we solved the nuclear norm minimization for ${\bm{X}}$ subject to $X_{ij} = M_{ij}$ on the selected entries. We declared ${\bm{M}}$ to be recovered if $\|{\bm{X_{\mathrm{opt}}}}-{\bm{M}}\|_F/\|{\bm{M}}\|_F<10^{-3}$. The results are shown for (a) $n=40$ and (b) $n=50$. The color of each cell reflects the empirical recovery rate (scaled between $0$ and $1$). White denotes perfect recovery in all experiments, and black denotes failure for all experiments.[]{data-label="fig:phase-trans-full"}](40by40.pdf "fig:"){width="8cm"} ![ For each $(n,m,r)$ triple, we repeated the following procedure $50$ times. A matrix ${\bm{M}}$ of rank $r$ and a subset of $m$ entries were selected at random. Then we solved the nuclear norm minimization for ${\bm{X}}$ subject to $X_{ij} = M_{ij}$ on the selected entries. We declared ${\bm{M}}$ to be recovered if $\|{\bm{X_{\mathrm{opt}}}}-{\bm{M}}\|_F/\|{\bm{M}}\|_F<10^{-3}$. The results are shown for (a) $n=40$ and (b) $n=50$. The color of each cell reflects the empirical recovery rate (scaled between $0$ and $1$). White denotes perfect recovery in all experiments, and black denotes failure for all experiments.[]{data-label="fig:phase-trans-full"}](50by50.pdf "fig:"){width="8cm"}
(a) (b)
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
For a second experiment, we generated random *positive semidefinite* matrices and tried to recover them from their entries using the nuclear norm heuristic. As above, we repeated the same procedure $50$ times for each $(n,m,r)$ triple. We generated ${\bm{M}}$, an $n\times n$ positive semidefinite matrix of rank $r$, by sampling an $n\times r$ factor ${\bm{M}}_F$ with i.i.d. Gaussian entries and setting ${\bm{M}}={\bm{M}}_F{\bm{M}}_F^*$. We sampled a subset $\Omega$ of $m$ entries uniformly at random. Then we solved the nuclear norm minimization problem $$\begin{array}{ll}
\textrm{minimize} & \quad {\operatorname{trace}}({\bm{X}})\\
\textrm{subject to} & \quad X_{ij} = M_{ij}, \quad (i,j) \in \Omega\\
& \quad {\bm{X}} \succeq 0
\end{array}.$$ As above, we declared ${\bm{M}}$ to be recovered if $\|{\bm{X_{\mathrm{opt}}}}-{\bm{M}}\|_F/\|{\bm{M}}\|_F<10^{-3}$. Figure \[fig:phase-trans-psd\] shows the results of these experiments for $n=40$ and $50$. The $x$-axis again corresponds to the fraction of the entries of the matrix that are revealed to the SDP solver, but, in this case, the number of measurements is divided by $D_n = n(n+1)/2$, the number of unique entries in a positive-semidefinite matrix and the dimension of the rank $r$ matrices is $d_r = nr - r(r-1)/2$. The color of each cell is chosen in the same fashion as in the experiment with full matrices. Interestingly, the recovery region is much larger for positive semidefinite matrices, and future work is needed to investigate if the theoretical scaling is also more favorable in this scenario of low-rank matrix completion.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![ For each $(n,m,r)$ triple, we repeated the following procedure $50$ times. A positive semidefinite matrix ${\bm{M}}$ of rank $r$ and a set of $m$ entries were selected at random. Then we solved the nuclear norm minimization subject to $X_{ij} = M_{ij}$ on the selected entries with the constraint that ${\bm{X}}\succeq 0$. The color scheme for each cell denotes empirical recovery probability and is the same as in Figure \[fig:phase-trans-full\]. The results are shown for (a) $n=40$ and (b) $n=50$.[]{data-label="fig:phase-trans-psd"}](40by40_psd.pdf "fig:"){width="8cm"} ![ For each $(n,m,r)$ triple, we repeated the following procedure $50$ times. A positive semidefinite matrix ${\bm{M}}$ of rank $r$ and a set of $m$ entries were selected at random. Then we solved the nuclear norm minimization subject to $X_{ij} = M_{ij}$ on the selected entries with the constraint that ${\bm{X}}\succeq 0$. The color scheme for each cell denotes empirical recovery probability and is the same as in Figure \[fig:phase-trans-full\]. The results are shown for (a) $n=40$ and (b) $n=50$.[]{data-label="fig:phase-trans-psd"}](50by50_psd.pdf "fig:"){width="8cm"}
(a) (b)
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Finally, in Figure \[fig:phase-trans-gaussian\], we plot the performance of the nuclear norm heuristic when recovering low-rank matrices from Gaussian projections of these matrices. In these cases, ${\bm{M}}$ was generated in the same fashion as above, but, in place of sampling entries, we generated $m$ random Gaussian projections of the data (see the discussion in Section \[sec:background\]). Then we solved the optimization $$\begin{array}{ll}
\textrm{minimize} & \quad \|{\bm{X}}\|_*\\
\textrm{subject to} & \quad {\cal A}({\bm{X}}) = {\cal A}({\bm{M}})
\end{array}.$$ with the additional constraint that ${\bm{X}}\succeq 0$ in the positive semidefinite case. Here $\mathcal{A}({\bm{X}})$ denotes a linear map of the form (\[eq:linearfunctional\]) where the entries are sampled i.i.d. from a zero-mean unit variance Gaussian distribution. In these experiments, the recovery regime is far larger than in the case of that of sampling entries, but this is not particularly surprising as each Gaussian observation measures a contribution from every entry in the matrix ${\bm{M}}$. These Gaussian models were studied extensively in [@Recht07].
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![ For each $(n,m,r)$ triple, we repeated the following procedure $10$ times. In (a), a matrix of rank $r$ was generated as in Figures \[fig:phase-trans-full\]. In (b) a positive semidefinite matrix of rank $r$ was generated as in Figures \[fig:phase-trans-psd\]. In both plots, we select a matrix $\mathcal{A}$ from the Gaussian ensemble with $m$ rows and $n^2$ (in (a)) or $D_n=n(n+1)/2$ (in (b)) columns. Then we solve the nuclear norm minimization subject to $\mathcal{A}({\bm{X}}) = \mathcal{A}({\bm{M}})$. The color scheme for each cell denotes empirical recovery probability and is the same as in Figures \[fig:phase-trans-full\] and \[fig:phase-trans-psd\]. []{data-label="fig:phase-trans-gaussian"}](40by40_gaussian.pdf "fig:"){width="8cm"} ![ For each $(n,m,r)$ triple, we repeated the following procedure $10$ times. In (a), a matrix of rank $r$ was generated as in Figures \[fig:phase-trans-full\]. In (b) a positive semidefinite matrix of rank $r$ was generated as in Figures \[fig:phase-trans-psd\]. In both plots, we select a matrix $\mathcal{A}$ from the Gaussian ensemble with $m$ rows and $n^2$ (in (a)) or $D_n=n(n+1)/2$ (in (b)) columns. Then we solve the nuclear norm minimization subject to $\mathcal{A}({\bm{X}}) = \mathcal{A}({\bm{M}})$. The color scheme for each cell denotes empirical recovery probability and is the same as in Figures \[fig:phase-trans-full\] and \[fig:phase-trans-psd\]. []{data-label="fig:phase-trans-gaussian"}](40by40_gaussian_psd.pdf "fig:"){width="8cm"}
(a) (b)
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Discussion {#sec:discussion}
==========
Improvements
------------
In this paper, we have shown that under suitable conditions, one can reconstruct an $n \times n$ matrix of rank $r$ from a small number of its sampled entries provided that this number is on the order of $n^{1.2} r \log n$, at least for moderate values of the rank. One would like to know whether better results hold in the sense that exact matrix recovery would be guaranteed with a reduced number of measurements. In particular, recall that an $n \times n$ matrix of rank $r$ depends on $(2n-r)r$ degrees of freedom; is it true then that it is possible to recover most low-rank matrices from on the order of $nr$—up to logarithmic multiplicative factors—randomly selected entries? Can the sample size be merely proportional to the true complexity of the low-rank object we wish to recover?
In this direction, we would like to emphasize that there is nothing in our approach that apparently prevents us from getting stronger results. Indeed, we developed a bound on the spectral norm of each of the first four terms $({{\cal P}_{T^\perp}}{{\cal P}_{\Omega}}{{\cal P}_T})\mathcal{H}^k(E)$ in the series (corresponding to values of $k$ equal to $0,
1, 2, 3$) and used a general argument to bound the remainder of the series. Presumably, one could bound higher order terms by the same techniques. Getting an appropriate bound on $\|({{\cal P}_{T^\perp}}{{\cal P}_{\Omega}}{{\cal P}_T}){\cal
H}^4(E)\|$ would lower the exponent of $n$ from $6/5$ to $7/6$. The appropriate bound on $\|({{\cal P}_{T^\perp}}{{\cal P}_{\Omega}}{{\cal P}_T})\mathcal{H}^5(E)\|$ would further lower the exponent to $8/7$, and so on. To obtain an optimal result, one would need to reach $k$ of size about $\log n$. In doing so, however, one would have to pay special attention to the size of the decoupling constants (the constant $C_D$ for two variables in Lemma \[teo:decoupling\]) which depend on $k$—the number of decoupled variables. These constants grow with $k$ and upper bounds are known [@delaPena1; @delaPena2].
Further directions
------------------
It would be of interest to extend our results to the case where the unknown matrix is approximately low-rank. Suppose we write the SVD of a matrix ${\bm{M}}$ as $${\bm{M}} = \sum_{1 \le k \le n} \sigma_k {\bm{u}}_k {\bm{v}}_k^*,$$ where $\sigma_1 \ge \sigma_2 \ge \ldots \ge \sigma_n \ge 0$ and assume for simplicity that none of the $\sigma_k$’s vanish. In general, it is impossible to complete such a matrix exactly from a partial subset of its entries. However, one might hope to be able to recover a good approximation if, for example, most of the singular values are small or negligible. For instance, consider the truncated SVD of the matrix ${\bm{M}}$, $${\bm{M}}_r = \sum_{1 \le k \le r} \sigma_k {\bm{u}}_k {\bm{v}}_k^*,$$ where the sum extends over the $r$ largest singular values and let ${\bm{M}}_\star$ be the solution to . Then one would not expect to have ${\bm{M}}_\star = {\bm{M}}$ but it would be of great interest to determine whether the size of ${\bm{M}}_\star -
{\bm{M}}$ is comparable to that of ${\bm{M}} - {\bm{M}}_r$ provided that the number of sampled entries is sufficiently large. For example, one would like to know whether it is reasonable to expect that $\|{\bm{M}}_\star - {\bm{M}}\|_*$ is on the same order as $\|{\bm{M}} - {\bm{M}}_r\|_*$ (one could ask for a similar comparison with a different norm). If the answer is positive, then this would say that approximately low-rank matrices can be accurately recovered from a small set of sampled entries.
Another important direction is to determine whether the reconstruction is robust to noise as in some applications, one would presumably observe $$Y_{ij} = M_{ij} + z_{ij}, \quad (i,j) \in \Omega,$$ where $z$ is a deterministic or stochastic perturbation. In this setup, one would perhaps want to minimize the nuclear norm subject to $\|{{\cal P}_{\Omega}}({\bm{X}}-{\bm{Y}})\|_F \le \epsilon$ where $\epsilon$ is an upper bound on the noise level instead of enforcing the equality constraint ${{\cal P}_{\Omega}}({\bm{X}}) = {{\cal P}_{\Omega}}({\bm{Y}})$. Can one expect that this algorithm or a variation thereof provides accurate answers? That is, can one expect that the error between the recovered and the true data matrix be proportional to the noise level?
Appendix
========
Proof of Theorem \[teo:rudelson2\]
----------------------------------
The proof of follows that in [@CR07] but we shall use slightly more precise estimates.
Let $Y_1, \ldots, Y_n$ be a sequence of independent random variables taking values in a Banach space and let $Y_\star$ be the supremum defined as $$\label{eq:Z}
Y_\star = \sup_{f \in {\cal F}} \, \sum_{i = 1}^n f(Y_i),$$ where ${\cal F}$ is a countable family of real-valued functions such that if $f \in {\cal F}$, then $-f \in {\cal F}$. Talagrand [@talagrand96ne] proved a concentration inequality about $Y_\star$, see also [@ledoux01co Corollary 7.8].
\[teo:LT\] Assume that $|f| \le B$ and ${\operatorname{\mathbb{E}}}f(Y_i) = 0$ for every $f$ in ${\cal
F}$ and $i = 1, \ldots, n$. Then for all $t \ge 0$, $$\label{eq:Talagrand}
{\operatorname{\mathbb{P}}}(|Y_\star - {\operatorname{\mathbb{E}}}Y_\star| > t) \le 3 \exp\left(-\frac{t}{K B} \log \left(1 + \frac{B t}
{\sigma^2 + B {\operatorname{\mathbb{E}}}Y_\star}\right)\right),$$ where $\sigma^2 = \sup_{f \in {\cal F}} \, \sum_{i = 1}^n {\operatorname{\mathbb{E}}}f^2(Y_i)$, and $K$ is a numerical constant.
We note that very precise values of the numerical constant $K$ are known and are small, see [@klein05co].
We will apply this theorem to the random variable $Z$ defined in the statement of Theorem \[teo:rudelson2\]. Put $\mathcal{Y}_{ab} = p^{-1}
(\delta_{ab} - p) \, {{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*}) \otimes {{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*})$ and $\mathcal{Y} =
\sum_{ab} \mathcal{Y}_{ab}$. By definition, $$\begin{aligned}
Z = \sup \,\, {\langle}{\bm{X}}_1, \mathcal{Y}({\bm{X}}_2){\rangle}& =
\sup \,\, \sum_{ab} {\langle}{\bm{X}}_1, \mathcal{Y}_{ab} ({\bm{X}}_2){\rangle}\\
& = \sup \,\, p^{-1} \sum_{ab} (\delta_{ab} - p) {\langle}{\bm{X}}_1,
{{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*}){\rangle}\, {\langle}{{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*}), {\bm{X}}_2{\rangle},\end{aligned}$$ where the supremum is over a countable collection of matrices ${\bm{X}}_1$ and ${\bm{X}}_2$ obeying $\|{\bm{X}}_1\|_F \le 1$ and $\|{\bm{X}}_2\|_F \le 1$. Note that it follows from $$\begin{aligned}
|{\langle}{\bm{X}}_1, \mathcal{Y}_{ab}({\bm{X}}_2){\rangle}| &= p^{-1}\, |\delta_{ab}-p|\,
|{\langle}{\bm{X}}_1,
{{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*}){\rangle}| \, |{\langle}{{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*}), {\bm{X}}_2{\rangle}| \\
&\le p^{-1} \, \|{{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*})\|_F^2 \le 2\mu_0 r/(\min(n_1,n_2) p) =
2\mu_0 \, nr/m\end{aligned}$$ (recall that $n = \max(n_1,n_2)$). Hence, we can apply Theorem \[teo:LT\] with $B = 2\mu_0 (nr/m)$. Also $$\begin{aligned}
{\operatorname{\mathbb{E}}}|{\langle}{\bm{X}}_1, \mathcal{Y}_{ab}({\bm{X}}_2){\rangle}|^2 & = p^{-1}(1-p) \, |{\langle}{\bm{X}}_1,
{{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*}){\rangle}|^2 \,
|{\langle}{\bm{X}}_2, {{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*}){\rangle}|^2 \\
& \le p^{-1}\, \|{{\cal P}_T}({{\bm{e}}_{a}{\bm{e}}_{b}^*})\|_F^2 \, |{\langle}{{\cal P}_T}({\bm{X}}_2), {{\bm{e}}_{a}{\bm{e}}_{b}^*}{\rangle}|^2\end{aligned}$$ so that $$\begin{aligned}
\sum_{ab} {\operatorname{\mathbb{E}}}|{\langle}{\bm{X}}_1, \mathcal{Y}_{ab}({\bm{X}}_2){\rangle}|^2 & \le (2\mu_0 \, nr/m) \sum_{ab} |{\langle}{{\cal P}_T}({\bm{X}}_2), {{\bm{e}}_{a}{\bm{e}}_{b}^*}{\rangle}|^2\\
& = (2 \mu_0 \, nr/m) \, \|{{\cal P}_T}({\bm{X}}_2)\|_F^2 \le 2\mu_0 nr/m.\end{aligned}$$ Since ${\operatorname{\mathbb{E}}}Z \le 1$, Theorem \[teo:LT\] gives $$P(|Z - {\operatorname{\mathbb{E}}}Z| > t) \le 3 \exp\left(-\frac{t}{KB} \log(1+ t/2)\right) \le 3
\exp\left(-\frac{t \log 2}{K B} \min(1, t/2)\right),$$ where we have used the fact that $\log(1+u) \ge (\log 2)\, \min(1,u)$ for $u \ge 0$. Plugging $t = \lambda \sqrt{\frac{\mu_0\, nr \log
n}{m}}$ and $B = 2\mu_0\, nr/m$ establishes the claim.
Proof of Lemma \[teo:E-max-indep2\]
-----------------------------------
We shall make use of the following lemma which is an application of well-known deviation bounds about binomial variables.
Let $\{\delta_i\}_{1 \le i \le n}$ be a sequence of i.i.d. Bernoulli variables with ${\operatorname{\mathbb{P}}}(\delta_i = 1) = p$ and $Y = \sum_{i = 1}^n
\delta_i$. Then for each $\lambda > 0$, $$\label{eq:binomial}
{\operatorname{\mathbb{P}}}(Y > \lambda \, {\operatorname{\mathbb{E}}}Y) \le \exp\left(-\frac{\lambda^2}{2 + 2\lambda/3}
\, {\operatorname{\mathbb{E}}}Y\right).$$
The random variable $\sum_{b} \delta_{ab} E_{ab}^2$ is bounded by $\|{\bm{E}}\|_\infty^2 \, \sum_b \delta_{ab}$ and it thus suffices to estimate the $q$th moment of $Y_* = \max Y_a$ where $Y_a = \sum_{b}
\delta_{ab}$. The inequality implies that $${\operatorname{\mathbb{P}}}(Y_* > \lambda np) \le n \, \exp\left(-\frac{\lambda^2}{2+2\lambda/3}
\, np\right),$$ and for $\lambda \ge 2$, this gives ${\operatorname{\mathbb{P}}}(Y_* > \lambda np) \le n \,
e^{-\lambda np/2}$. Hence $${\operatorname{\mathbb{E}}}Y_*^q = \int_0^\infty {\operatorname{\mathbb{P}}}(Y_* > t) \, q t^{q-1} \, dt \le
(2np)^q + \int_{2np}^\infty
n \, e^{-t/2} \, q t^{q-1} \, dt.$$ By integrating by parts, one can check that when $q \le np$, we have $$\int_{2np}^\infty
n \, e^{-t/2} \, q t^{q-1} \, dt \le nq \, (2np)^q \, e^{-np}.$$ Under the assumptions of the lemma, we have $nq \, e^{-np} \le 1$ and, therefore, $${\operatorname{\mathbb{E}}}Y_*^q \le 2 \, (2np)^q.$$ The conclusion follows.
Acknowledgments {#acknowledgments .unnumbered}
---------------
E. C. was partially supported by a National Science Foundation grant CCF-515362, by the 2006 Waterman Award (NSF) and by an ONR grant. The authors would like to thank Ali Jadbabaie, Pablo Parrilo, Ali Rahimi, Terence Tao, and Joel Tropp for fruitful discussions about parts of this paper. E. C. would like to thank Arnaud Durand for his careful proof-reading and comments.
[^1]: We emphasize that there is nothing special about ${\bm{M}}$ being square and all of our discussion would apply to arbitrary rectangular matrices as well. The advantage of focusing on square matrices is a simplified exposition and reduction in the number of parameters of which we need to keep track.
[^2]: Both the left and right singular vectors need to be uncorrelated with the standard basis. Indeed, the matrix ${\bm{e}}_1
{\bm{v}}^*$ has its first row equal to ${\bm{v}}$ and all the others equal to zero. Clearly, this rank-1 matrix cannot be recovered unless we basically see all of its entries.
[^3]: When $r \ge C' (\log n)^3$ for some positive constant $C'$, a better estimate is possible, namely, $\|{\bm{\sum_{1 \le k
\le r} {\bm{u}}_k {\bm{v}}_k^*}}\|_\infty \le C\, \sqrt{r\log
n}/n$.
[^4]: We would like to remark that one can often get better estimates; when $\omega_1 \neq \omega_2$, the bound $|P_{\omega_2 \omega_1}| \le 2\mu_0 r/n$ may be rather crude. Indeed, one can derive better estimates for the random orthogonal model, for example.
|
---
abstract: 'We describe a method to measure the relation in the non-local universe using dust-obscured QSOs. We present results from a pilot sample of nine 2MASS red QSOs with redshifts $0.14<z<0.37$. We find that there is an offset (0.8 dex, on average) between the position of our objects and the local relation for AGN, in the sense that the majority of red QSO hosts have lower velocity dispersions and/or more massive BHs than local galaxies. These results are in agreement with recent studies of AGN at similar and higher redshifts. This could indicate an unusually rapid growth in the host galaxies since $z\sim0.2$, if these objects were to land in the local relation at present time. However, the $z>0.1$ AGN (including our sample and those of previous studies) have significantly higher than those of local AGN, so a direct comparison is not straightforward. Further, using several samples of local and higher-$z$ AGN, we find a striking trend of an increasing offset with respect to the local relation as a function of AGN luminosity, with virtually all objects with log(L$_{5100}$/erg s$^{-1}$) $> 43.6$ falling above the relation. Given the relatively small number of AGN at $z>0.1$ for which there are direct measurements of stellar velocity dispersions, it is impossible at present to determine whether there truly is evolution in with redshift. Larger, carefully selected samples of AGN are necessary to disentangle the dependence of on mass, luminosity, accretion rates, and redshift.'
author:
- 'Gabriela Canalizo, Margrethe Wold, Kyle D. Hiner, Mariana Lazarova, Mark Lacy, Kevin Aylor'
bibliography:
- 'bibsigma.bib'
title: 'Probing the relation in the non-local universe using red QSOs'
---
Introduction
============
During the last decade, the astronomy community has come to realize that black hole (BH) activity plays an important role in galaxy formation, e.g., by regulating star formation through winds and outflows [@sca05; @cat05]. To understand how galaxies form, we therefore need to understand how supermassive BHs and galaxies co-evolve.
The most important observational tools for studying the co-evolution of BHs and galaxies are the empirical scaling relations between BH mass and global galaxy properties, such as the relation [@fm2000; @geb00a], a particularly tight relation between BH mass, , and the central, stellar velocity dispersion, $\sigma_*$. The relation suggests a causal connection between the BH and the bulge [although see e.g., @jahnke2011 for a counter argument], and is believed to hold clues to understanding the galaxy formation process. By studying how the BH mass relates to the mass of the bulge (as measured by $\sigma_*$), the galaxy evolution process can be examined across different redshift ranges and for different galaxy types.
Active galactic nuclei (AGN) have become instrumental in this work because they contain actively accreting black holes. Not only are they direct probes of the BH-galaxy co-evolution, but galaxies with AGN are the only galaxies in which we can measure black hole masses in the non-local universe. The method is referred to as the virial method, where the width of broad AGN emission lines such as H$\alpha$ or H$\beta$ is used to measure [e.g., @vestergaard2006]. Whereas measuring virial BH masses in AGN through broad emission lines is relatively straightforward, measuring the host galaxy velocity dispersion is hampered by the presence of the very bright nuclei which can completely overpower the light from the underlying galaxy.
Attempts have been made to use the width of \[OIII\] lines as a proxy for the bulge potential [@nelson2000; @shields2003; @salviander2007], but these studies suffer from a large scatter due to non-gravitational influences on the \[OIII\] line width [such as outflows, radio luminosity, accretion rates, etc; see, e.g., @netzer2007]. Studies of low-luminosity AGN at $z\lesssim0.1$ have shown that they follow the relation of local, more massive spheroids [e.g., @geb00b; @nelson2004; @greene2006a; @woo2010; @bennert2011a], and there is evidence that higher redshift AGN show evolution with regard to the local relation.
[@woo2006; @woo2008] and [@treu2007] studied two samples of Seyfert 1 galaxies at $z=0.36$ and $z=0.57$ and found an offset from the local relation suggesting that, for a fixed black hole mass, higher redshift spheroids have smaller velocity dispersions than local ones. This indicates that, in these objects, the galactic bulge is still forming around the black hole. [@hiner2012] find a similar offset with respect to the local relation in a sample of six post-starburst AGN at $z\sim0.3$. These results have serious implications and may provide important observational constraints for galactic evolution models which attempt to explain the co-evolution of black holes and bulges. The studies by @woo2008, @treu2007, and @hiner2012 suggest a very recent growth, within the last 5$-$6 billion years, of intermediate-mass bulges for a given black hole mass, in contrast to predictions made by detailed hydrodynamic simulations [e.g., @robertson2006b]. On the other hand, [@shen2008] study a large sample of AGN observed with the Sloan Digital Sky Survey (SDSS) at $z<0.452$ and find no evidence of evolution in the relation.
The study of the relation has been severely limited in QSOs due to the observational challenges such a project poses. However, studying in QSOs would be desirable since these are potentially the objects where the most rapid growth (in both BH and bulge) occurs. Studies of at $z>0$ need to include QSOs in order to probe the higher end of the relation and, potentially, episodes when either the bulge or the BH is catching up in its growth.
Measuring velocity dispersions from absorption lines in QSO host galaxies is observationally challenging. The relatively few studies of host galaxy stellar absorption line spectroscopy in the literature [e.g., @canalizo2001; @nolan2001; @jahnke2007; @wold2010] are not able to recover host spectra in the central regions of the galaxy. Using these spectra to measure velocity dispersions could lead to ambiguous results since it is difficult to determine whether the dispersions are being measured in the bulge of the galaxy or in the outskirts or some extended tidal feature where may not be representative of the mass of the bulge or that of the BH. Indeed, few attempts have been made to measure in the host galaxies of luminous QSOs [e.g., @wolf2008; @rothberg2012], but these studies have been in relation to the host structural properties rather than . Therefore, it would be ideal to find a class of objects with a natural “coronagraph” that would occult the QSO nucleus when measuring , but one that we could remove in order to measure from the width of the broad lines. As described below, we have found that dust-reddened QSOs fit the bill.
Red QSOs have some of the same characteristics as their blue counterparts, such as (at least some of the) strong broad emission lines and high bolometric luminosities, but with much redder continua. At present, there is not a clear definition for red QSOs, so that the different objects that are cataloged as red QSOs do not form a homogeneous class. The majority of red QSOs discovered to date, however, appear to be reddened by dust, and thus they are considered the dust-obscured equivalent of the blue QSO population [e.g., @cutri2002; @marble2003; @hall2002; @glikman2004; @white2003]. Even then, there are cases where the dust is intervening rather than intrinsic to the QSO [e.g., @gregg2002]. Here we focus on those objects where the dust is presumably near the nucleus.
The heavy extinction of the QSO nuclei at optical wavelengths makes dust reddened QSOs excellent candidates to study velocity dispersions and stellar populations in their host galaxies. The host galaxy spectrum suffers from much lower (if any) extinction than the nucleus, and stellar absorption features such as those around the MgIb and G-band regions are clearly visible in the blue region, being almost free of contamination from QSO light. At longer wavelengths, however, the contribution from the reddened QSO continuum increases and broad H$\alpha$ is clearly visible. Thus, can be measured from features in the underlying host galaxy and can be estimated from the width of H$\alpha$ using the [*same*]{} spectrum.
In this paper, we present a pilot study to test the feasibility of using these objects to measure . In § \[observations\] we describe the sample, imaging and spectroscopic observations, as well as the data reduction. In § \[analysis\] we describe our method to fit stellar velocity dispersions and to measure . In § \[results\] we present our results and consider potential biases in our sample, while in § \[discussion\] we discuss our results in the context of other studies of that use AGN. Throughout the paper, we have assumed a cosmological model with $H_{0} = 71$ kms$^{-1}$Mpc$^{-1}$, $\Omega_m=0.27$ and $\Omega_{\Lambda} = 0.73$ [@spergel2003], except where noted. We use the Vega magnitude system unless otherwise specified.
Observations and Data Reduction {#observations}
===============================
The Sample {#sample}
----------
Our sample is drawn from the sample of 29 red 2MASS QSOs studied by @marble2003, which in turn is drawn from the sample compiled by @cutri2002. We focused on the subsample from @marble2003 because these objects have Hubble Space Telescope ($HST$) imaging observations that are useful to measure the nuclear luminosities and to study the properties of the host galaxies. The objects in this sample have $M_{K}\lesssim-25$, $J-K_{s}>2.0$ and detections in each of the three 2MASS bands, $JHK_{s}$. They have a median redshift of $z=0.213$ and have been spectroscopically confirmed to contain an AGN [@smith2002].
A fraction of these objects have spectropolarimetry published by @smith2003. We avoided the objects that show a blue continuum, with broad emission features dominating the optical spectrum. Our final sample consists of nine of the remaining targets chosen at random within the observational constraints.
In Table \[table:sample\] we list the final sample. The redshifts listed were measured from stellar absorption lines in our Keck spectra (§ \[spectroscopy\]). $J-K_{s}$ colors and absolute $K_{s}$ magnitudes are quoted from @marble2003, and the latter assume $H_{0} = 75$ kms$^{-1}$Mpc$^{-1}$ and $q_{0}=0.5$.
[clcccccc]{} 1&005055.7$+$293328 & 0.1356 & $-$25.6 & 2.1 & 1800 & 0.21 & 0.50\
2&015721.0$+$171248 & 0.2139 & $-$27.0 & 2.7 & 7200 & 0.21 & 0.75\
3&022150.6$+$132741 & 0.1398 & $-$25.7 & 2.4 & 5400 & 2.86 & 6.96\
4&034857.6$+$125547 & 0.2112 & $-$26.7 & 3.3 & 3600 & 0.79 & 2.70\
5&163736.5$+$254302 & 0.2772 & $-$26.5 & 2.3 & 5400 & 0.21 & 0.88\
6&165939.7$+$183436 & 0.1709 & $-$26.5 & 2.2 & 5400 & 0.29 & 0.84\
7&225902.5$+$124646 & 0.1989 & $-$25.6 & 1.9 & 5400 & 0.42 & 1.36\
8&230442.4$+$270616 & 0.2370 & $-$25.4 & 2.1 & 7200 & 0.97 & 3.61\
9&232745.6$+$162434 & 0.3664 & $-$27.0 & 2.4 & 5400 & 1.29 & 6.51\
\[table:sample\]
Spectroscopy
------------
We obtained deep, medium resolution spectroscopic observations with the Echellette Spectrograph and Imager [ESI; @sheinis2002] on the Keck II telescope during three separate nights in October 2003, July 2004 and September 2004. We observed in the echellette mode, which provides a wavelength coverage from 3900 to 11000 Å in 10 orders (6 through 15). We used a 1 slit, which projected to $\sim$7 pixels (ranging from $\sim$8 pixels at the short-wavelength end to $\sim$6 pixels at the long-wavelength end) on the MIT-Lincoln Labs 2048$\times$4096 CCD detector. The spectral resolution is 11.4 km s$^{-1}$ pixel$^{-1}$. We placed the slit through the center of the host galaxies to ensure that the velocity dispersions that we measure correspond to those of the host galaxy bulges.
All targets were observed under clear weather conditions and subarcsecond seeing ($\sim 0\farcs6$ in V), except for 230442.4$+$270616 and 022150.6$+$132741, which were observed through clouds, and 232745.6 $+$162434, which was observed in high wind conditions. Exposure times for each target ranged from 1800 to 7200 s; specific times are listed in Table \[table:sample\].
We also observed a suite of template stars with stellar types ranging from F0 to M6 (listed on Table \[table:templates\]) and several spectrophotometric standards.
[lc]{} HD218140 & F0\
SAO90936 & F5\
HD11851 & F8\
BD$+$264556 & G0\
HD10995 & G2IV\
SAO91026 & G5 V\
SAO090989 & K0\
SAO91028 & K2\
HD11326 & K2III\
HD218113 & K5III\
SAO90990 & K5 V\
Ci20131 & M0\
HD11729 & M6\
SAO92712 & K0\
SAO92718 & G0\
\[table:templates\]
The spectra were reduced using a combination of IDL and IRAF scripts written to meet the specific needs of our program. After bias subtraction and flat fielding, each order was rectified independently in both spatial and wavelength directions using a modified version of the WMKONSPEC package[^1]. A star was observed at multiple positions across the slit in order to trace and rectify the spatial direction of the spectra. A wavelength solution, obtained from observations of a Cu-Ar lamp, was used to rectify the wavelength direction. Sky lines were removed by fitting the background in the two-dimensional spectra. The spectra were calibrated using spectrophotometric standards from @massey1988 observed with the slit at the parallactic angle. Each target had two or three individual exposures; we averaged the spatially corrected spectra using the IRAF task [*scombine*]{}.
We extracted spectra from the central few kpc of each spectrum, adjusting the size of the aperture to match approximately the effective radius, $r_{\rm eff}$, of each target (see § \[morphologies\]), as listed in Table \[table:sample\]. This corresponded to a slightly different aperture size for each echellette order, since the spatial scale varies for different orders. We then combined the spectra from all orders into a single spectrum for each target. We corrected the spectra for Galactic extinction, using the values given by @schlegel1998. Finally, we transformed the spectra to rest frame using redshifts measured from stellar absorption lines. The final spectra for the nine targets are shown in Figs. \[figure:spectra1\] and \[figure:spectra2\].
Imaging
-------
The imaging data were obtained from the $HST$ data archive (SNAP-9057; PI: D. Hines) and were originally published by @marble2003. The targets were observed with the PC chip of the WFPC2 in the F814W filter. Each target had two 400 s exposures. We combined the images with [*multidrizzle*]{}, using standard procedures.
Analysis
========
Velocity Dispersion {#sigma}
-------------------
The velocity dispersion in the host galaxies was measured by fitting broadened stellar templates to the galaxy spectra. A direct fitting algorithm based on that described by @barth02 was used. The algorithm has been extensively tested and utilized for AGN host galaxies by, e.g., @gh06 and @wold07. The model spectrum $M(x)$ which is fitted to the host galaxy is formed by convolving a stellar template $T(x)$ with a line-of-sight velocity broadening function, assumed to be a Gaussian with dispersion $\sigma_{v}$, adding a power-law continuum $C(x)$ and multiplying by a polynomial $P(x)$: $$M(x) = \left\{ \left[ T(x) \otimes G(x)\right] + C(x) \right\} \times P(x),$$ where $T(x)\otimes G(x)$ is the stellar template convolved with the Gaussian, $G(x)$. The $x$ coordinate is defined as $x=\ln \lambda$ so that velocity shifts are linear in $\ln \lambda$. The continuum is $C(x)=c_{0}+c_{1}x$ (note that this is a power-law as a function of wavelength), and $P(x)$ was chosen to be a 3rd degree polynomial $P(x)=c_{2}p_{0}(x)+c_{3}p_{1}(x)+c_{4}p_{2}(x)+c_{5}p_{3}(x)$, where $p_{n}(x)$ are Legendre polynomials. In order to find the best-fitting model, we varied the velocity dispersion and the six coefficients $c_{0}$ to $c_{5}$ using a downhill simplex method [@ps88] until a global minimum of the $\chi^{2}$ function was found.
Our object spectra are combinations of QSO and galaxy features, and the featureless continuum $C(x)$ describes the underlying AGN continuum. The polynomial $P(x)$ accounts for reddening between the template and the object, so the factor $C(x) \times P(x)$ which is added to the convolved stellar template describes the underlying reddened QSO spectrum. By comparing $C(x) \times P(x)$ with an unobscured QSO spectrum, we were able to obtain an estimate of the amount of reddening for each QSO. This is described in § \[L5100\].
Stellar templates $T(x)$ were formed from individual and various combinations of the 15 different stellar spectra observed with the same setup as the red QSOs (see Table \[table:templates\]). Average F, G, K and M spectra were made by normalizing to the flux in the 7000$-$7100 [Å]{} region and taking the average of all spectra within each of the four classes. Since the main source of uncertainty in measuring velocity dispersions typically comes from template mismatches [@barth02; @gh06], we ran a series of tests to determine the best template to fit for each region and thus minimize systematic uncertainties, as described in detail below.
Based on previous experience and descriptions of similar analyses in the literature [@barth02; @gh06; @wold07] we chose to estimate velocity dispersions from fits to two different spectral regions: (1) a blue region at 3900$-$4600 [Å]{} including CaK but excluding the CaH line (the region 3950$-$3985 Å around CaH was excluded because in some cases AGN contamination from H$\epsilon$ in emission complicated the fit), and (2) a red region at 5100$-$5550 [Å]{} with the b absorption line complex $\lambda\lambda$5167, 5173, 5184 masked out (5155$-$5220 [Å]{}). It is well-known that the correlation between \[Mg/Fe\] and velocity dispersion for elliptical galaxies [@jorgensen99; @kuntschner01] can cause problems with simultaneous fitting of the b absorption and the region redward of it [@barth02; @wold07]. The red region may also be affected by the AGN pseudo-continuum at 5050$-$5520 [Å]{}, affecting the fits to this region. This is seen in a couple of our objects, as well as emission from \[\] $\lambda\lambda$ 5198, 5200. We therefore chose to mask out the region 5155$-$5220 [Å]{} from our fits. Apart from this, the chosen red and blue regions are the two most reliable regions in the optical with strong enough stellar features to be used for velocity dispersion measurements.
The region redward of the b contains several iron absorption line features that are well fitted with our composite template. We also obtain good fits to the blue region, in particular to the CaK line at 3933 Å, H$\delta$-absorption at 4100 Å, and the G-band at 4300 [Å]{}. Generally, we do not see much AGN contamination in this region, except in one case (165939$+$183436), which consequently has larger uncertainties in the velocity dispersion. Some narrow AGN emission lines, such as H$\gamma$, H$\delta$, and \[\] $\lambda$4363, had to be masked from some of the fits, as well as corrupt spectral regions in a couple of cases. In cases where we could not achieve a good fit to the 4000Å break, but there were enough features longward of the break to constrain the fit, we restricted our fit to a region from 4000$-$4600 Å.
The galaxy spectra were shifted to the rest frame of the stellar templates, and the fitting algorithm initiated with a reasonable set of parameters (we fit seven free parameters including the velocity dispersion). The parameters were varied freely using the downhill simplex routine until a best fit was found. To improve on the best fit and to ensure that a global minimum was found, the fitting routine was started again, with initial model parameters equal to the first solution but perturbed by 10$-$20 % in random directions for each parameter. The confidence interval on $\sigma_{v}$ was thereafter found by varying the velocity dispersion in steps of 10 km s$^{-1}$ while letting the other six parameters float freely.
In order to minimize systematic uncertainties due to template mismatches, we used single stars as well as different combinations as stellar templates. For the red region, we find that, in general, we obtain the lowest reduced $\chi^{2}$ when using late stellar types as templates. Five of the objects were best fit in this region with a K5 template. This is not surprising since this region is generally dominated by features from later stellar types. This also consistent with the results found by Hiner et al. (in preparation), who measure stellar velocity dispersions in local galaxies by using templates constructed from a large suite of stellar types, accurately mimicking stellar populations of different ages [see also @hiner2012]. They find a one-to-one correlation in the velocity dispersions that they measure by using a single K star template versus more complex stellar templates in the b region (equivalent to our “red region”). Only three of the objects in our sample (005055.7+293328, 034857.6+125547, 225902.5+124646) are best fit by including a contribution from earlier stellar types. Although these fits produced the smallest reduced $\chi^{2}$, the resulting is within a few tens of km s$^{-1}$ of that measured with a K star template. In fact, the standard deviation in the values of measured from different templates is typically smaller than $\sim$20 km s$^{-1}$. Since we do not do a detailed analysis of the stellar populations in this work, we do not favor any particular composite template among those that yield the lowest reduced $\chi^{2}$ values. Therefore, for consistency, we chose to use the same composite template for these three objects as the one we used for the blue region, as described below. The remaining object, 015721.0$+$171248, shows significant QSO contamination in the red region and we were not able to obtain a reliable fit in this region.
The spectra of our objects in the blue region show features from both early- and late-type stars. Therefore, fits using single stellar types as templates consistently resulted in significantly larger reduced $\chi^{2}$ values than those obtained from fits using composite stellar templates. This is also consistent with the results by Hiner et al., who find that using single stellar types as templates for this spectral region result in systematic uncertainties of several tens of km s$^{-1}$. Thus, we tested a few templates consisting of combinations of stellar types with ratios by flux that would mimic intermediate-age populations (i.e., including features from older and younger populations) and chose the one that yield the lowest values of reduced $\chi^{2}$. Although, in principle, we could do a $\chi^{2}$ minimization to determine the best combination of stellar types for each template, we chose use the same template for every host galaxy in order to reduce the number of free parameters (and degeneracies) in the fits. The flux contribution at 7000 Å of each stellar type for this composite template is 40% K, 40% G, 15% F and 5% M-star (Fig. \[figure:master\_template\]).
The galaxy spectra in the red and the blue regions with the best-fit models overplotted are shown in Figs. \[figure:fit1\] and \[figure:fit2\], and the velocity dispersions obtained from these fits are listed in Table \[table:results\]. The reduced chi-squared, $\chi^{2}_{red}$, for each fit is also listed.
The confidence intervals on give the uncertainty related to the fitting procedure, with smaller confidence intervals for higher signal-to-noise spectra and for spectra with minimal AGN contribution in the fitting regions. Typically, we obtain 68% confidence intervals with a width of 10$-$20 km s$^{-1}$.
![Best-fit models (red trace) overplotted on the host galaxy spectra (black trace) in the two spectral regions described in the text. Top of each panel: Rest-frame host spectrum with a fitted broadened stellar template overplotted. Bottom: residual from the fit (blue trace).[]{data-label="figure:fit1"}](f3a.eps)
![Best-fit models (red trace) overplotted on the host galaxy spectra (black trace) in the two spectral regions described in the text. Top of each panel: Rest-frame host spectrum with a fitted broadened stellar template overplotted. Bottom: residual from the fit (blue trace).[]{data-label="figure:fit2"}](f3b.eps)
[lccccccc]{} 005055.7$+$293328 &$169_{-4}^{+6\phn}$ & 1.71 & 15.4 & $179_{-15}^{+7}$ & 0.69 & 18.3 & $176_{-17}^{+14}$\
015721.0$+$171248 &$250_{-8}^{+12}$& 6.75 & 53.0 & &&& $250_{-54}^{+54}$\
022150.6$+$132741 &$115_{-7}^{+9\phn}$ & 1.96 & 40.7 & $140_{-7}^{+7\phn}$ & 3.31 & 20.0 & $129_{-22}^{+22}$\
034857.6$+$125547 &$136_{-17}^{+9}$& 6.16 & 26.1 & $157_{-8}^{+9\phn}$ & 2.99 & 17.9 & $152_{-17}^{+17}$\
163736.5$+$254302 &$159_{-8}^{+2\phn}$ & 1.49 & 16.7 & $124_{-11}^{+14}$& 0.51 &8.4&$129_{-12}^{+14}$\
165939.7$+$183436 &$188_{-22}^{+13}$&0.50 & 22.1 & $150_{-26}^{+32}$& 0.20 & 16.8 & $159_{-25}^{+28}$\
225902.5$+$124646 &$198_{-2}^{+10}$& 0.87 & 25.8 & $190_{-12}^{+12}$& 0.69 & 27.8 & $194_{-20}^{+21}$\
230442.4$+$270616 &$165_{-4}^{+3\phn}$ & 1.53 & 35.3 & $125_{-7}^{+7\phn}$ & 0.76 &3.1& $127_{-7}^{+7\phn}$\
232745.6$+$162434 &$229_{-6}^{+7\phn}$ & 2.23 & 21.6 & $195_{-12}^{+11}$& 0.89 & 23.1 & $205_{-19}^{+19}$ \[table:results\]
In reality, the uncertainty in the velocity dispersion is larger than this because not all features in the stellar templates used match the host galaxy features equally well. In order to estimate the uncertainties related to template mismatches we fitted a range of stellar templates to the host galaxies and obtained a standard deviation of the measured values of . We list the standard deviations for each range of fits for each region in Table \[table:results\]. As mentioned above, the systematic uncertainties due to template mismatches are generally much smaller in the red region than in the blue region, in agreement with the results of Hiner et al. (in preparation).
The velocity dispersions measured from the blue and red regions agree within the uncertainties. In order to get representative velocity dispersions for each object, we average the two values weighted by their uncertainties and the goodness of fit (as given by $\chi^{2}_{red}$). The weighted averages with their uncertainties (including systematic uncertainties added in quadrature) are listed in the last column of Table \[table:results\].
Galaxy Morphologies {#morphologies}
-------------------
We analyzed the $HST$ imaging data with the two-dimensional modeling program GALFIT [@peng2002] to achieve three primary goals: (1) to obtain photometry of the nucleus to be used in the measurement of , (2) to determine the size and characteristics of the bulge component, if present, and (3) to search for signs of interaction in order to properly interpret our results.
It is imperative that the appropriate PSF is used when modeling galaxy morphologies. Empirical PSFs are ideal for this purpose, as they reflect the same instrumental conditions that were used for the observations. We searched the $HST$ archive for high signal-to-noise stars (typical peak counts were $\sim2100$) that were observed with the same instrument and filter and that fell near the same CCD position as the QSOs. The input PSF image must be large enough to encompass the wings of the function, so we chose a large region and interpolated across extraneous sources within the region. Since the PSF is undersampled in the PC image, we broadened it with a Gaussian of $\sigma = 0.8$ pixels. This increased the full-width at half-maximum (FWHM) of the PSF to be greater than 2 pixels, while preserving the encompassed flux.
The science images need to be large enough to accurately estimate the background sky level. This is especially important, because the edges of galaxy profiles are sensitive to the background sky level, and can affect, for example, the Sérsic index, $n$, of the profile fit to the galaxy. We cut each image to 15”.75 by 15”.75 (350 pixels x 0.045 arcsec pix$^{-1}$) centered on the science target, which is several times the size of the targets themselves. This large region sometimes included other objects, which we either masked from the fit or fit with an extra component. To match the resolution of the PSF, we also broadened the science images with the same Gaussian used on the PSF.
We followed an iterative process while fitting the morphologies of the galaxies. Each fit included the point source (a scaled PSF) and background sky level estimated by GALFIT. We began with simple models of single bulge or disk components for the host galaxies and added complexity as necessary. All but one of the galaxies required two or more components to model the host. Where the image showed multiple objects in the field, the secondary object was fit with its own component or masked from the fit. We generally used free index Sérsic profiles to model the bulges, but in many cases, GALFIT did not converge on a reasonable Sérsic index. To fit these hosts, we ran several iterations of GALFIT holding the index constant at values $n =$ 1, 2, 3, and 4. In each case the $n = 4$ fit the image the best. We adopted the best-fit models based on the $\chi^{2}$ values and residuals of the model subtracted image.
We report magnitudes for the best fits of each of the targets. Visual inspection of the residual images indicates that the point source and PSF were slightly mismatched. In order to estimate the uncertainty in the magnitudes, we also performed a direct PSF subtraction and determined the upper and lower limits for which the PSF was clearly over- and under-subtracted, respectively. The magnitudes measured by direct subtraction were, in each case, within 0.1 mag of the value obtained from the GALFIT fits, except for 005055.7$+$293328, for which the difference was 0.36 mag. Our measured magnitudes are systematically fainter than those reported by [@marble2003], with the differences ranging from less than 0.1 mag to 1.1 mag in the most extreme case (165939.7$+$183436). The differences likely arise from the fact that we fit simultaneously a PSF with the different components of the host galaxy (e.g., bulge and disk), so the nuclear fluxes that we measure are less likely to be contaminated by light from the host galaxy. Another reason for the differences in magnitudes may be the choice of PSFs: Whereas we selected empirical PSFs from the $HST$ archive, @marble2003 used one of the QSOs in their sample as the PSF to subtract from all the other QSOs. They chose 222202.2$+$195231 since it did not show significant extended emission after subtracting an artificial PSF created with the software package Tiny Tim [@krist2001].
In Table \[table:galfit\] we list the magnitudes of the PSF and S[é]{}rsic components used to fit each object. The magnitudes are apparent $HST$ F814W magnitudes, not corrected for Galactic extinction nor K-corrected, to allow for direct comparison with results published by [@marble2003]. Every object appears to have a clear bulge component. In Figs. \[figure:galfit\], \[figure:galfit\_b\] and \[figure:galfit\_c\] we show the image, the best fit model, and the residuals after subtraction of each of the targets. One-dimensional surface brightness profiles of each object, including all the components used to model them, are shown in Fig. \[figure:radial\].
[llcccc]{} 005055.7$+$293328 & PSF & 20.50 & & &\
& bulge & 19.27 & 0.22 & 0.52 & 4 (fixed)\
& disk? & 19.42 & 3.46 & 8.22 & 0.15\
& arms & 18.56 & 1.89 & 4.50 & 0.63\
015721.0$+$171248 & PSF & 20.45 & & &\
& bulge & 20.67 & 0.19 & 0.67 & 4 (fixed)\
& disk & 19.86 & 0.72 & 2.50 & 1 (fixed)\
022150.6$+$132741 & PSF & 21.17 & & &\
& bulge & 17.93 & 2.89 & 7.04 & 4 (fixed)\
& disk & 18.88 & 2.13 & 5.19 & 0.5\
034857.6$+$125547 & PSF & 21.57 & & &\
& bulge & 19.80 & 0.79 & 2.70 & 5.8\
& companion & 20.88 & 0.44 & 1.51 & 1 (fixed)\
& companion & 20.08 & 0.70 & 2.40 & 1 (fixed)\
163736.5$+$254302 & PSF & 21.37 & & &\
& bulge & 19.88 & 0.22 & 0.92 & 4.7\
& disk & 19.87 & 0.49 & 2.04 & 0.6\
165939.7$+$183436 & PSF & 19.11 & & &\
& bulge & 18.21 & 0.26 & 0.76 & 4 (fixed)\
225902.5$+$124646 & PSF & 19.40 & & &\
& bulge & 18.82 & 0.41 & 1.33 & 3.7\
& disk & 19.51 & 1.20 & 3.89 & 1 (fixed)\
230442.4$+$270616 & PSF & 20.70 & & &\
& bulge & 19.23 & 0.98 & 3.65 & 4 (fixed)\
& disk & 21.50 & 0.32 & 1.18 & 1 (fixed)\
232745.6$+$162434 & PSF & 22.12 & & &\
& bulge & 19.20 & 1.30 & 6.56 & 4 (fixed)\
& disk & 20.07 & 1.03 & 5.22 & 1 (fixed)\
\[table:galfit\]
The object 005055.7$+$293328 is hosted by a spiral galaxy and has a prominent dust lane in the central region. In order to obtain the best estimate for the nuclear flux we fit the host galaxy with multiple components so as to minimize the residuals after subtraction. The best fit included the spiral arms, disk, bulge, and an additional narrow S[é]{}rsic profile (not included in Table \[table:galfit\]) to account for the asymmetric light distribution due to the dust lane. While fitting all these components allows us to obtain a good fit for the PSF, it is likely that we are underestimating the flux and the effective radius of the bulge, since we are splitting it into two components.
For 163736.5+254302 and 165939.7+183436, we use additional components to fit the apparent companions. These components are not listed in Table \[table:galfit\], but they are shown in Fig. \[figure:radial\].
Black hole mass estimates
-------------------------
We estimated virial masses for the BHs in the sample by using the scaling relation for given by [@greene2010]: $$\begin{aligned}
{\ensuremath{M_\mathrm{BH}}}= (9.7 \pm 0.5) \times 10^6
\left(\frac{{\ensuremath{L_{\rm{5100 \AA}}}}}{10^{44}~{\rm erg~s^{-1}}} \right)^{0.519 \pm 0.07} \nonumber \\
\times \left(\frac{{\ensuremath{\mathrm{FWHM}_\mathrm{H{\alpha}}}}}{10^3~{\rm km s^{-1}}} \right)^{2.06 \pm 0.06}\end{aligned}$$
@greene2010 use the virial coefficient measured by @onken2004, i.e., $<f>$ = 5.5, which is a factor of 1.8 higher than the value assumed for a spherically symmetric broad line region [e.g., @kaspi2000]. This value is also consistent with that found by @woo2010 from a sample of reverberation mapped AGN. Although @graham2011 find a lower virial factor (i.e., 2.8), we adopt $<f>$ = 5.5 since this allows us to compare our results directly with those of previous studies (see §\[results\]).
In the scaling relation for virial masses, the size of the broad line region is a function of the FWHM of the broad component of H$\alpha$, FWHM$_{H\alpha}$, and the AGN continuum luminosity at rest frame 5100 Å, $L_{5100}$. Here we describe our procedure to measure each of these two quantities.
### Measuring FWHM$_{H\alpha}$
In order to measure the width of the broad line, we first had to do a detailed decomposition of the narrow component and the narrow $\lambda\lambda$ 6548, 6583 lines. We performed all the line fits using the [*specfit*]{} task [@Kriss94] in the IRAF STSDAS package, and following the method outlined by @glikman2007 and @GH04, as described below.
First, we created a narrow line model by fitting the $\lambda\lambda$6716, 6731 doublet. The line profiles often showed blue wings. Thus, we fit each line using two Gaussians, one of which was allowed to have skew (or asymmetry), although we found that skew was not needed in all cases. The continuum was fit simultaneously with a power law function. To ensure the same model fit for both lines, and to avoid unrealistic degeneracies, we constrained the model for each line of the doublet to be the same. This was done by matching the widths of each
component and the relative positions of the doublet. Because we used a two-Gaussian model, an extra constraint on the flux ratio was necessary to ensure the same total model for each line. @Garstang52 calculates the transition probability and intensity ratio of the doublet $^{4}$S - $^{2}$D transition in . He finds it can range from 0.5 to 0.8 depending on the parameters used in the calculation. However, he also cites a reasonable observed value of 0.74 based on NGC 7662. We adopted this observed value for our fits, as it produced the most reasonable fit of the range.
In some cases the doublet was contaminated by the broad emission or telluric absorption. In these cases we fit the \[\] $\lambda\lambda$4959, 5007 doublet to create a narrow line model. Similar techniques as with the doublet were used to fit the \[\] doublet. In this case the doublet flux ratio was constrained to 2.87 [@OF2006], which was a good fit for our data.
We then fit the region around using the narrow line model for $\lambda$6548 and $\lambda$6583. We constrained the relative position of the lines using the laboratory wavelengths for these lines. While @glikman2007 constrained the flux ratio to a theoretical value of 2.96, we found better fits when we let the flux of each line be freely fit. This often resulted in a flux ratio greater than 2.96. In addition, we included a narrow line component of using the fitted narrow line model and we used a power law to characterize the continuum. Finally, we included a broad Gaussian (with initial width of 5000 km s$^{-1}$) to model the broad line component. We first fixed the broad line to be symmetric, then added skew after its centroid position had been located by the fitting routine. We show the best fits with each of their components in Fig. \[figure:lineprofiles\].
[*Specfit*]{} uses a simplex method to minimize $\chi^{2}$. We adjusted input parameters and initial step sizes iteratively to probe different local $\chi^{2}$ minima and thus found the global $\chi^{2}$ minimum. The errors in the FWHM$_{H\alpha}$ resulting from the fit are typically $\sim$15 km s$^{-1}$. The true uncertainties, however, are much larger since they are dominated by the uncertainties in the modeling of the narrow component and the decomposition of the broad and narrow components.
In addition, although the AGN is much less extinguished in these wavelength regions than in the regions where we measure , there is still significant contribution from the host galaxy to the continuum around . We found that subtracting different stellar absorption lines (corresponding to different stellar templates or stellar populations) has no measurable effect on the value of FWHM$_{H\alpha}$. However, the shape of the stellar continuum around is different for different stellar populations, and this can potentially affect the fitting of the broad component, especially for the objects with the broadest emission lines. Thus the choice of stellar template to subtract from the AGN spectrum introduces a systematic uncertainty. In order to estimate this uncertainty, we measured FWHM$_{H\alpha}$ from the different template-subtracted emission lines and found the standard deviation for the different measurements. We added this uncertainty in quadrature to the fitting and systematic errors.
The final values for FWHM$_{H\alpha}$ and their uncertainties are listed in Table \[mbh\]. The FWHM$_{H\alpha}$ we measure for 225902.5$+$124646 is highly uncertain because H$\alpha$ has a very asymmetric profile, with very extended emission on the blue side of the line. This is likely indicative of an outflow, so that the we derive from it may not be reliable. We attempted a variety of ways to separate the core of the line from the outflow, by fitting the latter with multiple Gaussians blueshifted with respect to the core. In order to ensure that we obtain a conservative measure of , we chose the smallest value of FWHM$_{H\alpha}$ that would fit the line. The upper limit on the errors is given by the FWHM that would be obtained if a single Gaussian were used to fit the broad profile.
We note that some studies, including those of reverberation mapped AGN [e.g., @woo2010] use line dispersions rather than FWHM to calculate . However, these measurements are based on $H\beta$ rather than . For our study, we are unable to measure the line dispersion or FWHM of H$\beta$, since the AGN is still highly obscured at those wavelengths. Instead, we must use as a proxy for H$\beta$. The relations between these two lines in the literature are currently much better constrained for FWHM than for line dispersions. For example, the relation between FWHM$_{H\alpha}$ and FWHM$_{H\beta}$ used by @greene2010 is derived by fitting 162 objects
![One-dimensional surface brightness profiles of the $HST$ F814W images of the 2MASS QSOs, showing each of the components used to fit the objects with GALFIT. The black dashed lines in the residuals panels trace the one-sigma error bars in the observed data.[]{data-label="figure:radial"}](f5a.eps "fig:") ![One-dimensional surface brightness profiles of the $HST$ F814W images of the 2MASS QSOs, showing each of the components used to fit the objects with GALFIT. The black dashed lines in the residuals panels trace the one-sigma error bars in the observed data.[]{data-label="figure:radial"}](f5b.eps "fig:") ![One-dimensional surface brightness profiles of the $HST$ F814W images of the 2MASS QSOs, showing each of the components used to fit the objects with GALFIT. The black dashed lines in the residuals panels trace the one-sigma error bars in the observed data.[]{data-label="figure:radial"}](f5c.eps "fig:")\
![One-dimensional surface brightness profiles of the $HST$ F814W images of the 2MASS QSOs, showing each of the components used to fit the objects with GALFIT. The black dashed lines in the residuals panels trace the one-sigma error bars in the observed data.[]{data-label="figure:radial"}](f5d.eps "fig:") ![One-dimensional surface brightness profiles of the $HST$ F814W images of the 2MASS QSOs, showing each of the components used to fit the objects with GALFIT. The black dashed lines in the residuals panels trace the one-sigma error bars in the observed data.[]{data-label="figure:radial"}](f5e.eps "fig:") ![One-dimensional surface brightness profiles of the $HST$ F814W images of the 2MASS QSOs, showing each of the components used to fit the objects with GALFIT. The black dashed lines in the residuals panels trace the one-sigma error bars in the observed data.[]{data-label="figure:radial"}](f5f.eps "fig:")\
![One-dimensional surface brightness profiles of the $HST$ F814W images of the 2MASS QSOs, showing each of the components used to fit the objects with GALFIT. The black dashed lines in the residuals panels trace the one-sigma error bars in the observed data.[]{data-label="figure:radial"}](f5g.eps "fig:") ![One-dimensional surface brightness profiles of the $HST$ F814W images of the 2MASS QSOs, showing each of the components used to fit the objects with GALFIT. The black dashed lines in the residuals panels trace the one-sigma error bars in the observed data.[]{data-label="figure:radial"}](f5h.eps "fig:") ![One-dimensional surface brightness profiles of the $HST$ F814W images of the 2MASS QSOs, showing each of the components used to fit the objects with GALFIT. The black dashed lines in the residuals panels trace the one-sigma error bars in the observed data.[]{data-label="figure:radial"}](f5i.eps "fig:")\
, whereas @mcgill2008 derive their relation for line dispersions from only 19 objects. Thus, we use the relation for FWHM$_{H\alpha}$ rather than that for line dispersions. This choice does not affect our main results: if we were to use line dispersions rather than FWHM$_{H\alpha}$, the resulting would be, on average, 0.08 dex greater.
[clrcccc]{} 1&005055.7$+$293328 & $176_{-17}^{+14}$ &$2679^{+227}_{-175}$ & 1.64 & 2.52$\pm$1.24 & $1.19^{+0.40}_{-0.55}$\
2&015721.0$+$171248 & $250_{-54}^{+54}$ &$2196^{+27}_{-32}$ & 2.13 & 18.74$\pm$2.05 & $2.24^{+0.35}_{-0.72}$\
3&022150.6$+$132741 & $129_{-22}^{+22}$ &$4279^{+421}_{-492}$ & 1.76 & 1.78$\pm$0.24 & $2.61^{+1.06}_{-1.01}$\
4&034857.6$+$125547 & $152_{-17}^{+17}$&$4213^{+355}_{-342}$ & 1.68 & 4.16$\pm$0.95 & $3.93^{+1.47}_{-1.61}$\
5&163736.5$+$254302 & $129_{-12}^{+14}$ &$3178^{+371}_{-248}$ & 1.93 & 14.13$\pm$2.21 & $\phn4.15^{+1.87}_{-1.75}$\
6&165939.7$+$183436 & $159_{-25}^{+28}$ &$6742^{+3640}_{-142}$ & 0.39 & 0.94$\pm$0.24 & $4.79^{+9.31}_{-1.38}$\
7&225902.5$+$124646 & $194_{-20}^{+21}$ &$4457^{+5543}_{-100}$ & 0.84 & 4.03$\pm$0.55 & $4.34^{+27.9}_{-3.05}$\
8&230442.4$+$270616 & $127_{-7}^{+7\phn}$ &$6349^{+251}_{-265}$ & 0.99 & 1.71$\pm$0.47 & $5.77^{+1.62}_{-2.02}$\
9&232745.6$+$162434 & $205_{-19}^{+19}$&$2341^{+416}_{-161}$ & 1.93 & 19.16$\pm$5.57 & $2.59^{+1.55}_{-1.18}$\
\[mbh\]
### Measuring $L_{5100}$ {#L5100}
The AGN continuum flux at rest frame 5100 Å cannot be measured directly from our ESI spectra for two primary reasons: (1) The continua of red QSOs suffer from significant extinction at short wavelengths, and this extinction needs to be properly measured. (2) The spectrum in this wavelength region has a large contribution from the host galaxy that also needs to be accounted for.
In order to obtain the unobscured QSO flux at 5100 Å, we first measured the extinction for each object by modeling its spectrum. As described in §\[sigma\], we modeled the QSO continuum by including a polynomial (in $x = ln(\lambda)$ space) in the fitting procedure to obtain . The resulting polynomial, $C(x) \times P(x)$, is then a good representation of the reddened QSO. In order to estimate E($B-V$), we compared the polynomial to the Sloan Digitized Sky Survey (SDSS) composite QSO spectrum [@vandenberk2001] reddened with a Small Magellanic Cloud (SMC) reddening law [@prevot1984; @bouchet1985], with E($B-V$) as a free parameter. The values of E($B-V$) that we obtained using this procedure are listed in Table \[mbh\]. Two thirds of our objects span a range of E($B-V$) values comparable to those found by [@urrutia2009] in a sample selected from the FIRST-2MASS Red Quasar Surveys [@glikman2007], i.e., E($B-V$) $\lesssim 1.5$. The remaining three objects, however have E($B-V$) $\sim 2$. Typical errors in E($B-V$) derived from the fits are $\sim$0.1.
Whenever possible, we also estimated E($B-V$) using H$\alpha$/H$\beta$ ratios. Naturally, the values for these ratios become more uncertain with increasing extinction, so we only estimated E($B-V$) for the seven objects for which we could obtain reliable ratios. The resulting E($B-V$) values were generally consistent with the values obtained from the fitting procedure, although the two could differ by as much as E($B-V$)$\sim$0.3. We found no trend in the way the two values differed; the average E($B-V$) of the seven objects measured from the Balmer decrement was 1.49, compared to 1.45 as inferred from modeling the AGN continuum. Since there is no a priori reason why the continuum and broad emission lines should suffer the same amount of extinction, we only used the Balmer decrement estimates as a rough comparison to make sure the reddening estimates derived from the model were reasonable.
As a further test, we subtracted the corresponding reddened SDSS composite spectrum from the spectrum of each of our targets to recover the host galaxy spectrum. There were some QSO features that were over- or under-subtracted in the resulting host galaxy spectra. This is caused by the mismatches between the SDSS composite spectrum and the intrinsic spectra of our QSOs. Indeed, the greatest source of uncertainty in the final values of $L_{5100}$ comes from such mismatches. However, the overall spectral energy distribution (SED) of each host galaxy matched the shape of the SED that would be expected for a galaxy with the given set of stellar absorption features observed in that galaxy. Once we had determined the intrinsic reddening for each object, we scaled each continuum model, $C(x) \times P(x)$, to match the observed flux of the corresponding QSO nucleus, as measured from $HST$ WFPC2 F814W images using GALFIT (see Table \[table:galfit\]) and corrected for Galactic extinction using the values given by @schlegel1998. Thus we constructed continua of the reddened QSOs without galaxy contamination and taking into account flux loses due to the slit. Finally, we corrected these continua for intrinsic extinction (using the values listed in Table \[mbh\]) and measured the the flux of each QSO at rest-frame 5100 Å. As mentioned above, the greatest source of uncertainty comes from measuring E($B-V$) by making the assumption that the intrinsic continuum shape of our sources is equal to that of the SDSS composite QSO. The uncertainties for $\lambda L_{5100}$ listed in Table \[mbh\] include the uncertainty in the intrinsic power-law index, $\alpha_{\lambda}$ (conservatively assumed to be $\sim$0.05; [@vandenberk2001]) as well as the uncertainty in measuring E($B-V$) by the process we outline above.
The black hole masses derived from our measurements of FWHM$_{H\alpha}$ and $\lambda L_{5100}$ are listed in Table \[mbh\]. For completion, we also repeat the values of from Table \[table:results\] in Table \[mbh\]. As mentioned before, using line dispersions rather than FWHM to estimate black hole masses results in masses that are, on average, 0.08 dex greater than those presented in this table.
Results
=======
Our results are summarized in Fig. \[msigmaplot\], where the 2MASS red QSOs are plotted as black circles. We also plot objects from [@woo2010], which include the 24 local AGN that have published and from reverberation mapping (green squares), as well as the relation derived by @woo2010 from that sample (green dashed line). For reference, the relation for inactive local galaxies measured by [@mcconnell2011] is plotted as a blue dotted line.
The majority of the red QSOs are clearly above the local relations. We have specifically compared our sample to AGN only to avoid any potential biases introduced by comparing active to quiescent galaxies [@lauer2007], and we have used a virial coefficient consistent with that found by @woo2010 from the reverberation mapped AGN sample. [@woo2010] estimate an intrinsic scatter of 0.43 dex for their relation including all morphological types. In contrast, our objects are, on average, 0.80 dex above the @woo2010 relation, with six out of nine of them having log() above the intrinsic scatter. Comparison with the @mcconnell2011 relation for quiescent galaxies yields a similar, although smaller, discrepancy: our sample is, on average, 0.65 dex above the relation, and with five objects being above the intrinsic scatter.
When comparing our objects to those of @woo2010, it is pertinent to consider possible differences in the width of when measured from single epoch (SE) spectra versus rms spectra from reverberation mapped objects. @park2012 find that H$\beta$ is broader in rms spectra than in SE spectra, although the difference between the two decreases as the intrinsic width of the broad lines increases. @park2012 present a prescription to correct for this effect in relatively narrow-lined ($<$3000 [km s$^{-1}$ ]{}) objects. To get an idea of how much this effect might impact our results, we can assume that the correction for H$\beta$ holds for , and that it is a reasonable approximation for the objects in our sample with broader as well. With those assumptions, if we apply equation (8) of @park2012 to correct the FWHM$_{H\alpha}$ in Table \[mbh\], we find that the black hole masses of the objects in our sample decrease by 0.13 dex on average. This correction, however, is not enough to bring the objects to agreement with the local relation, as they would still be, on average, 0.67 dex above the @woo2010 relation. Even if the correction for and/or for objects with broader lines were greater, it is unlikely to be large enough to account for the offset we observe.
It is important to consider other potential biases that may be present in our sample. First, the morphology of the host galaxies could introduce a bias. It has been shown that the precise form of the relation has a dependence on the morphology of the galaxies in question [e.g., @graham2011], and that pseudobulges do not follow the same relation [@kormendy2011]. As can be seen in Table \[table:galfit\], every host galaxy in our sample has a bulge component that is best fit by a S[é]{}rsic of index $n\simeq4$. Although most of the galaxies also appear to have a disk component, the majority (7/9) are clearly dominated by the bulge. In order to investigate how the presence of the disk may affect our measurement of in the bulges, we tested whether they follow the Faber-Jackson relation [@faber1976]. In the left panel of Fig. \[figure:fj\], we plot velocity dispersions vs. absolute Cousins I magnitudes of the bulge components of the galaxies. The latter were obtained using the output magnitudes from GALFIT given in Table \[table:galfit\] corrected for Galactic reddening, applying k-corrections for the spectral type of each galaxy and passive evolution using the on-line calculator from @vandokkum2001. The dashed line in the left panel of Fig. \[figure:fj\] is the Faber-Jackson relation from @nigoche2010. @nigoche2010 use a sample of $\sim$90,000 SDSS-DR7 early-type galaxies to calculate the Faber-Jackson parameters (i.e., A and B in log = A - BM$_{r}$) for different magnitude ranges. Our objects span a magnitude range of $-20.6<M_{I}<-23.4$, or $-19.8<M_{r}<-22.6$ [AB magnitudes, using the transformation equations given by @jester2005]. In this magnitude range, @nigoche2010 find A = 0.152$\pm$0.009 and B = $-$0.976$\pm$0.193. Note that the relation of @nigoche2010 is measured in the SDSS r-filter, and we have simply transformed it to Cousins-I, without accounting for potential wavelength-dependent differences in the slope.
Two host galaxies fall clearly off the relation: those of 005055.7+293328 and 015721.0+171248 (objects \# 1 and 2, respectively), both of which have disks with luminosities comparable to, or greater than, their corresponding bulges. As described in § \[morphologies\], the host galaxy of 005055.7+293328 has prominent nearly edge-on disk with spiral arms and a dust lane that goes through the bulge. To mimic the dust lane, we fit this object with two separate components, so we almost certainly underestimate the flux contained in the bulge. @marble2003 fit a single component to the host galaxy of this object and find a M$_{I}$ = $-$22.3. This magnitude would bring the position of the object to agreement with the Faber-Jackson relation. The other object, 015721.0+171248, is the only one in the sample that has a disk that is more luminous than the bulge. The rotating component from a dynamically cold disk can bias toward higher values (when the disk is edge-on) or lower values . Although the disk of 015721.0+171248 has a lower inclination than that of 005055.7+293328, its velocity dispersion may still be somewhat increased by the dynamically cold component. More importantly, 015721.0+171248 is the only object for which there is significant QSO contamination in the stellar absorption line spectrum, so that we were not able to obtain a reliable fit of for the red region. We have marked both of these objects with hexagons in Figs. \[msigmaplot\] and \[deltambh\] to indicate that the measured in these objects may not be representative of normal bulges. Interestingly, these are the two objects that fall closest to the local AGN relation, and removing them from the analysis only accentuates the offset of the red QSO sample with respect to the local relation.
We also consider the position that the bulges in our sample occupy on the Fundamental Plane [FP; @djorgovski1987]. We obtain $<\mu>_{eff}$ as an output parameter from GALFIT, applying k-corrections and passive evolution to $z=0$ as above, as well as correcting from cosmological dimming. We compare it to the FP relation derived by @hyde2009 from a sample of $\sim$50,000 SDSS-DR4 (with parameters from DR6) early-type galaxies at $0<z<0.35$. Here we use their orthogonal fit to the plane in the SDSS $i-$band and we have transformed it to Cousins-I, using the corrections prescribed by @rothberg2012. The solid line in Fig. \[figure:fj\] (right panel) corresponds to the coefficients given in Table 2 of @hyde2009, and the dashed and dotted lines correspond to the values that enclose 68% and 95% of their SDSS early-type galaxies, respectively. While, on average, our objects fall below the FP, they are consistent with the scatter in the relation. The two objects that are off the Faber-Jackson relation also appear to be the ones with the largest offset from the FP.
Besides the effects of a potential dynamically cold component on , we need to consider the possibility that we are measuring in systems that are not dynamically relaxed. At least a third of the host galaxies in our sample show clear signs of interactions (Figs. \[figure:galfit\], \[figure:galfit\_b\], \[figure:galfit\_c\]). Many of the numerical simulations that are successful at reproducing the relation presuppose that AGN activity is triggered by mergers [e.g., @robertson2006a; @robertson2006b; @hopkins2006]. However, they predict the value after the system has reached dynamical equilibrium. To investigate the evolution of during mergers, [@stickley2012] use N-body simulations of merging galaxies to perform a flux-weighted measurement of through a diffraction slit. They find that, typically, tends to increase right after the first passage, followed by a period of oscillation about the value that eventually reaches once the system comes to dynamical equilibrium. Thus, the observed value of in our objects could potentially be different from their corresponding equilibrium value, depending on the precise merger stage at which we happen to be observing them. However, @stickley2012 estimate that, even in extreme cases, the observed value of should fall between 70% and 200% of the quiescent value, and the probability that the observed value of will actually be far from the equilibrium value is rather small. Therefore, even in the highly unlikely case that our entire sample is caught during an oscillation stage when has a lower value, the predicted quiescent value would not be high enough to bring our sample to agreement with the local relation. Moreover, using more sophisticated high-temporal resolution numerical simulations, Stickley and Canalizo (in preparation) find that the time at which the value of shows significant deviations from the equilibrium value preceeds the period of measurable accretion onto the black hole (i.e., the time when the object appears as a QSO).
Another possibility is that dust in the host galaxy may be obscuring a fraction of the stars and this may result in a biased measurement of . With the exception of 005055.7+293328, we find no clear evidence for significant amounts of dust in the host galaxies of the red QSOs. Their spectra show that, while the nucleus suffers from strong extinction, the host galaxies are not significantly reddened. As mentioned in § \[L5100\], after subtracting a reddened QSO spectrum, the overall SED of each host galaxy matches the shape of the SED that would be expected from the observed stellar absorption features. This indicates that there is no severe reddening in the host galaxies. Moreover, if the bulges of these galaxies were significantly reddened, they would not fall on the Faber-Jackson or FP relations (see Fig. \[figure:fj\]). However, we cannot rule out the possibility that a modest amount of dust reddenning (E($B-V$)$\lesssim$0.1) could be present in the hosts. Therefore, it is important to consider the effects that dust in the host galaxy, particularly if located close to the central regions of the bulge, would have on the measurement of . The simulations of [@stickley2012] include a toy model for dust consisting of a slab of gray attenuating material, and show that a lower value of can be measured if the light from stars closer to the central regions is more extinguished than that of stars at larger radii. The specific toy model used by @stickley2012 leads to a decrease on the measured value of $\lesssim$15%. Correcting for this factor would not bring the red 2MASS QSOs to the local relation. While more realistic models for dust attenuation are needed to truly quantify the effect of dust on the measurement of , we repeat that there is no evidence indicating that our host galaxies suffer from significant dust extinction.
Finally, we consider the bias that may be introduced from the sample selection. The objects were selected based on their near-infrared colors. The selection criterion that objects must have $J-K_{s}>2.0$ implies that their SEDs will be dominated by the AGN in the near-infrared; if they were dominated by the host galaxy, their colors would be bluer than this. Thus, the host galaxies of the most reddened AGN could possibly be biased toward lower luminosity (and lower mass). We investigate this potential bias by looking at the offset of the objects as a function of near-infrared colors and reddening. First, we plot our objects on the black hole mass - bulge luminosity relation, - L$_{\rm bulge}$, in Fig. \[figure:mlbul\], where we have transformed the bulge magnitudes, M$_{I}$ from Fig. \[figure:fj\] to M$_{R}$ assuming an average $R-I$ = 0.6 for the sample. The solid line in this figure is the local relation for inactive galaxies from @bettoni2003, as reported by @kim2008. Not surprisingly, our objects have overmassive BHs with respect to the local relation: they are, on average, 0.89 dex above the relation. Note that two of the objects that have the largest offsets with respect to the relation (objects 1 and 2) are also the two objects that do not follow the Faber-Jackson relation. If we do not include these two objects, then the sample is, on average, 0.69 dex above the relation.
If there were indeed a bias favoring less luminous galaxies for more reddened objects, we should see a trend of an increasing offset from the - $L_{bulge}$ relation, $\Delta$M$_{R,bulge}$, with increasing $J-K_{s}$ colors and with increasing reddening, E($B-V$). Figure \[figure:colors\] shows no such trends and, in fact, some of the most reddened objects also have the smallest offsets in magnitude. Therefore, we do not see evidence for a bias introduced by the color selection of our sample.
Discussion
==========
Our sample appears to have a true offset with respect to the local relation, in the sense that the black holes in red QSOs are overmassive compared to their host galaxies. [@woo2006; @woo2008] and [@treu2007] find similar offsets for samples of Seyfert 1 galaxies at $z\sim0.36$ and $z\sim0.57$; their results are plotted along with ours in Fig. \[msigmaplot\]. [@woo2008] investigate the relation at three different redshift bins ($<$0.1, 0.36, and 0.57), and conclude that there is evolution at the 95% confidence level. This is at least in qualitative agreement with several studies of the - relation at higher redshifts. Using 51 quasar hosts (mostly lensed), [@peng2006a; @peng2006b; @peng2006c] find evidence that the ratio / increases above $z=1$. [@bennert2011b] find similar results in a sample of $1<z<2$ X-ray selected, broad-line AGN by deriving stellar masses of the bulge component from multi-filter surface photometry in $HST$ images. The implication from these and other studies [e.g., @decarli2010] is that at $z \sim 2$ host bulges were undermassive relative to their BHs (compared to today), suggesting that BHs grew first and bulges have been playing “catch-up.”
While our results appear to be in line with all these studies, they also present additional challenges to the evolutionary picture. [@woo2008] find a potential trend in the evolution of , if indeed the higher redshift samples are the direct progenitors of the local AGN. They show this clearly in their Fig. 3, where they plot the offset in with respect to the local, quiescent sample of [@tremaine2002]. We repeat this plot, including our objects, in Fig. \[deltambh\], where we have calculated the offsets with respect to the local AGN relation from reverberation mapped objects [@woo2010], and we also include the local AGN samples of @bennert2011a and @greene2006a. The BH masses in the @greene2006a sample have been increased by log(1.8) = 0.255 to account for the difference in the virial coefficient used by @greene2006a compared to that used in the other studies. Thus, all the BH masses in Fig. \[deltambh\] assume a virial coefficient of $f\sim5.5$ [@onken2004; @woo2010].
Taken at face value, Fig. \[deltambh\] would indicate that there is a steep increase in the offset between redshifts of 0.1 and 0.2, and that most of the evolution has occurred in the last 3 Gyrs. Alternatively, in the context of models that find a non-causal origin for the scaling relations [e.g., @jahnke2011], most of the mass existing in the disks of these objects has been transferred to the bulge through mergers in the last 3 Gyrs. This would also be consistent with our result showing that the two objects with the most significant disk components are also the two objects with the largest offsets in the - L$_{\rm bulge}$ relation (Fig. \[figure:mlbul\]). However, the comparison between local and higher redshift AGN samples is not straightforward, as the latter have, on average, significantly higher than their local counterparts. Every object at $z>0.2$ has $> 10^{8}$ , whereas there is a dearth of objects in this mass range at lower redshifts. In Fig. \[deltambh\], we plot all the objects with $> 10^{8}$ with red symbols. If we consider the red objects only, the trend with redshift becomes less significant. The red objects at $z<0.1$ have an average offset of 0.40 dex, compared to 0.80 dex in our sample, and 0.66 dex and 0.61 dex, respectively, in the Seyfert 1 samples at $z=0.36$ and $z=0.57$ of @woo2006 [@woo2008].
The difference in range between the local and the higher-$z$ samples is due, at least in part, to the fact that they are drawn from flux-limited samples. In these samples, less luminous objects are not detected at higher redshifts and, since AGN luminosity depends partially on , objects with lower are less likely to be picked. @lauer2007 describe a potential bias that may be present in samples of AGN at the highest . They show that a bias may arise from the conspiratory nature of a steep luminosity function and intrinsic scatter in the relation, leading to a selection that favors higher mass BHs over lower mass BHs. Thus studies targeting luminous or massive AGN may find an offset from the local relation that is only due to this selection bias. On the other hand, @treu2007 run Monte Carlo simulations to investigate the effects on a hypothetical selection function log(/) $> 7.9$, which applies to the objects of @woo2006 [@woo2008] as well as our own. They compare the measured offset to a simulated input offset with respect to the local and find that the bias is negligible, unless the intrinsic scatter at higher redshifts increases dramatically (of the order of 1 dex). However, the fact that we find an offset from the local relation in all of the objects with $> 10^{8}$ , both in the local and the non-local universe, may indicate that there is still some bias at play.
In order to further assess potential biases, we examine the dependence of the offset on AGN luminosities and Eddington ratios (Figs. \[figure:luminosity\] and \[figure:eddington\]). For the reverberation mapped AGN of @woo2010, we use the values of L$_{5100}$ published by @park2012, @Bentz2006, @kaspi2000, and @denney2006. For consistency, we transform L$_{H\alpha}$ in the sample of @greene2006a to L$_{5100}$, using the relations given by @greene2005. The resulting plot is shown in Fig. \[figure:luminosity\]. We see a striking trend: The offset increases with luminosity, with virtually all the objects with log(L$_{5100}$/erg s$^{-1}$) $>43.6$ being above the relation, including those with $<$ 10$^{8}$ . Clearly, the majority (two thirds) of high luminosity AGN are in the non-local universe, and this could be driving the trend. However, we note that all of the high-luminosity objects in the local universe follow this trend as well.
One possible way to explain this trend might be if the radius-luminosity ($R_{BLR}-L$) relation somehow changed dramatically at high luminosities, if, for example, a point is reached where the broad line region is matter-bound so that the radius-luminosity relation would flatten. A different relation might also be expected if the SEDs of high luminosity objects are different from those of lower luminosity objects. The data points in Fig. 5 of @bentz2009 suggest that, indeed, some flattening of the $R_{BLR}-L$ relation may occur at the highest luminosities. However, even if that were the case, the potential flattening occurs at luminosities significantly higher than log(L$_{5100}$/erg s$^{-1}$) $= 43.6$. In fact, if we were to fit only the objects in the luminosity range of $43.6 <$ log(L$_{5100}$/erg s$^{-1}$) $<$ 45.3 (corresponding to the luminous objects of Fig. \[figure:luminosity\]), we would obtain a steeper relation, closer to the relation of @kaspi2000, and resulting in somewhat higher black hole masses in this range. Thus, variations in the $R_{BLR}-L$ relation cannot account for the trend we see in Fig. \[figure:luminosity\].
Using semi-analytical models of galaxy evolution, @portinari2011 find that more luminous QSOs tend to trace over-massive BHs with respect to the intrinsic relation [similar to the bias described by @lauer2007]. However, for objects at $z<0.5$, their simulations predict that the parameters of QSO host galaxies closely follow the distribution of those of inactive galaxies, with very small offsets ($\sim$0.1 dex) in host galaxy mass.
To measure Eddington ratios we obtain rough bolometric luminosities by assuming L$_{\rm bol}$ = 10 L$_{5100}$ [@woo2002]. The plot of $\Delta$ vs. L$_{\rm bol}$/L$_{\rm Edd}$ (Fig. \[figure:eddington\]) shows large scatter and no correlations. However, since Fig. \[figure:luminosity\] shows that there may be an offset in $\Delta$ at high AGN luminosities, it may be instructive to consider separately objects of lower luminosity (log (L$_{5100}$/erg s$^{-1}$) $<$ 43.5; blue symbols in Fig. \[figure:eddington\]) and those of higher luminosity (pink symbols). A trend appears to surface: objects with low L$_{\rm bol}$/L$_{\rm Edd}$ have more positive offsets with respect to than those with higher L$_{\rm bol}$/L$_{\rm Edd}$. We caution that the trends may be largely driven by the fact that we are dividing the objects in bins according AGN luminosity, e.g., a highly luminous AGN can have low only if it has a high accretion rate. Such trends of increasing L$_{\rm bol}$/L$_{\rm Edd}$ with decreasing have been observed in large flux-limited samples of AGN [e.g., @netzer2007]. However, we note that the trends we see in Fig. \[figure:eddington\] are in the offset with respect to and not just in .
A similar trend with respect to -L$_{\rm bulge}$ was observed by @kim2008 in a study of 45 $z\lesssim 0.35$ type 1 AGN. They find that, at a given bulge luminosity, objects with high L$_{\rm bol}$/L$_{\rm Edd}$ have lower . They suggest that this may be explained in terms of the growth phase of BH: If L$_{\rm bol}$/L$_{\rm Edd}$ decreases as a function of time during a black hole growth episode, then BH with the highest L$_{\rm bol}$/L$_{\rm Edd}$ have more “catching up to do” than those with lower L$_{\rm bol}$/L$_{\rm Edd}$. We searched for this trend by dividing the objects of Fig. \[figure:eddington\] in bins of stellar velocity dispersion (as a proxy for bulge luminosity; not shown in Fig. \[figure:eddington\]). Given the large scatter, we found no significant trends or correlations. We also searched for this trend in our sample by plotting $\Delta$M$_{R,bulge}$ (from Fig. \[figure:mlbul\]) against L$_{\rm bol}$/L$_{\rm Edd}$. Once again, we found a large scatter and no significant trends.
The combination of the potential biases in these samples with the trends that we have uncovered in this study prevents us from making any definitive statements on whether we see evolution with redshift in the relation. Clearly, better statistics are needed to disentangle the dependence of on mass, luminosity, accretion rates, and redshift. Studying samples with a wider range of these properties could help us to better understand the scatter and the different biases. For local AGN, [@bennert2011a] indicate a future study of $\sim$75 additional AGN with $> 10^{7}$ that can be used to probe the local relation at the highest and, likely, at high luminosities. In the non-local universe, we are conducting a study of additional red QSOs as well as other QSOs that are similarly suited for the study of , including objects in lower and lower luminosity ranges. We will present results from objects at $0.2<z<1$ in forthcoming papers [e.g., @hiner2012].
We are grateful to the referee for a very detailed report that helped improved the contents and the presentation of this paper. We thank Barry Rothberg for providing a copy of his manuscript before publication and Chien Peng for insightful discussions and suggestions. We also thank Alan Stockton and Michael Harrison for their assistance in obtaining some of the observations, and Roozbeh Davari for providing scripts for radial plots. Support for this program was provided by the National Science Foundation, under grant number AST 0507450. Additional support was provided by NASA through a grant from the Space Telescope Science Institute (Program AR-12626), which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555, and by the Norwegian Research Council, project number 191735. M.W. acknowledges funding from the Marie Curie Intra-European Fellowship. Some of the data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
[*Facilities:*]{} ,
[^1]: http://www2.keck.hawaii.edu/inst/nirspec-old/wmkonspec/index.html
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abstract: 'The spectral and spatial characteristics of two special categories of O stars found in the VFTS dataset are presented. One of them comprises very rapid rotators, including several more extreme than any previously known. These objects are distributed around the peripheries of the main 30 Doradus clusters, suggesting a runaway nature for which their radial velocities already provide preliminary supporting evidence. The other category consists of a large number of Vz stars, previously hypothesized on spectroscopic grounds to be on or very near the ZAMS. Their distribution is the inverse of that of the rapid rotators: the Vz are strongly concentrated to the ionizing clusters, plus a newly recognized band of recent and current star formation to the north, which provides strong circumstantial evidence for their extreme youth.'
author:
- 'Nolan R. Walborn,$^1$ Hugues Sana,$^2$ William D. Taylor,$^3$ Sergio Simón-Díaz,$^4$ and Christopher J. Evans$^5$'
title: 'Two Remarkable Spectroscopic Categories of Young O Stars from the VLT-FLAMES Tarantula Survey'
---
Introduction
============
The VLT-FLAMES Tarantula Survey (VFTS; Evans et al. 2011) has produced an unprecedented spectroscopic dataset for the stellar content of the 30 Doradus Nebula in the Large Magellanic Cloud, the largest in the Local Group. About 800 OB stars have been observed with a resolving power of $\sim$10$^4$ at the Very Large Telescope of the European Southern Observatory (program 182.D-0222). It is reasonable to expect that such an advance over previous material will entail both observational and theoretical progress on the formation and evolution of massive stars. Indeed, some early, unexpected empirical developments, resulting from the spectral classification of 167 O stars with no detected radial-velocity variations, are briefly described here; they will be amplified shortly by Walborn et al. (2012, in prep.), while the optimized classification atlas generated for this work will be presented by Sana et al. (2012, in prep.).
Rapid Rotators
==============
In this subsample, 17 O main-sequence stars have very high projected rotational velocities in the range of 300–600 km s$^{-1}$; the largest values previously known were just over 400 km s$^{-1}$ (Howarth & Smith 2001; Walborn et al. 2011). An additional 7 rapid rotators have giant luminosity classes indicated, although the actual luminosities of such extreme objects may require further investigation. The spectra of 6 luminosity class V stars are displayed in Figure 1, labeled with their detailed spectral types and preliminary $v\sin i$ measurements derived by SSD using a Fourier transform technique. (Definitive values will be derived by O. Ramírez-Agudelo et al. 2012 and C. Sabín-Sanjulián et al. 2012, in prep.). VFTS 285 is the current record holder. Remarkably, none of these spectra displays Balmer central stellar emission, even at H$\alpha$, indicating a possible absence of disks. A related case probably with a disk is discussed by Dufton et al. (2011).
![Rectified spectral intensity plots of extremely rapid rotators. The VFTS catalogue numbers are followed by the spectral types and $v\sin i$ measurements. VFTS 190 has an Onfp spectrum with emission wings at He II $\lambda$4686 (Walborn et al. 2010a). The central Balmer emission is nebular; “SB?” signifies a relative displacement of the stellar absorption lines, although radial-velocity variations have not been detected in the available data.[]{data-label="fig1"}](walbornfig1.ps){width="6.0in"}
The spatial distribution of the main-sequence rapid rotators is shown in Figure 2. It is seen that all but 4 of them are located about the peripheries of the clusters NGC 2060 and 2070. This circumstance immediately suggests a possible runaway nature of the class. The VFTS radial velocities of these stars already provide supporting evidence for that interpretation, containing a high fraction of large values compared to the rest of the sample (Sana et al., these proceedings; et al. 2012, in prep.). A current [*Hubble Space Telescope*]{} imaging program by D. Lennon et al. may determine their proper motions, thus enabling a full kinematical analysis. We emphasize that a population of high-mass, extreme-rotator runaways from a massive young cluster is a new phenomenon, which may be related to theoretical dynamical predictions and to the origin of gamma-ray bursts (Dale & Davies 2006; Allison et al. 2010). A possibly related situation for the Galactic cluster Westerlund 2 has been presented by Roman-Lopes, Barbá, & Morrell (2011).
![Distribution of the rapid rotators (VFTS numbers in red), on a logarithmic MPG/ESO/WFI $V$ image of 30 Doradus; VFTS 751 and 755 are separated by 284 in declination, or 71 pc in projection. North is up and east to the left. NGC 2060 is at the SW and NGC 2070 in the center. The unannotated red circles in the core are VFTS 465, 660, 654, and 706. The blue symbols correspond to O V((fc)) spectra (see Fig. 3 caption), which do not show any preferred locations.[]{data-label="fig2"}](walbornfig2.ps){width="4.0in"}
Zero-Age Main-Sequence Stars
============================
A class of O-type spectra with empirical evidence of subluminosity is described by Walborn (2009). Briefly, they display He II $\lambda4686$ absorption stronger than any other He II or He I line. Since emission in that particular He II line is a progressive luminosity effect (the Of phenomenon), these spectra, assigned luminosity class Vz, have been hypothesized to represent the inverse effect, i.e. lower luminosity, higher gravity, and/or lower mass-loss rate than for normal class V. Although such spectra have systematically been found in young regions, we were surprised to encounter no fewer than 35 definite plus another 10 possible examples in the VFTS subsample discussed here. The spectra of a few of them are reproduced in Figure 3 (along with examples of some other special class V categories mentioned in the caption). The physical parameters of this category have not been investigated previously, but they will be in the extensive VFTS sample.
![Rectified spectral intensity plots of Vz and some other subcategories of O-type main-sequence spectra, including V((fc)) with C III $\lambda$4650 emission lines comparable to N III $\lambda$4640 (Walborn et al. 2010b), He II wings stronger than He I (VFTS 638), and no N III or C III emission features whatever despite very high S/N (VFTS 849). Note that Vz spectra can be either ((f)) or ((fc)), or neither.[]{data-label="fig3"}](walbornfig3.ps){width="6.0in"}
The distribution of the definite Vz objects is shown in Figure 4. It is essentially the inverse of the rapid rotator distribution, with a strong concentration toward the main clusters NGC 2060 and 2070. In addition, several of the Vz stars lie in an approximately east-west band at the northern extreme of the field. The “isolated” WN stars R144, R146, and R147 (Feast, Thackeray, & Wesselink 1960) also lie in this band, as does one of the most luminous IR YSOs in the [*Spitzer*]{} images. Thus, this band evidently represents another, previously unrecognized region of recent and current star formation in 30 Dor, consistent with the presence of Vz stars there. This distribution provides strong circumstantial evidence for extreme youth of the Vz class, to be further investigated by quantitative analysis of the VFTS spectroscopic data.
NRW’s travel to Lac Taureau was supported by NASA through grant GO-12179.01 from STScI, which is operated by AURA, Inc. under contract NAS5-26555. Thanks to A. de Koter for comments that improved the clarity of the presentation.
Allison, R.J. et al. 2010, MNRAS, 407, 1098 Dale, J.E. & Davies, M.B. 2006, MNRAS, 366, 1424 Dufton, P.L. et al. 2011, ApJ, 743, L22 Evans, C.J. et al. 2011, A&A, 530, A108 Feast, M.W., Thackeray, A.D., & Wesselink, A.J. 1960, MNRAS, 121, 337 Howarth, I.D. & Smith, K.C. 2001, MNRAS, 327, 353 Roman-Lopes, A., Barbá, R.H., & Morrell, N.I. 2011, MNRAS, 416, 501 Walborn, N.R. 2009, STScI Symp Ser, 20, 167 Walborn, N.R. et al. 2010a, AJ 139, 1283 Walborn, N.R. et al. 2010b, ApJ, 711, L143 Walborn, N.R. et al. 2011, AJ, 142, 150
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abstract: 'The extended high-energy gamma-ray ($>$100 MeV) emission occurring after the prompt gamma-ray bursts (GRBs) is usually characterized by a single power-law spectrum, which has been explained as the afterglow synchrotron radiation. We report on the Fermi Large Area Telescope (LAT) observations of the $>$100 MeV emission from the very bright and nearby GRB 130427A, up to $\sim$100 GeV. By performing time-resolved spectral fits of GRB 130427A, we found a strong evidence of an extra hard spectral component above a few GeV that exists in the extended high-energy emission of this GRB. This extra spectral component may represent the first clear evidence of the long sought-after afterglow inverse Compton emission. Prospects for observations at the very high-energy gamma-rays, i.e., above 100 GeV, are described.'
address: |
Institute of Astronomy and Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan\
[email protected]
author:
- 'Pak-Hin Thomas Tam'
title: |
THE VERY BRIGHT AND NEARBY GRB 130427A:\
The Extra Hard Spectral Component and Implications for Very High-energy Gamma-ray Observations of Gamma-ray Bursts
---
Large Area Telescope Observations of Gamma-ray Bursts
=====================================================
Since 2008, the Fermi Large Area Telescope (LAT) has detected more than 35 gamma-ray bursts (GRBs)[@lat_grb_cat] and enabled us to establish the followings:
delayed onset of high-energy emission compared to the prompt keV-MeV emission phase,
extra spectral component seen at LAT energy range during the prompt emission phase for some GRBs,
the existence of the long-lasting MeV-GeV emission for many LAT GRBs.
It is commonly believed that the long-lasting MeV-GeV emission is dominated by synchrotron radiation.[@zou09; @kbd09; @gao09; @ghi10] On the other hand, the inverse Compton (IC) radiation of the forward shock electrons were expected to give rise to high-energy gamma-ray radiation as well.[@fan_piran_ic_review] However, the signature of IC radiation is missing: the spectrum of the extended emission above 100 MeV is usually characterized by a single power law of photon index $\sim$2 over a broad time ranges.[@lat_grb_cat]
Properties of GRB 130427A
=========================
The duration, $T_\mathrm{90}$, of GRB 130427A is about 138s. It has an energy fluence of 2$\times$10$^{-3}$ erg cm$^{-2}$, putting it as the GRB with the highest fluence in GBM[@gcn_gbm] and Konus-Wind[@gcn_kw] mission lives and the most luminous GRB at z$<$0.5. It also has the highest fluence in the LAT energy range during the prompt phase.[@gcn_lat] Twelve $>$10 GeV photons were detected in the first 700s after the burst onset, including a 95 GeV photon that arrived at 243s after the burst onset, corresponding to an intrinsic photon energy 128 GeV at z$=$0.34, breaking previous records of the highest energy photon from GRBs [^1].
The 100 MeV to 100 GeV emission from GRB 130427A lasts well beyond the prompt emission period until about one day after the GRB onset. This is the longest GeV afterglow emission ever recorded for a GRB. Furthermore, the GeV photons that GRB 130427A emits contain some of the most energetic gamma-rays from GRBs. Even compared to the four brightest LAT GRBs (GRB 080916C, GRB 090510, GRB 090902B, and GRB 090926A) known so far, GRB 130427A is still peculiar in that it radiated over a hundred $>$1 GeV photons and a dozen photons at energies above 10 GeV.[@tam130427a]
Light curve and time-resolved spectra
=====================================
The 100 MeV to 100 GeV light curve derived from likelihood analysis is shown in Figure \[lc\]. Details of the likelihood analyzes can be found in Ref. . One can see that the peak time ($\sim$20 s), peak flux ($\sim$2$\times$10$^{-3}$ photons cm$^{-2}$ s$^{-1}$) and the temporal decay index ($\alpha_\mathrm{L}\sim-$1.2) is not so much different from the other LAT GRBs (see Figure 14 in Ref. ).
Since GRB spectra are expected to change over time, *we performed the first time-resolved spectral analysis of the LAT data from GRB 130427A*.[@tam130427a] To extract general MeV–GeV spectral properties of GRB 130427A during the prompt and afterglow periods, only four time intervals were chosen. The results are shown in Figure \[SED\]. It can be clearly seen that the spectra are not well described by single power laws in some time intervals chosen here and there is a strong evidence for a spectral component below $\sim$2 GeV and another component above $\sim$2 GeV. We also tabulated the arrival times of $>$10 GeV photons.[@tam130427a] Fitting the data from 138 s to 80 ks with a broken power law gave an improvement over a power law at the $\sim$3 significance level. Comparison of spectral fits using single power law and broken power law are shown in Table \[spec\_comparism\].
*It is the first time that an extra hard component above a few GeV is seen from a GRB well after the end of the prompt emission phase and up to one day after the burst.* While it is unlikely that this component is produced by the synchrotron emission, we suggest that inverse-Compton processes provide a natural alternative explanation to the photons at tens of GeV energies that arrived deep in the afterglow period.[@fan130427a]
Very High-energy Gamma-ray Observations
=======================================
GRB 130427A
-----------
The burst occurred when it was day time at the MAGIC and H.E.S.S. telescope sites, thus no observation was possible. It was night time for VERITAS but the full moon has prevented any observation that night.
HAWC observed it with scaler mode while the elevation of the burst was $\sim$33$^\circ$. These observing conditions caused a reduced sensitivity of the instruments and higher energy threshold. No detection was found.[@gcn_hawc]
Future Prospects
----------------
Very high-energy gamma-ray observations of other GRBs have been carried out before but no detection have been found even though some observations started just several tens of seconds after the GRB onset.[@hess_grb; @magic_grb; @veritas_grb] This can be due to several factors: (a) the relatively high redshift of many GRBs, (b) delayed observations, and/or (c) the lack of a very-high energy gamma-ray component.[@xue09]
We have discovered an extra hard spectral component above a few GeV from GRB 130427A that exists from $\sim$100 s up to one day after the GRB onset. This means that the afterglow spectrum of a GRB may extend to the very high-energy (VHE) gamma-ray range, i.e., $>$100 GeV, thus we have shown that the factor (c) alone is not sufficient anymore for GRBs like GRB 130427A. More GRB observations using current or future generations of the Imaging Atmosphere Cherenkov Telescopes (IACTs), such as H.E.S.S. II,[@hess2_grb] MAGIC II, VERITAS, and CTA, not only minutes but hours after the prompt emission phase is thus crucial to study such a hard spectral component in more details, thanks to their much larger effective collection area.
As a final note, we would like to point out that by extrapolating the LAT spectrum at late times, i.e., 3 ks to 80 ks after the burst, to very high-energy gamma-ray range, the GRB flux is still a few times stronger than the Crab nebula flux at such energies and therefore can be easily detected with, e.g., one minute of H.E.S.S. observations.
Notes added in proof {#notes-added-in-proof .unnumbered}
====================
Very recently, the Fermi LAT Collaboration also reported on the temporally extended emission.[@lat_130427a] Using finer time bins between 138 s and 750 s after the burst, they found that the two spectral components reported here actually doimates at different times. At late times (i.e., 3 ks to 80 ks after the burst), emission above a few GeV can be clearly seen in the spectral energy distribution during this period as well (see their Fig. S1).
Acknowledgments {#acknowledgments .unnumbered}
===============
PHT is supported by the National Science Council of the Republic of China (Taiwan) through grant NSC101-2112-M-007-022-MY3.
[00]{} FERMI LAT Collab. (M. Ackermann [*et al*]{}.), [*Astrophys. J. Supp.*]{} [**209**]{}, 11 (2013)
Y.-C. Zou, Y.-Z. Fan and T. Piran, [*Mon. Not. R. Astron. Soc.*]{} [**396**]{}, 1163 (2009).
P. Kumar and R. Barniol Duran, [*Mon. Not. R. Astron. Soc.*]{} [**400**]{}, L75 (2009).
W. H. Gao, J. R. Mao, D. Xu and Y. Z. Fan, [*Astrophys. J. Lett.*]{} [**706**]{}, L33 (2009).
G. Ghisellini, G. Ghirlanda, L. Nava, and A. Celotti, [*Mon. Not. R. Astron. Soc.*]{} [**403**]{}, 926 (2010).
Y. Z. Fan and T. Piran, [*Front. Phys. China*]{} [**3**]{}, 306 (2008).
FERMI GBM Collab. (A. von Kienlin [*et al*]{}.), [*GCN Circ.*]{} 14473 (2013)
KONUS-WIND Collab. (S. Golenetskii, [*et al*]{}.), [*GCN Circ.*]{} 14487 (2013)
FERMI LAT Collab. (S. Zhu [*et al*]{}.), [*GCN Circ.*]{} 14508 (2013)
FERMI LAT Collab. (W. B. Atwood [*et al*]{}.), [*Astrophys. J.*]{} [**774**]{}, 76 (2013).
P. H. T. Tam, Q.-W. Tang, S.-J. Hou, R.-Y. Liu and X.-Y. Wang, [*Astrophys. J. Lett.*]{} [**771**]{}, L13 (2013).
Y.-Z. Fan, P. H. T. Tam, F.-W. Zhang, Y.-F. Liang, H.-N. He, B. Zhou, R.-Z. Yang, Z.-P. Jin and D.-M. Wei, [*Astrophys. J.*]{} [**776**]{}, 95 (2013).
D. Lennarz and I. Taboada, [*GCN Circ.*]{} 14549 (2013)
H.E.S.S. Collab. (A. Aharonian [*et al*]{}.), [*Astron. Astrophys.*]{} [**495**]{}, 505 (2009).
MAGIC Collab. (J. Albert [*et al*]{}.), [*Astrophys. J.*]{} [**667**]{}, 358 (2007).
VERITAS Collab. (V. A. Acciari [*et al*]{}.), [*Astrophys. J.*]{} [**732**]{}, 62 (2011).
R. R. Xue, P. H. Tam, S. J. Wagner, B. Behera, Y. Z. Fan and D. M. Wei, [*Astrophys. J.*]{} [**703**]{}, 60 (2009).
D. Lennarz, P. M. Chadwick, W. Domainko, R. D. Parsons, G. Rowell and P. H. T. Tam (for the H.E.S.S. Collab.) , Searching for TeV emission from GRBs: the status of the H.E.S.S. GRB programme, in [*Proc. 7th Huntsville Gamma-Ray Burst Symposium: GRB 2013*]{}, eConf Proceedings C1304143, 2013, p. 38.
FERMI LAT Collab. (M. Ackermann [*et al*]{}.), [*Science*]{}, in press, preprint\[arXiv:1311.5623\]
[^1]: Shortly before this oral presentation, photons with even higher energy from other GRBs were revealed using a new LAT event reconstruction scheme.[@pass8_grbs]
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abstract: |
We present near-IR photometry of a selected sample of southern hemisphere E+A galaxies. The sample includes 50 galaxies from nearby ($z \sim 0.05$) and distant ($z \sim 0.3$) clusters as well as E+A galaxies from the field ($z \sim 0.1$). We also observed 13 normal early-type galaxies from the field and from clusters to be compared with the E+A sample. The photometry includes $J$, $H$ and $K_s$ apparent magnitudes and colors. Observed colors are obtained from the apparent total magnitudes and compared to spectrophotometric models of galaxy evolution GISSEL96. There is an overall agreement between integrated colors of models and observed ones, for both the E+A located in clusters and in the field, at $z \lesssim 0.1$. However, large differences are found between colors predicted from models and those observed in E+A galaxies located in clusters at $z \sim 0.3$.
We also compute rest-frame colors for all the galaxies using two different sets of K-corrections, and obtain average colors for all the samples.
We investigate systematic properties of the E+A sample as a function of their environment. Results seem to indicate that cluster E+As (at low redshift) are bluer than field E+As at $z \sim 0.1$. Even this conclusion does not depend whether we use comoving or rest-frame colors, nor on the models used to obtain rest-frame colors; the difference is not significant enough, considering color dispersions between the samples. If differences are real, they could imply different stellar content for the E+A galaxies located in the field, compared to the cluster E+A.
author:
- Gaspar Galaz
title: 'E+A galaxies in the near-IR: Broad band photometry'
---
0[$H_0$ = 100 [*h*]{} km s$^{-1}$ Mpc$^{-1}$]{}
Introduction
============
The attention drawn to E+A galaxies (or post-starbursts) has increased since it was claimed that their fraction observed in clusters is correlated with the Butcher-Oemler effect [@butcher78; @dressler83; @rakos95]. The E+A galaxies present a peculiar spectrum in the optical: strong Balmer absorption lines, representative of a large population of A and B stars, but a [*lack*]{} of emission lines typical of blue, star forming galaxies, like \[OII\]$\lambda3727$, \[OIII\]$\lambda$5007, and H$\alpha$. Because of the further detection of metallic absorption lines such as Mg b $\lambda 5175$, Ca H & K $\lambda 3934$, 3968 and Fe $\lambda5270$, indicative of an old population dominated by G, K and M spectral types they were called E+A [@dressler83]. In addition, their spectra in the [*optical*]{} cannot be reproduced simply by a young stellar component without nebular emission lines: it is necessary to add an old population of stars, like the one present in quiescent elliptical galaxies [@liu96].
During the last 3 years, research on the E+A galaxies has seen a revival after the discovery of more such galaxies not only in nearby clusters, like Coma and others (e.g. see @caldwell97 and references therein), but also in the field; in particular those discovered during the Las Campanas Redshift Survey (LCRS, Shectman 1996) by @zabludoff96. An interesting feature is the apparent existence of two classes of E+A galaxies [@couch87]. One is formed by “blue” post-starburst galaxies and the other by redder, H$\delta$-strong (HDS) galaxies [@fabricant91]. These subclasses of E+As have colors and absorption line features related to the morphology: HDS E+As have in general a noticeable bulge and/or spheroidal component when compared with the blue class. In addition, the HDS class can be divided into 2 subclasses: bulge and disk HDSs (as observed in A665 and in Coma [@franx93]).
An almost unexplored domain of these E+A galaxies is their near-IR properties. In fact, no catalogue or systematic observations exist on this subject. Is the bright, red population, dominated mainly by giants and stars of the asymptotic giant branch (AGB) of the E+As, different from the red population of other elliptical galaxies, in particular the perturbed ellipticals? Is there any conspicuous signature in the near-IR colors of the E+A galaxies, as in the optical wavelengths? Do the cluster E+A galaxies have bluer colors than those in the field at these wavelengths? Are the near-IR colors of the E+A galaxies similar to the colors of normal galaxies, as predicted by spectrophotometric models?
In this paper, we investigate these questions with new data taken at Las Campanas Observatory, in Chile. The sample includes E+A galaxies from the field (most of them from the LCRS) and from nearby clusters as well as clusters at $z \sim 0.3$.
The paper is organized as follows. In §2 we present the galaxies selected for this work. In §3 we explain the observations, and in §4 the data reduction procedures. In §5 we present the results of the photometry, the apparent magnitudes, colors, and K-corrections. Both, [*observed*]{} and [*rest-frame*]{} colors are compared with colors obtained from spectrophotometric models of galaxy evolution in §6. In §7 we discuss the limitations of our results and the implications for some properties of the E+A galaxies from our near-IR colors. Conclusions are presented in §8.
The Sample
==========
All of the galaxies selected for this study have been spectroscopically classified as E+A galaxies from the analysis of their Balmer absorption lines (particularly the equivalent widths of H$\delta$ and H$\beta$) and the lack of nebular emission lines, representative of an ongoing stellar formation process. We have selected most of the southern E+A galaxies existent in the literature at present. The sample of galaxies is divided into 4 subsamples. The first subsample corresponds to 21 field E+A galaxies from the LCRS selected from a catalogue of $\sim 19000$ galaxies with redshift between $z \sim 0.07$ and $z \sim 0.18$ [@zabludoff96]. The second subsample contains 22 E+As from the nearby clusters DC2048-52, DC1842-63, DC0329-52, and DC0107-46 [@caldwell97] at $z \sim 0.05$. Seven E+A galaxies from rich clusters at $z \sim 0.31$ (AC103 and AC114) constitute the 3rd subsample [@couch87]. Some “control” galaxies were observed also (the 4th subsample, 13 galaxies). These galaxies have well-known properties and provide a reference sample to compare the observables of the E+A sample. The control sample includes mostly elliptical and lenticular galaxies (from clusters and the field), as well as a few galaxies between $z \sim 0.01$ to $z \sim 0.04$. Table \[samples\] summarizes the sample of 63 galaxies which have been observed.
Observations
============
All of the observations presented here were carried out at Las Campanas Observatory, Chile, during photometric conditions. Most of the images were obtained with the 40-inch Swope telescope, using a NICMOS3 HgCdTe array (256 $\times$ 256 pixels, 0.599 arcsec/pix, $2.5 \times 2.5$ arcmin FOV), in March, July, August and November 1998. We employed also the 100-inch du Pont telescope in September 1998, with a NICMOS3 detector, yielding a scale of 0.42 arcsec/pix ($1.8 \times 1.8$ arcmin FOV). We use the following filters: $J$, $H$, and $K$ short ($K_s$), centered at 1.24$\mu m$, 1.65$\mu m$, and 2.16$\mu m$, respectively, and bandwidths of 0.22$\mu m$, 0.30$\mu m$, and 0.33$\mu m$. For a detailed discussion of the photometric system see @persson98.
Between 5 and 8 standard stars from @persson98 were observed each night. The observation procedure for all the objects (standards included) was as follows. Each object was observed at several positions on the array, some amount of time in each position (I call this an [*observing sequence*]{} for a given object in a given filter). The amount of time depended on the magnitude of the object, on the sky brightness, and on the linearity regime of the array. The NICMOS3 becomes noticeably non-linear when the total counts (sky + object) exceed 17000 ADU. We exposed in each position between 60 and 120 seconds in $J$, between 30 and 60 seconds in $H$, and between 30 and 50 seconds in $K_s$. Total exposure times for a given observing sequence and filter varied between 10 and 45 minutes. For the standard stars (with magnitudes between $K_s \approx 10 - 12$), the exposure time ranged between 5 and 20 seconds in each position, for both telescopes. Typically, the number of non-redundant positions at which each object was observed varied between 4 and 10, depending on the size and/or magnitude of the object. A pre-reduction was made at the beginning of each observing sequence in order to estimate the total exposure time required to reach a minimum central S/N $\approx 12-15$. This pre-reduction consisted in the construction of a sky image averaging all of the stacked images for a given object (with a sigma-clipping rejection threshold), the subtraction of the sky image from each individual image, and the combination after registration of the individual sky-subtracted images.
Data Reduction
==============
The images have to be corrected for all of the instrumental effects, namely, non-linear deviations, dark-current contribution (dark subtraction), and pixel-to-pixel response differences (flat-field division).
For the determination of the linearity corrections, we take several dome-flats with different exposure time. Then, we plot the ratio of the average counts and the integration time of each frame, as a function of the average counts. After normalizing, we transform the count value I$_{in}$ of each pixel into I$_{out}$ = I$_{in}$ \[1 + C$\times$I$_{in}$\], and we solve for the [*constant*]{} C which proved to vary between $1.0 \times 10^{-6}$ and $5.0 \times 10^{-6}$. At 14000 counts, for example, which corresponds to $\sim 75\%$ of the whole range of the signal, the departure from linearity is only 2%.
At the beginning of the night we took series of 20 dark frames, the number of series depending on the number of different exposure times we used through the night. The flat-field images were constructed from (a) dome-flats and twilight sky-flats and (b) the raw science images of the night, which allowed us to construct [*super-flats*]{} from the combination of the individual frames. The useful images for these super-flats were those where the objects were faint and/or a small number of objects were observed. Because galaxy fields were in general uncrowded, it was always possible to construct super-flats. Typically, each super-flat was constructed from no less than 30 science images, for each filter. The results show that the super-flats allow to correct for fringes appearing in the stacked and combined images. The fringes are present in the 3 filters ($J$, $H$ and $K_s$), but are particularly prominent in the H images. Except for the presence of fringes, the dome- and sky-flats are quite similar to the super-flats (at the 0.6% level). However, given the better photon statistics of the super-flats and the fact that the fringes are better represented by the super-flats we chose the super-flats to remove the pixel-to-pixel variations. From the object frames, the dark and flat-fielding corrections were made on the images using the SQIID[^1] reduction package, implemented under IRAF[^2]. At this stage, a mask with the bad pixels was created (using the dark images) to correct those pixels by interpolation with their neighbor pixels in all the images.
Once the linearity corrections were done, as well as the dark correction and flat-fielding procedures in all the raw images, the procedure to obtain stacked and combined sky-subtracted images began. This part was done using DIMSUM[^3], also implemented under IRAF. The general procedure for a given [*observing sequence*]{} follows.
1. A scaling factor for each image was computed using a 5$\sigma$ iterated rejection method about the mean. The scaling factor is the median of the unrejected pixels, and is stored as a descriptor in the image header. This provides a first estimate for the sky level.
2. For each image in the observing sequence, a specified number of neighboring images of the sequence were selected. To construct a sky image, we selected only the neighboring images taken within $\pm 5$ min from the given image. This provides a sample of sky images within a short-period of time during which the variations of the sky level are not larger than 1% to 3% of the mean, in order to reduce the r.m.s. variation in the thermal emission background as well as the OH sky lines, characteristic of the NIR (see Figure \[sky\]). Furthermore, the higher background in $K_s$ produces higher shot noise even if this background did not vary with time. This is a key step. At each pixel a specified number of low and high values in the scaled images were rejected and the average of the remainder values was taken as the sky for that pixel. The resulting sky image was subtracted from the object image to create a sky subtracted object image.
3. Cosmic rays were found using a threshold algorithm applied to the ratio of the image and a median filtered image. The detected cosmic rays were replaced by the local median. A cosmic ray mask was created to record the location of the cosmic rays.
4. For a given observing sequence, a shift list is created to define the offsets between the images. To create this list it is first necessary to have at least one object in common among all of the images of the observing sequence (usually a star). Next we selected a set of additional objects to improve the determination of the relative shifts. These objects were used in constructing the shift list using a centroid-based algorithm.
5. A registration stage was done by shifting and combining the images of the sequence. A matching exposure map was also created, which allows to obtain a final mosaic properly weighted by the effective exposure time of each section of the mosaic.
6. To provide combined images free of “holes” arising from the sky subtraction, two different masks are created for each registered image of the observing sequence. The detailed procedure is explained in the DIMSUM package.
7. The sky subtraction is repeated as in the first pass, before the masking procedure, except that the pixels in the individual masks derived from (f) are ignored.
8. Finally, all the sky-subtracted images of an observing sequence are shifted and combined, and the different parts of the mosaic are scaled to an exposure time of 1 s.
Photometry
==========
Photometric calibrations
------------------------
The photometric calibrations were done using the faint standard stars from the list of @persson98. This list includes standard magnitudes in $J$, $H$, $K$ and $K_s$ for equatorial and southern photometric standard stars. Five to eight standards were observed every night at airmasses similar to those of the galaxies (no larger than 1.2). This minimizes the dimming and reddening due to the airmass contribution, especially in colors involving the $J$ filter.
Instrumental magnitudes were computed using the code SExtractor [@bertin96], which computes isophotal, isophotal corrected, and total magnitudes for all the objects detected above a given threshold. We have also computed instrumental aperture magnitudes using DAOPHOT, and verified that DAOPHOT aperture magnitudes for the standards do not differ by more than $0.5\%$ from the total magnitudes yielded by SExtractor. The aperture used for the standards in DAOPHOT is the maximum aperture after analyzing the shape of the grow curve for the instrumental magnitudes, and typically take radii values $\sim 6.0$ arcsec. We conclude that the instrumental magnitudes given by SExtractor are reliable, which we adopt hereafter.
The adopted photometric transformations between the instrumental and the calibrated magnitudes are: $$\begin{aligned}
\label{phot}
J & = & A_1 + j - 0.10 X \\ %- B_1 (j_s - k_s) \\
H & = & A_2 + h - 0.04 X \\ %- B_2 (h - k_s) \\
K_s & = & A_3 + k_s - 0.08 X %- B_3 (j_s - k_s),
%\addtocounter{equation}{1}\end{aligned}$$ where the $A_N$ coefficients ($N$ = 1, 2, 3) are the zero points, $X$ is the airmass, and the extinction coefficients are from @persson98. Note that we do not try to solve for airmass corrections night by night, as this can lead to spurious values for coefficients if the extinction is variable and non-gray. The latter is relevant at the filter passband edges where water vapor influences the effective width of the passband [@persson98].
The zero points $A_1$, $A_2$ and $A_3$ were determined on a nightly basis, and proved to vary between $1\%$ and $7\%$. We do not include color terms in these transformations (equations 1, 2, and 3) since they are smaller than 0.04 mag, a value close to the observational magnitude errors. We emphasize that all of the standards and the galaxies reported in this paper were observed during completely photometric nights. The photometric transformations have typical r.m.s. residuals of $\sim 0.02 - 0.05$ mag on both telescopes (see Figure \[stds\_error\]). This gives an internal error in the photometric calibrations around 2% to 5%. The main source of error are, in fact, the short-term sky fluctuations (in particular in $K_s$), which are of the order of 3% to 6% in time intervals spanning the longest exposure time of individual frames during each of the observing sequences (120 s in $J$, 60 s in $H$, and 50 s in $K_s$). Some galaxies were observed twice, even using the two telescopes, and therefore there is a good estimate of the global photometric errors, which proves to be around 7% for photometric nights. The observation of the same object during two photometric nights but with different telescope/instrument is the best way to estimate the photometric quality of the data (see §5.2), and the real dispersion of magnitudes.
Galaxy photometry
-----------------
Instrumental apparent magnitudes for all the galaxies were obtained on the registered and combined images using the code SExtractor [@bertin96]. Given the differences in size, shape and luminosity of the galaxies, the total magnitude is a better estimator than the aperture or isophotal magnitudes. SExtractor computes aperture magnitudes, isophotal magnitudes and “total” magnitudes for all of the objects detected above a given threshold. The total apparent instrumental magnitude for a given object is given by one of the two following approaches. (1) It is computed using an [*adaptive*]{} aperture magnitude or (2), using a [*corrected*]{} isophotal magnitude. In order to give the best estimate of the total magnitude, the adaptive aperture method is performed, except if a neighbor is suspected to bias the magnitude by more than 0.1 mag. If this happens, the corrected isophotal magnitude is taken as the total magnitude. This leads to the so-called MAG\_BEST magnitude, in the SExtractor output catalogue.
In order to check the calibrated magnitudes for our galaxies, and to have an idea of the accuracy of our total magnitudes, we observed some of the galaxies on two different nights, with the 40-inch and the 100-inch telescopes. For the 8 galaxies which were observed twice (2 for each subsample), we found r.m.s differences $\Delta J = 0.037$, $\Delta H = 0.042$ and $\Delta K_s = 0.061$. These differences, although large when taken at face value, represent the most realistic errors in the total magnitudes, since they were obtained with different instrumental set-ups during different observing runs.
Apparent magnitudes and K-corrections
-------------------------------------
Once the instrumental magnitudes are calculated using SExtractor, they are transformed to the standard system using the package PHOTCAL in IRAF. Table \[all\_calib\] shows the apparent calibrated total magnitudes for all of the galaxies of the 4 subsamples. The $K_s$ magnitude for galaxy \# 25, AC114\_89, was not computed since it was observed under possibly non-photometric conditions, and was marked with a NC (not calibrated). No internal reddening correction was applied to these magnitudes, nor a Galactic foreground extinction correction: both corrections are smaller than the photometric error and, in particular, are smaller than the uncertainty given by the K-correction, as we show in §5.6. The reddening by dust is $\Delta(J - H) \lesssim 0.03$ and $\Delta(H - K_s) \lesssim 0.02$, if we consider a simple screen model based on the reddening law of @cardelli89. If we consider a more complicated extinction model, following the star-dust mixture recipe by @wise96, the amount of reddening is similar. The correction due to Galactic extinction is also small for the 3 passbands ($\lesssim 0.03$), which proves to be well within the photometric uncertainties (for the Galactic reddening corrections in the near-IR photometric bands see @schlegel98). We emphasize, however, that Galactic and internal reddening corrections are systematic effects, while the photometric uncertainty is random. Given their small values, no attempt is made to correct for the extinctions. If we include foreground and internal extinction, the $J - H$ color redden probably no more than 0.03 – 0.05 (but see discussion at the end of §6).
Since the K-terms can significantly modify the intrinsic colors of the galaxies, they are critical in correcting the observed colors and magnitudes to the galaxy rest-frame. If $m_1$ and $m_2$ are the apparent magnitudes in the passbands 1 and 2, respectively, for a galaxy at a redshift $z$ and with a known spectral type $T$, then the rest-frame color for this galaxy is $$M_1 - M_2 = m_1 - m_2 - \left\{K_1(z,T) - K_2(z,T)\right\},
\label{rest_col}$$ where $M_1$ and $M_2$ are the corresponding absolute magnitudes, $K_1(z,T)$ and $K_2(z,T)$ are the K-corrections for the passband 1 and 2, respectively, for the galaxy with spectral type $T$ at redshift $z$. K-corrections are not included in the magnitudes and colors presented in Table \[all\_calib\] since they can have a wide range of values depending on the spectral energy distribution (SED) employed in their computation. When the SED is not available for a given object, it is common practice (although risky) to adopt K-terms from the correlation between spectral type and morphological classification, provided the latter is available.
In our case, we only have approximate morphological types for the E+As from the literature, nor do we have spectral information for the galaxies in the near-infrared part of the spectra. Even though most of our E+A galaxies are early types, we can ask the following. How will the near-IR K-corrections depend on the spectrophotometric model of galaxy evolution used to compute them? In order to study the model dependence in $J$, $H$ and $K$, take for example 2 SED models which provide (or allow us to compute) K-terms in the near-IR. The first K-terms were taken directly from @poggianti97, who computed K-corrections from the near-UV to the infrared. @poggianti97 provides K-corrections in several bands in the Johnson-Bessel & Brett photometric system, up to $z = 3$ as a function of morphological type. The values are computed according to an evolutionary synthesis model that reproduces the integrated galaxy spectrum in the range 1000-25000 Å, and uses the code of GISSEL93 [@bruzual93]. The models are instantaneous bursts with solar metallicity and Scalo IMF [@scalo86]. The age after the burst gives the SED which is compared with galaxies of known morphological type through colors. Note that the comparison is done in the optical part of the spectrum, mostly between 3000 and 8000 Å. The second set of K-corrections were derived using the model PEGASE[^4] [@fioc97] to generate synthetic spectra between 7000 Å and 30000 Å, and convolving these SEDs with the filter response functions (see Persson 1998), using the definition of the K-correction [@oke68]. Figure \[K\_poggianti\] shows the K-corrections in $J$, $H$ and $K$, calculated by @poggianti97 for 3 different morphological types, namely, E (solid line), Sa (dotted line) and Sc (dashed line). Note that the K-corrections in the near-IR are not necessarily small. Nevertheless, in most of the photometric bands they do not depend strongly on the spectral type or the morphological type. K-corrections are large (and negative) for the $K$ band, for $z \lesssim 0.5$, for all galaxy types (this makes galaxies to appear brighter than they really are). The average redshift in our sample is $\approx$ 0.08, and K-corrections in all the bands are less than 0.1 mag for most of the cases. The exceptions are the E+A galaxies in AC103 and AC104 (at $z \sim 0.3$). For these objects K-corrections can be larger and around $-0.3$ mag in $K$ for the late type galaxies. Figure \[K\_pegase\] shows the K-term calculated from PEGASE. These K-corrections are calculated from SEDs with solar metallicity and also instantaneous bursts. The IMF is from @scalo86. In PEGASE, the authors define their morphological types by directly comparing spectra generated from their models with @kennicutt92 optical spectra of nearby galaxies. @poggianti97, on the other hand, matches colors obtained from her model with observed colors of galaxies, taken from @persson79 and @bower92a [@bower92b].
Comparing Figure \[K\_poggianti\] and \[K\_pegase\], we conclude that although the K-corrections are quite different from one model to another, they are similar for $z \lesssim 0.1$. For $z \gtrsim 0.2$, differences are larger. K-corrections for different Hubble types are more similar if they are derived from PEGASE than from the models of Poggianti. Figure \[K\_diff\] shows the differences between these K-corrections for the two models, in the 3 passbands, and for the 3 Hubble types. Up to $z \sim 0.5$ the difference for the E type in $J$ and $H$ is less than 0.05 mag. However, the difference is $\sim
0.1$ mag for the later types. Equation (\[rest\_col\]) implies that the differences in $J - H$ will be less than 0.05 mag. However, this is not the case for colors involving the $K$ band ($J - K$ and $H - K$), due to the large difference in the K-corrections, for all the Hubble types, as shown also in Figure \[K\_diff\]. The difference in $K$ for the K-corrections reaches values $\sim 0.4$ mag at $z \sim 0.3$. This shows that K-correction uncertainty will have the largest impact on rest-frame colors. Other studies also show large differences between their K-corrections, although some of them are comparable to the values of this work, showing also large negative K-corrections in $K$ [@frogel78; @persson79; @bershady95]. For example, @bershady95 obtains type-averaged K-corrections in $K$, reaching $-0.33$ and $-0.60$ at $z = 0.14$ and $z = 0.30$, respectively. These values are larger than values from @poggianti97, but similar to those obtained from PEGASE (see Figure \[K\_poggianti\] and \[K\_pegase\]).
Comparison with models
======================
As shown in the preceding section, K-corrections in the near-IR can be very different depending on the spectrophotometric models used. Therefore, we do not use rest-frame colors, i.e. we do not de-redshift the data. Instead, we [*redshift*]{} current epoch SEDs. Although this approach is similar to work with rest-frame colors, it is more robust, since the SEDs of the current epoch models can be determined absolutely. In order to give an idea whether synthetic SEDs compare well with spectra of galaxies at the current epoch, we consider GISSEL96 models [@charlot96] and compare them with real, local galaxy spectra of known morphological types, given by @kennicutt92. As it is well-known, the age-metallicity degeneracy prevents us for deriving age and metallicity directly from colors, as was shown by @worthey94 [@ferreras98]. Therefore, we consider instantaneous bursts of fixed (solar) metallicity. Subsequent evolution is determined by adopting passive stellar evolution, measured in Gyrs and indicated by the label “age” for each model spectrum in Figure \[spectra\]. A simple $\chi^2$ test is used to determine the model spectra closest to the observed (Kennicutt) sample. We use a starting sample of 20 GISSEL96 spectra and 27 spectra representative of normal galaxies of known Hubble types [@galaz98]. Figure \[spectra\] shows the better spectral match between some Kennicutt spectra and the 20 selected GISSEL96 models.
The Hubble sequence fits well with an evolutionary sequence in the optical, but care has to be taken in the interpretation since more than one solution can be obtained from a synthetic set where both age and metallicity vary [@ronen99]. Even though metallicity can vary from one galaxy to another, it is realistic to set metallicity close to solar. Extremely metal-poor (Z $\lesssim 0.5$ Z$_\odot$) or metal-rich (Z $\gtrsim 1.5$ Z$_\odot$) cases are unlikely in this set of galaxies [@liu96]. Moreover, the fact that colors are obtained from [*integrated*]{} total apparent magnitudes, imply that colors are an average over the whole galaxy light and therefore likely to be representative of solar metallicity or lower in the luminosity weighted mean (see for example @edmunds92).
In order to compare the observed colors with models, we take the 20 spectra from GISSEL96 and we “redshift” them to several redshift values (from the rest-frame to $z = 0.5$). Afterwards, we compute synthetic colors using $J$, $H$ and $K_s$ passbands [@persson98] for the 20 synthetic spectra. Figure \[obs\_colors\] shows the color-color diagram for the E+A sample and the control sample (indicated as filled circles), as well as for the model spectra (indicated as open symbols) situated at different redshifts (as indicated by labels). We include 3 different evolving tracks in figure \[obs\_colors\], for instantaneous bursts after 1 Gyr, 3 Gyr and 16 Gyr indicated by circles, squares, and triangles, respectively. After 10 Gyr, the near-IR colors are almost independent of age, for a given redshift.
Figure \[obs\_colors\] shows that there is an overall agreement between near-IR colors of all subsamples and models, except for subsample 3. The average color $<H - K_s> = 0.41$ $(\sigma = 0.05)$ of subsample 1 (average redshift $<z> = 0.09$, $\sigma = 0.02$) agrees well with any model older than 3 Gyr at $z = 0.10$. However, the average $<J - H> = 0.66$ $(\sigma = 0.06)$ appears bluer than the same models by $\sim 0.1$ mag. Otherwise, $<J - H>$ is well fitted by a model with age $\lesssim 3$ Gyr at $z = 0.10$, but then $<H - K_s>$ of subsample 1 is redder by $\sim 0.1$ mag. These differences are twice the color dispersion for this subsample. Therefore we can conclude that colors of the models and the data do not differ by more than $2\sigma$. In subsample 2, the average colors $<H - K_s> = 0.29$ $(\sigma = 0.07)$ and $<J - H> = 0.69$ $(\sigma = 0.05)$, with average redshift $<z> = 0.046$ $(\sigma = 0.014)$, are well fitted by a model at $z = 0.05$ and 2.8 Gyr. Subsample 3, having $<H - K_s> = 0.61$ $(\sigma = 0.23)$, $<J - H> = 0.75$ $(\sigma = 0.25)$, and average redshift $<z> = 0.31$ $(\sigma = 0.01)$ is not fitted by the GISSEL96 models, even though the average color $<H - K_s>$ is closer to the $z = 0.3$ redshifted color of models. Subsample 4 (the control sample) matches the models colors well, despite the rather large scatter. This subsample has average colors $<H - K_s> = 0.29$ $\sigma = 0.08)$, $<J - H> = 0.74 (\sigma = 0.06)$, and average redshift $<z> = 0.030$ $(\sigma = 0.012)$. These average colors correspond to a model located at $z = 0.05$ and age 3 Gyr. This subsample shows a larger scatter in the color-color diagram. Most of these galaxies are nearby galaxies (from the PGC and NGC catalogues) and some galaxies from DC clusters [@caldwell97]. All have secure Hubble types, and most of them have known photometric properties in the optical (for $B$ and $R$ total magnitudes see Table \[all\_calib\]). The majority of these galaxies are well matched by the colors provided by the spectrophotometric models, for ages representative of early type galaxies. These galaxies have large apparent radii, and therefore, their photometry is more sensitive to color gradients. This is not a problem for more distant galaxies because of the poorer spatial resolution.
Now we compare color properties of subsample 1 (field E+As from the LCRS) and 2 (cluster E+As). From Figure \[obs\_colors\] it is apparent that subsample 1 has the same average $<J - H>$ (with a difference of 0.03), but a redder $<H - K_s>$ than subsample 2 (see above). The difference of 0.12 mag is $2.4\sigma$ and $\sim 1.7\sigma$ away from the intrinsic dispersion of subsample 1 and subsample 2, respectively. The expected color difference due to K-corrections between $<z> = 0.09$ (subsample 1) and $<z> = 0.046$ (subsample 2) is $\sim 0.06$ mag, for a 2 Gyr model (half of the 0.12 color difference between the two subsamples), which fits the average colors of both subsamples 1 and 2 better. Therefore, we can only conclude with a $\sim 1.5\sigma$ confidence level that E+A galaxies from the field are redder than cluster E+As. The fact that dust extinction is much more notorious in $J - H$ than in $H - K_s$ suggest that the color difference observed in $H - K_s$ between subsample 1 and subsample 2 is not due to differential internal dust extinction. However, because of the observed color dispersion, we cannot give a robust answer supporting stellar population differences instead of internal reddening differences due to extinction. We stress that our differences are only at $1.5\sigma$ significance level. It is worth noting that $J - H$ color would [*redden systematically*]{} $\sim 0.03 - 0.05$ if we account for foreground or internal extinction (see §5.3). This would make ages inferred from colors (see Figure \[obs\_colors\]) slightly older (0.5 to 1 Gyr), but in any case alter the results of the analysis, since changes are the same for all the galaxy samples.
Further analysis and discussion
===============================
Photometry uncertainties
------------------------
In order to interpret correctly the color properties of the observed galaxy sample, it is important to keep in mind the sources of uncertainty which affect the colors. The first source of uncertainty is of course the data acquisition itself. Given the nature of the near-IR imaging, the thermal variation of the sky affects the photometry for the faint objects, which require longer integration time than the brighter ones, sometimes much longer than the typical time of the sky fluctuations (see Figure \[sky\]). However, the nature of these variations is well understood and the fact that the sky fluctuations are sampled in [*real-time*]{} and subtracted for each image can reduce this error to 5% (see §3). The second important source of errors is the procedure employed to compute the magnitude. It is well known that total magnitudes depend on the cut level where the light contribution is null or not significant. In our case, SExtractor computes total magnitudes integrating all the light up to a given threshold above the sky (typically 1.5$\sigma$), and fitting elliptical isophotes to the profiles. An elliptical aperture for a given galaxy, defined by the elongation $\epsilon$ and position angle $\theta$, is computed from the 2nd order moment in the light distribution, above the isophotal threshold. The “first moment” $r_1$ is then computed[^5] within an aperture twice as large as the isophotal aperture, in order to reach the light distribution in the wings. This approach is very similar to the approach of @kron80. The parameter $r_1$ is then used to define the adaptive aperture where the total magnitude will be computed. The main axes of the ellipse are defined as $\epsilon k r_1$ and $k r_1 / \epsilon$, where $k$ is a value to be fixed by the user. We carried out some tests with both faint and bright galaxies and found that the value $k = 2.5$ allows us to include between $90\%$ and $95\%$ of the total flux without introducing additional noise within the aperture. Further details can be found in @arnouts96. This procedure ensures that not more than 5% of the light is lost. Another source of uncertainty is the photometric errors due to the transformation of the instrumental magnitudes to calibrated magnitudes. This process is well understood and in general the scatter is small. The errors of the zero points are $\sim 2\%$ to $\sim 7\%$.
The largest uncertainties (now for rest-frame colors) are due to the K-corrections. These uncertainties, as shown in the previous section, can be very large for galaxies with $z \gtrsim 0.25$, where the change in magnitude produced by the computation of K-corrections assuming one or another SED can reach differences as large as 30%, propagating these differences to the rest-frame colors (see Figure \[K\_diff\]). For galaxies with $z \lesssim 0.2$, differences are smaller: $\sim 10\%$ for $0.15 \lesssim z \lesssim 0.2$ and $\sim 5\%$ for $0 \le z \lesssim 0.15$. In order to compute reliable K-corrections, it is fundamental to obtain calibrated spectra at 9000 Å$\lesssim \lambda
\lesssim$ 25000 Å for different spectral types, including E+A galaxies. Of course, the nature of the uncertainties lies in the fact that K-corrections are expressed in term of the morphological type instead of the spectral type. The morphological type relies on a subjective classification procedure, often dependent on the passband through which the images are obtained (more or less sensitive to the star population which delineates the galaxy morphology) and is always strongly dependent on the image quality. On the other hand, there is no unique and reliable relationship between the spectral type and the morphological type of the galaxies. Even though this is approximately true for normal Hubble types [@folkes96; @galaz98], the dispersion can be large for some spectral types or active galaxies [@sodre99], leading to large uncertainties in the K-correction = $f(z$, T-type). However, we note that independently of what spectrophotometric models are used in obtaining rest-frame colors, the K-corrections in $K$ (or $K_s$ band), are only [*weakly*]{} dependent on the spectral type for $z \lesssim 0.2$ (see Figures \[K\_poggianti\], \[K\_pegase\], and \[K\_diff\]).
Implications from near-IR colors
--------------------------------
We now examine some color properties of the E+A galaxies observed in the near-IR, keeping in mind the limitations of the accuracy of our photometry, as discussed above.
Studying the position of the E+A galaxies in the $H - K_s$/$J - H$ plane shown in Figure \[obs\_colors\], we see that [*field*]{} galaxies located at $<z> \sim 0.09$ (subsample 1), have an average $J - H$ color similar to that of E+A galaxies located in nearby clusters ($<z> \sim 0.05$, subsample 2), but are slightly [*redder*]{} in the average $H - K_s$ color (see preceding section). The fact that the color difference of 0.12 mag in only significant at $\sim 1.5\sigma$ level prevents us from proposing a robust conclusion. However, we can now ask how the K-corrections can change this result. Here we examine the answer to this question using the two sets of K-corrections show in §5.3 the PEGASE and the Poggianti K-corrections.
Figure \[color\_average\] shows average rest-frame colors for our sample of galaxies computed using both sets of K-corrections. Also shown are the colors of the sample of elliptical galaxies from [@silva98]. We show the average colors for the [*cluster*]{} and the [*field*]{} galaxies separately. This Figure demonstrates that, although the color differences are small, the same trend is observed, independently of which set of K-corrections is used. The color difference in $<H - K_s>$ between field and cluster E+As is about 0.04 mag using PEGASE K-corrections and 0.15 mag using Poggianti K-corrections. Note that in Figure \[color\_average\] we compare cluster-field colors also for the LCRS sample (3 LCRS E+As belong to clusters). The field E+As from LCRS are also redder in $<H - K_s>$ than the LCRS cluster E+As.
As demonstrated by @persson83, stellar populations containing a large fraction of AGB stars (1 to 3 Gyr old), have redder $H - K$ color (but similar $J - H$ index), compared with populations that lack such stars. This might suggest that the E+A galaxies in the field have a larger fractions of AGB stars than those in clusters. We emphasize that, although the difference given by the K-correction between $z \sim 0.1$ and $z \sim 0.05$ for subsamples 1 and 2, respectively, does change the corresponding average colors, the observed color trend field/cluster does not change. Note that three out 21 LCRS E+As are embedded in clusters (LCRS \# 4, 11 and 20, These galaxies have an average color $<H - K_s> = 0.160 \pm 0.041$, using the PEGASE K-corrections, and $<H - K_s> = 0.260 \pm 0.005$, using the Poggianti K-corrections. These values are $35\%$ and $22\%$ bluer, respectively, than the average $H - K_s$ color for the LCRS E+A galaxies located in the field, and are consistent with the comparison field/cluster between subsamples 1 and 2.
Galaxies in more distant clusters (subsample 3), appear redder in $J - H$ (at 2$\sigma$ significance level) than those at lower redshift (compared with both subsamples 1 and 2). As discussed above, although for this subsample K-corrections are critical, the $J - H$ color does not change if one uses a different set of K-corrections. This could be interpreted as a temperature change of the first-ascent giant branch (FAGB) in the stellar populations of these $z \sim 0.3$ E+A galaxies (see @charlot96). A further spectroscopic analysis in the near-IR would settle this question, and also will help to disentangle possible significant extinction in the $J$ band.
One can also compare the integrated rest-frame colors between the E+A galaxies from subsample 2 with the control galaxies which [*also*]{} belong to these nearby clusters (e.g. galaxies \# 52, 53, 59, 60 and 61 in Table \[all\_calib\]. We note that the average $H - K_s$ color for both sets of galaxies is similar, and therefore any difference (in the mean) is observed between E+A galaxies and elliptical galaxies belonging to the same cluster. This is not the case if one compares the colors between subsamples 1 and 2, as shown before. We emphasize that the average $J - H$ color of the E+A galaxies of subsamples 1 and 2 is similar to the average $J - H$ of the control sample (at $1\sigma$ significance level, see Figure \[obs\_colors\]). It is worth noting that all the K-corrections used to obtain average rest-frame colors, as shown in Figure \[color\_average\], have been computed using solar metallicity models, assuming that K-corrections, for a given IMF, age and SFR scenario do not depend strongly on metallicity. We tested this assumption using GISSEL96 SEDs with different metallicity. Several tests were carried out for different ages, IMFs, and SFRs, and metallicity between the extreme values of \[Fe/H\] $=-1.65$ and \[Fe/H\] $= +1.00$. Differences in K-corrections between these two extreme metal-poor and metal-rich models can reach up to 0.3 mag in $J$ at $z = 0.3$, for a large range of fundamental parameters (age, IMF, and SFR). For more modest metallicity differences between models (probably more realistic), variations in K-corrections, for the different near-IR photometric bands, are between 0.15 mag for $J$ and 0.05 mag for $H$ and $K_s$, at $z = 0.3$. For smaller redshifts, these differences are even smaller. Figure \[diff\_metal\] shows K-correction differences in near-IR bands, as a function of redshift, for two SEDs (shown in the inset) with different metallicity (\[Fe/H\] $= -0.30$ and \[Fe/H\] $= +0.1$), derived from instantaneous bursts with the same age and IMF (in both cases Scalo IMF). These K-corrections differences imply $J - H$ and $H - K_s$ colors shifts no larger than 0.08 mag up to $z \sim 0.3$, given that differences in K-corrections, due to different metallicity have the same sign. We conclude that for a typical interval of metallicity found in the field and in clusters, the effect of varying metallicity should not be significant on the K-correction uncertainties, and hence on rest-frame colors. However, for more accurate estimates of near-IR colors from broad band photometry, especially at higher redshift ($z \gtrsim 0.5$), metallicity does play a significant role on the K-corrections.
Summary and conclusions
=======================
The E+A galaxies reported here include 32 galaxies from clusters and 18 galaxies from the field. In addition, 13 nearby galaxies which do not present post-starburst activity, were observed (5 located in clusters at $z \sim 0.05$ and 8 located in the field at very low redshift). All the galaxies have been observed in the near-IR bands $J$, $H$ and $K_s$ during photometric nights at Las Campanas Observatory. Total apparent magnitudes and colors were derived. The color-color diagram $H - K_s$/$J - H$ of the observed galaxies is compared to the expected corresponding colors of spectrophotometric models of galaxy evolution, at different redshifts. The models are those generated by GISSEL96 [@charlot96]. There is an overall agreement between these expected colors and the observed ones, for the E+A located in nearby clusters ($<z> \sim 0.05$) and for E+As located in the field ($<z> \sim 0.1$). The comparison of the colors of these two samples shows that even though cluster E+As appear bluer than field E+As, the color difference is only significant at $\sim 1.5\sigma$ level, and therefore we cannot strongly affirm that stellar population differences are observed between these two populations.
The colors of the E+A galaxies located in more distant clusters $z = 0.3$, on the other hand, do not agree with the color expected from models. In the mean, they appear bluer than expected in $H - K_s$ (by $\sim 0.3$mag) and redder in $J - H$ (by $\sim 0.15$ mag). The possible interpretation of the failure is strong internal reddening (mostly in the $J$ band), not considered in models.
In order to derive a more complete comparison with models, rest-frame colors were also obtained using two different sources of K-corrections: one based on the work of @poggianti97, and the other computed using the spectrophotometric model PEGASE [@fioc97]. We have shown that such K-corrections can be significant for $z \sim 0.2$ in the $K$ bands (or any band centered at 2$\mu m$), although they are not a strong function of spectral type. In addition, large differences exist in the K-corrections between these two models, having a large impact on the derived quantities, like rest-frame colors for high redshift galaxies. We have compared average rest-frame colors of E+A galaxies located in the field and in clusters. Results show that average rest-frame near-IR colors of E+A galaxies located in clusters at $z \sim 0.05$ [@caldwell97] and field E+As located at $z \sim 0.1$ (from the LCRS, Zabludoff 1996), follow the same color trend in $J - H$ and $H - K_s$ observed in the comoving color-color diagram: E+A galaxies located in nearby clusters appear bluer than field E+As ($z \sim 0.1$).
As well as comparing the observed colors with the GISSEL96 colors at different redshifts, the models do not fit the rest-frame colors of the E+A galaxies observed in clusters at $z \sim 0.3$. Their $H -K_s$ colors appear bluer (using the K-corrections of PEGASE), or redder (using the K-corrections of Poggianti) compared to the models. Their $J - H$ color index, although not particularly sensitive to one or the other K-correction, is also redder than the colors predicted by the models. The color of control galaxies, most of them ellipticals at $z \lesssim 0.01$, and the others from clusters at $z \sim 0.05$, agree with the near-IR colors predicted by models.
Integrated colors between the [*field*]{} E+As and the [*cluster*]{} E+As of the LCRS (LCRS \# 4, 11 and 20; see Table \[all\_calib\]) are similar, although those in clusters seem to be slightly ($\sim 25\%$) bluer in $H - K_s$ than the average color. This result is the same for both sets of rest-frame colors, the set corrected by the PEGASE K-corrections and the set corrected by the Poggianti K-corrections (see Figure \[color\_average\]). On the other hand, the corresponding $J - H$ color is similar for the cluster and field E+As in subsample 1.
In order to build more robust results, more field and cluster E+A galaxies have to be observed between $z = 0.1 - 1.0$. Spectroscopic observations of normal and E+A galaxies at different redshifts in the near-IR are necessary to (1) obtain calibrated SEDs and realistic K-corrections, and (2) to compare the spectra of the E+A galaxies with those of normal galaxies in the whole spectral range 3500 Å $\lesssim \lambda \lesssim 25000$ Å. We expect to continue this research by imaging new E+A galaxies in the near-IR at higher redshift, as well as obtaining near-IR spectra in order to construct a useful and larger database of normal and post-starburst galaxies in a large spectral range. In order to increase the number of E+A galaxies, some field galaxies already classified as E+As, are being observed in the near-IR at Las Campanas. Some of these galaxies belong to the ESO-Sculptor Survey [@delapparent97], and results will be published soon. Other wide-field surveys will provide a wealth of data for E+A galaxies at $0.01 \lesssim z \lesssim 0.2$, like SLOAN [@loveday98; @fan98], and the 2DF survey [@colless98], whose data are expected to become available to the public. In a forthcoming paper, we shall investigate systematic properties on the surface photometry and colors of the E+A galaxies.
I would like to thank the anonymous referee for useful comments and suggestions on how to improve this paper. I thank Ron Marzke, Eric Persson, and Ann Zabludoff for fruitful discussions on the nature of the E+A galaxies. I acknowledge Mauro Giavalisco for his help with the DIMSUM software. I also acknowledge Mario Hamuy, René Méndez, Mark Phillips, Miguel Roth, and Bill Kunkel in helping to improve the preliminary version of this paper. It is a pleasure to thank all the staff at Las Campanas Observatory. This research has made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This work is made possible through the fellowship \# C-12927, under agreement between Fundación Andes and Carnegie Institution of Washington.
[99]{} Arnouts, S. 1996, PhD thesis, Université Paris VII. Bershady, M. 1995, , 109, 87 Bertin, E., & Arnouts, S. 1996, , 117, 393 Bower, R., Lucey, J., & Ellis, R. 1992a, , 254, 589 Bower, R., Lucey, J., & Ellis, R. 1992b, , 254, 601 Bruzual, G., & Charlot, S. 1993, , 405, 538 (GISSEL93). Butcher, H., & Oemler, A. 1978, , 219, 18 Caldwell, N., & Rose, J. 1997, , 113, 492 Cardelli, J., Clayton, G., & Mathis, J. 1989, , 345, 245 Charlot, S., Worthey, G., & Bressan, A. 1996, , 457, 625 (GISSEL96) Colless, M. 1998, in Wide Field Surveys in Cosmology, 14th IAP meeting, Paris. Publisher: Editions Frontières. Couch, W., & Sharples, R. 1987, , 229, 423 de Lapparent, V., Galaz, G., Arnouts, S., Bardelli, S., & Ramella, M. 1997, The Messenger, 89, 21 de Vaucouleurs, G., de Vaucouleurs, A., Corwin, H., Buta, R., Paturel, G., & Fouqué, P. 1991, [*Third Reference Catalogue of Bright Galaxies*]{}, Springer-Verlag, Berlin, Heidelberg, New York (RC3) de Vaucouleurs, G., de Vaucouleurs, A., & Corwin, H. 1976, [*Second Reference Catalogue of Bright Galaxies (Austin: University of Texas Press)*]{} Dressler, A., & Gunn, J. 1983, , 270, 7 Edmunds, M. G. 1992, in The Stellar Populations of Galaxies, ed. B. Barbuy & A. Renzini (Dordrecht: Kluwer Academic), 277 Fabricant, D., McClintock, J., & Bautz, M. 1991, , 381, 33 Fairall, A., Willmer, C., Calderón, J., Latham, D., da Costa, L., Pellegrini, P., Nunes, M., Focardi, P., & Vettolani, G. 1992, , 103, 11 Fan, X., Strauss, M., Gunn, J., Hennessy, G., Ivezic, Z., Knapp, G., Lupton, R., Newberg, H., & Schneider, D. 1998, American Astronomical Society Meeting no. 193, 02.05 Ferreras, I., Charlot, S., & Silk, J. 1998, , in press Fioc, M., & Rocca-Volmerange, B. 1997, , 326, 950 (PEGASE) Folkes, S., Lahav, O., & Maddox, S. 1996, , 283, 651 Franx, M. 1993, , 407, L5 Galaz, G., & de Lapparent, V. 1998, , 332, 459 Frogel, J., Persson, E., Matthews, K., & Aaronson, M. 1978, , 220, 75 Kennicutt R. 1992, , 79, 255 Kron, R. 1980, , 43, 305 Liu, C., & Green, R. 1996, , 468, L63 Loveday, J., & Pier, J. 1998, in Wide Field Surveys in Cosmology, 14th IAP meeting, Paris. Publisher: Editions Frontières. Loveday, J. 1996, , 278, 1025 Oke, J., & Sandage, A. 1968, , 154, 21 Persson, E., Murphy, D., Krzeminski, W., Roth, M., & Rieke, M. 1998, , 116, 2475 Persson, E., Cohen, J., Matthews, K., Frogel, J., & Aaronson, M. 1983, , 266, 105 Persson, E., Frogel, J., & Aaronson, M. 1979, , 39, 61 Poggianti, B. 1997, , 122, 399 Prugniel, P., & Hérédeau, P. 1998, , 128, 299 Rakos, K., & Schombert, J. 1995, , 439, 47 Ronen, S., Aragon-Salamanca, A., & Lahav, O. 1999, , 303, 284 Scalo, J. 1986, , 11, 1 Schelegel, D., Finkbeiner, P. & Davis, M. 1998, , 500, 525 Shectman, S., Landy, S., Oemler, A., Tucker, D., Lin, H., Kirshner, R., & Schechter, P. 1996, , 470, 172 Silva, D., & Bothun, G. 1998, , 116, 85 Sodre, L., & Stasinska, G. 1999, astro-ph/9903130 Spellman, K., Madore, B., & Helou, G. 1989, , 101, 360 Wise, M., & Silva, D. 1996, , 461, 155 Worthey, G. 1994, , 95, 107 Zabludoff, A., Zaritsky, D., Lin, H., Tucker, D., Hashimoto, Y., Shectman, S., Oemler, A., & Kirshner, R. 1996, , 466, 104
[lllccclcc]{}
1 & 1 & g515 & 15:24:26 & +08:09:06 & 0.0870 & Abell 665 & 0 & (1)\
2 & 1 & dc204852\_26 & 20:49:52 & $-$53:02:58 & 0.0397 & ACO 3716 & $-$2 & (2)\
3 & 1 & dc184263\_39m & 18:42:49 & $-$63:12:28 & 0.0144 & DC1842-63 & $-$3 & (2)\
4 & 1 & dc204852\_100 & 20:51:49 & $-$52:44:45 & 0.0493 & ACO 3716 & $-$2 & (2)\
5 & 1 & dc204852\_148 & 20:49:13 & $-$52:33:51 & 0.0429 & ACO 3716 & $-$2 & (2)\
6 & 1 & dc204852\_39 & 20:50:01 & $-$52:59:56 & 0.0489 & ACO 3716 & $-$2 & (2)\
7 & 1 & dc204852\_45 & 20:52:10 & $-$52:56:09 & 0.0484 & ACO 3716 & $-$2 & (2)\
8 & 1 & dc204852\_104 & 20:51:07 & $-$52:43:34 & 0.0493 & ACO 3716 & 0 & (2)\
9 & 1 & dc204852\_149 & 20:48:30 & $-$52:33:07 & 0.0569 & ACO 3716 & 0 & (2)\
10 & 1 & dc204852\_192 & 20:51:56 & $-$52:03:45 & 0.0473 & ACO 3716 & $-$5 & (2)\
11 & 1 & dc204852\_77 & 20:52:54 & $-$52:47:28 & 0.0452 & ACO 3716 & $-$2 & (2)\
12 & 1 & dc204852\_174 & 20:51:46 & $-$52:16:09 & 0.0448 & ACO 3716 & $-$5 & (2)\
13 & 1 & dc204852\_184 & 20:54:00 & $-$52:08:15 & 0.0469 & ACO 3716 & $-$2 & (2)\
14 & 1 & dc204852\_216 & 20:49:24 & $-$51:56:56 & 0.0490 & ACO 3716 & $-$2 & (2)\
15 & 1 & dc204852\_231 & 20:51:40 & $-$51:45:22 & 0.0459 & ACO 3716 & $-$2 & (2)\
16 & 1 & dc032952\_135a & 03:29:31 & $-$52:27:18 & 0.0519 & ACO 3128 & $-$2 & (2)\
17 & 1 & dc032952\_156a & 03:31:15 & $-$52:22:28 & 0.0604 & ACO 3128 & $-$2 & (2)\
18 & 1 & dc010746\_30b & 01:10:51 & $-$45:51:52 & 0.0267 & ACO 2877 & $-$5 & (2)\
19 & 1 & dc032952\_82a & 03:31:09 & $-$52:36:49 & 0.0576 & ACO 3128 & $-$5 & (2)\
20 & 1 & dc032952\_158b & 03:29:35 & $-$52:39:58 & 0.0500 & ACO 3128 & 0 & (2)\
21 & 1 & dc010746\_22m & 01:08:23 & $-$46:09:09 & 0.0200 & ACO 2877 & 0 & (2)\
22 & 1 & dc010746\_45m & 01:09:07 & $-$45:44:29 & 0.0300 & ACO 2877 & 0 & (2)\
23 & 2 & ac103\_132 & 20:57:18 & $-$64:38:48 & 0.3047 & AC 103 & 0 & (3)\
24 & 2 & ac114\_22 & 22:58:50 & $-$34:48:13 & 0.3354 & AC 114 & 0 & (3)\
25 & 2 & ac114\_89 & 22:58:49 & $-$34:46:57 & 0.3169 & AC 114 & 0 & (3)\
26 & 2 & ac103\_03 & 20:56:55 & $-$64:40:11 & 0.3118 & AC 103 & 0 & (3)\
27 & 2 & ac103\_106 & 20:56:47 & $-$64:40:56 & 0.3091 & AC 103 & 0 & (3)\
28 & 2 & ac103\_280 & 20:57:26 & $-$64:42:11 & 0.3111 & AC 103 & 0 & (3)\
29 & 2 & ac103\_145 & 20:57:07 & $-$64:38:29 & 0.3105 & AC 103 & $-$2 & (3)\
30 & 3 & lcrs01 & 11:01:19 & $-$12:10:18 & 0.0746 & Field & 1 & (4)\
31 & 3 & lcrs17 & 10:13:52 & $-$02:55:47 & 0.0609 & Field & 0 & (4)\
32 & 3 & lcrs21 & 11:15:24 & $-$06:45:13 & 0.0994 & Field & 0 & (4)\
33 & 3 & lcrs13 & 11:19:52 & $-$12:52:39 & 0.0957 & Field & 1 & (4)\
34 & 3 & lcrs14 & 13:57:01 & $-$12:26:47 & 0.0704 & Field & 0 & (4)\
35 & 3 & lcrs12 & 12:05:59 & $-$02:54:32 & 0.0971 & Field & 1 & (4)\
36 & 3 & lcrs03 & 12:09:05 & $-$12:22:37 & 0.0810 & Field & 1 & (4)\
37 & 3 & lcrs16 & 12:19:55 & $-$06:14:01 & 0.0764 & Field & 1 & (4)\
38 & 3 & lcrs15 & 14:40:44 & $-$06:39:54 & 0.1137 & Field & 0 & (4)\
39 & 3 & lcrs06 & 11:53:55 & $-$03:10:36 & 0.0884 & Field & 0 & (4)\
40 & 3 & lcrs08 & 14:32:03 & $-$12:57:31 & 0.1121 & Field & $-$2 & (4)\
41 & 3 & lcrs07 & 22:41:09 & $-$38:34:35 & 0.1141 & Field & 0 & (4)\
42 & 3 & lcrs20 & 00:38:44 & $-$38:57:12 & 0.0632 & Cluster & $-$2 & (4)\
43 & 3 & lcrs18 & 00:22:46 & $-$41:33:37 & 0.0598 & Field & 0 & (4)\
44 & 3 & lcrs05 & 01:58:01 & $-$44:37:14 & 0.1172 & Field & $-$2 & (4)\
45 & 3 & lcrs19 & 02:07:49 & $-$45:20:50 & 0.0640 & Field & 0 & (4)\
46 & 3 & lcrs11 & 01:14:49 & $-$41:22:30 & 0.1216 & Cluster & 0 & (4)\
47 & 3 & lcrs02 & 02:17:39 & $-$44:32:47 & 0.0987 & Field & 2 & (4)\
48 & 3 & lcrs09 & 01:17:38 & $-$41:24:23 & 0.0651 & Field & 0 & (4)\
49 & 3 & lcrs10 & 02:11:43 & $-$44:07:39 & 0.1049 & Field & 0 & (4)\
50 & 3 & lcrs04 & 04:00:00 & $-$44:35:16 & 0.1012 & Cluster & 1 & (4)\
51 & 4 & pgc35435 & 11:30:05 & $-$11:32:47 & 0.0178 & Field & $-$3 & (5)\
52 & 4 & dc204852\_116 & 20:51:19 & $-$52:40:41 & 0.0441 & ACO 3716 & $-$5 & (2)\
53 & 4 & dc204852\_66 & 20:51:45 & $-$52:51:19 & 0.0410 & ACO 3716 & $-$5 & (2)\
54 & 4 & pgc60102 & 17:20:28 & $-$00:58:46 & 0.0304 & Field & $-$2 & (6)\
55 & 4 & eso290-IG\_050& 23:06:46 & $-$44:15:06 & 0.0290 & Field & $-$2 & (7)\
56 & 4 & pgc62615 & 18:57:41 & $-$52:31:46 & 0.0280 & Field & 2 & (8)\
57 & 4 & pgc57612 & 16:15:04 & $-$60:54:26 & 0.0183 & Field & $-$5 & (9)\
58 & 4 & ngc6653 & 18:44:39 & $-$73:15:47 & 0.0172 & Field & $-$5 & (9)\
59 & 4 & dc204852\_115 & 20:51:21 & $-$52:39:17 & 0.0440 & ACO 3716 & $-$5 & (2)\
60 & 4 & dc204852\_126 & 20:51:44 & $-$52:37:57 & 0.0489 & ACO 3716 & $-$2 & (2)\
61 & 4 & dc204852\_38 & 20:50:05 & $-$53:00:28 & 0.0454 & ACO 3716 & $-$2 & (2)\
62 & 4 & ngc6328 & 17:23:41 & $-$65:00:37 & 0.0142 & Field & 2 & (6)\
63 & 4 & pgc62765 & 19:05:59 & $-$42:21:59 & 0.0193 & Field & $-$2 & (6)\
[llccccccccccc]{}
&&&&&&&&&&&&\
1 & g515 & 13.86 0.02 & 13.11 0.05 & 12.75 0.05 & 0.75 0.05 & 0.36 0.07 & 1.11 0.05 & 0.0870 & & & 1 & 0\
2 & dc204852\_26 & 13.82 0.03 & 13.16 0.04 & 12.99 0.03 & 0.66 0.05 & 0.17 0.05 & 0.83 0.04 & 0.0397 & 16.99 & 15.38 & 1 & $-$2\
3 & dc184263\_39m & 11.01 0.02 & 10.27 0.06 & 10.05 0.03 & 0.74 0.06 & 0.22 0.07 & 0.96 0.04 & 0.0144 & & & 1 & $-$3\
4 & dc204852\_100 & 14.62 0.04 & 13.95 0.06 & 13.64 0.03 & 0.67 0.07 & 0.31 0.07 & 0.98 0.05 & 0.0493 & 17.61 & 16.19 & 1 & $-$2\
5 & dc204852\_148 & 14.44 0.04 & 13.74 0.06 & 13.44 0.04 & 0.70 0.07 & 0.30 0.07 & 1.00 0.06 & 0.0429 & 17.57 & 16.01 & 1 & $-$2\
6 & dc204852\_39 & 14.50 0.03 & 13.81 0.05 & 13.48 0.03 & 0.69 0.06 & 0.33 0.06 & 1.02 0.04 & 0.0489 & 17.77 & 16.16 & 1 & $-$2\
7 & dc204852\_45 & 15.04 0.04 & 14.39 0.05 & 14.03 0.03 & 0.65 0.06 & 0.36 0.06 & 1.01 0.05 & 0.0484 & & & 1 & $-$2\
8 & dc204852\_104 & 14.99 0.05 & 14.25 0.05 & 13.95 0.04 & 0.74 0.07 & 0.30 0.06 & 1.04 0.06 & 0.0493 & & & 1 & 0\
9 & dc204852\_149 & 13.91 0.03 & 13.24 0.05 & 12.91 0.04 & 0.67 0.06 & 0.33 0.06 & 1.00 0.05 & 0.0569 & & & 1 & 0\
10 & dc204852\_192 & 13.83 0.05 & 13.13 0.04 & 12.80 0.05 & 0.70 0.06 & 0.33 0.06 & 1.03 0.07 & 0.0473 & 16.98 & 15.40 & 1 & $-$5\
11 & dc204852\_77 & 14.88 0.03 & 14.16 0.04 & 13.90 0.04 & 0.72 0.05 & 0.26 0.06 & 0.98 0.05 & 0.0452 & & & 1 & $-$2\
12 & dc204852\_174 & 14.84 0.03 & 14.15 0.05 & 13.88 0.03 & 0.69 0.06 & 0.27 0.06 & 0.96 0.04 & 0.0448 & 18.09 & 16.43 & 1 & $-$5\
13 & dc204852\_184 & 14.29 0.02 & 13.60 0.04 & 13.25 0.04 & 0.69 0.04 & 0.35 0.06 & 1.04 0.04 & 0.0469 & 17.36 & 15.78 & 1 & $-$2\
14 & dc204852\_216 & 13.87 0.04 & 13.18 0.03 & 12.88 0.02 & 0.69 0.05 & 0.30 0.04 & 0.99 0.04 & 0.0490 & & & 1 & $-$2\
15 & dc204852\_231 & 13.58 0.03 & 12.88 0.03 & 12.58 0.03 & 0.70 0.04 & 0.30 0.04 & 1.00 0.04 & 0.0459 & 16.72 & 15.18 & 1 & $-$2\
16 & dc032952\_135a & 14.34 0.02 & 13.52 0.04 & 13.09 0.04 & 0.82 0.04 & 0.43 0.06 & 1.25 0.04 & 0.0519 & 18.09 & 16.21 & 1 & $-$2\
17 & dc032952\_156a & 13.22 0.04 & 12.48 0.03 & 12.15 0.03 & 0.74 0.05 & 0.33 0.04 & 1.07 0.05 & 0.0604 & 16.61 & 14.93 & 1 & $-$2\
18 & dc010746\_30b & 14.99 0.07 & 14.42 0.04 & 14.21 0.03 & 0.57 0.08 & 0.21 0.05 & 0.78 0.08 & 0.0267 & 17.90 & 16.41 & 1 & $-$5\
19 & dc032952\_82a & 14.96 0.03 & 14.35 0.03 & 14.18 0.03 & 0.61 0.04 & 0.17 0.04 & 0.78 0.04 & 0.0576 & 17.81 & 16.42 & 1 & $-$5\
20 & dc032952\_158b & 14.13 0.03 & 13.41 0.02 & 13.02 0.04 & 0.72 0.04 & 0.39 0.04 & 1.11 0.05 & 0.0500 & 17.26 & 15.76 & 1 & 0\
21 & dc010746\_22m & 14.49 0.04 & 13.87 0.04 & 13.60 0.02 & 0.62 0.06 & 0.27 0.04 & 0.89 0.04 & 0.0200 & & & 1 & 0\
22 & dc010746\_45m & 14.98 0.03 & 14.32 0.04 & 14.16 0.03 & 0.66 0.05 & 0.16 0.05 & 0.82 0.04 & 0.0300 & 17.37 & 16.24 & 1 & 0\
23 & ac103\_132 & 18.45 0.08 & 18.24 0.07 & 17.23 0.06 & 0.21 0.10 & 1.01 0.09 & 1.22 0.10 & 0.3047 & & 19.34 & 2 & 6\
24 & ac114\_22 & 18.26 0.08 & 17.57 0.06 & 16.76 0.07 & 0.69 0.10 & 0.81 0.09 & 1.50 0.11 & 0.3354 & & 19.85 & 2 & 0\
25 & ac114\_89 & 17.79 0.09 & 17.24 0.07 & NC NC & 0.55 0.11 & & & 0.3169 & & 19.78 & 2 & 0\
26 & ac103\_03 & 16.33 0.08 & 15.44 0.08 & 15.09 0.05 & 0.89 0.11 & 0.35 0.09 & 1.24 0.09 & 0.3118 & 19.95 & 18.12 & 2 & 0\
27 & ac103\_106 & 17.15 0.09 & 16.34 0.07 & 15.76 0.06 & 0.81 0.11 & 0.58 0.09 & 1.39 0.11 & 0.3091 & & & 2 & 0\
28 & ac103\_280 & 17.21 0.06 & 16.23 0.07 & 15.76 0.07 & 0.98 0.09 & 0.47 0.10 & 1.45 0.09 & 0.3111 & 21.00 & 18.93 & 2 & 0\
29 & ac103\_145 & 17.20 0.08 & 16.31 0.08 & 15.90 0.07 & 0.89 0.11 & 0.41 0.11 & 1.30 0.10 & 0.3105 & & 19.66 & 2 & 3\
30 & lcrs01 & 16.18 0.04 & 15.57 0.05 & 15.10 0.04 & 0.61 0.06 & 0.47 0.06 & 1.08 0.06 & 0.0746 & & 17.05 & 3 & 1\
31 & lcrs17 & 15.83 0.03 & 15.19 0.03 & 14.75 0.05 & 0.64 0.04 & 0.44 0.06 & 1.08 0.06 & 0.0609 & & 16.99 & 3 & 0\
32 & lcrs21 & 15.55 0.03 & 14.94 0.04 & 14.53 0.04 & 0.61 0.05 & 0.41 0.06 & 1.02 0.05 & 0.0994 & & 16.93 & 3 & 0\
33 & lcrs13 & 14.49 0.03 & 13.67 0.02 & 13.29 0.03 & 0.82 0.04 & 0.38 0.04 & 1.20 0.04 & 0.0957 & & 12.97 & 3 & 1\
34 & lcrs14 & 14.90 0.03 & 14.20 0.05 & 13.77 0.03 & 0.70 0.06 & 0.43 0.06 & 1.13 0.04 & 0.0704 & & 16.05 & 3 & 0\
35 & lcrs12 & 15.02 0.03 & 14.35 0.04 & 13.82 0.03 & 0.67 0.05 & 0.53 0.05 & 1.20 0.04 & 0.0971 & & 16.78 & 3 & 1\
36 & lcrs03 & 14.11 0.04 & 13.47 0.03 & 13.04 0.03 & 0.64 0.05 & 0.43 0.04 & 1.07 0.05 & 0.0810 & & 15.03 & 3 & 1\
37 & lcrs16 & 15.35 0.04 & 14.75 0.04 & 14.41 0.03 & 0.60 0.06 & 0.34 0.05 & 0.94 0.05 & 0.0764 & & 16.69 & 3 & 1\
38 & lcrs15 & 15.84 0.05 & 15.16 0.05 & 14.73 0.04 & 0.68 0.07 & 0.43 0.06 & 1.11 0.06 & 0.1137 & & 17.19 & 3 & 0\
39 & lcrs06 & 15.64 0.05 & 15.09 0.04 & 14.72 0.03 & 0.55 0.06 & 0.37 0.05 & 0.92 0.06 & 0.0884 & & 16.81 & 3 & 0\
40 & lcrs08 & 15.63 0.04 & 15.01 0.03 & 14.55 0.04 & 0.62 0.05 & 0.46 0.05 & 1.08 0.06 & 0.1121 & & 17.87 & 3 & $-$2\
41 & lcrs07 & 13.62 0.05 & 12.89 0.03 & 12.45 0.04 & 0.73 0.06 & 0.44 0.05 & 1.17 0.06 & 0.1141 & & 15.00 & 3 & 0\
42 & lcrs20 & 14.48 0.03 & 13.89 0.03 & 13.53 0.05 & 0.59 0.04 & 0.36 0.06 & 0.95 0.06 & 0.0632 & & 15.96 & 3 & $-$2\
43 & lcrs18 & 14.70 0.04 & 14.02 0.03 & 13.62 0.03 & 0.68 0.05 & 0.40 0.04 & 1.08 0.05 & 0.0598 & & 16.09 & 3 & 0\
44 & lcrs05 & 15.36 0.05 & 14.80 0.03 & 14.32 0.05 & 0.56 0.06 & 0.48 0.06 & 1.04 0.07 & 0.1172 & & 16.73 & 3 & $-$2\
45 & lcrs19 & 14.95 0.03 & 14.24 0.04 & 13.90 0.03 & 0.71 0.05 & 0.34 0.05 & 1.05 0.04 & 0.0640 & & 16.42 & 3 & 0\
46 & lcrs11 & 15.48 0.04 & 14.78 0.04 & 14.38 0.03 & 0.70 0.06 & 0.40 0.05 & 1.10 0.05 & 0.1216 & & 16.96 & 3 & 0\
47 & lcrs02 & 14.95 0.03 & 14.28 0.03 & 13.95 0.03 & 0.67 0.04 & 0.33 0.04 & 1.00 0.04 & 0.0987 & & 16.36 & 3 & 2\
48 & lcrs09 & 15.98 0.05 & 15.30 0.04 & 14.96 0.03 & 0.68 0.06 & 0.34 0.05 & 1.02 0.06 & 0.0651 & & 17.47 & 3 & 0\
49 & lcrs10 & 15.29 0.04 & 14.65 0.05 & 14.27 0.04 & 0.64 0.06 & 0.38 0.06 & 1.02 0.05 & 0.1049 & & 16.68 & 3 & 0\
50 & lcrs04 & 14.49 0.04 & 13.80 0.05 & 13.41 0.03 & 0.69 0.06 & 0.39 0.06 & 1.08 0.05 & 0.1012 & & 15.68 & 3 & 1\
51 & pgc35435 & 11.75 0.03 & 10.98 0.04 & 10.66 0.02 & 0.77 0.05 & 0.32 0.04 & 1.09 0.04 & 0.0178 & 13.75 & & 4 & $-$3\
52 & dc204852\_116 & 12.62 0.06 & 11.92 0.05 & 11.65 0.02 & 0.70 0.08 & 0.27 0.05 & 0.97 0.06 & 0.0441 & 15.84 & 14.06 & 4 & $-$5\
53 & dc204852\_66 & 14.45 0.05 & 13.62 0.04 & 13.36 0.03 & 0.83 0.06 & 0.26 0.05 & 1.09 0.06 & 0.0410 & 17.48 & 15.88 & 4 & $-$5\
54 & pgc60102 & 12.96 0.06 & 12.13 0.03 & 11.65 0.04 & 0.84 0.07 & 0.47 0.05 & 1.31 0.07 & 0.0304 & 15.36 & & 4 & $-$2\
55 & eso290-IG\_050& 13.46 0.03 & 12.74 0.03 & 12.39 0.03 & 0.72 0.04 & 0.35 0.04 & 1.07 0.04 & 0.0290 & 15.18 & 14.21 & 4 & $-$2\
56 & pgc62615 & 12.65 0.04 & 11.92 0.04 & 11.63 0.04 & 0.73 0.06 & 0.29 0.06 & 1.02 0.06 & 0.0280 & & & 4 & 2\
57 & pgc57612 & 10.99 0.03 & 10.22 0.03 & 10.10 0.03 & 0.77 0.04 & 0.11 0.04 & 0.88 0.04 & 0.0183 & 13.30 & 11.33 & 4 & $-$5\
58 & ngc6653 & 11.53 0.04 & 10.79 0.01 & 10.59 0.04 & 0.74 0.04 & 0.20 0.04 & 0.94 0.06 & 0.0172 & & & 4 & $-$5\
59 & dc204852\_115 & 14.98 0.03 & 14.33 0.03 & 14.04 0.03 & 0.65 0.04 & 0.29 0.04 & 0.94 0.04 & 0.0440 & 18.13 & 16.53 & 4 & $-$5\
60 & dc204852\_126 & 15.01 0.04 & 14.29 0.04 & 14.01 0.04 & 0.72 0.06 & 0.28 0.06 & 1.00 0.06 & 0.0489 & 18.21 & 16.60 & 4 & $-$2\
61 & dc204852\_38 & 13.49 0.05 & 12.90 0.03 & 12.56 0.04 & 0.59 0.06 & 0.34 0.05 & 0.93 0.06 & 0.0454 & 16.73 & 15.12 & 4 & $-$2\
62 & ngc6328 & 11.33 0.03 & 10.57 0.04 & 10.24 0.04 & 0.77 0.05 & 0.32 0.06 & 1.09 0.05 & 0.0142 & 13.17 & 11.45 & 4 & 2\
63 & pgc62765 & 11.42 0.04 & 10.68 0.04 & 10.36 0.03 & 0.74 0.06 & 0.32 0.05 & 1.06 0.05 & 0.0193 & & & 4 & $-$2\
[^1]: Simultaneous Quad-color Infrared Imaging Device software package, developed by Michael Merrill and John Mac Kenty.
[^2]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
[^3]: DIMSUM is the Deep Infrared Mosaicing Software package developed by Peter Eisenhardt, Mark Dickinson, Adam Stanford, and John Ward, and is available via ftp from [ftp://iraf.noao.edu/iraf/contrib/dimsumV2/dimsum.tar.Z]{}
[^4]: Projet d’Etude des GAlaxies par Synthèse Evolutive.
[^5]: $r_1$ is defined as $r_1 = \frac{\sum_r r I(r)}{\sum_r I(r)}$.
|
---
abstract: 'Topological quantum computation encodes quantum information nonlocally by nucleating non-Abelian anyons separated by distances $L$, typically spanning the qubit device size. This nonlocality renders topological qubits exponentially immune to dephasing from *all* sources of classical noise with operator support local on the scale of $L$. We perform detailed analytical and numerical analyses of a time-domain Ramsey-type protocol for noisy Majorana-based qubits that is designed to validate this coveted topological protection in near-term devices such as the so-called ‘tetron’ design. By assessing dependence of dephasing times on tunable parameters, e.g., magnetic field, our proposed protocol can clearly distinguish a bona fide Majorana qubit from one constructed from semilocal Andreev bound states, which can otherwise closely mimic the true topological scenario in local probes. In addition, we analyze leakage of the qubit out of its low-energy manifold due to classical-noise-induced generation of quasiparticle excitations; leakage limits the qubit lifetime when the bulk gap collapses, and hence our protocol further reveals the onset of a topological phase transition. This experiment requires measurement of two nearby Majorana modes for both initialization and readout—achievable, for example, by tunnel coupling to a nearby quantum dot—but no further Majorana manipulations, and thus constitutes an enticing pre-braiding experiment. Along the way, we address conceptual subtleties encountered when discussing dephasing and leakage in the context of Majorana qubits.'
author:
- 'Ryan V. Mishmash'
- Bela Bauer
- Felix von Oppen
- Jason Alicea
bibliography:
- 'MajoranaNoise.bib'
title: |
Dephasing and leakage dynamics of noisy Majorana-based qubits:\
Topological versus Andreev
---
Introduction {#sec:intro}
============
Developing a scalable quantum computing architecture that can withstand decoherence to the extent necessary for real-world applications poses an enormous scientific and technological undertaking. One path forward pursues Majorana-based topological qubits. These promise to provide robust protection against decoherence at the hardware level by storing quantum information nonlocally, specifically in a fermion-parity degree of freedom in a topological phase [@Kitaev01_PhysU_44_131; @sarma_majorana_2015]. Since 2012, remarkable experimental progress [@mourik_signatures_2012; @deng_anomalous_2012; @das_zero-bias_2012; @churchill_superconductor-nanowire_2013; @lee_spin-resolved_2013; @finck_anomalous_2013; @albrecht_exponential_2016; @deng_majorana_2016; @zhang_quantized_2018; @vaitiekenas_effective_2018; @deng_nonlocality_2018; @moor_electric_2018; @suominen_zero-energy_2017; @nichele_scaling_2017] (see Ref. [@lutchyn_majorana_2018] for a recent review) has established the potential existence of Majorana zero modes in devices following the popular recipe [@lutchyn_majorana_2010; @oreg_helical_2010] of engineering topological superconductivity [@alicea_new_2012] by interfacing $s$-wave superconductors with spin-orbit-coupled semiconducting nanowires subjected to modest magnetic fields.
Most of these laboratory efforts have, however, focused on end-of-wire tunneling conductance spectroscopy, a local measurement which may have difficulty differentiating bona fide well-separated Majorana zero modes in a topological phase [@Kitaev01_PhysU_44_131] from near-zero-energy Andreev bound states (ABSs) arising in a trivial phase. One might naively expect that the latter should not ‘stick’ near zero energy upon variation of system parameters such as magnetic field, chemical potential, etc. Important theoretical and numerical work [@kells_near-zero-energy_2012; @prada_transport_2012; @liu_andreev_2017; @moore_two-terminal_2018; @setiawan_electron_2017; @moore_quantized_2018; @liu_distinguishing_2018; @vuik_reproducing_2018; @penaranda_quantifying_2018; @reeg_zero-energy_2018; @stanescu_illustrated_2018] has nevertheless shown that seemingly quite reasonable spatial variations in the potentials can trap ABSs—even deep in the trivial phase—that exhibit a high propensity to reside near zero energy (at least within the energy resolution of transport experiments). Andreev states, i.e., ordinary (complex) fermionic modes, can always be described in terms of two Majorana operators. Oftentimes near-zero-energy ABSs are *semilocal* in that they consist of two Majorana operators separated in space by a finite distance, albeit one much less than the physical wire length [@moore_two-terminal_2018; @moore_quantized_2018; @vuik_reproducing_2018; @penaranda_quantifying_2018; @stanescu_illustrated_2018]; the corresponding coupling strength between the constituent Majorana operators can thus be very small. This property endows semilocal ABSs with the (rather unfortunate) ability to masquerade as true Majorana zero modes even with respect to more detailed local-probe characteristics such as $2e^2/h$ zero-bias-peak conductance quantization [@setiawan_electron_2017; @moore_quantized_2018] and $4\pi$ Josephson periodicity [@chiu_fractional_2018]. ABSs induced by local potential inhomogeneities arguably describe much of the experimental data as compellingly as true Majorana zero modes. While states encoded through semilocal ABSs might still furnish qubits with some degree of protection [@vuik_reproducing_2018], such a situation is clearly suboptimal.
Various schemes have been proposed to distinguish Majorana zero modes from Andreev levels: (1) Detecting the bulk topological phase transition accompanying the onset of Majorana modes—e.g., through bulk-gap closure and reopening [@stanescu_close_2012; @mishmash_approaching_2016]—surely provides an unambiguous fingerprint; see Ref. [@grivnin_concomitant_2018] for an experiment in this direction. Recently studied multi-terminal N-S-N devices provide an appealing platform for revealing the bulk phase transition via non-local conductance measurements [@AndreevRectifier; @danon_nonlocal_2019; @menard_conductance-matrix_2019; @zhang_quantum_2019]. (2) Majorana modes generate correlations in conductance measured at the two ends of a wire, whereas ABSs localized to each end generically produce no such correlations [@moore_two-terminal_2018]. (3) Sensitivity of the energy to local perturbations, such as a quantum dot [@clarke_experimentally_2017; @prada_measuring_2017; @deng_nonlocality_2018] or sharp edge potential [@liu_distinguishing_2018] is expected to differ sharply for ABSs and Majorana modes due to the nonlocal, and hence energetically rigid, nature of the latter (see also Ref. [@yavilberg_differentiating_2019]). (4) The current-phase relation in a Cooper-pair transistor has been shown to also differ qualitatively for these two scenarios [@schrade_andreev_2018]. (5) On the more challenging end, demonstration of inherently robust non-Abelian braiding operations is possible only with true Majorana zero modes.
We add to this list by analyzing in detail a time-domain protocol that not only unambiguously differentiates ABSs from Majorana modes, but further reveals important device characteristics (notably qubit lifetimes and precise Majorana-hybridization energies), exposes the topological phase transition, and clearly elucidates the enormous benefit of Majorana-based qubits insofar as robustly *protecting* quantum information. The main idea is to use *noise sensitivity* of a prototype qubit to validate (or invalidate) its topological nature. Similar ideas were originally put forth in Sec. IV of Ref. [@aasen_milestones_2016] in a related context. As we will see, in our framework ABSs that arise over a restricted parameter regime before the onset of well-separated Majorana modes in fact become a feature rather than a bug; their presence allows one to benchmark the quality of a bona fide topological qubit within a single device.
The remainder of this paper is organized as follows. Section \[sec:executive\] provides an executive summary of our main results. Section \[sec:models\] introduces the microscopic models that we employ and also outlines our numerical simulation strategy. The Ramsey-like protocol that we use to probe the qubit dynamics is detailed in Sec. \[sec:protocol\]. Section \[sec:analytics\] analytically estimates dephasing and leakage times for the qubit, while Sec. \[NumericalResults\] extracts these quantities from explicit time-dependent numerical simulations. We conclude with a discussion in Sec. \[sec:discussion\]. Appendices provide supplemental calculational details.
![Schematics of the topological tetron qubit (top) and a non-topological qubit arising from Andreev bound states (ABSs) residing near zero energy (bottom). Both cases consist of two parallel semiconducting nanowires (gray) proximity coupled to $s$-wave superconductors (orange and green, in the topological and non-topological scenarios, respectively); a trivial superconducting bridge connects the two parallel superconducting segments. As described in the text, the spatial separation of the low-energy Majorana modes $\gamma_{1,2,3,4}$ is qualitatively distinct in the two scenarios. The two leftmost such modes, i.e., $\gamma_1$ and $\gamma_3$, are coupled to a nearby quantum dot (QD), allowing initialization and readout of the respective Majorana parity $i\gamma_1\gamma_3 = \pm1$. \[fig:qubits\]](Qubits.pdf){width="0.7\columnwidth"}
Executive Summary {#sec:executive}
=================
For concreteness, we focus on the so-called ‘tetron’ Majorana qubit design shown in Fig. \[fig:qubits\] [@plugge_majorana_2017; @karzig_scalable_2017; @knapp_dephasing_2018] (equivalent for our purposes to the ‘loop qubit’ [@knapp_dephasing_2018]). This prototype topological qubit consists of two parallel semiconducting nanowires (gray) each proximitized by an $s$-wave superconductor and tuned to the topological phase with an external magnetic field (orange denotes topological superconductors). The concomitant four Majorana zero modes—defined through *instantaneous* eigenstates of the microscopic, time-dependent Hamiltonian—are denoted $\gamma_{1,2,3,4}$. A trivial superconductor (blue) bridges the two wires. In a real physical implementation, the entire mesoscopic device is floating so as to protect against quasiparticle poisoning events from the outside via charging energy [@karzig_scalable_2017]. For the non-topological ABS qubit [^1] that we wish to contrast against, $\gamma_{1,2,3,4}$ instead represent a maximally localized set of Majorana operators that weakly hybridize at the left end of the device as illustrated in the bottom of Fig. \[fig:qubits\]. Unless specified otherwise our discussion below pertains to both types of qubits.
Assuming that we are working on time scales much less than the characteristic poisoning time, and additionally that the system is confined to the low-energy, nearly degenerate ground-state manifold with high probability—e.g., the temperature and characteristic noise frequencies are well below the bulk excitation gap (see below)—we can encode logical qubit states by $|0\rangle \equiv |i\gamma_1\gamma_2 = +1\rangle$ and $|1\rangle \equiv |i\gamma_1\gamma_2 = -1\rangle$. \[Here global fermion parity is fixed to $(i\gamma_1\gamma_2)(i\gamma_3\gamma_4) = +1$, so that specifying $i\gamma_1\gamma_2$ automatically specifies $i\gamma_3\gamma_4$.\] In this basis, we can identify Pauli operators $Z \equiv i\gamma_1\gamma_2$, $X \equiv i\gamma_1\gamma_3$, and $Y \equiv -i\gamma_1\gamma_4$. Suppose that $\gamma_1$ and $\gamma_2$ couple with hybridization energy $\varepsilon_{12}(t)$ that is time-dependent due to noise, and that $\gamma_3$ and $\gamma_4$ similarly couple with energy $\varepsilon_{34}(t)$. The time-averaged couplings decay asymptotically with the intra-wire Majorana separation as an oscillatory exponential: $\varepsilon_{12} \sim \cos(k_\mathrm{top} L_\mathrm{top})e^{-L_\mathrm{top}/\xi_\mathrm{top}}$ and $\varepsilon_{34} \sim \cos(k_\mathrm{bot} L_\mathrm{bot})e^{-L_\mathrm{bot}/\xi_\mathrm{bot}}$, where $k$ is related to the Fermi wave vector of the wire, $\xi$ is the effective superconducting coherence length [@das_sarma_splitting_2012], and ‘top’/‘bot’ refers to the top and bottom wires of the tetron. For the topological qubit, $L_{\rm top/bot}$ are given by the wire length $L$—thus taking full advantage of the exponential suppression. By contrast, for the ABS qubit $L_{\rm top/bot}$ is on the scale of the coherence length or smaller, and hence the splitting need not conform to such an exponential. In principle, couplings between Majoranas on different wires, such as $\varepsilon_{14}$, $\varepsilon_{23}$, etc., can also be present. However, since the separation between these Majorana modes generally exceeds that between Majorana modes on the same wire, these couplings will be significantly smaller than the intra-wire hybridization energies—at least if the superconducting bridge connecting the two wires is sufficiently long and well-gapped. In this case, their primary effect is to reduce the amplitude of qubit oscillations and slightly shift the qubit precession frequency while leaving the qualitative features unaffected. We can therefore safely ignore such subdominant terms and write the Hamiltonian governing the dynamics of the qubit as $$\begin{aligned}
H(t) &=& \frac{i}{2}[\varepsilon_{12}(t)\gamma_1\gamma_2 + \varepsilon_{34}(t) \gamma_3\gamma_4]
\nonumber \\
&=& \frac{1}{2}[\varepsilon_{12}(t) + \varepsilon_{34}(t)]Z \equiv \frac{1}{2}E(t) Z.
\label{qubitH}\end{aligned}$$ Finally, a quantum dot (ignored thus far) sits proximate to the left ends of the nanowires, thereby allowing a joint parity readout [@gharavi_readout_2016; @karzig_scalable_2017] of nearby Majoranas $\gamma_1$ and $\gamma_3$, i.e., the Pauli $X$ operator.
Our proposed protocol is conceptually very simple: (1) Initialize the system into an $X$ eigenstate by measuring $i\gamma_1\gamma_3$; (2) let the system evolve unitarily for a wait time $t$; and finally (3) re-measure $X$. Assuming the unitary evolution is governed by Eq. , which ignores inter-wire Majorana hybridization as well as higher-energy excitations, the qubit precesses about the $z$ axis on the equator of the Bloch sphere. Let us write the instantaneous qubit splitting defined through Eq. as $E(t) = \hbar \omega_0 + \delta E(t)$. Here $\omega_0$ is the *time-averaged* qubit precession frequency while $\delta E(t)$ encodes the effects of classical noise and is responsible for dephasing. Note that we assume here that $\delta E(t)$ fluctuates around zero mean. Taking the initial state to be an eigenstate of $X = i\gamma_1\gamma_3$ with eigenvalue $+1$, noise averaging the $X$ readout measurement gives $$\begin{aligned}
\overline{\langle \psi(t) | X | \psi(t) \rangle} = \cos(\omega_0 t) f(t/T_2), \label{eq:X_decay}\end{aligned}$$ where the envelope function $f(t/T_2)$ decays on a time scale given by the *dephasing time* $T_2$. This experiment therefore simultaneously probes $\omega_0$ (and thus reveals the mean Majorana hybridization energy) as well as $T_2$—both critical device characteristics that differ starkly for topological and ABS qubits. We note that the described protocol is similar in spirit to the so-called Ramsey sequence [@vion_manipulating_2002; @ithier_decoherence_2005]. The main differences are that we (1) do not include an external driving field transverse to $Z$ and (2) initialize and read out the qubit with projective $X$ measurements in contrast to utilizing $\pi/2$ pulses and $Z$ initialization/readout [^2].
A particularly advantageous feature of Majorana-based topological qubits is that, due to their inherently nonlocal encoding, the splitting $E$ remains exponentially small—and thus also ‘exponentially flat’—in response to changes in *all* local Hamiltonian couplings (e.g., chemical potential, Zeeman field, etc.). This property endows topological qubits with exponential protection from all classical noise sources, yielding the following nontrivial scaling relation that connects time-averaged splitting and dephasing [@aasen_milestones_2016]: $$\begin{aligned}
\frac1{\omega_0} \sim T_2^a \sim e^{L/\xi}. \label{eq:scaling}\end{aligned}$$ Here $a$ is an order-one number dependent on details of the noise. In other words, as the topological qubit becomes ‘perfect’, the splitting vanishes and the dephasing time diverges in a correlated fashion. For comparison, the transmon qubit [@koch_charge-insensitive_2007] benefits similarly from an exponential protection from charge noise, but not other noise sources; additionally, there the qubit splitting need not be exponentially small. The pre-braiding protocol sketched above and analyzed in detail below is capable of probing the scaling relation in Eq. —which can for practical purposes be taken as a *definition* of a topologically protected quantum memory. Note also that as the qubit quality improves (i.e., as $L/\xi$ increases) the splitting $\hbar \omega_0$ becomes more difficult to resolve in transport [@albrecht_exponential_2016; @vaitiekenas_flux-induced_2018] yet easier to resolve in the time-domain measurements employed in our protocol.
The trivial ABS qubit exhibits qualitatively different behavior. Firstly, since the requisite ‘accidental’ low-energy Andreev bound states would emerge due to some local, nonuniversal features in the potential landscapes near the ends of the wires, the ABS qubit would clearly not exhibit exponential protection against arbitrary local noise sources. Moreover, neither $\omega_0$ nor $T_2$ will vary appreciably with wire length $L$, in sharp contrast to the scaling relation in Eq. that uniquely identifies the topological qubit.
While it would be ideal to perform experiments at different $L$, keeping all other parameters fixed in this process has its practical challenges. We nevertheless argue that one can compellingly distinguish the two scenarios in a *single* prototype qubit at fixed $L$ by carrying out the protocol at different field strengths $B$. Numerous numerical simulations [@liu_andreev_2017; @moore_two-terminal_2018; @moore_quantized_2018; @vuik_reproducing_2018; @penaranda_quantifying_2018; @stanescu_illustrated_2018] indicate a tendency for low-energy Andreev bound states to form over an extended field interval below the onset of a topological phase hosting true Majorana zero modes. In such a scenario, upon increasing $B$ from zero, the system first realizes an ABS qubit, then encounters a topological phase transition at a critical field $B_c$, and finally forms a topological qubit.
The behavior of $\omega_0$ and $T_2$ during this evolution reveals a wealth of information. In the ABS regime there is likely no clear universal behavior present, though the measured coherence times would provide a useful baseline. When $B$ approaches $B_c$, the bulk gap becomes close to zero. Here even low-frequency noise can efficiently excite the qubit out of the computational subspace, causing a precipitous reduction in the qubit lifetime. (Below we denote the characteristic ‘leakage’ time associated with such excitations by $T_\mathrm{leak}$.) Hence, our protocol provides a novel means of detecting the topological phase transition. As $B$ increases beyond $B_c$, the gap re-opening and concomitant appearance of robust Majorana zero modes rapidly boost the qubit’s coherence time—ideally to values exceeding the ABS-qubit lifetime by orders of magnitude. The scaling relation in Eq. , which should now be viewed in terms of field-induced variation of $\xi$ at fixed $L$, becomes operative in this topological regime. The topological-qubit frequency $\omega_0$ additionally exhibits characteristic oscillations with magnetic field, reflecting oscillatory overlap of the Majorana wave functions [@das_sarma_splitting_2012], in turn yielding out-of-phase oscillations in the dephasing time $T_2$ [@aasen_milestones_2016]. Further increasing $B$ eventually suppresses the gap for the topological phase and hence increases $\xi$, thereby diminishing $1/\omega_0$ and $T_2$ in accordance with Eq. . Note also that for sufficiently long topological wires, the exponentially long dephasing time $T_2$ is eventually cut off by the leakage time $T_\mathrm{leak}$ (which is set by the $L$-independent excitation gap).
![Qubit coherence time defined as $\min(T_2, T_\mathrm{leak})$ (top) and qubit precession period $2\pi/\omega_0$ (bottom) as a function of Zeeman energy $V_z$ for tetron qubits constructed from the spinful nanowire model of Eq. at different wire lengths $L$. For $V_{zc}' < V_z < V_{zc}$, the qubit is of the non-topological ABS variety (Fig. \[fig:qubits\], bottom panel), while for $V_z > V_{zc}$, the qubit is topological (Fig. \[fig:qubits\], top panel). In the top panel, solid curves indicate that the qubit lifetime is *dephasing limited* ($T_2 < T_\mathrm{leak}$), while dashed curves indicate that the qubit lifetime is *leakage limited* ($T_\mathrm{leak} < T_2$). These calculations were performed using the quasi-analytic methods of Sec. \[sec:analytics\]; for more details, including the specific parameters chosen, please see Sec. \[sec:spinful\]. \[fig:Vzscan\_summary\]](Vz_scan-summary.pdf){width="\columnwidth"}
The above points are demonstrated in Fig. \[fig:Vzscan\_summary\], where we show concrete calculations of the qubit coherence time defined as $\min(T_2, T_\mathrm{leak})$ and qubit precession period $2\pi/\omega_0$ for noisy tetron qubits modeled within the canonical single-band description of proximitized spin-orbit-coupled nanowires [@lutchyn_majorana_2010; @oreg_helical_2010]. Here, the horizontal axis denotes the Zeeman energy, related to the magnetic field $B$ through $V_z = \frac12 g \mu_B B$, with $g$ the effective Landé $g$-factor (assumed equivalent for both wires of the tetron) and $\mu_B$ the Bohr magneton. In these plots, and in all simulations in this paper, we neglect dependence of the noise on magnetic field and other parameters. We stress that although this assumption will undoubtedly be violated to some degree in experiment, the non-monotonicity and sheer magnitude of the effects illustrated in Fig. \[fig:Vzscan\_summary\] are unlikely to be modified significantly by such non-universal noise properties. For more details of the model Hamiltonian, means of calculation, and chosen physical parameters for the data in Fig. \[fig:Vzscan\_summary\], see Secs. \[sec:Hamis\], \[sec:analytics\], and \[sec:spinful\] below.
Many previous studies have explored the influence of noise on Majorana-based qubits from various perspectives (see, e.g., Refs. [@goldstein_decay_2011; @rainis_majorana_2012; @schmidt_decoherence_2012; @pedrocchi_majorana_2015; @hu_majorana_2015; @pedrocchi_monte_2015; @rahmani_optimal_2017; @ritland_optimal_2018; @knapp_dephasing_2018; @knapp_modeling_2018; @li_four-majorana_2018; @munk_fidelity_2019]). We emphasize that here we are proposing to use noise to our *advantage* in verifying the topological nature of a prototype Majorana qubit, and as a byproduct are able to extract crucial device characteristics that would otherwise be difficult to obtain. Time-domain experiments of the type that we propose certainly entail significant challenges, but they are arguably a prerequisite for successful demonstration of exponentially accurate braiding gates.
Microscopic models and theoretical formulation {#sec:models}
==============================================
Microscopic Hamiltonians {#sec:Hamis}
------------------------
We exploit two different microscopic Hamiltonians for the one-dimensional (1D) topological superconductors comprising our prototype tetron qubits. For a truly minimal description we appeal to Kitaev’s model [@Kitaev01_PhysU_44_131] for a 1D spinless $p$-wave superconductor, given by the Hamiltonian $$\begin{aligned}
H_{\rm K} = \sum_{i=1}^L\left[-\mu c_i^\dagger c_i - \frac12\left(J c_i^\dagger c_{i+i} + \Delta c_i c_{i+1} + \mathrm{H.c.}\right)\right]. \label{eq:Kitaev}\end{aligned}$$ Here $\mu$ is the chemical potential, $J \geq 0$ is the nearest-neighbor hopping strength, and $\Delta$ is the $p$-wave pairing strength (assumed real). The topological phase occurs for $|\mu| < J$, wherein a chain with open boundary conditions harbors a pair of Majorana zero modes (MZMs), one at each end.
For a more realistic treatment, we consider the one-band spinful model of a Rashba spin-orbit coupled nanowire proximitized with an $s$-wave superconductor and subjected to a Zeeman field [@lutchyn_majorana_2010; @oreg_helical_2010]. The Hamiltonian reads $$\begin{aligned}
H_{\rm NW} &= \int_0^L dx\bigg{[}\psi^\dagger \bigg{(}-\frac{\hbar^2\partial_x^2}{2m} - \mu + V(x) \nonumber \\
&- i \alpha \sigma^y \partial_x + V_z \sigma^z\bigg{)}\psi + \Delta(x) (\psi_\uparrow \psi_\downarrow + \mathrm{H.c.})\bigg{]},
\label{eq:nanowire}\end{aligned}$$ where $m$ is the effective electron mass, $\mu$ is the chemical potential, $\alpha$ is the Rashba spin-orbit strength, $V_z = \frac12 g \mu_B B$ is the Zeeman energy, $\Delta(x)$ is the induced $s$-wave pairing amplitude, and Pauli matrices $\sigma^{y,z}$ act in spin space. Note that we explicitly included a spatially varying external electric potential $V(x)$ and also allowed the pairing potential $\Delta(x)$ to depend on position—these features are essential for creating conditions that trap near-zero-energy ABS’s that mimic MZM’s. A uniform system with $V(x) = 0$ and $\Delta(x) = \Delta_0$ resides in the topological phase provided $h > \sqrt{\Delta_0^2 + \mu^2}$. For numerical evaluation we discretize Eq. in a standard way; when essential, we will specify the lattice spacing used in this discretization procedure below.
At fixed global fermion parity, we microscopically model tetron qubits via $$\begin{aligned}
H = H_\mathrm{top} + H_\mathrm{bot}, \label{eq:microQubit}\end{aligned}$$ where $H_{\mathrm{top}}$ and $H_{\mathrm{bot}}$ take the form of either Eqs. or and respectively describe the top and bottom topological superconductors in the qubit. Classical noise is readily included by endowing parameters in $H$ with stochastic time dependence. In principle, one should include a ‘bridge’ term that couples the two superconductors (see Fig. \[fig:qubits\]). We assume for simplicity that the bridge yields only small quantitative effects and thus ignore its presence in our calculations. Both of our microscopic models describe noninteracting electrons in a non-number-conserving formulation and thus fail to capture charging-energy effects that would be present in a laboratory realization of the tetron (see Sec. \[sec:executive\]). This simplification is assumed for obvious computational tractability reasons; however, the effects of $1/f$ charge noise, as was argued to be the dominant physical source of tetron dephasing in Ref. [@knapp_dephasing_2018], can be captured at a rough qualitative level by taking the global chemical potentials to fluctuate stochastically in Eqs. and .
Majorana-operator reformulation {#sec:MajoranaReform}
-------------------------------
In our microscopic numerical simulations of tetrons, we find it most convenient to work in the language of *local* Majorana operators $a_i$—with $\{a_i,a_j\} = 2\delta_{ij}$, $a_i^\dagger = a_i$, and $(a_i)^2 = 1$—by decomposing the local Dirac fermions in the usual way, e.g., $c_i = (a_{2i-1} - ia_{2i})/2$ in the Kitaev model context [@Kitaev01_PhysU_44_131]. In this basis, the (quadratic) microscopic tetron Hamiltonian is represented by an $N\times N$ anti-symmetric matrix $A = -A^T$: $$H = \frac{i}{4}\sum_{i,j=1}^N A_{ij} a_i a_j = \frac{i}{4}\,\mathbf{a}^T\,A\, \mathbf{a}, \label{eq:H-A}$$ with $N$ the total number of local Majorana operators; for spinless (spin-1/2) models with $N_\mathrm{sites}$ physical sites, $N = 2N_\mathrm{sites}$ ($N = 4N_\mathrm{sites}$). It is a relatively straightforward numerical exercise (see, e.g., Refs. [@wimmer_algorithm_2012; @bravyi_complexity_2017; @bauer_dynamics_2018]) to bring $A$ into so-called ‘canonical form’ via an orthogonal rotation $U$: $$B = U^T\,A\,U = \bigoplus_{k=1}^{N/2} \varepsilon_k i \sigma^y, \label{eq:B}$$ such that $$H = \frac{i}{4}\,\mathbf{b}^T\,B\,\mathbf{b} = \frac{i}{2} \sum_{k=1}^{N/2} \varepsilon_k b_{2k-1} b_{2k}. \label{eq:Hcanon}$$ The canonical Majorana modes $b_i = [\mathbf{b}]_i$ are related to the original local Majoranas $a_i = [\mathbf{a}]_i$ via the orthogonal transformation $\mathbf{b} = U^T\,\mathbf{a}$, and $\varepsilon_k \geq 0$ are the energies (or instantaneous energies when the Hamiltonian depends on time).
Due to Wick’s theorem, any quadratic system can be completely described by the real, anti-symmetric covariance matrix $$M_{ij} = \frac{-i}{2}\langle[a_i, a_j]\rangle, \label{eq:Mdef}$$ where the square brackets denote a commutator and $\langle \dots \rangle = {\rm Tr} \{\rho \dots\}$ a quantum expectation value taken with respect to density matrix $\rho$; if $\rho = |\psi\rangle\langle\psi|$ represents a pure state, one finds $M^2=-1$. As an example, the ground state of $H$ corresponds to a covariance matrix $$M_\mathrm{g.s.} = U\,M_y\,U^T \label{eq:Mgs},$$ with $$M_y = \bigotimes_{k=1}^{N/2} i\sigma^y \label{eq:M0}$$ a reference covariance matrix in the canonical basis of the modes $b_i$ which encodes $i b_{2k-1} b_{2k} = -1~\forall\,k=1,\dots,N/2$. It is straightforward to construct more general states by appropriately toggling elements of $M_y$ (see Refs. [@bravyi_complexity_2017; @bauer_dynamics_2018] and Sec. \[sec:protocol\]).
Finally, the Schrödinger equation for Gaussian states takes the form of the following ordinary matrix differential equation for the covariance matrix: $$\frac{dM(t)}{dt} = [A(t), M(t)], \label{eq:dMdt}$$ where $A(t)$ specifies the (possibly time-dependent) Hamiltonian \[cf. Eq. \] and the square brackets denote the matrix commutator. Any physical observable can then be computed with knowledge of $M(t)$; relevant examples of such measurements will be described in the next section.
Ramsey-type protocol {#sec:protocol}
====================
Qubit definition {#sec:qubitdef}
----------------
In what follows we assume for concreteness that the devices in Fig. \[fig:qubits\] possess even global fermion parity. We will encode the qubit in the subspace consisting of the two lowest-lying *instantaneous* even-parity eigenstates of the microscopic Hamiltonian $H(t)$ that depends on time due to noise. These presumed nearly degenerate states could arise because the system either realizes bona fide topological phases yielding four well-separated Majorana modes (upper panel of Fig. \[fig:qubits\]), or exhibits a pair of ‘accidentally’ low-energy Andreev bound states (lower panel of Fig. \[fig:qubits\]). Since we will explore platforms that contain both types of near-degeneracies depending on parameters, it is useful to treat them in a common framework. To this end we introduce a quartet of *maximally localized* (in real space) Majorana operators $\gamma_{1,2,3,4}(t)$ that span the instantaneous qubit subspace at time $t$. In the topological case residual overlap between Majorana operators splits the degeneracy by an energy that is exponentially small in their separation; in the low-energy Andreev-bound-state scenario, pairs of Majorana operators sit in close proximity yet happen to hybridize weakly.
For a more precise definition of the qubit, suppose that $$P(t) \equiv [i\gamma_1(t)\gamma_2(t)][i\gamma_3(t)\gamma_4(t)] = +1$$ in the lowest-lying even-fermion-parity states, and define Pauli operators $$Z_t = i \gamma_1(t)\gamma_2(t),~~~ X_t = i \gamma_1(t) \gamma_3(t).$$ Our logical qubit states are then the minimum-energy many-body states with $Z_t = \pm 1$, i.e., $$\ket{0_t} \equiv \ket{Z_t = +1},~~~\ket{1_t} \equiv \ket{Z_t = -1}.$$ The subscript $t$ (suppressed in Sec. \[sec:executive\]) is included here and below as a reminder that these states are defined with respect to Majorana operators extracted from the instantaneous Hamiltonian at time $t$, rather than with respect to a fixed basis.
Some discussion is warranted regarding the definition of the maximally localized Majorana operators $\gamma_{1,2,3,4}(t)$ in our microscopic numerical simulations. The canonical modes $b_i$ of the quadratic Hamiltonian $H$ defined above \[cf. Eqs. and \] are not unique since arbitrary SO(2) rotations $O_k^{2\times2}=\exp(i\theta_k \sigma^y)$ within each $2\times2$ block of Eq. leave the matrix $B$ invariant: $$B = O\,B\,O^T,\quad O = \bigoplus_{k=1}^{N/2} O_k^{2\times2}. \label{eq:Btilde}$$ Thus, Majorana modes $\tilde{b}_i = [\tilde{\mathbf{b}}]_i$ defined through $$\tilde{\mathbf{b}} = O^T\,\mathbf{b} = \tilde{U}^T\mathbf{a}, \label{eq:btilde}$$ with $\tilde{U} = U\,O$, also form a canonical set with the same instantaneous energies, i.e., $$H = \frac{i}{4}\,\tilde{\mathbf{b}}^T\,B\,\tilde{\mathbf{b}} = \frac{i}{2} \sum_{k=1}^{N/2} \varepsilon_k \tilde{b}_{2k-1} \tilde{b}_{2k}. \label{eq:Hcanontilde}$$ The Majorana modes $\gamma_{1,2,3,4}(t)$ depicted in Fig. \[fig:qubits\] refer to $\tilde{b}_{1,2,3,4}$ with $O_{k=1,2}^{2\times2}$ chosen such that the associated wave functions maximally localize at a given time $t$. Specifically, we first find a set of canonical near-zero-energy modes $b_{1,2,3,4}$ by bringing $A$ into canonical form using the software package presented in Ref. [@wimmer_algorithm_2012]; in the local basis of the $a_i$, these modes are represented by column vectors of the matrix $U$, say $U_{i,j=1,2,3,4}$. We then optimize the $O_{k=1,2}^{2\times2}$ (each characterized by an angle $0\leq\theta_k<2\pi$) such that for the resulting $\tilde{U} = U\,O$ the 4^th^ moments $\sum_{j=1}^2\sum_{i=1}^N |\tilde{U}_{ij}|^4$ and $\sum_{j=3}^4\sum_{i=1}^N |\tilde{U}_{ij}|^4$ (for $k=1,2$, respectively) are maximized; now, in the local $a_i$ basis, these maximally localized modes $\tilde{b}_{1,2,3,4}$ are given by column vectors of $\tilde{U}$, i.e., $\tilde{U}_{i,j=1,2,3,4}$. We order these modes according to their real-space locations as shown in Fig. \[fig:qubits\], thereby giving the desired $\gamma_{1,2,3,4}(t)$ [^3].
As an aside, the numerical procedure spelled out above is tailored to the situation where the two superconductors comprising the qubit decouple—which again we assume for simplicity is the case throughout. The Majorana modes $\gamma_{1,2}(t)$ ($\gamma_{3,4}(t)$) then have support entirely on the top (bottom) wire of the tetron. In the more realistic scenario in which a nonzero ‘bridge’ Hamiltonian couples the wires, a physically plausible definition of $\gamma_{1,2,3,4}(t)$ involves instead maximizing the single 4^th^ moment $\sum_{j=1}^4\sum_{i=1}^N |\tilde{U}_{ij}|^4$ via a single orthogonal rotation $O^{4\times4}$ acting on the $b_{1,2,3,4}$. In the latter case $\gamma_{1,2,3,4}(t)$ are not generally eigenmodes of the Hamiltonian.
Protocol details
----------------
Section \[sec:executive\] summarized our time-domain Ramsey-type protocol for the physical implementation of the tetron. In the ‘ideal’ case the protocol involves (1) initializing the system into, say, the $X_{t = 0} = +1$ eigenstate at time $t = 0$ via energy-level spectroscopy on the nearby quantum dot which tunnel couples to $\gamma_1$ and $\gamma_3$, (2) letting the system evolve freely for time $t$ under the influence of classical noise, and (3) reading out $X_t = \pm1$ with the same quantum dot used for initialization. (For practical purposes, we will sometimes depart from this ‘ideal’ protocol by initializing and measuring with respect to a fixed basis.) In the remainder of this section we elaborate on these steps to set the stage for our analytic treatment of dephasing and noise in Sec. \[sec:analytics\] and our full time-dependent microscopic simulations of tetron qubits in Sec. \[NumericalResults\].
### Initialization
We start with initialization via the proximate quantum dot. At the level of our analysis, we assume that the quantum dot is ‘perfect’ in the sense that a measurement projects exactly and instantaneously onto the qubit state $$\ket{\psi(0)} = \frac{1}{\sqrt{2}}(\ket{0_{t = 0}} + \ket{1_{t = 0}}) \label{eq:psit0}$$ with $X_{t = 0} = +1$ at time $t = 0$. Under this highly idealized assumption, we need not explicitly include the quantum dot in our analysis. In our microscopic simulations of noisy tetron qubits in Sec. \[NumericalResults\], we use the *time-averaged* Hamiltonian—specified by an anti-symmetric matrix $A_0$—to define the relevant $\gamma_{1,2,3,4}$ used for initialization, which we denote $\gamma^{(0)}_{1,2,3,4}$. \[Obtaining the modes instead using the initial Hamiltonian $A(t=0) \neq A_0$ for each noise realization, as implied by Eq. , leads to only negligible quantitative differences in the results presented there; see Sec. \[sec:fixedvinst\] for more discussion on this point.\] Specifically, we bring $A_0$ into canonical form as described above in Sec. \[sec:models\] \[see Eq. \]; we subsequently find the corresponding set of maximally localized near-zero-energy Majorana modes, which are encoded in column vectors of an orthogonal matrix $\tilde{U}_0$. Next, we construct a reference covariance matrix $M_y(t=0)$ \[cf. Eqs. and \] in the basis of these modes $\tilde{b}_i^{(0)}$ corresponding to, say, $X_0 = i\gamma^{(0)}_1\gamma^{(0)}_3 = +1$ and $i\gamma^{(0)}_2\gamma^{(0)}_4 = -1$. The global fermion parity thus reads $(i\gamma^{(0)}_1\gamma^{(0)}_2)(i\gamma^{(0)}_3\gamma^{(0)}_4) = +1$, which for all presented simulations coincides with the parity of the absolute ground state of $A_0$. (Recall that modes $\tilde{b}_{i=1,2,3,4}$ within the low-energy manifold correspond to the maximally localized modes $\gamma_{1,2,3,4}$, while the remainder $\tilde{b}_{i\neq1,2,3,4}$ are equivalent to the respective original canonical modes $b_i$.) Finally, the initial covariance matrix in the $a_i$ basis \[cf. Eq. \] reads $M(t=0) = \tilde{U}_0\,M_y(t=0)\,\tilde{U}_0^T$.
### Unitary evolution {#sec:UnitaryEvolution}
Once initialized, the system undergoes unitary time evolution with respect to the noisy Hamiltonian $H(t)$. After time $t$ the initial state becomes $$\begin{aligned}
\ket{\psi(t)} &=& U(t) \ket{\psi(0)}
\nonumber \\
&\equiv& a_t \ket{0_t} + b_t \ket{1_t} + c_t \ket{v_t},
\label{psit}\end{aligned}$$ where $U(t)$ is the time evolution operator (we discuss the second line shortly). Our microscopic numerical simulations are instead carried out in the Heisenberg picture, for which unitary evolution is encoded in Eq. ; there noise is included through time dependence in $A(t)$, which is essentially the Hamiltonian expressed in the Majorana basis. Appendix \[app:noise\] details our treatment of noise (e.g., the procedure we employ for generating individual noise trajectories for a given realization) and numerical solution of Eq. .
We stress that in general $\ket{0_t} \neq U(t) \ket{0_{t = 0}}$ since $\ket{0_t}$ is defined through instantaneous $H(t)$ eigenstates (and similarly for $\ket{1_t}$) [^4]. In the second line of Eq. we thus expressed the time-evolved state in terms of qubit states $|0_t\rangle$ and $\ket{1_t}$ at time $t$, *and* a ket $\ket{v_t}$ that signifies dynamically generated excited states. Generation of weight on the latter via non-zero $c_t$ corresponds to ‘leakage’ of the qubit away from the computational subspace. Leakage can arise from an odd or even number of fermionic excitations, thus respectively flipping or preserving $P(t)$. Using Eq. we can define a qubit leakage time $T_{\rm leak}$ by writing $$\overline{|a_t|^2} + \overline{|b_t|^2} \equiv 1-f(t/T_{\rm leak}), \label{eq:a2pb2}$$ where the overline indicates noise averaging. On the right side, $f(t/T_{\rm leak})$ is a system-dependent function that vanishes at $t = 0$ and grows—thus shifting weight onto excited states—on a characteristic time scale $T_{\rm leak}$. An alternative leakage metric can be extracted from $$\overline{\bra{\psi(t)}P(t)\ket{\psi(t)}} \equiv 1- \tilde f(t/\tilde T_{\rm leak})
\label{Tleaktilde}$$ with $\tilde f$ a different function that grows on a time scale $\tilde T_{\rm leak}$. The latter measures the weight on excited states containing only odd numbers of fermionic excitations, and thus provides an upper bound on the leakage time defined above, i.e., $T_{\rm leak} \leq \tilde T_{\rm leak}$.
At this point it is worth commenting on an alternative scheme wherein one defines the qubit in terms of a fixed basis of maximally localized Majorana operators obtained from, say, the time-averaged Hamiltonian $A_0$ or, for a given noise realization, the initial Hamiltonian $A(t=0)$. For an extreme case, suppose that $H(t)$ supports exact instantaneous zero-energy Majorana modes at any time $t$, but that *adiabatic* noise causes the locations of the Majorana modes to vary with time. The system will then evolve out of the low-energy subspace spanned by the fixed Majorana operators, whereas the qubit subspace should clearly be perfectly preserved—as captured by tracking instantaneous Majorana operators. In general we expect that employing a fixed basis *overestimates* qubit errors, with the difference being most pronounced when the noise is ‘slow’ and of ‘large’ amplitude. We will quantify such effects later in Sec. \[sec:fixedvinst\].
### Readout {#sec:readout}
The final step of the protocol involves readout. We again assume that the nearby quantum dot can perform this task, i.e., measure $X_t = \pm1$, instantaneously and without error. Under this assumption the relevant physical quantity is the noise-averaged expectation value of $X_t$ with respect to the time-evolved state: $$\overline{\bra{\psi(t)} X_t \ket{\psi(t)}} = \cos(\omega_0 t) f(t/T_2),$$ which defines the qubit precession frequency $\omega_0$ and dephasing time $T_2$. Compared to Eq. , the $t$ subscript on the left side explicitly indicates that the expectation value is (ideally) taken with respect to the instantaneous Majoranas at time $t$.
To calculate the quantum expectation value $\langle\psi(t)|X|\psi(t)\rangle$ for a given noise realization in our microscopic simulations, we basically invert the initialization procedure described above. Specifically, we conjugate $M(t)$ with $\tilde{U}$ (which defines the measurement basis; see below) to arrive at $$M_y(t) = \tilde{U}^T\,M(t)\,\tilde{U}, \label{eq:M0tildeOft}$$ from which any $\langle i\gamma_i\gamma_j\rangle$ can be read off directly. Measurements involving a product of more than two operators can be evaluated with use of Wick’s theorem; an important example is $P(t)$, which (partially) probes leakage out of the computational subspace \[recall Eq. \].
As with initialization, there exists a choice as to precisely what is meant by the Majorana operators $\gamma_{1,2,3,4}$ being measured. Namely, these could be defined as (maximally localized) near-zero-energy modes with respect to, say, the *fixed* time-averaged Hamiltonian, $A_0$, or the *instantaneous* Hamiltonian, $A(t)$ [^5] Operationally, to measure in the basis corresponding to the time-averaged Hamiltonian such that $\gamma_{1,2,3,4} = \gamma^{(0)}_{1,2,3,4}$, we use Eq. with $\tilde{U} = \tilde{U}_0$ (the same orthogonal matrix used in the initialization procedure). On the other hand, to measure in the instantaneous basis we need to find a set of maximally localized modes $\gamma_{1,2,3,4} = \gamma_{1,2,3,4}(t)$ at every time $t$, which are encoded in a time-dependent orthogonal matrix $\tilde{U} = \tilde{U}(t)$ to be used in Eq. . In our microscopic simulations, we implement both instantaneous and fixed measurement bases. Employing a fixed basis is computationally cheaper as it does not require bringing the Hamiltonian into canonical form and finding maximally localized modes at each time step. Additionally, there is an inherent sign ambiguity in the canonical modes (and thus also maximally localized modes) upon bringing $A(t)$ into canonical form numerically, i.e., the Hamiltonian is invariant upon taking $b_{2k-1}\to-b_{2k-1}$ and $b_{2k}\to-b_{2k}$ for a given $k$ in Eq. . Therefore, measurements in the instantaneous basis such as $\langle i\gamma_1(t)\gamma_3(t)\rangle$ will be plagued by $\gamma_1(t)$ and $\gamma_3(t)$ having arbitrary relative sign for different $t$. For these reasons, we use the fixed basis defined by the time-averaged Hamiltonian for the data presented in Secs. \[NumericalResults\]. Note that measuring in the fixed basis corresponding to the time-averaged Hamiltonian plausibly mimics ‘imperfect’ quantum dot readout that is slow on the time scale of the typical noise correlation time. In Sec. \[sec:fixedvinst\], we circumvent the sign ambiguity problem in the instantaneous basis by noise averaging instead $\langle i\gamma_1(t)\gamma_3(t)\rangle^2$ and are thereby able to directly compare the two measurement approaches (and thus qubit encodings).
Analytical estimation of qubit time scales {#sec:analytics}
==========================================
In this section we use standard techniques to derive analytic formulas for the dephasing and leakage times. These estimates will be compared with time scales extracted from our time-dependent numerical simulations in Sec. \[NumericalResults\].
Dephasing time {#sec:T2_analytic}
--------------
We first restrict attention to the low-energy subspace, neglecting the quasiparticle continuum above the superconducting gap. This approximation is justified provided noise is slow compared to the gap scale. Leakage of the qubit into excited states will be discussed in the next subsection.
Within the low-energy subspace, we model the qubit by the minimal two-level Hamiltonian in Eq. (\[qubitH\]). Time dependence in the instantaneous energy splitting $E(t)$ arises due to stochastic temporal variations in parameters in the ‘parent’ microscopic Hamiltonian. We denote fluctuations in these microscopic parameters about their time-averaged values by $\lambda_i(t)$, which are assumed uncorrelated with one another. This set can include variations in the electrochemical potential, magnetic field, etc. in different spatial regions of the device. For example, if only the global chemical potential varies in time by an amount $\delta\mu(t)$, then we simply have $\lambda_1(t) = \delta \mu(t)$; if instead uncorrelated chemical potential variations arise in the left and right halves of the wires, then we have $\lambda_1(t) = \delta \mu_L(t)$ and $\lambda_2(t) = \delta\mu_R(t)$; and so on. Assuming the fluctuations are weak, we Taylor expand the energy to second order in $\lambda_i$’s, yielding $$E(t) \approx E_0 + \sum_i \lambda_i(t)E'_i + \frac{1}{2} \sum_{i,j} \lambda_i(t)\lambda_j(t) E''_{ij},
\label{E_expansion}$$ where we defined short-hand notation $$E'_i = \frac{dE}{d\lambda_i}|_{\lambda_i = 0},~~~E''_{ij} = \frac{d^2E}{d\lambda_i d\lambda_j}|_{\lambda_{i,j} = 0}.$$
To noise average we will assume that each $\lambda_i$ exhibits Gaussian noise correlations with mean and variance $$\begin{aligned}
\overline{\lambda_i(t)} &= 0,
\label{lambda_ave}
\\
\overline{\lambda_i(t)\lambda_j(t')} &= \delta_{ij}S_i(t-t').
\label{noise_corr}\end{aligned}$$ We further take $S_i(t)$ to be Gaussian as a function of time, i.e., $$S_i(t) = D_i^2 e^{-t^2/(2\tau_i)^2},
\label{noise_power}$$ corresponding to a power spectrum $$\tilde S_i(\omega) = D_i^2 \sqrt{\frac{4\pi}{\kappa_i^2}}\, e^{-(\omega/\kappa_i)^2}.
\label{eq:Sw_Gaussian}$$ In Eqs. and , $D_i$ and $\tau_i \equiv 1/\kappa_i$ respectively denote the fluctuation amplitude and characteristic noise correlation time for fluctuator $\lambda_i$; note that the frequency scale $\kappa_i$ plays the role of a high-frequency cutoff in the power spectrum. The qubit precession frequency $\omega_0$ then follows from $$\hbar \omega_0 = \overline{E(t)} = E_0 + \frac{1}{2} \sum_i D_i^2 E''_{ii}.
\label{omega0_shifted}$$ Notice that noise generically shifts the mean energy splitting $\hbar \omega_0$ away from the noise-free value $E_0$.
We could in principle also adopt the more realistic case of $1/f$ noise, which leads to similar qualitative conclusions (see, e.g., Ref. [@aasen_milestones_2016] for a discussion in a related context). However, in the microscopic models discussed below, $1/f$ noise requires the introduction of a high-frequency cutoff which leads to corrections that make quantitative comparisons between theoretical and numerical estimates challenging. In the two-level system approximation used here, it is straightforward to evaluate the readout for the Ramsey-type protocol described in the preceding section. For a particular noise realization one obtains $$\begin{aligned}
Q(t) \equiv \bra{\psi(t)} X_t \ket{\psi(t)} = \cos\left(\int_0^t dt' E(t')/\hbar \right).
\label{Pt}\end{aligned}$$ Appendix \[app:averaging\] noise averages $Q(t)$; in the long-time limit $t \gg \tau_i$ we find $$\overline{ Q(t \gg \tau_i)} \approx \cos(\omega_0t)e^{-t/T_2}
\label{eq:T2analytic_func}$$ with dephasing time $$T_2 = \frac{\hbar^2}{\sqrt{\pi}}\left[\sum_i\tau_i\left(D_i E_i'\right)^2 + \frac{1}{2}\sum_{ij}\frac{\tau_i \tau_j}{\sqrt{\tau_i^2 +
\tau_j^2}}\left(D_i E_{ij}''D_j\right)^2 \right]^{-1}.
\label{T2analytic}$$ (Appendix \[app:averaging\] also examines the short-time limit $t \ll \tau_i$, which is sometimes relevant for our numerical simulations.) Upon varying parameters, e.g., to transition between an ABS and topological qubit, we expect that $T_2$ will vary by far most dramatically through the $E_i'$ and $E_{ij}''$ factors. These factors quantify the rigidity of the qubit splitting with respect to fluctuations and can be efficiently derived from microscopics through exact diagonalization of *noise-free* static Hamiltonians, in principle containing various bells and whistles. We will carry out such an analysis in Sec. \[NumericalResults\] in parallel with our explicit time-dependent qubit simulations.
Leakage time {#sec:Tleak_analytic}
------------
We will estimate the qubit leakage time by studying just a single noisy wire in the tetron using Fermi’s golden rule. For analytical tractability, we will further employ a *fixed basis* for Majorana modes in the device; as stressed in Sec. \[sec:UnitaryEvolution\] the analytical estimates that follow should thus be viewed as an *upper bound* on the true leakage time that would be obtained by following the instantaneous basis of maximally localized Majorana modes.
Let us describe the system within the spinful-wire model (extension to the Kitaev-chain description is straightforward). We write the full microscopic Bogoliubov-de Gennes Hamiltonian as $H=H_0+\delta H(t)$, where $H_0$ is given by Eq. and $\delta H(t)$ incorporates noise. The static part of the Hamiltonian is diagonalized upon decomposing the fermion $\psi_\sigma(x)$ with spin $\sigma$ at position $x$ as $$\psi_\sigma(x) = \Phi_{1\sigma}(x) \gamma_1^{(0)} + \Phi_{2\sigma}(x) \gamma_2^{(0)} + \sum_{E>0} \varphi_{E\sigma}(x) f_E.
\label{psi_decomposition}$$ Here $\Phi_{1,2\sigma}$ denote the maximally localized Majorana wave functions associated with the fixed-basis Majorana operators $\gamma_{1,2}^{(0)}$ derived from $H_0$, while $\varphi_{E\sigma}$ denote wave functions for above-gap excitations created by operators $f_E^\dagger$. After this decomposition $H_0$ becomes $$H_0 = \frac{i}{2} \varepsilon_{12} \gamma_1^{(0)} \gamma_2^{(0)} + \sum_{E>0} E f_E^\dagger f_E$$ with $\varepsilon_{12}$ the noise-free splitting of states with $i\gamma_1 \gamma_2 = \pm 1$.
We write the noise terms as $$\delta H(t) = \sum_i \lambda_i(t) \int_0^L dx \psi^\dagger \Lambda_i(x) \psi$$ for some matrices $\Lambda_i(x)$ dependent on the nature of the parameter fluctuations $\lambda_i(t)$. For instance, if $\lambda_1(t) = \delta \mu(t)$ represents a global chemical potential fluctuation, then $\Lambda_1 = -I$. Note that we neglect noise in the pairing channel for simplicity, though such terms could be readily incorporated if desired. Rewriting $\delta H(t)$ using Eq. yields $$\delta H(t) = \sum_{E>0} \{[A_{1E}(t) \gamma_1^{(0)} + A_{2E}(t) \gamma_2^{(0)}]f_E + \mathrm{H.c.}\} + \cdots.$$ The ellipsis denotes pieces involving two Majorana operators or two $f_E$ operators, while the coefficients above read $$A_{1,2E}(t) = \sum_i \lambda_i(t) \int_0^L dx \Phi_{1,2}^\dagger(x) \Lambda_{i}(x) \varphi_{E}(x).$$ Next we introduce the complex fermion $f_0 = (\gamma_1 + i \gamma_2)/2$ and define $A_{\pm,E}(t) = A_{1E}(t) \pm i A_{2E}(t)$ so that $$\delta H(t) = \sum_{E>0} \{[A_{-,E}(t) f_0 + A_{+,E}(t) f_0^\dagger]f_E + \mathrm{H.c.}\} + \cdots.$$
According to Fermi’s golden rule, the noise-averaged leakage rates $\Gamma_\pm$ out of the ground states with $f_0^\dagger f_0=0$ ($\Gamma_-$) and $f_0^\dagger f_0=1$ ($\Gamma_+$) are given by $$\Gamma_\pm = \frac{2\pi}{\hbar} \sum_{E>0} \int_{\omega,\omega'} \overline{A_{\pm,E}(\omega)A_{\pm,E}^*(\omega')}\delta(E\mp \varepsilon_{12}-\hbar\omega).$$ The noise correlations in Eqs. through yield $$\begin{aligned}
\overline{A_{\pm,E}(\omega)A_{\pm,E}^*(\omega')} &=2\pi \delta(\omega-\omega')
\nonumber \\
&\times 2 \sqrt{\pi}\sum_j\tau_j D_j^2 e^{-(\omega \tau_j)^2}B_{\pm,j}(E) ,\end{aligned}$$ where $$\begin{aligned}
B_{\pm,j}(E) =
\bigg{|}\int_x[\Phi_1^\dagger(x) \pm i \Phi_2^\dagger(x)]\Lambda_j(x) \varphi_E(x)\bigg{|}^2.
\label{B}\end{aligned}$$ Feeding this expression into $\Gamma_\pm$ leads to $$\Gamma_\pm = 2 \sqrt{\pi}\sum_j\frac{\tau_j D_j^2}{\hbar^2}\sum_{E>0} B_{\pm,j}(E) e^{-[(E\mp \varepsilon_{12}) \tau_j/\hbar]^2} .$$ Suppose now that $E_g$ is the bulk gap and $\rho(E)$ is the density of states for the above-gap excitations; assuming the coefficients $B_{\pm,j}$ vary smoothly with energy, we approximate the leakage rates as $$\begin{aligned}
\Gamma_\pm &\approx 2 \sqrt{\pi}\sum_j\frac{\tau_j D_j^2}{\hbar^2}B_{\pm,j}(E_g)
\nonumber \\
&\times
\int_{E_g}^\infty dE\rho(E) e^{-[(E\mp \varepsilon_{12}) \tau_j/\hbar]^2} .\end{aligned}$$
Leakage will be most important in the vicinity of the topological phase transition where the bulk gap approaches zero. In this regime, the excitation spectrum can be viewed as arising from gapped-out counterpropagating bulk Majorana fermions with mode velocity $v$ and dispersion $$E(k) = \sqrt{(\hbar vk)^2 + E_g^2}.
\label{Ek}$$ The density of states follows as $$\rho(E) = \frac{L}{\pi \hbar v}\frac{E}{\sqrt{E^2-E_g^2}}$$ for $E>E_g$ and vanishes otherwise. \[We have implicitly assumed periodic boundary conditions in writing Eq. , though the choice of boundary conditions should not significantly influence $\rho(E)$.\] Assuming $\varepsilon_{12} \ll E_g$, we then obtain $$\Gamma_\pm \approx \frac{L}{\hbar v}\sum_j\frac{ D_j^2}{\hbar}B_{\pm,j}(E_g) e^{-(E_g \tau_j/\hbar)^2}.
\label{GammaFinal}$$
The coefficients $B_{\pm,j}(E_g)$ depend on details of the wave functions and noise sources. We will use scaling arguments to roughly quantify these factors in three regimes:
$(i)$ *Ultra-short-range noise*. Suppose first that a noise source acts near one of the two Majorana modes, but only on a very local region of size $\xi^{\rm noise}_{j} \ll \xi$ (again, $\xi$ denotes the spatial extent of the Majorana wave function). In this case $\Lambda_j(x)$ in Eq. has appreciable weight only over a distance $\xi^{\rm noise}_j$ so that $$B_{\pm,j}(E_g) \propto \left(\frac{\xi^{\rm noise}_j}{\sqrt{\xi L}}\right)^2 = \left(\frac{\xi^{\rm noise}_j}{\xi}\right)^2 \frac{\xi}{L}.$$ The square root in the denominator simply reflects normalization factors in the Majorana and above-gap-excitation wave functions. Note that $B_{+,j} = B_{-,j}$ here, which implies approximately equal [^6] leakage rates out of the low-energy states with $i\gamma_1 \gamma_2 = \pm 1$.
$(ii)$ *Short-range noise*. If $\gamma_{1,2}$ are separated by a sufficiently long distance $L_{12}$, one can envision a short-range-noise scenario wherein $\xi < \xi^{\rm noise}_j < L_{12}$. That is, the noise source acts over the entire window of one of the Majorana modes, but does not influence its partner. (This scenario is reasonable for bona fide Majorana zero modes separated by the length of the wire, but is less natural for accidental Andreev bound states.) Here we get $$B_{\pm,j}(E_g) \propto \left(\frac{\xi}{\sqrt{\xi L}}\right)^2 = \frac{\xi}{L},$$ again yielding equal leakage rates out of the $i\gamma_1 \gamma_2 = \pm 1$ states.
$(iii)$ *Long-range noise*. Finally, consider a long-range-correlated noise source with $\xi^{\rm noise}_j > L_{12}$. The integral in Eq. now picks up weight from *both* Majorana wave functions, yielding similar scaling to the previous case but with unequal leakage rates for the $i\gamma_1\gamma_2 = \pm 1$ states: $$B_{\pm,j}(E_g) \propto (1\pm \eta_j) \frac{\xi}{L}$$ for $-1< \eta_j <1$. This regime offers the interesting possibility of a ‘dark state’: when $\eta_j = \pm1$ the transition rate for one of the two low-energy states vanishes.
The three cases above can be put on equal footing by writing $$B_{\pm,j}(E_g) \approx c_j (1\pm \eta_j)\left(\frac{\xi^{\rm noise}_j}{\xi}\right)^{b_j} \frac{\xi}{L},$$ the proportionality constant $c_j$, exponent $b_j$, and $\eta_j$ are determined by properties of noise source $j$. We thereby obtain leakage times $$\begin{aligned}
T_{\rm leak}^\pm &= 1/\Gamma_\pm \nonumber \\
&\approx \hbar^2\bigg{[}\sum_i c_i \frac{\xi}{v} D_i^2 (1\pm \eta_i)\left(\frac{\xi^{\rm noise}_i}{\xi}\right)^{b_i}e^{-(E_g \tau_i/\hbar)^2} \bigg{]}^{-1}.
\label{Tleak_analytic}\end{aligned}$$ Comparison to the dephasing time in Eq. (first term) is instructive. We see that the time scale $\xi/v$ appearing in $T_{\rm leak}^\pm$ plays the role of $\tau_i$ in $T_2$; similarly, the exponential factor $e^{-(E_g \tau_i/\hbar)^2}$ plays the role of the ‘energetic rigidity’ $E_i'$ characterizing the qubit states. Upon approaching the topological phase transition—either from the Andreev-bound-state or true Majorana-zero-mode regime—collapse of the bulk gap $E_g$ readily allows leakage to dominate the qubit lifetime.
Numerical Results {#NumericalResults}
=================
We are now in position to present results on both quasi-analytic evaluation of the dephasing and leakage times (using the formulas derived in Secs. \[sec:T2\_analytic\] and \[sec:Tleak\_analytic\]) as well as full microscopic simulations of noisy tetron dynamics for both the Kitaev and spinful nanowire models.
Kitaev tetron {#sec:kitaev}
-------------
We first consider a tetron qubit built from two parallel Kitaev chains \[Eq. \] each consisting of $L$ physical sites. The time-averaged chemical potentials of the two wires are offset by a small amount $\mu_\mathrm{offset}$ to break degeneracies which would occur for identical wires; that is, $\mu^{(0)}_{\mathrm{top}/\mathrm{bot}} = \mu \pm \mu_\mathrm{offset}$. We adopt a noise model consisting of two independent fluctuators acting on the respective chemical potentials such that $\mu_{\mathrm{top}/\mathrm{bot}}(t) = \mu^{(0)}_{\mathrm{top}/\mathrm{bot}} + \delta\mu_{\mathrm{top}/\mathrm{bot}}(t)$. Each fluctuator $\delta\mu_{\mathrm{top}/\mathrm{bot}}(t)$ obeys Gaussian noise correlations with a Gaussian noise power spectrum as specified in Eqs. through . For simplicity, we take the amplitude of typical fluctuations $D$ and noise correlation time $\tau\equiv1/\kappa$ equal for both fluctuators: $D_{\mathrm{top}/\mathrm{bot}} = \delta\mu_{\mathrm{top}/\mathrm{bot}}^\mathrm{typ} = \delta\mu_\mathrm{typ}$ and $\kappa_{\mathrm{top}/\mathrm{bot}} = \kappa$.
![Energy spectra of the Kitaev tetron as a function of $\mu$ for the configuration used in our microscopic numerical simulations, cf. Figs. \[fig:noise\_averaging\], \[fig:comp\_Kitaev\], \[fig:meas\_basis\_mu0.15\], \[fig:meas\_basis\_mu0.8\], and \[fig:single\_realization\]. Here, each wire of the tetron consists of an $L=10$ site Kitaev chain with $J_\mathrm{top/bot}=J=1$ and $\Delta_\mathrm{top/bot}=\Delta=0.4$; a slight chemical potential shift of $\mu_\mathrm{offset}=0.01$ is introduced between the two wires to break any associated degeneracies. The low-energy qubit manifold is highlighted in orange, corresponding to the energies $\varepsilon_{1,2}$ in Eq. . In the inset, we plot the spatial dependence of the wave function square modulus for the maximally localized Majorana modes $\gamma_{1,2,3,4}$ (see Sec. \[sec:qubitdef\]) at $\mu=0.15$. \[fig:kitaev\_setup\]](spectra-and-wfs_kitaev.pdf){width="0.9\columnwidth"}
![Example single noise realizations (top) and noise-averaged data (bottom) for the (quantum-averaged) quantity $\langle i\gamma_1\gamma_3 \rangle$ (with $\gamma_{1,3}=\gamma_{1,3}^{(0)}$) simulated in the Kitaev tetron of Fig. \[fig:kitaev\_setup\], here at $\mu=0.15$. The top panel shows the results of ten independent noise realizations (‘noisy’; light curves) with $\kappa=0.05$ and $\delta\mu_\mathrm{typ}=0.02$ (one global $\mu$ fluctuator on each wire of the tetron), as well as for a system without noise (‘clean’; black curve). In the bottom panel, we present the noise-averaged signal (‘numerics’; solid blue) and a fit (‘fit’; dashed orange) guided by the theoretical prediction of Eq. . The theoretical prediction itself (and its envelope) obtained using Eqs. - is also plotted (‘theory’; dashed gray). The orange and gray vertical dotted lines indicate the respective $T_2$ times. \[fig:noise\_averaging\]](ig1g3_realizations.pdf "fig:"){width="0.9\columnwidth"} ![Example single noise realizations (top) and noise-averaged data (bottom) for the (quantum-averaged) quantity $\langle i\gamma_1\gamma_3 \rangle$ (with $\gamma_{1,3}=\gamma_{1,3}^{(0)}$) simulated in the Kitaev tetron of Fig. \[fig:kitaev\_setup\], here at $\mu=0.15$. The top panel shows the results of ten independent noise realizations (‘noisy’; light curves) with $\kappa=0.05$ and $\delta\mu_\mathrm{typ}=0.02$ (one global $\mu$ fluctuator on each wire of the tetron), as well as for a system without noise (‘clean’; black curve). In the bottom panel, we present the noise-averaged signal (‘numerics’; solid blue) and a fit (‘fit’; dashed orange) guided by the theoretical prediction of Eq. . The theoretical prediction itself (and its envelope) obtained using Eqs. - is also plotted (‘theory’; dashed gray). The orange and gray vertical dotted lines indicate the respective $T_2$ times. \[fig:noise\_averaging\]](ig1g3_avg.pdf "fig:"){width="0.9\columnwidth"}
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Given such a setup, it is straightforward to numerically evaluate the analytic (perturbative) estimates for the dephasing and leakage times as derived above in Sec. \[sec:analytics\]. In addition, we perform full microscopic numerical experiments of the noisy tetron dynamics by generating $N_\mathrm{real}$ independent noise realizations, evolving under the time-dependent Schrödinger equation \[which in terms of the covariance matrix is given by Eq. \] for each realization, and finally noise-averaging the resulting measurements. (Details of our noise generation procedure and subsequent solution of the evolution equation can be found in Appendix \[app:noise\].) As spelled out above in Sec. \[sec:protocol\], for our Ramsey-type protocol we initialize the system into a state corresponding to fixed $X_0 = i\gamma^{(0)}_1\gamma^{(0)}_3 = +1$, where the four near-zero-energy Majorana modes $\gamma^{(0)}_{1,2,3,4}$ are defined with respect to the time-averaged Hamiltonian $A_0$. Throughout the evolution we monitor the expectation values $\langle i\gamma_1\gamma_3\rangle$, $\langle i\gamma_1\gamma_3\rangle^2$ (see Sec. \[sec:readout\]), and $\langle(i\gamma_1\gamma_2)(i\gamma_3\gamma_4)\rangle$ with $\gamma_{1,2,3,4}$ defined in terms of either the fixed, time-averaged basis used for initialization \[i.e., $\gamma_{1,2,3,4} = \gamma^{(0)}_{1,2,3,4}$\] or the basis derived from the instantaneous Hamiltonian $A(t)$ \[i.e., $\gamma_{1,2,3,4} = \gamma_{1,2,3,4}(t)$\]. Finally, we also track the overlap of the evolving wave function onto the low-energy subspace, for both the fixed and instantaneous basis, respectively corresponding to $|a_0|^2 + |b_0|^2$ and $|a_t|^2 + |b_t|^2$ from Eq. .
Within a topological phase, both the qubit precession period $2\pi/\omega_0$ and dephasing time $T_2$ are expected to be exponentially long in the wire length $L$. Simulating the full microscopic dynamics over these time scales thus becomes prohibitive for modest $L$ even for noninteracting electronic models. Evaluating the relevant time scales using the analytic perturbative predictions, on the other hand, faces no such limitations as this procedure merely requires computing derivatives of the lowest-energy Majorana hybridization energies, which are obtainable with a sparse eigensolver applied to the Hermitian matrix $iA$.
We present data obtained for numerical experiments performed on a tetron built from two Kitaev wires each containing $L = 10$ physical sites, and with varyious $\mu, \kappa$ but all other parameters fixed according to $J_{\mathrm{top}/\mathrm{bot}}=J=1$, $\Delta_{\mathrm{top}/\mathrm{bot}}=\Delta=0.4$, $\mu_\mathrm{offset}=0.01$, and—when noise is present—$\delta\mu_\mathrm{typ}=0.02$. (With $\hbar=J=1$, energies are in units of $J$, times are in units of $\hbar/J$, and frequencies, e.g., $\kappa$, are in units of $J/\hbar$.) Figure \[fig:kitaev\_setup\] shows the energy spectra for this qubit configuration as a function of $\mu$ in the noise-free limit. Note that for this system size the topological ‘phase transition’ near $\mu = 1$ is significantly smeared by finite-size effects. Within the topological regime at $|\mu | \lesssim 1$, however, fairly well-formed Majorana zero modes arise. The inset plots the spatial profiles of the near-zero-energy maximally localized Majorana modes $\gamma_{1,2,3,4} = \gamma^{(0)}_{1,2,3,4}$ at $\mu = 0.15$.
Figure \[fig:noise\_averaging\] contains simulation results for a noisy system with $\mu = 0.15$ and $\kappa = 0.05$. The top panel shows the evolution of $\langle i\gamma_1\gamma_3\rangle$—defined here in terms of the fixed basis shown in the inset of Fig. \[fig:kitaev\_setup\]—for ten independent noise realizations. Also shown for comparison are data for a noiseless run (thick black curve). The bottom panel shows the results of averaging $N_\mathrm{real} = O(10^3)$ noise realizations [^7], a fit of this noise-averaged numerical data to the functional form $\cos(\omega_0 t) e^{-t/T_2}$, as well as the time dependence predicted [^8] by the (small-noise-amplitude) analytic calculations of Sec. \[sec:T2\_analytic\] \[cf. Eqs. and \]. We see excellent agreement between the numerical data and analytical prediction. These parameters yield negligible leakage (as determined by the relevant measurements discussed above; not shown) on the time scale of the dephasing time. That is, $T_\mathrm{leak} \gg T_2$.
Finally, in Fig. \[fig:comp\_Kitaev\], we present results of a comprehensive study varying $\mu$ at different values of $\kappa=0.001, 0.01, 0.05$, and $0.1$. The bottom panels plot the qubit precession period $2\pi/\omega_0$ versus $\mu$ as obtained by both the microscopic numerical simulations (‘numerics’) and analytic prediction (‘theory’) \[see Eq. and note that the predicted qubit precession frequency is independent of $\kappa$\]. The top panels similarly show the corresponding coherence times $T_2$ and, in the rightmost plot, $T_\mathrm{leak}$. Numerical $T_2$ data corresponds to fits of the noise-averaged, fixed-basis $\langle i\gamma_1\gamma_3\rangle$ data to an oscillatory exponential, precisely as in Fig. \[fig:noise\_averaging\]. For the analytic $T_2$ predictions, we show the time scales predicted both in the limit $t \gg 1/\kappa$ (‘$T_2$ theory’) and $t \ll 1/\kappa$ (‘$T_2$ theory (transient)’). In the transient case, the decay is expected to follow a Gaussian instead of an exponential \[see Eqs. and in Appendix \[app:averaging\]\]. As a guide, in the left two panels where the noise is ‘slow’, the horizontal green lines indicate the time scale $\tau = 1/\kappa$. Coherence times falling below that scale indicate importance of the transient regime. For consistency, however, we fit the data for all numerical experiments to an exponential form. Nonetheless, when the coherence times fall below $\tau = 1/\kappa$, the extracted time scales from the numerics track the $t \ll 1/\kappa$ prediction reasonably well. To summarize, when the dephasing time measured in the numerics is much larger than $1/\kappa$, it matches the (long time) analytic prediction very well; and when it dips below $1/\kappa$, it follows the (short time, ‘transient’) analytic prediction.
As in the example presented in Fig. \[fig:noise\_averaging\], for all of the data points with $\kappa \leq 0.05$ in Fig. \[fig:comp\_Kitaev\], we detect negligible leakage out of the low-energy qubit subspace over the time scale of dephasing. Only for $\kappa=0.1$ at $\mu\sim1$ upon exiting the topological phase do we observe appreciable leakage. The quantities $T_\mathrm{leak}$ in the rightmost panel of Fig. \[fig:comp\_Kitaev\] were extracted by fitting $|a_t|^2 + |b_t|^2$ \[see Eqs. -\] to the (phenomenological) form $Ae^{-t/T_\mathrm{leak}} + (1-A)$; only data points for which $A>0.1$ are plotted. We see from the energy spectra in Fig. \[fig:kitaev\_setup\] that the dip in leakage times for $\mu\lesssim1$ coincides with the finite-size crossover into the trivial phase on this small $L=10$ site Kitaev tetron, i.e. the point where the gap is minimal. On the other hand, for small $\kappa \leq 0.05$ the minimum finite-size gap exceeds $\kappa$ so that the qubit lifetime is governed by the dephasing time $T_2$ for all $\mu$.
Spinful nanowire tetron {#sec:spinful}
-----------------------
![Energy spectra (top) and representative near-zero-energy Majorana wave functions (bottom) for the continuum-limit $L=2~\mu\mathrm{m}$ spinful tetron configuration analyzed in Fig. \[fig:Vzscan\_summary\]. The chosen parameters were inspired by Ref. [@moore_two-terminal_2018] (see text for all details) to give a sizable window of a non-topological ABS regime for $V_{zc}' < V_z < V_{zc}$ before the onset of the topological phase transition at $V_{zc}$. In the bottom panels, we show the square modulus of the wave functions for the maximally localized Majorana modes $\gamma_{1,2,3,4}$ (summing both spin components) at $V_z=0.6~\mathrm{meV}$ (partially-separated ABS qubit) and $V_z=1.4~\mathrm{meV}$ (topological qubit). \[fig:spinful\_setup\_cont\]](spectra-and-wfs_spinful-cont.pdf){width="0.9\columnwidth"}
![Energy spectra (top) and representative near-zero-energy Majorana wave functions (bottom) for the $L=0.8~\mu\mathrm{m}$ spinful tetron configuration studied in Fig. \[fig:comp\_spinful\]; this data is analogous to Fig. \[fig:spinful\_setup\_cont\] but with some parameter changes—see text for all details. Here we discretize the spinful nanowire model of Eq. onto $N_\mathrm{site}=22$ sites per wire to make subsequent full microscopic simulations of the noisy dynamics numerically tractable, while still maintaining several of the qualitative features of the continuum-limit model of Figs. \[fig:Vzscan\_summary\] and \[fig:spinful\_setup\_cont\]. \[fig:spinful\_setup\_disc\]](spectra-and-wfs_spinful-disc.pdf){width="0.9\columnwidth"}
We now turn to tetrons assembled from two spinful nanowires governed by the Hamiltonian defined in Eq. . Guided by the data presented in Ref. [@moore_two-terminal_2018], we focus on a set of parameters that yields low-energy edge ABSs over a sizable window of Zeeman fields $V_{zc}' < V_z < V_{zc}$ before entering a topological regime with bona fide Majorana zero modes at larger fields $V_z > V_{zc}$. Above in Fig. \[fig:Vzscan\_summary\], we presented data of qubit coherence times and precession periods for such a system as a function of $V_z$ evaluated numerically according to the analytical estimates of Sec. \[sec:analytics\]. The corresponding (time-averaged) system parameters are as follows: $\Delta(x)=\frac{\Delta_0}{2}\left[\tanh\left(\frac{x-x_0}{\ell_\Delta}\right)+1\right]$ with $\Delta_0=0.25~\mathrm{meV}$, $\ell_\Delta=0.03~\mu\mathrm{m}$, and $x_0=0.3~\mu\mathrm{m}$; $V(x)=\frac{V_0}{2}\left[-\tanh\left(\frac{x-x_0}{\ell_V}\right)+1\right]$ with $V_0=3.8\Delta_0$ and $\ell_V=0.03~\mu\mathrm{m}$; $m=0.03m_e$; $\mu=V_0=3.8\Delta_0$ (with top/bottom wire offset $\mu_\mathrm{offset}=0.01~\mathrm{meV}$); and $\alpha=500~\mathrm{meV}\,\mathrm{\AA}$. For our discretization, we take a lattice spacing $a\approx0.01~\mu\mathrm{m}$ corresponding to $N_\mathrm{sites}=100,200,300,400$ sites (per wire) for the sequence of sizes $L=1,2,3,4~\mu\mathrm{m}$. In Fig. \[fig:spinful\_setup\_cont\], we show additional data for the $L=2~\mu\mathrm{m}$ system. The top panel plots the energy spectrum versus $V_z$ while the bottom panels illustrate the spatial profiles of the maximally localized near-zero-energy Majorana modes $\gamma_{1,2,3,4}$ at $V_z = 0.6~\mathrm{meV}$ (non-topological ABS qubit) and $1.4~\mathrm{meV}$ (topological qubit). The chosen noise configuration consists of four independent fluctuators: one global, spatially constant chemical potential ($\mu$) and Zeeman field ($V_z$) fluctuator per wire each with Gaussian correlated noise with inverse correlation times $\kappa=0.05~\mathrm{meV}/\hbar$ and typical fluctuation amplitudes $\delta\mu_\mathrm{typ}=\delta V_{z,\mathrm{typ}}=0.01~\mathrm{meV}$.
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Given these parameters, the $T_2$ and $\omega_0$ values plotted in Fig. \[fig:Vzscan\_summary\] were obtained by numerically evaluating Eqs. and via sparse diagonalization of the Hermitian matrix $iA$ and computing the requisite energy derivatives. (Only the leading-order terms were included in the evaluations for the more conceptual Fig. \[fig:Vzscan\_summary\], while for Figs. \[fig:comp\_Kitaev\] and \[fig:comp\_spinful\] the second-order terms were also calculated and are in fact necessary to obtain quantitative agreement with the microscopic numerics.) For $T_\mathrm{leak}$ we take a simplified approach which focuses entirely on the predicted exponential dependence: $T_\mathrm{leak} \sim e^{{(E_g/\hbar\kappa)}^2}$. Specifically, we evaluate Eq. at fixed $c_i = 1$, $\eta_i = 0$, $b_i = 0$, $v = \alpha/\hbar~(= 500~\mathrm{meV}\,\mathrm{\AA}/\hbar)$, and $\xi = 200~\mathrm{nm}$ (recall $D_i = \delta\mu_\mathrm{typ} = \delta V_{z,\mathrm{typ}} = 0.01~\mathrm{meV}$ and $\kappa_i = \kappa = 0.05~\mathrm{meV}/\hbar$). While the analysis of Sec. \[sec:Tleak\_analytic\] considered only a single nanowire, it can be safely applied to the noisy tetron considered here as the wires are taken to be decoupled and the spatial profiles of all noise sources have support on only one wire or the other (two global fluctuators per wire). In evaluating Eq. , we thus consider only two fluctuators, and for the excitation gap $E_g$, we take an average of the excitation gaps \[$\varepsilon_{3,4}$ from Eq. \] for the top and bottom wires (recall the finite $\mu_\mathrm{offset}$). This procedure ignores the complicated dependencies of the (qualitatively less important) subexponential factors on the wave function amplitudes, noise profile details, etc., but it faithfully captures the dominant exponential dependence and thus suffices for our purposes here.
As highlighted in Sec. \[sec:executive\], Fig. \[fig:Vzscan\_summary\] demonstrates several important points: First, within the ABS regime ($V_{zc}' < V_z < V_{zc}$) the qubit coherence time and precession frequency exhibit weak, nonuniversal dependence on wire length $L$. Second, the qubit lifetime drops sharply near the topological phase transition ($V_z \sim V_{zc}$) due to noise-induced leakage out of the qubit subspace, leading to a leakage-limited qubit. Third, deep in the topological phase the system forms an exponentially protected, dephasing-limited topological qubit (though for still longer wires the topological qubit will eventually become leakage limited as the bulk excitation gap is roughly independent of $L$). Fourth, within the topological phase the dephasing time and qubit precession period oscillate out of phase and eventually decrease upon increasing $V_z$, reflecting an enlargment of the coherence length $\xi$ at ‘large’ $V_z$.
Performing full microscopic simulations of such noisy spinful tetrons containing hundreds of lattice sites in a manner that parallels the Kitaev simulations presented in Fig. \[fig:comp\_Kitaev\] is numerically intractable. We can, however, construct a small-$L$ ‘toy’ system far from the continuum limit of Eq. that is amenable to simulations and exhibits qualitatively similar features to the systems considered in Fig. \[fig:Vzscan\_summary\]. For this system, we choose a set of parameters \[including functional forms of the external potentials $V(x)$ and $\Delta(x)$\] identical to those used in Figs. \[fig:Vzscan\_summary\] and \[fig:spinful\_setup\_cont\] with the following exceptions: now $\alpha=400~\mathrm{meV}\,\mathrm{\AA}$, $\ell_V=0.08~\mu\mathrm{m}$, and $L=0.8~\mu\mathrm{m}$ discretized into $N_\mathrm{sites}=22$ (per wire) such that the ‘lattice constant’ $a = \frac{L}{N_\mathrm{sites}-1} \approx 0.038~\mu\mathrm{m}$. In Fig. \[fig:spinful\_setup\_disc\], we show the corresponding energy spectra as a function of external Zeeman field $V_z$ as well as the wave function amplitudes for the maximally localized Majorana modes $\gamma_{1,2,3,4}$ at $V_z = 0.6~\mathrm{meV}$ (ABS qubit) and $1.4~\mathrm{meV}$ (‘topological’ qubit). For the noise setup, we again take one global $\mu$ and $V_z$ fluctuator on each wire of the tetron with typical fluctuation amplitudes $\delta\mu_\mathrm{typ}=\delta V_{z,\mathrm{typ}}=0.01~\mathrm{meV}$ and identical inverse correlation times $\kappa$. Using these parameters, we show in Fig. \[fig:comp\_spinful\] data for a spinful tetron analogous to the Kitaev tetron results of Fig. \[fig:comp\_Kitaev\]. Here, as in Fig. \[fig:Vzscan\_summary\], we sweep over $V_z$; the four panels correspond to $\kappa=0.001,0.01,0.05,0.1~\mathrm{meV}/\hbar$. The extraction of the numerical (fitted) time scales (‘numerics’) and evaluation of the analytical time scales (‘theory’) were carried out in exactly the same manner as in Fig. \[fig:comp\_Kitaev\]. Again, in the two leftmost panels at small $\kappa=0.001,0.01~\mathrm{meV}/\hbar$ we (1) see the importance of the transient regime when comparing numerical $T_2$ times to the theoretical predictions and (2) observe no detectable leakage out of the ground state manifold on the simulated time scales for all $V_z$. As seen in Fig. \[fig:comp\_spinful\], the crossover from the ABS regime to topological regime occurs near $V_z\approx0.9~\mathrm{meV}$ where the excitation gap is minimal. In this vicinity, for $\kappa=0.05~\mathrm{meV}/\hbar$, we see that the numerically obtained leakage times become on the order of or smaller than the theoretically predicted dephasing times. In turn the *numerical* dephasing times drop below the analytic prediction: the noise-averaged signal $\overline{\langle i\gamma_1\gamma_3\rangle}$ is now damped by both dephasing effects (cf. Fig. \[fig:noise\_averaging\]) as well as leakage effects in which the amplitude of oscillations for *individual noise realizations* also decay in time (see also Fig. \[fig:meas\_basis\_mu0.8\]). While our theoretical analysis in Sec. \[sec:analytics\] treated the effects of dephasing and leakage independently, our full microscopic simulations as presented here are able to capture both effects simultaneously and faithfully. (The same physics is also occurring near the crossover out of the topological phase in the rightmost panel of Fig. \[fig:comp\_Kitaev\].)
Finally, at $\kappa=0.1~\mathrm{meV}/\hbar$ in the rightmost panel of Fig. \[fig:comp\_spinful\], we observe an intricate interplay between dephasing and leakage. Here, $\kappa$ is sufficiently large to cutoff the numerically extracted $T_2$ time below the pure dephasing limit even in the ABS regime for $V_z\lesssim0.9~\mathrm{meV}$. In the vicinity of the crossover we see a sharp reduction in the leakage time due to gap closing, thereby limiting the qubit lifetime as discussed above in the context of Fig. \[fig:Vzscan\_summary\]. (Note that for the particular qubit simulated in Fig. \[fig:comp\_spinful\], due to e.g. the extremely short wire length, even pure dephasing effects cause a reduction in coherence times in the ‘topological’ regime relative to the ABS one; there is thus a concomitant drop near the corresponding crossover which is unrelated to the leakage-induced qubit lifetime reduction due to gap closing.) Within the ‘topological’ phase, for points corresponding to a minimal excitation gap \[for example at $V_z=1.3,1.6,1.9~\mathrm{meV}$ (cf. Fig. \[fig:spinful\_setup\_cont\])\], $\kappa=0.1~\mathrm{meV}/\hbar$ is again sufficiently large to cause leakage to largely dictate the qubit coherence time. (In principle, leakage-limited lifetimes deep within the topological phase could also occur for sufficiently fast noise in the continuum limit analytical study of Fig. \[fig:Vzscan\_summary\]; however, studying these effects quantitatively in that framework would require a less crude evaluation of the analytical $T_\mathrm{leak}$ prediction.) On the other hand, for other values of $V_z$ with excitation gaps in excess of $\hbar\kappa=0.1~\mathrm{meV}$, the qubit lifetime is limited by dephasing, and the numerically extracted time scales match the analytic $T_2$ prediction impressively well.
Estimation of coherence times for realistic nanowire models {#sec:realistic}
-----------------------------------------------------------
![Dephasing analysis of ‘slab geometry’ nanowire. The upper panel shows the low-energy spectrum for a single wire, while the lower panel shows the derivative of the qubit splitting $E$ with respect to gate voltage $V_g$. Given some knowledge about the noise environment, this quantity can be directly related to the dephasing time $T_2$ via Eq. . For a detailed list of parameters used in these simulations, see main text. We find that also for this more realistic, multi-band model of a nanowire, the qualitative features such as exponential dependence of dephasing time on the length of the wire as well as the correlation between splitting amplitude and dephasing time are preserved.[]{data-label="fig:slab"}](slab_spectra.pdf){width="\columnwidth"}
A key advantage of the approach outlined in Sec. \[sec:T2\_analytic\] to estimate dephasing times is that it relies purely on the derivative of the Majorana splitting with respect to the fluctuating variables. Such simple spectral properties can be obtained using sparse diagonalization techniques and are thus easy to evaluate also for models where a computation of the full dynamics would be prohibitively expensive. To further illustrate this point, we perform such an estimate for a more sophisticated model of a nanowire that include a realistic electrostatic potential and several subbands. We closely follow the approach of Ref. [@antipov2018], which considers a rectangular wire of dimensions $L_x \times L_y \times L_z$ proximitized on the top by a superconductor of thickness $d_z$; here, $z$ is the direction transverse to the superconductor and $x$ is the long direction of the wire, i.e., $L_x \gg L_y, L_z$. We simplify the model in two key ways: First, we ignore the effect of subbands in the $y$ direction. And second, instead of treating the superconductor explicitly, we directly induce superconductivity in the semiconductor through a mean-field pairing term. We perform simulations with fixed $L_z = 60{\rm\ nm}$ and variable $L_x$.
The first-quantized normal-state Hamiltonian of the system can be written as $$\begin{aligned}
\label{eq:H_N}
H_{\rm N} =& -\frac{1}{2m^*} \left( \partial_x^2 + \partial_z^2 \right) \!-\! \alpha \hat k_x \sigma_y \\
& \!+\!\phi(z) \!+\! \frac{\mu_B g B}{2} \sigma_x, \nonumber\end{aligned}$$ with $m^*$ the effective mass, $\alpha$ the strength of spin-orbit coupling, $\phi(z)$ the electrostatic potential, $g$ the $g$-factor of the semiconducting material, $B$ the external magnetic field, and $\sigma_\alpha$ the Pauli matrices acting on spin. We use $m^* = 0.026$, $g=-15$, $\alpha = 0.05{\rm\ eV\ nm}$. Using a Nambu-space notation, where $\tau_\alpha$ are Pauli matrices acting in particle-hole space, we can write the Hamiltonian of the superconducting system as $$H = H_{\rm N} \tau_z - \Delta \sigma_y \otimes \tau_y,$$ where $\Delta$ denotes the strength of superconducting pairing, which we set to $\Delta = 0.1{\rm\ meV}$. The electrostatic potential $\phi(z)$ is obtained from a self-consistent Thomas-Fermi calculation, where the boundary condition near the superconductor is set to an assumed band offset of $300{\rm\ meV}$, while the boundary condition at the other end is tuned via an electrostatic gate with an applied voltage $V_g$. We use a gate voltage of $V_g = -0.288{\rm V}$, which tunes the system close to the bottom of a band and thus favors the formation of a topological phase.
The spectrum is evaluated by employing a finite-difference approximation, with a regular discretization of $a = 2{\rm\ nm}$, and using a shift-and-invert eigensolver. Results are shown in Fig. \[fig:slab\]. Here, we evaluate the derivative of the energy splitting $\varepsilon$ with respect to the applied gate voltage $V_g$. Using Eq. , this estimate can be combined with an estimate for the noise correlation time as well as typical fluctuation amplitude to obtain an estimate for the $T_2$ dephasing time. We note that this calculation can in principle be extended to more realistic models of the system, including explicit treatment of the superconductor and even full three-dimensional models of the device. However, major uncertainy enters the estimates through the properties characterizing the noise, which can only be extracted from experiments.
Effects of fixed versus instantaneous measurement basis {#sec:fixedvinst}
=======================================================
In this section, we investigate in more quantitative detail the effects of encoding and measuring the qubit in the basis of maximally localized Majorana modes associated with the *instantaneous* Hamiltonian, as opposed to using a *fixed* basis throughout the time evolution. In our numerical simulations presented in Sec. \[NumericalResults\], we used the fixed basis of maximally localized Majorana modes $\gamma_{1,2,3,4}=\gamma_{1,2,3,4}^{(0)}$ corresponding to the *time-averaged* Hamiltonian specified by $A_0$ to initialize the qubit. Furthermore, for technical reasons described in Sec. \[sec:readout\], we often employed this same basis for readout of quantities such as $\langle i\gamma_1\gamma_3\rangle$. For a given noise realization, the initial Hamiltonian will differ from the time-averaged one, i.e., $A(t=0) \neq A_0$, and hence this procedure introduces a sort of ‘quench’ in the dynamics at $t=0$, the effects of which become more pronounced for large typical fluctuation amplitudes $D_i$. For the small, seemingly reasonable $D_i$ chosen above this effect is negligible. We expect, however, the difference in measurement basis choice to manifest itself most clearly when $D_i$ becomes large. Here, it is worth reiterating an important point made in Sec. \[sec:UnitaryEvolution\]: Fluctuations, especially those of large amplitude, may significantly alter the maximally localized Majorana modes (e.g., their locations), but in the purely adiabatic limit the qubit subspace is still preserved provided that one tracks and measures in the instantaneous basis. On the other hand, measuring in a fixed basis throughout the evolution will naturally underestimate qubit coherence times, as the noisy dynamics will generically induce a ‘leakage’ out of the fixed basis over time, even if the qubit subspace is perfectly preserved by the instantaneous basis.
Initializing using the time-averaged basis $\gamma_{1,2,3,4}^{(0)}$ is clearly problematic for addressing these matters at large $D_i$; e.g., even at $t=0$ the system may have measurable leakage out of the qubit subspace derived from the instantaneous basis. For this section, we thus employ a different initialization procedure from that used in Sec. \[NumericalResults\]: For each noise realization, we find a maximally localized set of near-zero-energy Majorana modes $\gamma_{1,2,3,4}(t=0)$ derived from $A(t=0)$; we subsequently set $i\gamma_1(t=0)\gamma_3(t=0)=+1$ and $i\gamma_2(t=0)\gamma_4(t=0)=\pm1$, with the latter chosen such that the global fermion parity is always fixed to $+1$. We then compare the results of two different measurement bases: (1) the initial basis defined by the $\gamma_{1,2,3,4}(t=0)$ (which is fixed for each noise realization) and (2) the instantaneous basis defined by $\gamma_{1,2,3,4}(t)$. \[For small enough $D_i$, we have checked that using the former initial/fixed measurement basis produces results which are numerically indistinguishable from the initialization/readout procedure used in Sec. \[NumericalResults\] based on the $\gamma_{1,2,3,4}^{(0)}$.\]
Figures \[fig:meas\_basis\_mu0.15\] and \[fig:meas\_basis\_mu0.8\] present examples of noise-averaged time dynamics of a Kitaev tetron comparing the results of the two different measurement bases. We focus on the following two quantities: $\overline{\langle i\gamma_1\gamma_3\rangle^2}$ and $\overline{|a|^2} + \overline{|b|^2}$, where for the initial/fixed measurement basis $\gamma_i = \gamma_i(t=0)$ and $|a|^2 + |b|^2 = |a_{t=0}|^2 + |b_{t=0}|^2$, while for the instantaneous basis $\gamma_i = \gamma_i(t)$ and $|a|^2 + |b|^2 = |a_{t}|^2 + |b_{t}|^2$. (We consider $\langle i\gamma_1\gamma_3\rangle^2$ instead of $\langle i\gamma_1\gamma_3\rangle$ as the latter would be polluted by the sign ambiguity described in Sec. \[sec:readout\] in the case of the instantaneous basis; see also Fig. \[fig:single\_realization\].) The noise model is the same as that considered in Sec. \[sec:kitaev\]: one global $\mu$ fluctuator on each wire of the tetron (for a total of two fluctuators), each characterized by typical amplitude $\delta\mu_\mathrm{typ}$ and inverse correlation time $\kappa$. We perform simulations with $\delta\mu_\mathrm{typ}=0.02,0.05,0.1$ and $\kappa=0.05,0.1$. Figure \[fig:meas\_basis\_mu0.15\] corresponds to $\mu=0.15$ which is deep in the topological phase, while Fig. \[fig:meas\_basis\_mu0.8\] corresponds to $\mu=0.8$ near the crossover into the trivial phase (see Fig. \[fig:kitaev\_setup\]). The top panels show the time dependence of $\overline{\langle i\gamma_1\gamma_3\rangle^2}$ , while the bottom panels show $|a|^2 + |b|^2$; solid blue (dashed orange) curves represent measurements in the initial/fixed (instantaneous) basis. In the case of pure dephasing, $\overline{\langle i\gamma_1\gamma_3\rangle^2}$ is expected to take the following form (for times $t\gg1/\kappa$): $\frac12\left[\cos(2\omega_0 t)e^{-4t/T_2} + 1\right]$, i.e., relative to $\overline{\langle i\gamma_1\gamma_3\rangle}$ \[cf. Eq. \] it oscillates at twice the frequency and approaches $\frac12$ (instead of 0) at four times the decay rate.
For relatively small $\delta\mu_\mathrm{typ}=0.02$, we see in the first row of data presented in Figs. \[fig:meas\_basis\_mu0.15\] and \[fig:meas\_basis\_mu0.8\] that the chosen measurement basis negligibly impacts the results [^9]. This conclusion holds both in the case of (nearly) pure dephasing dynamics ($\mu=0.15$ at $\kappa=0.05,0.1$ and $\mu=0.8$ at $\kappa=0.05$) and dynamics involving both dephasing and leakage ($\mu=0.8$ at $\kappa=0.1$). Note that in the latter case, $\overline{\langle i\gamma_1\gamma_3\rangle^2}$ approaches a value less than $\frac12$ as the oscillations die out—leakage causes the traces for individual noise realizations to themselves decay in oscillation amplitude over time (in contrast to Fig. \[fig:noise\_averaging\]).
![Measurements of $\overline{\langle i\gamma_1\gamma_3\rangle^2}$ and $\overline{|a|^2} + \overline{|b|^2}$ in both the fixed/initial (solid blue) and instantaneous (dashed orange) basis for a Kitaev tetron at $\mu=0.15$, $\kappa=0.05,0.1$ (increasing left to right), and $\delta\mu_\mathrm{typ}=0.02,0.05,0.1$ (increasing top to bottom); see text for details of other chosen parameters. Here we initialize the system using a procedure based on the *initial* Hamiltonian encoded by $A(t=0)$. For large enough $\delta\mu_\mathrm{typ}$, a noticeable difference in the choice of measurement basis is observed on a time scale on the order of $\tau=1/\kappa$ (vertical green dotted lines). \[fig:meas\_basis\_mu0.15\]](inst_vs_fixed_L10p0_mu0p15_kappa0p05_dmutyp0p02_noquench.pdf "fig:"){width="0.49\columnwidth"} ![Measurements of $\overline{\langle i\gamma_1\gamma_3\rangle^2}$ and $\overline{|a|^2} + \overline{|b|^2}$ in both the fixed/initial (solid blue) and instantaneous (dashed orange) basis for a Kitaev tetron at $\mu=0.15$, $\kappa=0.05,0.1$ (increasing left to right), and $\delta\mu_\mathrm{typ}=0.02,0.05,0.1$ (increasing top to bottom); see text for details of other chosen parameters. Here we initialize the system using a procedure based on the *initial* Hamiltonian encoded by $A(t=0)$. For large enough $\delta\mu_\mathrm{typ}$, a noticeable difference in the choice of measurement basis is observed on a time scale on the order of $\tau=1/\kappa$ (vertical green dotted lines). \[fig:meas\_basis\_mu0.15\]](inst_vs_fixed_L10p0_mu0p15_kappa0p1_dmutyp0p02_noquench.pdf "fig:"){width="0.49\columnwidth"}
![Measurements of $\overline{\langle i\gamma_1\gamma_3\rangle^2}$ and $\overline{|a|^2} + \overline{|b|^2}$ in both the fixed/initial (solid blue) and instantaneous (dashed orange) basis for a Kitaev tetron at $\mu=0.15$, $\kappa=0.05,0.1$ (increasing left to right), and $\delta\mu_\mathrm{typ}=0.02,0.05,0.1$ (increasing top to bottom); see text for details of other chosen parameters. Here we initialize the system using a procedure based on the *initial* Hamiltonian encoded by $A(t=0)$. For large enough $\delta\mu_\mathrm{typ}$, a noticeable difference in the choice of measurement basis is observed on a time scale on the order of $\tau=1/\kappa$ (vertical green dotted lines). \[fig:meas\_basis\_mu0.15\]](inst_vs_fixed_L10p0_mu0p15_kappa0p05_dmutyp0p05_noquench.pdf "fig:"){width="0.49\columnwidth"} ![Measurements of $\overline{\langle i\gamma_1\gamma_3\rangle^2}$ and $\overline{|a|^2} + \overline{|b|^2}$ in both the fixed/initial (solid blue) and instantaneous (dashed orange) basis for a Kitaev tetron at $\mu=0.15$, $\kappa=0.05,0.1$ (increasing left to right), and $\delta\mu_\mathrm{typ}=0.02,0.05,0.1$ (increasing top to bottom); see text for details of other chosen parameters. Here we initialize the system using a procedure based on the *initial* Hamiltonian encoded by $A(t=0)$. For large enough $\delta\mu_\mathrm{typ}$, a noticeable difference in the choice of measurement basis is observed on a time scale on the order of $\tau=1/\kappa$ (vertical green dotted lines). \[fig:meas\_basis\_mu0.15\]](inst_vs_fixed_L10p0_mu0p15_kappa0p1_dmutyp0p05_noquench.pdf "fig:"){width="0.49\columnwidth"}
![Measurements of $\overline{\langle i\gamma_1\gamma_3\rangle^2}$ and $\overline{|a|^2} + \overline{|b|^2}$ in both the fixed/initial (solid blue) and instantaneous (dashed orange) basis for a Kitaev tetron at $\mu=0.15$, $\kappa=0.05,0.1$ (increasing left to right), and $\delta\mu_\mathrm{typ}=0.02,0.05,0.1$ (increasing top to bottom); see text for details of other chosen parameters. Here we initialize the system using a procedure based on the *initial* Hamiltonian encoded by $A(t=0)$. For large enough $\delta\mu_\mathrm{typ}$, a noticeable difference in the choice of measurement basis is observed on a time scale on the order of $\tau=1/\kappa$ (vertical green dotted lines). \[fig:meas\_basis\_mu0.15\]](inst_vs_fixed_L10p0_mu0p15_kappa0p05_dmutyp0p1_noquench.pdf "fig:"){width="0.49\columnwidth"} ![Measurements of $\overline{\langle i\gamma_1\gamma_3\rangle^2}$ and $\overline{|a|^2} + \overline{|b|^2}$ in both the fixed/initial (solid blue) and instantaneous (dashed orange) basis for a Kitaev tetron at $\mu=0.15$, $\kappa=0.05,0.1$ (increasing left to right), and $\delta\mu_\mathrm{typ}=0.02,0.05,0.1$ (increasing top to bottom); see text for details of other chosen parameters. Here we initialize the system using a procedure based on the *initial* Hamiltonian encoded by $A(t=0)$. For large enough $\delta\mu_\mathrm{typ}$, a noticeable difference in the choice of measurement basis is observed on a time scale on the order of $\tau=1/\kappa$ (vertical green dotted lines). \[fig:meas\_basis\_mu0.15\]](inst_vs_fixed_L10p0_mu0p15_kappa0p1_dmutyp0p1_noquench.pdf "fig:"){width="0.49\columnwidth"}
![Data analogous to that presented in Fig. \[fig:meas\_basis\_mu0.15\] but here taken at $\mu=0.8$. Now leakage out of the instantaneous low-energy qubit subspace is observed at $\kappa=0.1$ (as detected by $\overline{|a_t|^2} + \overline{|b_t|^2}$). For all other data at $\mu=0.15$ in Fig. \[fig:meas\_basis\_mu0.15\] and $\kappa=0.05$ here, leakage out of the instantanous qubit subspace is negligible. Measuring in the fixed basis, on the other hand, generically leads to an effective (topologically unprotected) ‘leakage’ out of the fixed-basis manifold, which can become significant for large $\delta\mu_\mathrm{typ}$ thereby underestimating qubit coherence times. \[fig:meas\_basis\_mu0.8\]](inst_vs_fixed_L10p0_mu0p8_kappa0p05_dmutyp0p02_noquench.pdf "fig:"){width="0.49\columnwidth"} ![Data analogous to that presented in Fig. \[fig:meas\_basis\_mu0.15\] but here taken at $\mu=0.8$. Now leakage out of the instantaneous low-energy qubit subspace is observed at $\kappa=0.1$ (as detected by $\overline{|a_t|^2} + \overline{|b_t|^2}$). For all other data at $\mu=0.15$ in Fig. \[fig:meas\_basis\_mu0.15\] and $\kappa=0.05$ here, leakage out of the instantanous qubit subspace is negligible. Measuring in the fixed basis, on the other hand, generically leads to an effective (topologically unprotected) ‘leakage’ out of the fixed-basis manifold, which can become significant for large $\delta\mu_\mathrm{typ}$ thereby underestimating qubit coherence times. \[fig:meas\_basis\_mu0.8\]](inst_vs_fixed_L10p0_mu0p8_kappa0p1_dmutyp0p02_noquench.pdf "fig:"){width="0.49\columnwidth"}
![Data analogous to that presented in Fig. \[fig:meas\_basis\_mu0.15\] but here taken at $\mu=0.8$. Now leakage out of the instantaneous low-energy qubit subspace is observed at $\kappa=0.1$ (as detected by $\overline{|a_t|^2} + \overline{|b_t|^2}$). For all other data at $\mu=0.15$ in Fig. \[fig:meas\_basis\_mu0.15\] and $\kappa=0.05$ here, leakage out of the instantanous qubit subspace is negligible. Measuring in the fixed basis, on the other hand, generically leads to an effective (topologically unprotected) ‘leakage’ out of the fixed-basis manifold, which can become significant for large $\delta\mu_\mathrm{typ}$ thereby underestimating qubit coherence times. \[fig:meas\_basis\_mu0.8\]](inst_vs_fixed_L10p0_mu0p8_kappa0p05_dmutyp0p05_noquench.pdf "fig:"){width="0.49\columnwidth"} ![Data analogous to that presented in Fig. \[fig:meas\_basis\_mu0.15\] but here taken at $\mu=0.8$. Now leakage out of the instantaneous low-energy qubit subspace is observed at $\kappa=0.1$ (as detected by $\overline{|a_t|^2} + \overline{|b_t|^2}$). For all other data at $\mu=0.15$ in Fig. \[fig:meas\_basis\_mu0.15\] and $\kappa=0.05$ here, leakage out of the instantanous qubit subspace is negligible. Measuring in the fixed basis, on the other hand, generically leads to an effective (topologically unprotected) ‘leakage’ out of the fixed-basis manifold, which can become significant for large $\delta\mu_\mathrm{typ}$ thereby underestimating qubit coherence times. \[fig:meas\_basis\_mu0.8\]](inst_vs_fixed_L10p0_mu0p8_kappa0p1_dmutyp0p05_noquench.pdf "fig:"){width="0.49\columnwidth"}
![Data analogous to that presented in Fig. \[fig:meas\_basis\_mu0.15\] but here taken at $\mu=0.8$. Now leakage out of the instantaneous low-energy qubit subspace is observed at $\kappa=0.1$ (as detected by $\overline{|a_t|^2} + \overline{|b_t|^2}$). For all other data at $\mu=0.15$ in Fig. \[fig:meas\_basis\_mu0.15\] and $\kappa=0.05$ here, leakage out of the instantanous qubit subspace is negligible. Measuring in the fixed basis, on the other hand, generically leads to an effective (topologically unprotected) ‘leakage’ out of the fixed-basis manifold, which can become significant for large $\delta\mu_\mathrm{typ}$ thereby underestimating qubit coherence times. \[fig:meas\_basis\_mu0.8\]](inst_vs_fixed_L10p0_mu0p8_kappa0p05_dmutyp0p1_noquench.pdf "fig:"){width="0.49\columnwidth"} ![Data analogous to that presented in Fig. \[fig:meas\_basis\_mu0.15\] but here taken at $\mu=0.8$. Now leakage out of the instantaneous low-energy qubit subspace is observed at $\kappa=0.1$ (as detected by $\overline{|a_t|^2} + \overline{|b_t|^2}$). For all other data at $\mu=0.15$ in Fig. \[fig:meas\_basis\_mu0.15\] and $\kappa=0.05$ here, leakage out of the instantanous qubit subspace is negligible. Measuring in the fixed basis, on the other hand, generically leads to an effective (topologically unprotected) ‘leakage’ out of the fixed-basis manifold, which can become significant for large $\delta\mu_\mathrm{typ}$ thereby underestimating qubit coherence times. \[fig:meas\_basis\_mu0.8\]](inst_vs_fixed_L10p0_mu0p8_kappa0p1_dmutyp0p1_noquench.pdf "fig:"){width="0.49\columnwidth"}
As we increase $\delta\mu_\mathrm{typ}$ in the middle ($\delta\mu_\mathrm{typ}=0.05$) and bottom ($\delta\mu_\mathrm{typ}=0.1$) rows of Figs. \[fig:meas\_basis\_mu0.15\] and \[fig:meas\_basis\_mu0.8\], we see that the difference in measurement basis becomes more pronounced. In addition, at fixed $\delta\mu_\mathrm{typ}$ this difference is most severe for slow noise; this point is particularly apparent in the bottom row of Fig. \[fig:meas\_basis\_mu0.8\] comparing $\kappa=0.05$ to $\kappa=0.1$. Furthermore, for both considered measurements, we note that the difference becomes manifest only after a time scale on the order of the noise correlation time $\tau=1/\kappa$ (indicated by vertical green dotted lines). Ultimately, we can clearly see from this data that the relevant coherence time diagnostics decay more quickly when evaluated in a fixed basis [^10]. These results are fully consistent with the qualitative picture sketched above in Sec. \[sec:protocol\].
Discussion {#sec:discussion}
==========
In this paper we explored a Ramsey-type protocol that probes qubit dynamics in proximitized nanowire devices—which can support either a topological Majorana-based qubit or a trivial ABS qubit depending on parameters. These two scenarios are challenging to distinguish in local probes, e.g., transport, yet display vastly different noise sensitivity as quantified by the qubit dephasing time revealed by our protocol. The required measurements are more challenging than transport but yield correspondingly more detailed information including the qubit lifetime, time-domain detection of the qubit splitting, and the presence of a topological phase transition. In our study we employed both analytical estimates of dephasing and leakage times as well as explicit numerical simulations of the noisy qubit dynamics. An appealing feature of the analytical estimates is that they can be readily evaluated (modulo uncertainties in noise details that require experimental input) even in state-of-the-art microscopic models. Some proof-of-concept simulations were presented in Sec. \[sec:realistic\], and it would be valuable to further quantify the ‘rigidity’ of the qubit splitting to fluctuations in future modeling efforts. Another feature highlighted by our study is the distinction between fixed-basis and instantaneous Majorana modes in a noisy environment. We argued on general grounds, and confirmed in our explicit simulations, that examining the former underestimates the true qubit lifetime, with the effect becoming increasingly prominent as the noise becomes slower and of larger amplitude.
In actual experiments, additional imperfections could obfuscate some of the features of the dephasing time discussed in this manuscript. For example, due to long-range inhomogeneities (on a scale longer than the superconducting coherence length) in the electrostatic potential and other system parameters, a nanowire may not undergo a topological phase transition simultaneously at all positions. Instead, the critical field may vary in space, so that some regions can enter the topological phase earlier than others. In such cases, the minimal gap when sweeping the magnetic field does not scale inversely with the system size, but instead inversely with the size of the largest contiguous region undergoing the phase transition. However, the analysis of Sec. \[sec:Tleak\_analytic\] shows that the relevant regions would be those whose critical states significantly overlap with the Majorana wavefunctions—i.e., regions proximate to the ends of the wire. Thus, the leakage time near the phase crossover regime will be determined by the properties of those critical regions.
Moreover, throughout this paper (e.g., in Fig. \[fig:Vzscan\_summary\]) we have ignored magnetic-field-induced suppression of the bulk superconducting gap. Such effects are clearly important in present-day experiments [@deng_majorana_2016; @zhang_quantized_2018] and may ultimately limit the feasibility of our proposed fixed-length study in those devices. Nevertheless, we are hopeful that future devices will harbor a larger window of field strengths over which one can scrutinize trends in splittings and coherence times.
In practice, the qubit frequency and the dephasing time, which are both tuned exponentially via the wire length and the coherence length, need to fall into an appropriate window for the effects discussed in this paper to be observable. If the wires are too long, the lifetime of the qubit may become limited by error processes not included here, for example quasi-particle poisoning, which render the effects we discuss unobservable. Estimates for the quasi-particle poisoning times vary widely, but can easily exceed one microsecond [@albrecht2017transport]. Conversely, if the wires are too short, both the qubit precession period and its lifetime may be shorter than time-domain experiments can resolve. This time scale is mostly limited by how quickly the coupling between the qubit and the measurement dot can be tuned, which likely limits the resolvable time scales to about one nanosecond. Given these constraints, the ideal regime where the effects discussed in this paper can be observed still encompasses several orders of magnitude in time scale, and thus a significant window of wire lengths. As future direction, it would be interesting to develop protocols that can probe an even broader range of time scales.
Finally, we focused entirely on the minimal ‘tetron’ qubit design for simplicity. A natural future direction is to extend our study to hexon devices that allow much more flexibility, including additional readout channels, measurement-based braiding, etc. Quantifying the role of noise in such higher-level applications would be an important and illuminating exercise. All in all, it is a remarkable feature of hardware-based topological qubits that we can directly simulate—in the time domain and with a single simulation—the effects of device-level noise on topologically protected quantum information.
The authors thank Andrey Antipov, William Cole, Torsten Karzig, Enrico Rossi, and Mike Zaletel for useful discussions. This work was supported by CRC 183 of Deutsche Forschungsgemeinschaft (F.v.O.); QuantERA project TOPOQUANT (F.v.O.); sabbatical support from IQIM, an NSF physics frontier center funded in part by the Moore Foundation (F.v.O.); the Army Research Office under Grant Award W911NF17-1-0323 (J.A.); the NSF through grant DMR-1723367 (J.A.); the Caltech Institute for Quantum Information and Matter, an NSF Physics Frontiers Center with support of the Gordon and Betty Moore Foundation through Grant GBMF1250 (R.V.M. and J.A.); the Walter Burke Institute for Theoretical Physics at Caltech (R.V.M. and J.A.); and the Gordon and Betty Moore Foundation’s EPiQS Initiative, Grant GBMF8682 (J.A.). Part of this work was performed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611 (R.V.M.).
Details of noisy time-dependent simulations and noise generation {#app:noise}
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We perform time evolution following the Gaussian covariance matrix formalism described in Sec. \[sec:MajoranaReform\]. (See also Refs. [@kraus2010generalized; @bravyi_complexity_2017]; for a self-contained overview, see Ref. [@bauer_dynamics_2018].) Our numerical implementation is based on the package in Julia [@rackauckas_differentialequations.jl_2017; @DifferentialEquations.jl].
To generate a noise realization of some fluctuating quantity $X(t)$ with a given noise power spectrum or correlation function, we use techniques based on fast Fourier transformation. This approach scales superlinear in the desired total time of the simulation ($T_\mathrm{sim}$ below). While for many types of noise (e.g., $1/\omega$ and $1/\omega^2$) more efficient ways of generating noise trajectories are known, in our context we never found this step to be a computational bottleneck. In addition, this approach has the advantage of being very general.
Consider a random variable $X(t)$ with $\overline{X(t)} = 0$ and $\overline{X^2(t)} = (\delta X_{\rm typ})^2$, and correlation function $\overline{X(t) X(t')} = S(|t - t'|)$. At each time, the variable is drawn from a Gaussian distribution, $P[X(t)] \sim e^{-X^2/2(\delta X_{\rm typ})^2}$. We can define the noise in frequency space $X(\omega)$ as well as the noise power spectrum $S(\omega) \propto \overline{|X(\omega)|^2}$ as $$\begin{aligned}
X(\omega) &= \int_{-\infty}^{\infty} dt\ e^{-i \omega t} X(t), \\
S(\omega) &= \int_{-\infty}^{\infty} dt\ e^{-i \omega t} S(t).\end{aligned}$$
For a simulation up to some finite time $T_{\rm sim}$, we generate a discrete noise trajectory up to some time $T_{\rm noise} \gg T_{\rm sim}$ with a timestep $\Delta t = T_{\rm noise}/N$, i.e., the trajectory is defined on timeslices $t_n = n \Delta t$, $n = 0, \ldots, N-1$. It is important that $\Delta t \ll \omega_{\rm max}$, where $\omega_{\rm max}$ is the highest relevant frequency occurring in the noise. In the example of Gaussian noise characterized by a high-frequency cutoff $\kappa$, it is natural to set $\omega_{\rm max}$ to some sufficiently large multiple of $\kappa$.
To numerically generate a single noise trajectory, we first create a white noise trajectory $Y(t_n)$ by drawing a sample on each time point from an independent and identical Gaussian ensemble of variance 1. We then perform a discrete Fourier transformation, $$\begin{aligned}
Y(\omega_k) &= \sum_{n=0}^{N-1} Y(t_n) e^{-i \omega_k t_n}, &\omega_k &= k \frac{2\pi}{T_{\rm noise}}.\end{aligned}$$ We can then obtain our desired noise trajectory in Fourier space by taking $$\begin{aligned}
X(\omega_k) &= \left[ \frac{S(\omega_k)}{\Delta t} \right]^{1/2} Y(\omega_k)\end{aligned}$$ and finally performing an inverse Fourier transformation to obtain $X(t_n)$: $$\begin{aligned}
X(t_n) = \frac{1}{N} \sum_{k=0}^{N-1} X(\omega_k) e^{i \omega_k t_n}.\end{aligned}$$ In all these steps, it is convenient to choose $N$ even so that the Fourier transformation is real. Furthermore, it is in many cases important to be able to sample $X(t)$ for arbitrary $t$, for example when integrating the Schrödinger equation with an adaptive time step. To this end, we perform a linear interpolation between timesteps. If $\Delta t$ is chosen sufficiently small, this will not incur significant numerical error.
In Fig. \[fig:single\_realization\], we show the time dependence of all quantum-averaged quantities considered in this work for a single noise realization of our Ramsey-type protocol over one qubit precession period for the Kitaev tetron system of Secs. \[sec:kitaev\] and \[sec:fixedvinst\]. We also show data necessary to derive the instantaneous qubit splitting [@Kitaev01_PhysU_44_131]: the two lowest-lying instantaneous energies $\varepsilon_{1,2}$ and the determinant of the instantaneous orthogonal transformation $U$ \[see Eqs. -\]. The left (right) panel corresponds to the initialization procedure based on the time-averaged (initial) Hamiltonian $A_0$ \[$A(t=0)$\] used in Sec. \[sec:kitaev\] (Sec. \[sec:fixedvinst\]). Quantities with the subscript ‘fixed’ denote measurement in a fixed basis throughout the evolution \[e.g., $\gamma_i = \gamma_i^{(0)}$ in the left panel and $\gamma_i = \gamma_i(t=0)$ in the right panel\], while quantities with the subscript ‘inst’ denote measurement in the instantaneous basis derived from $A(t)$ \[e.g., $\gamma_i = \gamma_i(t)$\]. The data shown for $\langle i\gamma_1\gamma_3 \rangle_\mathrm{inst} = \langle i\gamma_1(t)\gamma_3(t) \rangle$ is polluted by the numerical sign ambiguity described in Sec. \[sec:readout\]; we show it here over a single realization to illustrate this point. Comparing the envelope of $\langle i\gamma_1\gamma_3 \rangle_\mathrm{inst}$ to $\langle i\gamma_1\gamma_3 \rangle_\mathrm{fixed}$ demonstrates that for these parameters the choice of instantaneous versus fixed measurement basis does not have any impact on the results down to the level of each individual realization. Furthermore, comparing the dephasing diagnostics data in the top panels of Fig. \[fig:single\_realization\] reveals—again for these chosen parameters—that the choice of initialization procedure makes negligible difference. For the leakage diagnostics in the bottom panel, the measurements are very close to unity, and we thus show an appropriately zoomed in view. Note the observable high-frequency components in these measurements due to the small ‘quench’ at $t=0$ when using the $A_0$-based initialization procedure.
To calculate noise-averaged quantities, we run $N_\mathrm{real} = O(10^3)$ independent such individual realizations. While this approach is relatively numerically demanding (compared, for example, to the master-equation based approach of Ref. [@hu_majorana_2015]), it has multiple advantages: For one, our techniques are completely general in terms of what noise models can be simulated and what physical quantities can be measured / noise averaged (in contrast to the methods of Ref. [@hu_majorana_2015] which are more limiting in these respects); furthermore, our simulation strategy very closely mimics the actual experimental procedure we propose (although in experiment noise and quantum averaging cannot be distinguished).
Noise-averaging analysis {#app:averaging}
========================
Here we provide details on the noise-averaging of Eq. assuming the Gaussian noise correlations specified in Eqs. through . We will specifically evaluate $\overline{\exp\left(i\int_0^tdt'E(t')\right)}$ with $E(t')$ given by the harmonic approximation in Eq. and $\hbar = 1$ for notational simplicity; the noise average of $Q(t)$ follows straightforwardly from this quantity. It is convenient to discretize time (for intermediate stages of the calculation) and write $$\begin{aligned}
\overline{e^{i\int_0^tdt'E(t')}} &= \frac{1}{Z}\int \mathcal{D}\lambda_i(t) e^{i \Delta t\sum_{t' = 0}^t E(t') }
\nonumber \\
&\times e^{-\frac{1}{2}\sum_{t' t''}\sum_i S_i^{-1}(t'-t'')\lambda_i(t')\lambda_i(t'')}.\end{aligned}$$ Here $\Delta t$ is the time interval used for discretization, the second line is the weighting factor that gives the desired noise correlations, and $Z$ is a normalization defined as $$Z = \int \mathcal{D}\lambda_i(t) e^{-\frac{1}{2}\sum_{t' t''}\sum_i S_i^{-1}(t'-t'')\lambda_i(t')\lambda_i(t'')}.$$ Next we introduce $$\begin{aligned}
M_{ij}(t',t'') = M^0_{ij}(t',t'') + \delta M_{ij}(t',t''),\end{aligned}$$ where $$\begin{aligned}
M^0_{ij}(t',t'') &= \delta_{ij}S^{-1}_i(t'-t'')
\\
\delta M_{ij}(t',t'') &=
\begin{cases}
-i\Delta t \delta_{t',t''} E_{ij}'',~~~ 0< t' < t \\
0, ~~~~~~~~\text{otherwise},
\end{cases}\end{aligned}$$ and also $$\begin{aligned}
v_i(t') = \begin{cases}
i\Delta t E_i',~~~ 0< t' < t \\
0, ~~~~~~~~\text{otherwise}.
\end{cases}\end{aligned}$$ These definitions allow us to write $$\begin{aligned}
\overline{e^{i\int_0^tdt'E(t')}} &= \frac{e^{i E_0 t}}{Z}\int \mathcal{D}\lambda_i(t) e^{\sum_i \sum_{t'} v_i(t') \lambda_i(t')}
\nonumber \\
&\times e^{-\frac{1}{2} \sum_{ij} \sum_{t't''} \lambda_i(t')M_{ij}(t',t'')\lambda_j(t'')}.\end{aligned}$$ Note that the $t',t''$ sums are unrestricted above. We can now perform the Gaussian integration to obtain $$\begin{aligned}
\overline{e^{i\int_0^tdt'E(t')}} &= e^{i E_0 t}e^{-\frac{1}{2} \ln\left(\frac{\det M}{\det M^0}\right)}
\nonumber \\
&\times e^{\frac{1}{2} \sum_{ij} \sum_{t't''}v_i(t')M^{-1}_{ij}(t',t'') v_j(t'')}.
\label{GI}\end{aligned}$$
To proceed we first expand the log term in Eq. to second order in $\delta M$: $$\begin{aligned}
\ln\left(\frac{\det M}{\det M^0}\right) &= \ln \det[I + (M^0)^{-1} \delta M]
\nonumber \\
&\approx {\rm Tr}[(M^0)^{-1} \delta M] -\frac{1}{2} {\rm Tr}[(M^0)^{-1} \delta M]^2
\nonumber \\
&= -i t\sum_i D_i^2 E_{ii}''
\nonumber \\
&+ \frac{1}{2} \sum_{ij}(E_{ij}'')^2 \int_0^t dt' dt'' S_i(t'-t'')S_j(t''-t').\end{aligned}$$ The third line above encodes the leading shift in the qubit precession frequency \[cf. Eq. \]. At long times, $t \gg \tau_i$, the integrals in the fourth line become $$\begin{aligned}
\int_0^t dt' dt'' S_i(t'-t'')S_j(t''-t')
\approx t\frac{2 \sqrt{\pi}\tau_i \tau_j}{\sqrt{\tau_i^2 +
\tau_j^2}}(D_iD_j)^2.
\label{int1}\end{aligned}$$ In the second line of Eq. we simply replace $M^{-1} \approx (M^0)^{-1}$; the next-leading correction provides a higher-order shift to the qubit precession frequency compared to that captured above. This approximation yields $$\begin{aligned}
&\sum_{ij} \sum_{t't''}v_i(t')M^{-1}_{ij}(t',t'') v_j(t'')
\nonumber \\
&\approx - \sum_i (E_i')^2 \int_0^t dt' dt'' S_i(t'-t'') \approx t 2\sqrt{\pi} \sum_i \tau_i(D_i E_i')^2,
\label{int2}\end{aligned}$$ where on the far right side we again assumed the long-time limit.
Putting everything together, we find $$\begin{aligned}
\overline{e^{i\int_0^tdt'E(t')}} &=e^{i\left(E_0 + \frac{1}{2}\sum_i D_i^2 E_{ii}''\right)t}
\nonumber \\
&\times e^{-\sqrt{\pi}\left[ \sum_i\tau_i\left(D_i E_i'\right)^2 + \frac{1}{2}\sum_{ij}\frac{\tau_i \tau_j}{\sqrt{\tau_i^2 +
\tau_j^2}}\left(D_i E_{ij}''D_j\right)^2\right]t}\end{aligned}$$ at $t \gg \tau_i$. Upong restoring explicit $\hbar$’s, this result indeed recovers the qubit precession frequency and dephasing time quoted in Eqs. and .
The short-time limit, $t \ll \tau_i$, can be easily treated as well. Here we can simply write $$\begin{aligned}
\int_0^t dt' dt'' S_i(t'-t'')S_j(t''-t') &\approx t^2 S_i(0)S_j(0) = (D_i D_j t)^2
\\
\int_0^t dt' dt'' S_i(t'-t'') &\approx t^2 S_i(0)= (D_i t)^2\end{aligned}$$ in Eqs. and , so that $$\begin{aligned}
\overline{e^{i\int_0^tdt'E(t')}} &=e^{i\left(E_0 + \frac{1}{2}\sum_i D_i^2 E_{ii}''\right)t}
\nonumber \\
&\times e^{-\frac{1}{2}\left[ \sum_i\left(D_i E_i'\right)^2 + \frac{1}{2}\sum_{ij}\left(D_i E_{ij}''D_j\right)^2\right]t^2}.\end{aligned}$$ The noise-averaged $Q(t)$ is then $$\overline{Q(t \ll \tau_i)} \approx \cos(\omega_0 t)e^{-(t/T_2^{\rm short~time})^2}
\label{eq:Pt_short-time}$$ with (restoring $\hbar$’s) $$T_2^{\rm short~time} = 2\hbar^2\left[ \sum_i\left(D_i E_i'\right)^2 + \frac{1}{2}\sum_{ij}\left(D_i E_{ij}''D_j\right)^2\right]^{-1}.
\label{eq:T2_short-time}$$
[^1]: For more parallel terminology one can view the ABS qubit as a superficial, hardly topological tetron (SHT-tetron).
[^2]: A related sequence involving $Z$ initialization, two $\pi/2$ rotations about the $x$ axis buttressing a wait time $t$, followed by a final $Z$ measurement was proposed in Ref. [@aasen_milestones_2016], as appropriate for the qubit design presented therein.
[^3]: This method for obtaining maximally localized near-zero-energy Majorana modes working entirely in the local Majorana representation parallels that described in Ref. [@moore_two-terminal_2018] using the more traditional Bogoliubov-de Gennes (BdG) framework.
[^4]: $\ket{0_t}$ and $\ket{1_t}$ are the two lowest-energy eigenstates of the instantaneous Hamiltonian in the same global parity sector as the evolving state $\ket{\psi(t)}$ (assumed even in this discussion), which may or may not coincide with the parity of the absolute instantaneous ground state.
[^5]: An alternative choice similar to the former case could be to use modes derived from the fixed *initial* Hamiltonian $A(t=0) \neq A_0$ for a given noise realization. We take this approach in Sec. \[sec:fixedvinst\].
[^6]: We took $\varepsilon_{12}/E_g = 0$ to arrive at Eq. ; corrections from nonzero $\varepsilon_{12}/E_g$ can still give different leakage rates even with short-range-correlated noise.
[^7]: Error bars are typically on the order of the symbol size or smaller for all data that we present.
[^8]: Throughout, when evaluating the energy splitting $E$ in Eqs. and for a given microscopic model, for simplicity we take $E = \varepsilon_1 + \varepsilon_2$ \[with $\varepsilon_{1,2} \geq 0$, cf. Eq. \] without enforcing a fixed global parity.
[^9]: The residual oscillations in $\overline{\langle i\gamma_1\gamma_3\rangle^2}$ at long times are a convergence artifact associated with noise averaging the square.
[^10]: Although for the qubit and noise model considered here, it is difficult to find a regime where clear ‘Ramsey oscillations’ of $\braket{i\gamma_1\gamma_3}$ persist *and* the basis choice gives rise to a clear difference.
|
---
abstract: |
Let the circle act on a 4-dimensional compact almost complex manifold $M$ with a discrete fixed point set. Let us call the fixed point data of $M$ by the collection of multisets of weights at the fixed points. In this paper, we give a complete classification of the fixed point data of $M$ in terms of a multigraph encoding it. We prove that the multigraph can be achieved from a multigraph for a semi-free action followed by a combination of 4 types of operations on it. Moreover, we prove the converse; for any multigraph $\Gamma$ obtained from our minimal multigraph followed by the operations, we construct a 4-dimensional compact almost complex $S^1$-manifold $M$ with a discrete fixed point set such that $\Gamma$ encodes the fixed point data of $M$.
A special case of our classification result is when $M$ is complex or symplectic. In this case, the Todd genus of $M$ is 1 and our classification result generalizes as a particular case the classification of the fixed point data of complex manifolds by Carrell-Howard-Kosniowski [@CHK] and the classification of the fixed point data of symplectic manifolds by Ahara-Hattori [@AhHa], Audin [@Au], and Karshon [@Ka].
As an application of our result, we determine all possible Chern numbers of a 4-dimensional compact almost complex $S^1$-manifold $M$ with a discrete fixed point set, answering negatively a question by Sabatini [@S] if $c_1^2[M] \leq 3c_2[M]$ holds for any such manifold. We also provide a sharp lower bound on the number of fixed points in terms of the Todd genus.
As another application, given any possible fixed point data for a semi-free circle action on a compact almost complex manifold with a discrete fixed point set, we construct a compact connected almost complex $S^1$-manifold with the fixed point data.
address: 'Department of Mathematics, Pusan National University, Pusan, Korea'
author:
- Donghoon Jang
title: Circle actions on four dimensional almost complex manifolds with discrete fixed point sets
---
[^1]
Introduction {#s1}
============
The purpose of this paper is to classify the fixed point data of a circle action on a 4-dimensional compact almost complex manifold with a discrete fixed point set and discuss several applications. By the fixed point data, we mean a collection of multisets of the weights at the fixed points. The fixed point data encodes information of a manifold; the Chern numbers, the Euler characteristic, the Hirzebruch $\chi_y$-genus, the signature, the Todd genus, etc.
First, we discuss the known classification results of circle actions on 4-dimensional manifolds having discrete fixed point sets in different categories. In complex case, Carrell, Howard, and Kosniowski proved that a holomorphic vector field on a complex surface can be achieved from either $\mathbb{CP}^2$ or a Hirzebruch surface by blow-ups [@CHK]. In symplectic case, a Hamiltonian $S^1$-space can also be obtained from $\mathbb{CP}^2$ or a Hirzebruch surface by blow-ups [@AhHa], [@Au], [@Ka]. Note that for both complex and symplectic manifolds, the case that the fixed point set is not discrete is also dealt in the papers.
For oriented manifolds, Fintushel classified circle actions on 4-dimensional compact oriented manifolds in terms of orbit data [@F]. The classification of the fixed point data of a circle action on a 4-dimensional compact oriented manifold with a discrete fixed point set is given in [@J4]; also see [@P]. For an oriented manifold, the sign of each weight at a fixed point is not well-defined, while for an almost complex manifold it is well-defined. Therefore, the classification results for oriented manifolds do not imply results for almost complex manifolds.
Throughout the paper, we assume that any circle action on an almost complex manifold preserves the almost complex structure. Let the circle act on a compact almost complex manifold $M$. Let $p$ be an isolated fixed point. Then the local action of $S^1$ near $p$ can be described by
$g \cdot (z_1,\cdots,z_n)=(g^{w_{p,1}}z_1,\cdots,g^{w_{p,n}}z_n)$,
for all $g \in S^1 \subset \mathbb{C}$ and $z_1,\cdots,z_n \in \mathbb{C}$, where $\dim M=2n$ and $w_{p,i}$ are non-zero integers, for $1 \leq i \leq n$. The non-zero integers $w_{p,i}$ are called the **weights** at $p$. The **index** $n_p$ of $p$ is the number of negative weights at $p$.
\[d11\]
(1) A **multigraph** is a pair $(V,E)$, where $V$ is a set of vertices and $E$ is a set of (unordered) edges.
(2) A multigraph $(V,E)$ is called **directed** if there are maps $i:E \to V$ and $t:E \to V$ giving the initial and terminal vertices of each edge.
(3) A multigraph $(V,E)$ is called **labeled** if there is a map $w$ from $E$ to the set $\mathbb{N}^+$ of positive integers. For any edge $e$, $w(e)$ is called the **label** of $e$.
(4) Let $\Gamma=(V,E)$ be a directed multigraph. For a vertex $v$ of $\Gamma$, the **index** $n_v$ of $v$ is the number of edges whose terminal vertex is $v$. The **Todd genus** $T(\Gamma)$ of $\Gamma$ is the number of vertices of index 0.
(5) Let $\Gamma=(V,E)$ be a labeled directed multigraph. Let $e$ be an edge of $\Gamma$. We say that $(i(e),t(e))$ is **$w(e)$-edge**. Alternatively, we say that $(t(e),i(e))$ is $(-w(e))$-edge.
For an action of a group $G$ on a manifold $M$, denote by $M^G$ the set of points in $M$ that are fixed by the $G$-action. That is, $M^G=\{m \in M \, | \, g \cdot m = m \textrm{ for all } g \in G\}$.
\[d12\] [@JT] For a circle action on a compact almost complex manifold $M$ with a discrete fixed point set, we say that a labeled directed multigraph $\Gamma$ **describes** $M$ if the following hold:
(1) The vertex set of $\Gamma$ is the fixed point set $M^{S^1}$ of $M$.
(2) The multiset of the weights at $p$ is $\{w(e)\,|\,i(e)=p\} \cup \{-w(e)\,|\,t(e)=p\}$ for all $p \in M^{S^1}$.
(3) For each edge $e$, the two endpoints $i(e)$ and $t(e)$ lie in the same connected component of the isotropy submanifold $M^{\mathbb{Z}/(w(e))}$. Here, $\mathbb{Z}/(w(e))$ acts on $M$ as a subgroup of $S^1$.
Suppose that a labeled directed multigraph $\Gamma$ describes a compact almost complex $S^1$-manifold $M$ with a discrete fixed point set. Let $p$ be a vertex (fixed point). If an edge $e$ with label $w$ has initial vertex $p$, Definition \[d12\] means that the fixed point $p$ has weight $+w$. If an edge $e$ with label $w$ has terminal vertex $q$, Definition \[d12\] means that the fixed point $q$ has weight $-w$. It follows that if $\dim M = 2n$, $\Gamma$ is $n$-regular; every vertex has $n$-edges. Since a multigraph describing $M$ encodes the fixed point data of $M$, the multigraph encodes the same information as the fixed point data of $M$.
A group action on a manifold is called **semi-free**, if the action is free outside the fixed point set. For a semi-free circle action on an almost complex manifold with a discrete fixed point set, this means that every weight at any fixed point is either $+1$ or $-1$. Accordingly, we introduce a notion for a multigraph for a semi-free action.
\[d13\] A labeled directed multigraph $\Gamma$ is called **semi-free** if
1. every edge of $\Gamma$ has label $1$; and
2. for every edge $e$ of $\Gamma$, the index of the initial vertex of $e$ is one less than the index of the terminal vertex of $e$, i.e., $n_{i(e)}+1=n_{t(e)}$.
A **semi-free multigraph** is a labeled directed multigraph that is semi-free.
By definition, if $\Gamma$ is a 2-regular semi-free multigraph and $T(\Gamma)=k$, $\Gamma$ has $k$ vertices of index 0, $2k$ vertices of index 1, and $k$ vertices of index 2. Figure \[fig1\] is an example of a connected 2-regular semi-free multigraph $\Gamma$, with $T(\Gamma)=2$.
In this paper, we prove that for a circle action on a 4-dimensional compact almost complex manifold $M$ with a discrete fixed point set, there exists a multigraph describing $M$ that can be achieved from a 2-regular semi-free multigraph followed by 4 types of operations on it. Pictures for the operations 1, 2, 3, and 4 are provided in Figures \[fig2\], \[fig3\], \[fig4\], and \[fig5\], respectively. Moreover, we prove the converse that for any multigraph $\Gamma$ obtained from a 2-regular semi-free multigraph followed by the 4 types of operations on it, there exists a 4-dimensional compact almost complex $S^1$-manifold with a discrete fixed point set that is described by $\Gamma$. Our main result is the following:
\[t11\] Let the circle act effectively on a 4-dimensional compact almost complex manifold $M$ with a discrete fixed point set. Then a labeled directed multigraph describing $M$ can be constructed in the following way: begin with a disjoint union of connected 2-regular semi-free multigraphs $\Gamma_j$, $1 \leq j \leq k$, for some positive integer $k$. Then, apply a combination of the following 4 operations on each $\Gamma_j$:
(1) Suppose that $(p_{i-1},p_i)$ is $a$-edge and $(p_i,p_{i+1})$ is $b$-edge, for some vertices $p_{i-1}$, $p_i$, $p_{i+1}$ and for some positive integers $a$, $b$. Then replace $p_i$ with $(a+b)$-edge $(p_i',p_i'')$ (Figure \[fig2\]).
(2) Suppose that $(p_i,p_{i-1})$ is $c$-edge, $(p_{i+1},p_i)$ is $d$-edge, and $(p_{i+1},p_{i+2})$ is $c$-edge, for some vertices $p_{i-1}$, $p_i$, $p_{i+1}$, $p_{i+2}$ and for some positive integers $c$, $d$. Then replace the label of the $(p_{i+1},p_i)$-edge with $d+c$ (Figure \[fig3\]).
(3) Suppose that $(p_{i-1},p_i)$ is $e$-edge, $(p_i,p_{i+1})$ is $f$-edge, and $(p_{i+2},p_{i+1})$ is $e$-edge, for some vertices $p_{i-1}$, $p_i$, $p_{i+1}$, $p_{i+2}$, and for some positive integers $e$, $f$. Then replace the label of the $(p_i,p_{i+1})$-edge with $f+e$ (Figure \[fig4\]).
(4) Suppose that $(p_{i-1},p_{i-2})$ is $g$-edge, $(p_{i-1},p_i)$ is $h$-edge, $(p_{i},p_{i+1})$ is $g$-edge, and $(p_{i+2},p_{i+1})$ is $h$-edge, for some vertices $p_{i-1}$, $p_i$, $p_{i+1}$, $p_{i+2}$, and for some positive integers $g$, $h$. Then replace $(p_{i-1},p_i)$-edge and $(p_i,p_{i+1})$-edge with $(p_{i-1},p_{i+1})$-edge, removing $p_i$; the new $(p_{i-1},p_{i+1})$-edge has label $(g+h)$ (Figure \[fig5\]).
The Todd genus of $M$ is $\textrm{Todd}(M)=\sum_{j=1}^k T(\Gamma_j)$. Moreover, given any 2-regular labeled directed multigraph $\Gamma$ constructed from a 2-regular semi-free multigraph followed by the 4 operations in this theorem, there exists a 4-dimensional compact connected almost complex manifold $M$ equipped with a circle action having a discrete fixed point set that is described by $\Gamma$.
[.9]{}
\(A) [$p_1$]{}; (B) \[above right=of A\] [$p_2$]{}; (C) \[above right=of B\] [$p_3$]{}; (D) \[below right=of C\] [$p_4$]{}; (E) \[below right=of D\] [$p_5$]{}; (F) \[above right=of E\] [$p_6$]{}; (G) \[above right=of F\] [$p_7$]{}; (H) \[below right=of G\] [$p_8$]{}; (A) \[->\] edge node\[right\] [$1$]{} (B); (B) \[->\] edge node \[right\] [$1$]{} (C); (D) \[->\] edge node \[right\] [$1$]{} (C); (E) \[->\] edge node \[right\] [$1$]{} (D); (E) \[->\] edge node \[right\] [$1$]{} (F); (F) \[->\] edge node \[right\] [$1$]{} (G); (H) \[->\] edge node \[right\] [$1$]{} (G); (A) \[->\] edge node \[pos=.3, below\] [$1$]{} (H);
[.4]{}
\(a) [$p_{i-1}$]{}; (b) \[above right=of a\] [$p_i$]{}; (c) \[above left=of b\] [$p_{i+1}$]{}; (a) \[->\] edge node\[right\] [$a$]{} (b); (b) \[->\] edge node \[right\] [$b$]{} (c);
[.4]{}
\(a) [$p_{i-1}$]{}; (b) \[above right=of a\] [$p_{i}'$]{}; (c) \[above=of b\] [$p_i''$]{}; (d) \[above left=of c\] [$p_{i+1}$]{}; (a) \[->\] edge node\[right\] [$a$]{} (b); (b) \[->\] edge node \[left\] [$a+b$]{} (c); (c) \[->\] edge node \[left\] [$b$]{} (d);
[.4]{}
\(a) [$p_{i-1}$]{}; (b) \[below left=of a\] [$p_i$]{}; (c) \[below right=of b\] [$p_{i+1}$]{}; (d) \[above right=of c\] [$p_{i+2}$]{}; (b) \[->\] edge node\[right\] [$c$]{} (a); (c) \[->\] edge node \[right\] [$d$]{} (b); (c) \[->\] edge node \[right\] [$c$]{} (d);
[.4]{}
\(a) [$p_{i-1}$]{}; (b) \[below left=of a\] [$p_i$]{}; (c) \[below right=of b\] [$p_{i+1}$]{}; (d) \[above right=of c\] [$p_{i+2}$]{}; (b) \[->\] edge node\[right\] [$c$]{} (a); (c) \[->\] edge node \[right\] [$d+c$]{} (b); (c) \[->\] edge node \[right\] [$c$]{} (d);
[.4]{}
\(a) [$p_{i+2}$]{}; (b) \[above left=of a\] [$p_{i+1}$]{}; (c) \[below left=of b\] [$p_{i}$]{}; (d) \[below right=of c\] [$p_{i-1}$]{}; (a) \[->\] edge node\[right\] [$e$]{} (b); (c) \[->\] edge node \[right\] [$f$]{} (b); (d) \[->\] edge node \[right\] [$e$]{} (c);
[.4]{}
\(a) [$p_{i+2}$]{}; (b) \[above left=of a\] [$p_{i+1}$]{}; (c) \[below left=of b\] [$p_{i}$]{}; (d) \[below right=of c\] [$p_{i-1}$]{}; (a) \[->\] edge node\[right\] [$e$]{} (b); (c) \[->\] edge node \[pos=.3, right\] [$f+e$]{} (b); (d) \[->\] edge node \[right\] [$e$]{} (c);
[.4]{}
\(a) at (3, 1) [$p_{i+2}$]{}; (b) at (1, 2) [$p_{i+1}$]{}; (c) at (0, 0) [$p_{i}$]{}; (d) at (1, -2) [$p_{i-1}$]{}; (e) at (3, -1) [$p_{i-2}$]{}; (a) \[->\] edge node\[below\] [$h$]{} (b); (c) \[->\] edge node \[right\] [$g$]{} (b); (d) \[->\] edge node \[right\] [$h$]{} (c); (d) \[->\] edge node \[above\] [$g$]{} (e);
[.4]{}
\(a) at (3, 1) [$p_{i+2}$]{}; (b) at (1, 2) [$p_{i+1}$]{}; (d) at (1, -2) [$p_{i-1}$]{}; (e) at (3, -1) [$p_{i-2}$]{}; (a) \[->\] edge node\[below\] [$h$]{} (b); (d) \[->\] edge node \[right\] [$g+h$]{} (b); (d) \[->\] edge node \[above\] [$g$]{} (e);
The proof of Theorem \[t11\] is given in Section \[s3\]. The main ideas of the proof are the existence of a multigraph describing $M$ (Lemma \[l26\]), the relationship between the weights at fixed points lying in the same connected component of $M^{\mathbb{Z}_w}$ (Lemma \[l21\]), and the equivariant plumbing technique (Theorem \[t31\]).
Note that for an effective circle action on a 4-dimensional compact almost complex manifold $M$ with a discrete fixed point set, Theorem \[t11\] does not imply that a multigraph describing $M$ is unique. Let $w>1$ be an integer. If a fixed point $p$ has weight $+w$, a fixed point $q$ has weight $-w$, and $p$ and $q$ lie in the same connected component of $M^{\mathbb{Z}_w}$, then for any multigraph describing $M$, (3) of Definition \[d13\] means that $(p,q)$ must be $w$-edge. However, if $\pm 1$ occur as weights at some fixed points, there are many possibilities for drawing edges with label 1. For instance, suppose that a manifold $M$ is described by Figure \[fig1\]. Then $M$ is also described by a disjoint union of two copies of Figure \[fig6\]. However, a multigraph describing $M$ is unique, up to redrawing edges of label 1. Moreover, a multigraph can describe two different manifolds; see Remark \[r34\].
From now on, we shall discuss applications of Theorem \[t11\]. The first application is when a manifold is complex or symplectic. To discuss, let $M$ be a 4-dimensional compact connected manifold, complex or symplectic. Let the circle act on $M$, preserving the given structure. Suppose that the fixed point set is non-empty and discrete. If $M$ is complex, in [@CHK], Carrell, Howard, and Kosniowski classified the fixed point data of $M$. If $M$ is symplectic, the fixed point dats is classified by Ahara-Hattori [@AhHa], Audin [@Au], and Karshon [@Ka]. When $M$ is complex or symplectic, the Todd genus of $M$ is equal to 1. The Todd genus $\textrm{Todd}(M)$ of $M$ is equal to the number of fixed points of index 0; see Theorem \[t21\]. Therefore, Theorem \[t11\] implies that in this case the fixed point data of $M$ can be achieved from a single 2-regular semi-free multigraph $\Gamma$ with $T(\Gamma)=1$ (Figure \[fig6\]) followed by a combination of the 4 operations in Theorem \[t11\].
\[t12\] Let $M$ be a 4-dimensional compact connected manifold, complex or symplectic. Let the circle act on $M$ preserving the complex or symplectic structure. Suppose that the fixed point set is non-empty and discrete. Then a multigraph describing $M$ can be constructed by beginning with a 2-regular semi-free multigraph $\Gamma$ with $T(\Gamma)=1$ (Figure \[fig6\]) followed by a combination of the 4-operations in Theorem \[t11\].
\(A) [$p_1$]{}; (B) \[above left=of A\] [$p_4$]{}; (C) \[above right=of A\] [$p_2$]{}; (D) \[above right=of B\] [$p_3$]{}; (A) \[->\] edge node\[right\] [$1$]{} (B); (A) \[->\] edge node \[right\] [$1$]{} (C); (B) \[->\] edge node \[right\] [$1$]{} (D); (C) \[->\] edge node \[right\] [$1$]{} (D);
The proof of Theorem \[t12\] and the comparison of the classifications of the fixed point datum in almost complex, complex, and symplectic cases are given in Section \[s4\].
The second application of Theorem \[t11\] is the classification of the Chern numbers of a 4-dimensional compact almost complex $S^1$-manifold with a discrete fixed point set. In [@S], Sabatini asked what possible values are for the Chern numbers of such a manifold.
\[q16\] [@S] Let the circle act on a compact almost complex manifold $M$ with a discrete fixed point set. Then what are the possible values for the Chern numbers of $M$?
In particular, Sabatini asked if the following inequality holds, as it holds if $M$ admits a symplectic structure and the circle action is Hamiltonian.
\[q17\] [@S] Let the circle act on a 4-dimensional compact almost complex manifold $M$ with a discrete fixed point set. Then does $c_1^2[M] \leq 3c_2[M]$ hold?
Here, $c_1$ and $c_2$ are the first and second equivariant Chern classes of $M$, respectively, and $[M]$ is the fundamental class of $M$. In [@S], Sabatini proved that for a 4-dimensional compact almost complex $S^1$-manifold $M$ with a discrete fixed point set, $c_1^2[M]=10 N_0-N_1$ and $c_2[M]=2N_0+N_1$, where $N_i$ denotes the number of fixed points of index $i$. Therefore, proving $c_1^2[M] \leq 3c_2[M]$ is equivalent to proving $N_0 \leq N_1$. We give a complete answer to Question \[q16\] in dimension 4, answering Question \[q17\] negatively by constructing a manifold $M$ with $N_1 < N_0$.
\[c115\] Let the circle act on a 4-dimensional compact almost complex manifold $M$ with a non-empty discrete fixed point set. Then $c_1^2[M]=10n_0-n_1$ and $c_2[M]=2n_0+n_1$ for some positive integers $n_0$ and $n_1$.
Conversely, given any positive integers $n_0$ and $n_1$, there exists a circle action on a 4-dimensional compact connected almost complex manifold $M$ with a discrete fixed point set such that $c_1^2[M]=10n_0-n_1$ and $c_2[M]=2n_0+n_1$.
Corollary \[c115\] is obtained as a consequence of the following corollary of Theorem \[t11\], which establishes a lower bound on the number of fixed points. Moreover, the bound is sharp.
\[c19\] Let the circle act on a 4-dimensional compact almost complex manifold $M$ with a non-empty fixed point set. Then there are at least 3 fixed points. If in addition the fixed point set is discrete, then $\mathrm{Todd}(M)>0$ and there are at least $2 \cdot \mathrm{Todd}(M)+1$ fixed points. Moreover, for any positive integers $N_0$ and $N_1$, there exists a 4-dimensional compact connected almost complex manifold $M$ with Todd genus $N_0$, equipped with a circle action having precisely $2N_0+N_1$ fixed points, $N_0$ fixed points of index 0, $N_1$ fixed points of index 1, and $N_0$ fixed points of index 2.
In fact, three fixed points can only occur in dimension 4; see [@J2]. The proof of Corollary \[c19\] is given in Section \[s5\].
Another application of Theorem \[t11\] is the classification of a semi-free circle action on a compact almost complex manifold with a discrete fixed point set. In [@TW], Tolman and Weitsman proved that if the circle acts semi-freely on a $2n$-dimensional compact almost copmlex manifold with a discrete fixed point set, $N_i=N_0 \cdot {n \choose i}$ for $0 \leq i \leq n$. As a consequence, they proved that if the circle acts symplectically and semi-freely on a compact symplectic manifold $M$ with a non-empty discrete fixed point set, the action must be Hamiltonian (and hence the Todd genus of $M$ is equal to 1), and the fixed point data is the same as that of a diagonal action on the product of $S^2$’s, on each of which the circle acts by rotation. In particular, there are exactly $2^n$ fixed points. Feldman [@Fe] and Li [@L] reproved this. In this paper, we prove the converse of the above statement; given any positive integers $k$ and $n>1$, we construct a semi-free circle action on a $2n$-dimensional compact connected almost complex manifold $M$ with a discrete fixed point set such that $\textrm{Todd}(M)=k$, and $M$ has precisely $k \cdot 2^n$ fixed points. Therefore, our result can be viewed as a generalization of the result for semi-free symplectic circle actions by [@TW] to almost complex manifolds.
\[t13\] Let the circle act semi-freely on a $2n$-dimensional compact almost complex manifold $M$ with a discrete fixed point set. Then there are $\textrm{Todd}(M) \cdot 2^n$ fixed points. Conversely, let $k$ be a positive integer. Then for any $n>1$, there exists a $2n$-dimensional compact connected almost complex manifold $M$ equipped with a semi-free circle action having a discrete fixed point set, whose Todd genus is $k$. Moreover, there are $k \cdot 2^n$ fixed points.
The proof of Theorem \[t13\] is given in Section \[s6\]. In Theorem \[t13\], we exclude dimension 2 simply because in dimension 2 if there is a fixed point, the manifold is $S^2$ and we can only have 2 fixed points, that is, rotation of the 2-sphere.
The paper is organized as follows. In Section \[s2\], we review background and preliminary results. In Section \[s3\], we prove of our main result, Theorem \[t11\]. In Section \[s4\], we prove Theorem \[t12\] and compare circle actions on 4-dimensional almost complex, complex, and symplectic manifolds that have discrete fixed point sets. In Section \[s5\], we prove Corollary \[c115\] and Corollary \[c19\] that determine the Chern numbers and provide a lower bound on the number of fixed points. In Section \[s6\], we prove Theorem \[t13\] which generalizes semi-free symplectic circle actions with discrete fixed points sets to semi-free circle actions on almost complex manifolds. In Section \[s7\], we discuss when a multigraph behaves like a manifold, that is, given a labeled directed multigraph $\Gamma$, under which conditions on $\Gamma$ there exists a compact almost complex $S^1$-manifold that is described by $\Gamma$.
Acknowledgements {#acknowledgements .unnumbered}
================
The author would like to thank Mikiya Masuda and Susan Tolman for fruitful conversations. The author is supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2018R1D1A1B07049511).
Background and preliminaries {#s2}
============================
An **almost complex manifold** $(M,J)$ is a manifold $M$ with a fiberwise linear complex structure $J_m$ on each tangent space $T_mM$. A circle action on an almost complex manifold $(M,J)$ is said to **preserve the almost complex structure** if $dg \circ J=J \circ dg$ for all $g \in S^1$.
Let $M$ be a compact almost complex manifold. The Hirzebruch $\chi_y$-genus is the genus belonging to the power series $\frac{x(1+ye^{-x(1+y)})}{1-e^{-x(1+y)}}$. The Hirzebruch $\chi_y$-genus of $M$ contains three information; $\chi_{-1}(M)=\chi(M)$ is the Euler characteristic of $M$, $\chi_0(M)=\textrm{Todd}(M)$ is the Todd genus of $M$, and $\chi_1(M)=\textrm{sign}(M)$ is the signature of $M$. In [@HBJ], Hirzebruch, Berger, and Jung proved that for a circle action on a compact complex manifold with a discrete fixed point set, the Hirzebruch $\chi_y$-genus is rigid under the circle action. Following the idea, Li extended the result to almost complex manifolds.
\[t21\] [@HBJ], [@L] Let the circle act on a $2n$-dimensional compact almost complex manifold $M$ with a discrete fixed point set. For each integer $i$ such that $0 \leq i \leq n$,
$\displaystyle \chi^i(M)=\sum_{p \in M^{S^1}} \frac{\sigma_i (t^{w_{p,1}}, \cdots, t^{w_{p,n}})}{\prod_{j=1}^n (1-t^{w_{p,j}})} = (-1)^i N_i = (-1)^i N_{n-i}$,
where $\chi_y(M)=\sum_{i=0}^n \chi^i(M) \cdot y^i$ is the Hirzebruch $\chi_y$-genus of $M$, $t$ is an indeterminate, $\sigma_i$ is the $i$-th elementary symmetric polynomial in $n$ variables, and $N_i$ is the number of fixed points of index $i$. Moreover, $\chi^0(M)=\chi_0(M)$ is equal to the Todd genus of $M$.
For a circle action on a compact almost complex manifold with a discrete fixed point set, each time weight $w$ occurs, there occurs weight $-w$.
[@H], [@L] Let the circle act on a compact almost complex manifold $M$ with a discrete fixed point set. For each integer $w$,
$\displaystyle \sum_{p \in M^{S^1}} N_p(w)=\sum_{p \in M^{S^1}} N_p(-w)$.
Here, for an integer $w$, $N_p(w)=|\{i \ | \ w_{p,i}=w, 1 \leq i \leq n\}|$ denotes the number of times weight $w$ occurs at $p$.
In addition, the smallest positive weight satisfies a stronger property.
\[p22\] [@JT] Let the circle act on a $2n$-dimensional compact almost complex manifold $M$ with a discrete fixed point set. Let $a$ be the smallest positive weight that occurs at any fixed point. Given any $j \in \{0,1,\dots,n-1\}$, the number of times the weight $+a$ occurs at fixed points of index $j$ is equal to the number of times the weight $-a$ occurs at fixed points of index $j+1$. That is, for any $j \in \{0,1,\dots,n-1\}$,
$\displaystyle \sum_{p \in M^{S^1}, n_p=j} N_p(a)=\sum_{p \in M^{S^1},n_p=j+1} N_p(-a)$,
where $n_p$ is the index (the number of negative weights) of $p$, and for an integer $w$, $N_p(w)=|\{i \ | \ w_{p,i}=w, 1 \leq i \leq n\}|$ is the number of times weight $w$ occurs at $p$.
Applying Proposition \[p22\] to the smallest positive weight, it follows that there must exist two fixed points whose indices differ by 1.
[@J5] \[c23\] Let the circle act on a $2n$-dimensional compact almost complex manifold $M$ with a non-empty discrete fixed point set. Then there exists $0 \leq i \leq n-1$ such that both of $N_i$ and $N_{i+1}$ are not zero, where $N_i$ is the number of fixed points of index $i$.
Alternatively, since $\chi^i(M)=(-1)^iN_i$ for $0 \leq i \leq n$, Corollary \[c23\] means that there exists $0 \leq i \leq n-1$ such that both of $\chi^i(M)$ and $\chi^{i+1}(M)$ are not zero, where $\chi_y(M)=\sum_{i=0}^n \chi^i(M) \cdot y^i$ is the Hirzebruch $\chi_y$-genus of $M$.
For a directed multigraph, by a self-loop we mean an edge $e$ whose initial vertex and terminal vertex coincide, i.e., $i(e)=t(e)$. Recall that the index $n_v$ of a vertex $v$ is the number of edges whose terminal vertex is $v$. In [@JT], using Proposition \[p22\], Tolman and the author proved that for a compact almost complex $S^1$-manifold $M$ with a discrete fixed point set, there exists a labeled directed multigraph describing $M$ that does not have any self-loops.
\[l26\] [@JT] Let the circle act on a compact almost complex manifold $M$ with a discrete fixed point set. Then there exists a labeled, directed multigraph describing $M$ such that, for each edge $e$, the index of $i(e)$ in the isotropy submanifold $M^{\mathbb{Z}/(w(e))}$ is one less than the index of $t(e)$ in $M^{\mathbb{Z}/(w(e))}$. In particular, the multigraph has no self-loops.
We shall explore a multigraph satisfying the conditions in Lemma \[l26\] in dimension 4. Consider a circle action on a 4-dimensional compact almost complex manifold $M$ with a discrete fixed point set. Without loss of generality, by quotienting out by the subgroup that acts trivially, suppose that the action is effective. Suppose that a fixed point $p$ has weight $+w$ for some positive integer $w$ such that $w>1$. The group $\mathbb{Z}_w$ acts on $M$ as a subgroup of $S^1$. The set $M^{\mathbb{Z}_w}$ of points in $M$ that are fixed by the $\mathbb{Z}_w$-action is a union of smaller dimensional compact almost complex submanifolds. Let $F$ be a connected component of $M^{\mathbb{Z}_w}$ that contains $p$. In general, $\dim F=2m$ if and only if $p$ has exactly $m$ weights that are divisible by $w$. Since the action is effective, $\dim M=4$, and $p$ has weight $w>1$, it follows that $\dim F=2$. The circle action on $M$ restricts to a circle action $C$ on $F$. The $C$-action on $F$ has $p$ as a fixed point. This implies that $F$ is the 2-sphere, since among compact oriented Riemann surfaces, only $S^2$ admits a circle action with a fixed point. Since any circle action on $S^2$ has 2 fixed points, the $C$-action on $F$ has another fixed point $q$ that has weight $-w$ as a weight of the $C$-action; hence the original circle action on $M$ also has weight $-w$ at $q$. Draw an edge from $p$ to $q$ with label $w$. Therefore, by (5) of Definition \[d11\], $(p,q)$ is $w$-edge. Alternatively, $(q,p)$ is $(-w)$-edge. In [@Ka], the $(p,q)$-edge is called $\mathbb{Z}_w$-sphere.
By Proposition \[p22\], the number of times weight $+1$ occurs at fixed points of index $i$ is equal to the number of times weight $-1$ occurs at fixed points of index $i+1$; therefore, if a fixed point $p$ has weight $+1$, draw an edge from $p$ to a fixed point $q$ that has weight $-1$ and $n_p+1=n_q$. This is how we draw a 2-regular labeled directed multigraph describing $M$ that has no self-loops by Lemma \[l26\].
The weights at fixed points lying in the same connected component of $M^{\mathbb{Z}_w}$ are equal modulo $w$;
\[l21\] [@T], [@GS] Let the circle act on a compact almost complex manifold $M$. Let $p$ and $p'$ be fixed points which lie in the same connected component of $M^{\mathbb{Z}_{w}}$, for some positive integer $w$. Then the $S^{1}$-weights at $p$ and at $p'$ are equal modulo $w$.
Lemma \[l21\] states that if $p$ and $p'$ lie in the same connected component of $M^{\mathbb{Z}_w}$, there exists a bijection $\pi:\{1,2,\cdots,n\} \to \{1,2,\cdots,n\}$ such that $w_{p,i} \equiv w_{p',{\pi(i)}} \mod w$ for all $i$, where $\dim M=2n$.
The following lemma shows that we can equivariantly glue together two 4-dimensional compact connected almost complex manifolds equipped with $\mathbb{T}^2$-actions. The latter part of the lemma will be used to prove Theorem \[t11\]. For a $\mathbb{T}^2$-action on an almost complex manifold $M$, denote by $\Sigma_M^{\mathbb{T}^2}$ the collection of multisets of the $\mathbb{T}^2$-weights at the fixed points of $M$.
\[l27\] For $i=1,2$, let $M_i$ be a 4-dimensional compact connected almost complex manifold equipped with an effective $\mathbb{T}^2$-action. Then we can perform equivariant connected sum along free orbits of $M_i$ to construct a 4-dimensional compact connected almost complex manifold $M$ equipped with an effective $\mathbb{T}^2$-action, such that the action has $\Sigma_{M_1}^{\mathbb{T}^2} \sqcup \Sigma_{M_2}^{\mathbb{T}^2}$ as the collection of multisets of the $\mathbb{T}^2$-weights at the fixed points of $M$.
In addition, suppose that the following hold:
1. For $i=1,2$, for any fixed point $p$ in $M_i$, $w_{p,j} \neq 0$ for $j=1,2$, where $(v_{p,1},v_{p,2})$ are the $\mathbb{T}^2$-weights at $p$, and $v_{p,j}=(w_{p,j}, x_{p,j}) \in \mathbb{Z}^2$, for $j=1,2$.
2. For $i=1,2$, a labeled directed multigraph $\Gamma_i$ describes $M_i$ for the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M_i$.
Then the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M$ is described by the disjoint union of the two multigraphs $\Gamma_1$ and $\Gamma_2$.
For each $i=1,2$, take an equivariant tubular neighborhood $U_i \approx D^2 \times \mathbb{T}^2$ of a free orbit of the $\mathbb{T}^2$-action on $M_i$. For each $i=1,2$, the almost complex structure $J_i$ on $M_i$ is $\mathbb{T}^2$-invariant, and hence on $U_i$ and on the boundary of $U_i$. Denote by $B_i =\partial U_i \approx S^1 \times \mathbb{T}^2$ the boundary of $U_i$.
For $i=1,2$, the complex structure on $B_i$ is classified by a continuous map from $B_i$ to $GL(4,\mathbb{R})/GL(2,\mathbb{C})$, which is homotopy equivalent to $S^2 \sqcup S^2$. Since $\mathbb{T}^2$ acts on $B_i$ freely and $J_i$ invariantly, the orbit space is $S^1$ and the map factors through $B_i \to S^1 \to GL(4,\mathbb{R})/GL(2,\mathbb{C})$. Choose a circle $S$ in the interior of $D^2$ of $U_2 \approx D^2 \times \mathbb{T}^2$ and let $C_2 \approx S \times \mathbb{T}^2 \subset U_2$.
We glue $B_1$ and $C_2$ equivariantly. Denote by $\phi$ the equivariant diffeomorphism between $B_1$ and $C_2$. By reversing the orientation of $M_2$ if necessary, the complex structure $J'$ on $C_2$ induced by $J_1$ via the equivariant diffeomorphism $\phi$ and the complex structure $J_2$ on $B_2$ lie in the same component of $GL(4,\mathbb{R})/GL(2,\mathbb{C}) \approx S^2 \sqcup S^2$. Therefore, between $C_2$ and $B_2$, we can perturb the almost complex structure $J_2$ on $B_2$ to agree with the induced complex structure $J'$ on $C_2$. Therefore, when we take the equivariant connected sum of $M_1$ and $M_2$ by gluing $B_1$ and $C_2$ equivariantly, the resulting manifold $M$ admits an almost complex structure.
On the other hand, since the equivariant connected sum is along free orbits of $M_1$ and $M_2$, it follows that the fixed point set of the $\mathbb{T}^2$-action on $M$ is the union $M_1^{\mathbb{T}^2} \sqcup M_2^{\mathbb{T}^2}$ of the fixed point sets of $M_1$ and $M_2$, and the fixed point data of the $\mathbb{T}^2$-action on $M$ is $\Sigma_{M_1}^{\mathbb{T}^2} \sqcup \Sigma_{M_2}^{\mathbb{T}^2}$. This proves the first claim.
The second claim follows from the construction of $M$; suppose that
1. for $i=1,2$, for any fixed point $p$ in $M_i$, $w_{p,j} \neq 0$ for $j=1,2$, where $(v_{p,1},v_{p,2})$ are the $\mathbb{T}^2$-weights at $p$, and $v_{p,j}=(w_{p,j}, x_{p,j}) \in \mathbb{Z}^2$, for $j=1,2$; and
2. for $i=1,2$, a labeled directed multigraph $\Gamma_i$ describes $M_i$ for the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M_i$.
For each fixed point $p \in M^{\mathbb{T}^2}$, we have $p \in M_i^{\mathbb{T}^2}$ for some $i=1,2$; hence if $\{v_{p,1},v_{p,2}\}$ are the $\mathbb{T}^2$-weights at $p$, then $\{w_{p,1},w_{p,2}\}$ are the weights at $p$ for the action for the first $S^1$ of $\mathbb{T}^2$ on $M$, where $v_{p,j}=(w_{p,j},x_{p,j})$ for some integers $w_{p,j}$, $x_{p,j}$ for $j=1,2$. Since $w_{p,j} \neq 0$ for all $p \in M^{\mathbb{T}^2}$ and $j=1,2$, the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M$ has the same fixed point set as the $\mathbb{T}^2$-action $M$, i.e., $M^{S^1}=M^{\mathbb{T}^2}$. Hence if $\Gamma_i$ describes $M_i$ for the action of the first $S^1$-factor of $\mathbb{T}^2$-action on $M_i$ for $i=1,2$, then $\Gamma_1 \sqcup \Gamma_2$ describes $M$ for the action of the first $S^1$-factor of $\mathbb{T}^2$-action on $M$.
Proof of the main result {#s3}
========================
In this section, we prove our main theorem, Theorem \[t11\]. For a technical reason to prove Theorem \[t11\], we need the following definition.
\[d31\] Let the circle act on a 4-dimensional compact almost complex manifold $M$ with a discrete fixed point set. We say that a 2-regular labeled directed multigraph $\Gamma$ **describes $M$ in the strong sense** if the following hold.
1. The multigraph $\Gamma$ describes $M$.
2. If $(p,q)$ is $w$-edge for some positive integer $w$, then $p$ and $q$ lie in the same 2-sphere $S^2$, where the circle action on $M$ restricts to act on this $S^2$ with fixed points $p$ and $q$ that have $S^1$-weights $w$ and $-w$, respectively.
We need Definition \[d31\] to deal with edges with label 1. To discuss, consider an effective circle action on a 4-dimensional compact almost complex manifold with a discrete fixed point set. Let $w>1$ be an integer. If two fixed points $p$ and $q$ lie in the same 2-sphere $S^2$ on which the circle action on $M$ restricts to act with fixed points $p$ and $q$ that have weights $w$ and $-w$, respectively, then Definition \[d12\] means that $(p,q)$ must be $w$-edge for any multigraph describing $M$. Now, let $w=1$. On the other hand, if a fixed point $p$ has weight $+1$ and two fixed points $q$ and $q'$ both have weight $-1$ and satisfy $n_p+1=n_q=n_{q'}$, then $M^{\mathbb{Z}/w}=M$ and hence we have a choice that $(p,q)$ is 1-edge or $(p,q')$ is 1-edge. However, when we prove Theorem \[t11\], a manifold we construct will satisfy that the manifold is described by a multigraph in the strong sense; there exists a unique fixed point $q$ that lie in the same $S^2$ on which the circle action on $M$ restricts to act with fixed points $p$ and $q$ that have weights $+1$ and $-1$, respectively.
Let $\Gamma$ be any multigraph obtained from a 2-regular semi-free multigraph by the 4 operations in Theorem \[t11\]. To prove the existence of a 4-dimensional almost complex $S^1$-manifold which is described by $\Gamma$, the key idea is the equivariant plumbing technique. Masuda [@M] used the equivariant plumbing to construct a 4-dimensional compact almost complex manifold equipped with a $\mathbb{T}^2$-action. We shall review and reformulate the theorem and the proof.
\[t31\] Let $v_0,v_1,\cdots,v_{k+1}$ be a sequence of vectors in $\mathbb{Z}^2$ in counterclockwise order with $v_k=v_0$ and $v_{k+1}=v_1$. Suppose that
1. Each successive pair $v_i$ and $v_{i+1}$ is a basis of $\mathbb{Z}^2$, for $1 \leq i \leq k$.
2. For $1 \leq i \leq k$, there exists an integer $a_i$ such that $v_{i+1}=-a_i v_i-v_{i-1}$ for each $1 \leq i \leq k$.
Then there exists a 4-dimensional compact, connected almost complex manifold $M$ equipped with a $\mathbb{T}^2$-action having $k$ fixed points $p_1,\cdots,p_k$ such that the $\mathbb{T}^2$-weights at $p_i$ are $\{v_i,-v_{i-1}\}$ for $1 \leq i \leq k$.
In addition, suppose that $v_{i,1} \neq 0$ for all $i$, where $v_i=(v_{i,1},v_{i,2}) \in \mathbb{Z}^2$. Then as the first $S^1$-factor of $\mathbb{T}^2$, the $S^1$-action on $M$ has the same fixed point set as the $\mathbb{T}^2$-action, and the $S^1$-weights at $p_i$ are $\{v_{i,1},-v_{i-1,1}\}$ for $1 \leq i \leq k$. Moreover, if we let $
\Gamma$ be a labeled directed multigraph such that $(p_i,p_{i+1})$ is $v_{i,1}$-edge for $1 \leq i \leq k$, then $\Gamma$ describes $M$ in the strong sense.
Let $a$ be an integer. Let $O(a)$ be the holomorphic line bundle over $\mathbb{CP}^1$ whose self-intersection number of the zero section is $a$. That is, the quotient of $(\mathbb{C}^2-\{0\}) \times \mathbb{C}$ by the $C^*$-action given by
$g \cdot (z_1,z_2,w)=(gz_1,gz_2,g^aw)$,
for all $g \in C^*$ and $(z_1,z_2,w) \in (\mathbb{C}^2-\{0\}) \times \mathbb{C}$. Denote by $[z_1,z_2,w]$ the equivalence class of $(z_1,z_2,w)$. Let $\mathbb{T}^2$ act on $O(a)$ by
$t \cdot [z_1,z_2,w]=[z_1,t^{u_1}z_2,t^{u_2}w]$
for all $t=(t_1,t_2) \in \mathbb{T}^2$, for some $u_1,u_2 \in \mathbb{Z}^2$ that form a basis of $\mathbb{Z}^2$. Let $D_a(u_1,u_2)$ be the disk bundle of $O(a)$ with this $\mathbb{T}^2$-action. The action has two fixed points $q_1=[1,0,0]$ and $q_2=[0,1,0]$, and the $\mathbb{T}^2$-weights at the fixed points $q_1$ and $q_2$ are $\{u_1,u_2\}$ and $\{-au_1+u_2,-u_1\}$, respectively.
For $D_{a_i}(v_i,-v_{i-1})$, the $\mathbb{T}^2$-weights at $q_{i,2}$ are $\{-a_i v_i-v_{i-1},-v_i \}$ and for $D_{a_{i+1}}(v_{i+1},-v_i)$, the $\mathbb{T}^2$-weights at $q_{i+1,1}$ are $\{v_{i+1},-v_i\}$, for $1 \leq i \leq k$. By the assumption, for $1 \leq i \leq k$, there exists an integer $a_i$ such that $v_{i+1}=-a_i v_i-v_{i-1}$. Hence the $\mathbb{T}^2$-weights at $q_{i,2}$ and $q_{i+1,1}$ agree for $1 \leq i \leq k$. Therefore, for every $1 \leq i \leq k$, we can equivariantly plumb two manifolds $D_{a_i}(v_i,-v_{i-1})$ and $D_{a_{i+1}}(v_{i+1},-v_i)$ at $q_{i,2}$ and $q_{i+1,1}$. Then we get a 4-dimensional compact connected manifold $N$ with the $\mathbb{T}^2$-action. As in the proof of Theorem 5.1 of [@M], by pasting $N$ and $D^2 \times \mathbb{T}^2$ along the boundary $\partial N \simeq S^1 \times \mathbb{T}^2$ of $N$, we get a 4-dimensional closed connected almost complex $\mathbb{T}^2$-manifold $M$; see the detailed proof of Theorem 5.1 in [@M] that $M$ admits a $\mathbb{T}^2$-invatiant almost complex structure. Moreover, $M$ has $k$ fixed points, and the $\mathbb{T}^2$-weights at each fixed point $p_i=q_{i,1}$ are $\{v_i,-v_{i-1}\}$ for each $1 \leq i \leq k$. This proves the first part.
Suppose that $v_{i,1} \neq 0$ for all $i$. The last two claims for the restriction of $\mathbb{T}^2$ on the first $S^1$-factor follow from the construction. From the construction, the $S^1$-weights at $p_i$ are $\{v_{i,1},-v_{i-1,1}\}$ for $1 \leq i \leq k$. Moreover, from the construction it follows that for any $1 \leq i \leq k$, $p_i$($=q_{i,1}$) and $p_{i+1}$($=q_{i,2}=q_{i+1,1}$) lie in the same 2-sphere $S^2=\mathbb{CP}^1=\{[z_1,z_2,0]|(z_1,z_2) \in \mathbb{C}^2-\{0\}\} \subset D_{a_i}(v_i,v_{i-1})$, where the first $S^1$-factor of $\mathbb{T}^2$ restricts to act on this $S^2$ by $t_1 \cdot [z_1,z_2,0]=[z_1, t_1^{v_{i,1}}z_2,0]$ for all $t_1 \in S^1$, with fixed points $p_i$ and $p_{i+1}$ that have $S^1$-weights $v_{i,1}$ and $-v_{i,1}$, respectively. Therefore, if we draw a 2-regular labeled directed multigraph $\Gamma$ so that $(p_i,p_{i+1})$ is $v_{i,1}$-edge for $1 \leq i \leq k$, then $\Gamma$ describes $M$ in the strong sense for the action of the first $S^1$-factor of $\mathbb{T}^2$.
There is a slight difference between constructions of $\mathbb{T}^2$-manifolds in Theorem \[t31\] and Theorem 5.1 of [@M]. In Theorem 5.1 of [@M], Masuda also took $v_i$ in counterclockwise order but then used the dual basis $\{u_1^{(i)},u_2^{(i)}\}$ of $\{v_{i-1},v_i\}$ to construct a $\mathbb{T}^2$-manifold. Let $\{u_1^{(i)},u_2^{(i)}\}$ be the dual basis of $\{v_{i-1},v_i\}$. Then Masuda used $D_{a_i}(u_1^{(i)},u_2^{(i)})$ to construct a manifold, where the $\mathbb{T}^2$-weights at $q_{i,1}$ and $q_{i,2}$ are $\{u_1^{(i)},u_2^{(i)}\}$ and $\{u_2^{(i)}-a_iu_1^{(i)},-u_1^{(i)}\}$, respectively. For $1 \leq i \leq k$, Condition (2) of Theorem \[t31\] can be written as
$\begin{pmatrix} v_{i} & v_{i+1} \end{pmatrix} = \begin{pmatrix} v_{i-1} & v_{i} \end{pmatrix} \begin{pmatrix} 0 & -1 \\ 1 & -a_i \end{pmatrix}$
Since $\{u_1^{(i)},u_2^{(i)}\}$ is the dual basis of $\{v_{i-1},v_i\}$, for each $1 \leq i \leq k$, the relationship between $\{u_1^{(i+1)}, u_2^{(i+1)}\}$ and $\{u_1^{(i)}, u_2^{(i)}\}$ are then $$\label{e1}
\begin{pmatrix} u_1^{(i+1)} \\ u_2^{(i+1)} \end{pmatrix} = \begin{pmatrix} -a_i & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} u_1^{(i)} \\ u_2^{(i)} \end{pmatrix}$$ Since Equation holds, the $\mathbb{T}^2$-weights at $q_{i,2}$ and $q_{i+1,1}$ agree; hence we can equivariantly plumb two manifolds $D_{a_i}(u_1^{(i)},u_2^{(i)})$ and $D_{a_{i+1}}(u_1^{(i+1)},u_2^{(i+1)})$ at $q_{i,2}$ and $q_{i+1,1}$ for each $1 \leq i \leq k$.
On the other hand, instead of taking the dual basis of $\{v_i,v_{i-1}\}$ if we take $u_1^{(i)}=v_i$ and $u_2^{(i)}=-v_{i-1}$, then Equation \[e1\] also holds. Therefore, in Theorem \[t31\] we directly used $D_{a_i}(v_i,-v_{i-1})$ to construct a $\mathbb{T}^2$-manifold without introducing $u_m^{(i)}$’s. Since the $\mathbb{T}^2$-weights at $q_{i,2}$ and $q_{i+1,1}$ also agree in our case, we can perform equivariant plumbing. Therefore, the proofs for the arguments in Theorem 5.1 of [@M] apply smoothly for the rest of Theorem \[t31\] (such as, by pasting $N$ and $D^2 \times \mathbb{T}^2$ along the boundary $\partial N \simeq S^1 \times \mathbb{T}^2$ of $N$, we get a 4-dimensional closed connected almost complex $\mathbb{T}^2$-manifold $M$, and $M$ admits a $\mathbb{T}^2$-invariant almost complex structure, etc).
In Theorem \[t31\], it is important that for each pair $(v_i,v_{i+1})$, $v_i$ and $v_{i+1}$ are in counterclowise order; if we go from $v_i$ to $v_{i+1}$ in counterclockwise order, the angle between them must be strictly less than $\pi$. Then our task is to show that if we construct a manifold $M'$ using Theorem \[t31\] which is described by a multigraph $\Gamma'$, where $\Gamma'$ is obtained by performing one of the operations in Theorem \[t11\] from another multigraph $\Gamma$ that describes a manifold $M$ and $M$ is constructed by Theorem \[t31\] with $v_i$ in counterclockwise order, then $M'$ also satisfies in its construction that $v_i'$ are in counterclockwise order. Because of this, we introduce the following definition.
Let $v_0,v_1,\cdots,v_{k+1}$ be a sequence of vectors in $\mathbb{Z}^2$ in counterclockwise order with $v_k=v_0$ and $v_{k+1}=v_1$ such that the following hold.
(a) Each successive pair $v_i$ and $v_{i+1}$ is a basis of $\mathbb{Z}^2$, for $1 \leq i \leq k$.
(b) For any $1 \leq i \leq k$, there exists an integer $a_i$ such that $v_{i+1}=-a_i v_i-v_{i-1}$.
(c) For any $1 \leq i \leq k$, we have $v_{i,1} \neq 0$, where $v_i=(v_{i,1},v_{i,2}) \in \mathbb{Z}^2$.
Suppose that using the $v_i$ and $a_i$, by Theorem \[t31\] we construct a 4-dimensional compact, connected almost complex manifold equipped with a $\mathbb{T}^2$-action having fixed points $p_i$, $1 \leq i \leq k$. Let $\Gamma$ be a labeled directed multigraph describing $M$ in the strong sense for the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M$, as in Theorem \[t31\]. We say that $M$ **satisfies Property A** if the following hold, by reversing the order the index of $p_i$, $v_i$, and $a_i$ if necessary:
(1) At each fixed point $p_i$, the $S^1$-weights $\{v_{i,1},-v_{i-1,1}\}$ at $p_i$ are pairwise prime.
(2) Suppose that $(p_i,p_{i-1})$ is $c$-edge, $(p_{i+1},p_i)$ is $d$-edge, and $(p_{i+1},p_{i+2})$ is $c$-edge, for some $i$ and for some positive integers $c$ and $d$ (Figure \[fig3\]). Then $v_{i+1}=-v_{i-1}$.
(3) Suppose that $(p_{i-1},p_i)$ is $e$-edge, $(p_{i},p_{i+1})$ is $f$-edge, and $(p_{i+2},p_{i+1})$ is $e$-edge, for some $i$ and for some positive integers $e$ and $f$ (Figure \[fig4\]). Then $v_{i+1}=-v_{i-1}$.
To prove Theorem \[t11\], we first prove that for any 2-regular semi-free multigraph $\Gamma$, we can construct a semi-free circle action on a compact almost complex manifold $M$ which is described by $\Gamma$ in the strong sense and $M$ satisfies Property A. This corresponds to our minimal model (multigraph) in Theorem \[t11\]. Figure \[fig9’\] describes the construction in Lemma \[l32\].
\[l32\] Let $\Gamma$ be a connected 2-regular semi-free multigraph. Then by Theorem \[t31\] we can construct a 4-dimensional compact connected almost complex manifold $M$ equipped with a $\mathbb{T}^2$-action, such that the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M$ is semi-free, and $\Gamma$ describes $M$ in the strong sense for this $S^1$-action. Moreover, $M$ satisfies Property A.
Let $T(\Gamma)=k$, that is, the number of vertices of index 0. Since $\Gamma$ is semi-free, $\Gamma$ has $4k$ vertices; $k$ vertices of index 0, $2k$ vertices of index 1, and $k$ vertices of index 2. In Theorem \[t31\], take $v_{4j+1}=(1,0)$, $v_{4j+2}=(1,1)$, $v_{4j+3}=(-1,0)$, $v_{4j+4}=(-1,-1)$ for each $0 \leq j \leq k-1$, and $a_i=0$ for all $1 \leq i \leq 4k$. Then $v_i$ and $a_i$ satisfy Conditions (1) and (2) in Theorem \[t31\]. Let $M$ be the 4-dimensional compact, connected almost complex manifold equipped with a $\mathbb{T}^2$-action constructed. Then the $\mathbb{T}^2$-action on $M$ has $4k$ fixed points.
Consider the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M$. Since $v_{i,1} \neq 0$ for all $1 \leq i \leq 4k$, where $v_i=(v_{i,1},v_{i,2}) \in \mathbb{Z}^2$, the second claim of Theorem \[t31\] implies that the $S^1$-action on $M$ has the same fixed point set as the $\mathbb{T}^2$-action on $M$. Moreover, the $S^1$-weights at each fixed point $p_i$ are $\{v_{i,1},-v_{i-1,1}\}$, $1 \leq i \leq 4k$. This means that the $S^1$-weights at $p_{4j+1}$, $p_{4j+2}$, $p_{4j+3}$, $p_{4j+4}$ are $\{1,1\}$, $\{1,-1\}$, $\{-1,-1\}$, $\{-1,1\}$, respectively, for $0 \leq j \leq k-1$. Since every $S^1$-weight at any fixed point is either $+1$ or $-1$, the $S^1$-action is semi-free and $M$ satisfies Condition (1) of Property A.
By the construction, $v_{i+2}=-v_i$ for any $1 \leq i \leq 4k-2$, $v_1=-v_{4k-1}$, and $v_2=-v_{4k}$. It follows that $M$ satisfies Property A. The last claim of Theorem \[t31\] implies that $\Gamma$ describes $M$ in the strong sense.
[1]{}
\(A) [$p_{4j+1}$]{}; (B) \[above right=of A\] [$p_{4j+2}$]{}; (C) \[above right=of B\] [$p_{4j+3}$]{}; (D) \[below right=of C\] [$p_{4j+4}$]{}; (E) \[below right=of D\] [$p_{4j+5}$]{}; (F) \[above right=of E\] [$p_{4j+6}$]{}; (A) \[->\] edge node\[left\] [$1, (1,0)$]{} (B); (B) \[->\] edge node \[left\] [$1$, $v_{4j+2}=(1,1)$]{} (C); (D) \[->\] edge node \[right\] [$1$, $v_{4j+3}=(-1,0)$]{} (C); (E) \[->\] edge node \[left\] [$1$, $v_{4j+4}=(-1,-1)$]{} (D); (E) \[->\] edge node \[right\] [$1, (1,0)$]{} (F);
The next lemma states that from an $S^1$-manifold described by a multigraph $\Gamma$ such that $\Gamma$ is obtained from Theorem \[t11\], we can construct another $S^1$-manifold described by a multigraph, which is obtained from $\Gamma$ by applying Operation 1 in Theorem \[t11\]; see Figure \[fig2\]. Both manifolds satisfy Property A.
\[l33\] Let $\Gamma$ be a connected 2-regular labeled directed multigrph. Assume that by Theorem \[t31\] we can construct a 4-dimensional compact connected almost complex manifold $M$ equipped with a $\mathbb{T}^2$-action having a discrete fixed point set such that the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M$ is described by $\Gamma$ in the strong sense and $M$ satisfies Property A. Suppose that a fixed point $p_i$ of $M$ has $S^1$-weights $\{-a,b\}$ for some positive integers $a$ and $b$. Then by Theorem \[t31\] we can construct a 4-dimensional compact connected almost complex manifold $M'$ equipped with a $\mathbb{T}^2$-action having a discrete fixed point set such that the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M'$ is described by $\Gamma'$ in the strong sense, where $\Gamma'$ (Figure \[fig2-2\]) is obtained from $\Gamma$ (Figure \[fig2-1\]) by replacing the vertex $p_i$ with $(a+b)$-edge. Moreover, $M'$ satisfies Property A.
Suppose that by Theorem \[t31\] we construct a 4-dimensional compact connected almost complex manifold $M$ equipped with a $\mathbb{T}^2$-action having a discrete fixed point set such that $\Gamma$ describes $M$ in the strong sense for the action of the first $S^1$-factor of $\mathbb{T}^2$, and $M$ satisfies Property A. Let $v_j=(v_{j,1},v_{j,2})$ and $a_j$ be the elements of $\mathbb{Z}^2$ and $\mathbb{Z}$ that are used to construct $M$ in Theorem \[t31\], $1 \leq j \leq k$, where $k$ is the number of fixed points of $M$. Then each fixed point $p_j$ of $M$ has $\mathbb{T}^2$-weights $\{v_j,-v_{j-1}\}$ for all $1 \leq j \leq k$. Next, by using Theorem \[t31\], construct a 4-dimensional compact connected almost complex manifold $M'$ equipped with a $\mathbb{T}^2$-action having $k+1$ fixed points, by taking the following values for $v_j'$ and $a_j'$.
1. $v_1'=v_1$, $\cdots$, $v_{i-2}'=v_{i-2}$, $v_{i-1}'=v_{i-1}$, $v_i'=v_{i-1}+v_i$, $v_{i+1}'=v_i$, $\cdots$, $v_{k+1}'=v_{k}$, $v_{k+2}'=v_1'$.
2. $a_1'=a_1$, $\cdots$, $a_{i-2}'=a_{i-2}$, $a_{i-1}'=a_{i-1}-1$, $a_i'=-1$, $a_{i+1}'=a_i-1$, $a_{i+2}'=a_{i+1}$, $\cdots$, $a_{k+1}'=a_k$.
Since $v_j$ and $a_j$ satisfy Conditions (1) and (2) of Theorem \[t31\], $v_j'$ and $a_j'$ also satisfy Conditions (1) and (2) of Theorem \[t31\]. It is easy to check Condition (1) of Theorem \[t31\]. For Condition (2), for instance, since $v_{i-1}'=v_{i-1}$, $v_{i-2}'=v_{i-2}$, and $a_{i-1}'=a_{i-1}-1$, we have $v_i'=v_{i-1}+v_i=v_{i-1}-a_{i-1}v_{i-1}-v_{i-2}=-(a_{i-1}-1) v_{i-1}-v_{i-2}=-a_{i-1}' v_{i-1}'-v_{i-2}'$. Similarly, one can check that $v_j'$ and $a_j'$ satisfy Condition (2) of Theorem \[t31\]. By the construction, the fixed points $p_j'$ of $M'$ have the following $\mathbb{T}^2$-weights.
1. If $j < i$, $p_j'$ have the same $\mathbb{T}^2$-weights $\{v_j,-v_{j-1}\}$ as $p_j$.
2. The fixed point $p_{i}'$ has $\mathbb{T}^2$-weights $\{v_{i-1}+v_{i},-v_{i-1}\}$.
3. The fixed point $p_{i+1}'$ has $\mathbb{T}^2$-weights $\{v_{i},-v_{i-1}-v_{i}\}$.
4. If $j > i+1$, the $\mathbb{T}^2$-weights of $p_j$ and $p_{j+1}'$ are the same.
Since $v_{i-1,1}=a$ and $v_{i,1}=b$, we have $v_{i,1}'=v_{i-1,1}+v_{i,1}=a+b$. Therefore, by the last claim of Theorem \[t31\], $p_i'$ and $p_{i+1}'$ lie in the same 2-sphere $S^2$, where the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M$ restricts to act on this $S^2$ with fixed points $p_i'$ and $p_{i+1}'$ that have $S^1$-weights $a+b$ and $-a-b$, respectively. Therefore, if we let $\Gamma'$ be a multigraph that is obtained from $\Gamma$ by replacing the vertex $p_i$ with $(a+b)$-edge $(p_i',p_{i+1}')$, $\Gamma'$ describes $M'$ in the strong sense.
From now on, we check that $M'$ also satisfies Property A.
First, we check Condition (1) of Property A. Since $M$ satisfies Property A, the $S^1$-weights $-a$ and $b$ at $p_i$ are relatively prime. Therefore, the $S^1$-weights $-a$ and $a+b$ at $p_i'$ are relatively prime, and the $S^1$-weights $-a-b$ and $b$ at $p_{i+1}'$ are relatively prime. Since other fixed points $p_j'$ have the same weights as $p_m$ for some $m$, it follows that the $S^1$-weights at each fixed point in $M'$ are relatively prime.
Now we check Conditions (2) and (3) of Property A. This means that we check Property A for the edge $(p_{i}',p_{i+1}')$. And for $(p_{i}',p_{i+1}')$-edge we check Condition (2) of Property A if $(p_{i-1}',p_{i-2}')$ is $(a+b)$-edge (Figure \[fig10-2\]), since then $(p_{i-1}',p_{i-2}')$ is $(a+b)$-edge, $(p_{i-1}',p_i')$ is $a$-edge, and $(p_{i}',p_{i+1}')$ is $(a+b)$-edge so that these edges are in the situation of Condition (2) of Property A. Similarly, we check Condition (3) of Property A if $(p_{i+3}',p_{i+2}')$ is $(a+b)$-edge (Figure \[fig11-2\]), since then $(p_{i}',p_{i+1}')$ is $(a+b)$-edge, $(p_{i+1}',p_{i+2}')$ is $b$-edge, and $(p_{i+3}',p_{i+2}')$ is $(a+b)$-edge so that these edges are in the situation of Condition (3) of Property A.
First, suppose that $(p_{i-1}',p_{i-2}')$ is $(a+b)$-edge. This means that in $\Gamma$ we have $(p_{i-1},p_{i-2})$ is $(a+b)$-edge (Figure \[fig10-1\]). We show that $v_{i}'=-v_{i-2}'$. For this, consider the original $v_j$. We have $v_{i-2,1}=-a-b$, $v_{i-1,1}=a$, and $v_{i,1}=b$. Since $v_{i}=-a_{i-1} \cdot v_{i-1}-v_{i-2}$, it follows that $v_{i,1}=-a_{i-1} \cdot v_{i-1,1}-v_{i-2,1}$ and hence $b=v_{i,1}=-a_{i-1} \cdot v_{i-1,1}-v_{i-2,1}=-a_{i-1} \cdot a-(-a-b)$, which means that $a_{i-1}=1$. Therefore, $a_{i-1}'=a_{i-1}-1=0$. Hence $v_i'=-a_{i-1}' \cdot v_{i-1}'-v_{i-2}'=-v_{i-2}'$. Therefore, Condition (2) of Property A holds for all edges in $\Gamma'$, since other $v_i'$ are the same as $v_m$ for some $m$ and $M$ satisfies Condition (2) of Property A.
Second, suppose that $(p_{i+3}',p_{i+2})'$ is $(a+b)$-edge. This means that in $\Gamma$ we have $(p_{i+2},p_{i+1})$ is $(a+b)$-edge (Figure \[fig11-1\]). We show that $v_{i+2}'=-v_i'$. For this, consider $v_j$. We have $v_{i-1,1}=a$, $v_{i,1}=b$, and $v_{i+1,1}=-a-b$. Since $v_{i+1}=-a_{i} \cdot v_{i}-v_{i-1}$, it follows that $v_{i+1,1}=-a_i \cdot v_{i,1}-v_{i-1,1}$ and hence $-a-b=v_{i+1,1}=-a_{i} \cdot v_{i,1}-v_{i-1,1}=-a_i \cdot b-a$, which means that $a_i=1$. Since $a_{i+1}'=a_i-1$, $a_{i+1}'=0$ and hence $v_{i+2}'=-a_{v+1}' \cdot v_{i+1}'-v_i'=-v_i'$. Therefore, Condition (3) of Property A holds for all edges in $\Gamma'$, since other $v_i'$ are the same as $v_m$ for some $m$ and $M$ satisfies Condition (3) of Property A.
Therefore, $M'$ satisfies Property A.
[.4]{}
\(a) [$p_{i-1}$]{}; (d) \[above left=of a\] [$p_{i-2}$]{}; (b) \[above right=of a\] [$p_i$]{}; (c) \[above left=of b\] [$p_{i+1}$]{}; (a) \[->\] edge node\[right\] [$a$]{} (b); (b) \[->\] edge node \[right\] [$b$]{} (c); (a) \[->\] edge node\[left\] [$a+b$]{} (d);
[.4]{}
\(a) [$p_{i-1}'$]{}; (b) \[above right=of a\] [$p_{i}'$]{}; (c) \[above=of b\] [$p_{i+1}'$]{}; (d) \[above left=of c\] [$p_{i+2}'$]{}; (e) \[above left=of a\] [$p_{i-2}'$]{}; (a) \[->\] edge node\[right\] [$a$]{} (b); (b) \[->\] edge node \[left\] [$a+b$]{} (c); (c) \[->\] edge node \[left\] [$b$]{} (d); (a) \[->\] edge node\[left\] [$a+b$]{} (e);
[.4]{}
\(a) [$p_{i-1}$]{}; (b) \[above right=of a\] [$p_i$]{}; (c) \[above left=of b\] [$p_{i+1}$]{}; (d) \[below left=of c\] [$p_{i+2}$]{}; (a) \[->\] edge node\[right\] [$a$]{} (b); (b) \[->\] edge node \[right\] [$b$]{} (c); (d) \[->\] edge node\[left\] [$a+b$]{} (c);
[.4]{}
\(a) [$p_{i-1}'$]{}; (b) \[above right=of a\] [$p_{i}'$]{}; (c) \[above=of b\] [$p_{i+1}'$]{}; (d) \[above left=of c\] [$p_{i+2}'$]{}; (e) \[below left=of d\] [$p_{i+3}'$]{}; (a) \[->\] edge node\[right\] [$a$]{} (b); (b) \[->\] edge node \[left\] [$a+b$]{} (c); (c) \[->\] edge node \[left\] [$b$]{} (d); (e) \[->\] edge node\[left\] [$a+b$]{} (d);
The following lemma corresponds to Operation 2 and Figure \[fig3\] in Theorem \[t11\].
\[l34\] Let $\Gamma$ be a connected 2-regular labeled directed multigrph. Assume that by Theorem \[t31\] we can construct a 4-dimensional compact connected almost complex manifold $M$ equipped with a $\mathbb{T}^2$-action having a discrete fixed point set such that the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M$ is described by $\Gamma$ in the strong sense and $M$ satisfies Property A. Suppose that $(p_2,p_1)$ is $c$-edge, $(p_3,p_2)$ is $d$-edge, and $(p_3,p_4)$ is $c$-edge, for some vertices $p_1$, $p_2$, $p_3$, $p_4$ and for some positive integers $c$, $d$. Then by Theorem \[t31\] we can construct a 4-dimensional compact connected almost complex manifold $M'$ equipped with a $\mathbb{T}^2$-action having a discrete fixed point set such that the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M'$ is described by $\Gamma'$ in the strong sense, where $\Gamma'$ (Figure \[fig3-2\]) is obtained from $\Gamma$ (Figure \[fig3-1\]) by replacing the label $d$ of the edge $(p_3,p_2)$ with $d+c$. Moreover, $M'$ satisfies Property A.
Suppose that by Theorem \[t31\] we construct a 4-dimensional compact connected almost complex manifold $M$ equipped with a $\mathbb{T}^2$-action having a discrete fixed point set such that $\Gamma$ describes $M$ in the strong sense for the action of the first $S^1$-factor of $\mathbb{T}^2$, and $M$ satisfies Property A. Let $v_j=(v_{j,1},v_{j,2})$ and $a_j$ be the elements of $\mathbb{Z}^2$ and $\mathbb{Z}$ that are used to construct $M$ in Theorem \[t31\], $1 \leq j \leq k$, where $k$ is the number of fixed points of $M$. By using Theorem \[t31\], construct a 4-dimensional compact connected almost complex manifold $M'$ equipped with a $\mathbb{T}^2$-action having $k$ fixed points, by taking the following values for $v_j'$ and $a_j'$.
1. $v_1'=v_1$, $v_2'=v_1+v_2$, $v_3'=v_3$, $\cdots$, $v_{k+1}'=v_1'$.
2. $a_1'=a_1-1$, $a_2'=0$, $a_3'=a_1+1$, $a_4'=a_4$, $\cdots$, $a_k'=a_k$, $a_{k+1}'=a_1-1$.
Since $v_j$ and $a_j$ satisfy Conditions (1) and (2) of Theorem \[t31\], $v_j'$ and $a_j'$ also satisfy the conditions of Theorem \[t31\]. By the construction, the fixed points $p_j'$ of $M'$ have the following $\mathbb{T}^2$-weights.
1. The fixed point $p_2'$ has $\mathbb{T}^2$-weights $\{v_2+v_1,-v_1\}$.
2. The fixed point $p_3'$ has $\mathbb{T}^2$-weights $\{v_3,-v_2-v_1\}$.
3. If $j \neq 2,3$, the $\mathbb{T}^2$-weights at $p_j$ and $p_j'$ are equal.
Since $v_{1,1}=-c$ and $v_{2,1}=-d$, we have $v_{2,1}'=v_{1,1}+v_{2,1}=-c-d$. Therefore, by the last claim of Theorem \[t31\], $p_2'$ and $p_{3}'$ lie in the same 2-sphere $S^2$, where the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M$ restricts to act on this $S^2$ with fixed points $p_2'$ and $p_{3}'$ that have $S^1$-weights $-c-d$ and $c+d$, respectively. Therefore, if we let $\Gamma'$ be a multigraph which is obtained from $\Gamma$ by replacing the label $d$ of $(p_3,p_2)$-edge with $(d+c)$, then $\Gamma'$ describes $M'$ in the strong sense.
We check that $M'$ satisfies Property A.
First, we check Condition (1) of Property A. The fixed points $p_2$ and $p_3$ have $S^1$-weights $\{c,-d\}$ and $\{d,c\}$, respectively. Since $M$ satisfies Condition (1) of Property A, $c$ and $d$ are relatively prime. The fixed points $p_2'$ and $p_3'$ have $S^1$-weights $\{c,-c-d\}$ and $\{c+d,c\}$. Since $p_i'$ have the same weights as $p_i$ for $i \neq 2,3$ and $M$ satisfies Condition (1) of Property A, it follows that $M'$ also satisfies Condition (1) of Property A.
Second, we check Condition (2) of Property A. Since $v_1'=v_1$, $v_3'=v_3$, and $a_2'=0$, we have $v_3'=-a_2' \cdot v_2'-v_1'=-v_1=-v_1'$. Next, we check Condition (2) of Property A for the edge $(p_3',p_2')$. We only need to check if $(p_4',p_5')$ is $(c+d)$-edge, since then $(p_3',p_2')$ is $(c+d)$-edge, $(p_3',p_4')$ is $c$-edge, and $(p_4',p_5')$ is $(c+d)$-edge so that these edges are in the situation of Condition (2) of Property A. Suppose that $(p_4',p_5')$ is $(c+d)$-edge. This means that $(p_4,p_5)$ is $(c+d)$-edge. We show that $v_4'=-v_2'$. We have $v_{2,1}=-d$, $v_{3,1}=c$, and $v_{4,1}=c+d$. Since $v_4=-a_3 \cdot v_3-v_2$, it follows that $v_{4,1}=-a_3 \cdot v_{3,1}-v_{2,1}$ and hence $c+d=v_{4,1}=-a_3 \cdot v_{3,1}-v_{2,1}=-a_3 \cdot c+d$, which implies that $a_3=-1$. Since $a_3'=a_3+1$, it follows that $a_3'=0$ and hence $v_4'=-a_3' \cdot v_3'-v_2'=-v_2'$. Therefore, Condition (2) of Property A holds for all edges in $\Gamma'$, since other $v_i'$ are the same as $v_i$ and $M$ satisfies Condition (2) of Property A.
Third, we check Condition (3) of Property A. This means that we check Condition (3) of Property A for the edge $(p_3',p_2')$, and we only need to check if $(p_{k-1}',p_1')$ is $(c+d)$-edge, since then $(p_{k-1}',p_1')$ is $(c+d)$-edge, $(p_2',p_1')$ is $c$-edge, and $(p_3',p_2')$ is $(c+d)$-edge so that these edges are in the situation of Condition (3) of Property A. Suppose that $(p_{k-1}',p_1')$ is $(c+d)$-edge. This means that $(p_{k-1},p_1)$ is $(c+d)$-edge. We show that $v_2'=-v_{k-1}'$. We have $v_{k-1,1}=c+d$, $v_{1,1}=-c$, and $v_{2,1}=-d$. Since $v_2=-a_1 \cdot v_1-v_{k-1}$, $-d=v_{2,1}=-a_1 \cdot v_{1,1}-v_{k-1,1}=a_1 \cdot c-c-d$ and hence $a_1=1$. Since $a_1'=a_1-1$, it follows that $a_1'=0$ and hence $v_2'=-a_1' \cdot v_1'-v_{k-1}'=-v_{k-1}'$. Therefore, Condition (3) of Property A holds for all edges in $\Gamma'$, since other $v_i'$ are the same as $v_i$ and $M$ satisfies Condition (3) of Property A.
The following lemma corresponds to Operation 3 and Figure \[fig4\] in Theorem \[t11\]. The proof of Lemma \[l35\] is analogous to Lemma \[l34\].
\[l35\] Let $\Gamma$ be a connected 2-regular labeled directed multigrph. Assume that by Theorem \[t31\] we can construct a 4-dimensional compact connected almost complex manifold $M$ equipped with a $\mathbb{T}^2$-action having a discrete fixed point set such that the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M$ is described by $\Gamma$ in the strong sense and $M$ satisfies Property A. Suppose that $(p_1,p_2)$ is $e$-edge, $(p_2,p_3)$ is $f$-edge, and $(p_4,p_3)$ is $e$-edge, for some vertices $p_1$, $p_2$, $p_3$, $p_4$ and for some positive integers $e$, $f$. Then by Theorem \[t31\] we can construct a 4-dimensional compact connected almost complex manifold $M'$ equipped with a $\mathbb{T}^2$-action having a discrete fixed point set such that the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M'$ is described by $\Gamma'$ in the strong sense, where $\Gamma'$ (Figure \[fig4-2\]) is obtained from $\Gamma$ (Figure \[fig4-1\]) by replacing the label $f$ of the edge $(p_2,p_3)$ with $f+e$. Moreover, $M'$ satisfies Property A.
Suppose that by Theorem \[t31\] we construct a 4-dimensional compact connected almost complex manifold $M$ equipped with a $\mathbb{T}^2$-action having a discrete fixed point set such that $\Gamma$ describes $M$ in the strong for the action of the first $S^1$-factor of $\mathbb{T}^2$, and $M$ satisfies Property A. Let $v_j=(v_{j,1},v_{j,2})$ and $a_j$ be the elements of $\mathbb{Z}^2$ and $\mathbb{Z}$ that are used to construct $M$ in Theorem \[t31\], $1 \leq j \leq k$, where $k$ is the number of fixed points of $M$. By using Theorem \[t31\], construct a 4-dimensional compact connected almost complex manifold $M'$ equipped with a $\mathbb{T}^2$-action having $k$ fixed points, by taking the following values for $v_j'$ and $a_j'$.
1. $v_1'=v_1$, $v_2'=v_1+v_2$, $v_3'=v_3$, $\cdots$, $v_{k+1}'=v_1'$.
2. $a_1'=a_1-1$, $a_2'=0$, $a_3'=a_1+1$, $a_4'=a_4$, $\cdots$, $a_k'=a_k$, $a_{k+1}'=a_1-1$.
Since $v_j$ and $a_j$ satisfy Conditions (1) and (2) in Theorem \[t31\], the $v_j'$ and $a_j'$ also satisfy Conditions (1) and (2) in Theorem \[t31\]. By the construction, the fixed points $p_j'$ of $M'$ have the following $\mathbb{T}^2$-weights.
1. The fixed point $p_2'$ has the $\mathbb{T}^2$-weights $\{v_2+v_1,-v_1\}$.
2. The fixed point $p_3'$ has the $\mathbb{T}^2$-weights $\{v_3,-v_2-v_1\}$.
3. If $j \neq 2,3$, the $\mathbb{T}^2$-weights of $p_j$ and $p_j'$ are equal.
Since $v_{1,1}=e$ and $v_{2,1}=f$, we have $v_{2,1}'=v_{2,1}+v_{1,1}=f+e$. Therefore, by the last claim of Theorem \[t31\], and $p_2'$ and $p_{3}'$ lie in the same 2-sphere $S^2$, where the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M$ restricts to act on this $S^2$ with fixed points $p_2'$ and $p_{3}'$ that have $S^1$-weights $f+e$ and $-f-e$, respectively. Therefore, if we let $\Gamma'$ be a multigraph which is obtained from $\Gamma$ by replacing the label $f$ of $(p_2,p_3)$-edge with $(f+e)$, then $\Gamma'$ describes $M'$ in the strong sense.
We check that $M'$ satisfies Property A.
First, we check Condition (1) of Property A. The fixed points $p_2$ and $p_3$ have $S^1$-weights $\{-e,f\}$ and $\{-e,-f\}$, respectively. Since $M$ satisfies Condition (1) of Property A, $e$ and $f$ are relatively prime. The fixed points $p_2'$ and $p_3'$ have $S^1$-weights $\{-e,e+f\}$ and $\{-e,-e-f\}$. Since $p_i'$ have the same $S^1$-weights as $p_i$ for $i \neq 2,3$ and $M$ satisfies Condition (1) of Property A, it follows that $M'$ also satisfies Condition (1) of Property A.
Second, we check Condition (2) of Property A. We check (2) of Property A for the edge $(p_2',p_3')$, and we only need to check if $(p_{1}',p_{k-1}')$ is $(e+f)$-edge, since then $(p_{1}',p_{k-1}')$ is $(e+f)$-edge, $(p_1',p_2')$ is $e$-edge, and $(p_2',p_3')$ is $(e+f)$-edge so that these edges are in the situation of (2) of Property A. Suppose that $(p_{1}',p_{k-1}')$ is $(e+f)$-edge. This means that $(p_{1},p_{k-1})$ is $(e+f)$-edge. We show that $v_2'=-v_{k-1}'$. We have $v_{k-1,1}=-e-f$, $v_{1,1}=e$, and $v_{2,1}=f$. Since $v_2=-a_1 \cdot v_1-v_{k-1}$, it follows that $v_{2,1}=-a_1 \cdot v_{1,1}-v_{k-1,1}$ and this implies that $a_1=1$. Since $a_1'=a_1-1$, we have $a_1'=0$ and hence $v_2'=-a_1' \cdot v_1'-v_{k-1}'=-v_{k-1}'$. Therefore, Condition (2) of Property A holds for all edges in $\Gamma'$, since other $v_i'$ are the same as $v_i$ and $M$ satisfies Condition (2) of Property A.
Third, we check (3) of Property A. We check Condition (3) of Property A for the edge $(p_2',p_3')$. We only need to check if $(p_5',p_4')$ is $(e+f)$-edge, since then $(p_2',p_3')$ is $(e+f)$-edge, $(p_4',p_3')$ is $e$-edge, and $(p_5',p_4')$ is $(e+f)$-edge so that these edges are in the situation of (3) of Property A. Suppose that $(p_5,p_4)$ is $(e+f)$-edge. This means that $(p_5,p_4)$ is $(e+f)$-edge. We show that $v_4'=-v_2'$. We have $v_{2,1}=f$, $v_{3,1}=-e$, and $v_{4,1}=-e-f$. Since $v_4=-a_3 \cdot v_3-v_2$, we have $v_{4,1}=-a_3 \cdot v_{3,1}-v_{2,1}$ and this implies that $a_3=-1$. Since $a_3'=a_3+1$, we have $a_3'=0$ and hence $v_4'=-a_3' \cdot v_3'-v_2'=-v_2'$. Therefore, Condition (3) of Property A holds for all edges in $\Gamma'$, since other $v_i'$ are the same as $v_i$ and $M$ satisfies Condition (3) of Property A.
Instead of Lemma \[l35\], we can also do the following. Let $\Gamma$ be a multigraph describing a 4-dimensional compact almost complex manifold $M$ equipped with a circle action having a discrete fixed point set. Next, reverse the circle action. By reversing the circle action we mean that each $g \in S^1$ acts on $M$ by $g^{-1} \cdot p$ for any $p \in M^{S^1}$. Reversing the circle action reverses the sign of every weight at a fixed point. This means that reversing the circle action corresponds to reversing the direction of each edge in $\Gamma$. Now, perform Operation 2 on the reversed multigraph; let $\Gamma'$ be the multigraph obtained. By Lemma \[l34\] there exists a 4-dimensional compact almost complex manifold $M'$ equipped with a circle action that is described by $\Gamma'$. And then reverse the circle action again; this reverses the direction of every edge of $\Gamma'$. This whole procedure corresponds to Operation 3; compare Figure \[fig3\] and Figure \[fig4\].
The following lemma corresponds to Operation 4 and Figure \[fig5\] in Theorem \[t11\].
\[l36\] Let $\Gamma$ be a connected 2-regular labeled directed multigrph. Assume that by Theorem \[t31\] we can construct a 4-dimensional compact connected almost complex manifold $M$ equipped with a $\mathbb{T}^2$-action having a discrete fixed point set such that the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M$ is described by $\Gamma$ in the strong sense and $M$ satisfies Property A. Suppose that $(p_2,p_1)$ is $g$-edge, $(p_2,p_3)$ is $h$-edge, $(p_3,p_4)$ is $g$-edge, and $(p_5,p_4)$ is $h$-edge, for some vertices $p_1$, $p_2$, $p_3$, $p_4$, $p_5$ and for some positive integers $g$, $h$. Then by Theorem \[t31\] we can construct a 4-dimensional compact connected almost complex manifold $M'$ equipped with a $\mathbb{T}^2$-action having a discrete fixed point set such that the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M'$ is described by $\Gamma'$ in the strong sense, where $\Gamma'$ (Figure \[fig5-2\]) is obtained from $\Gamma$ (Figure \[fig5-1\]) by replacing $(p_2,p_3)$-edge and $(p_3,p_4)$-edge with $(p_2,p_4)$-edge (which is $(p_2',p_3')$-edge) with label $(g+h)$, removing $p_3$. Moreover, $M'$ satisfies Property A.
Suppose that by Theorem \[t31\] we construct a 4-dimensional compact connected almost complex manifold $M$ equipped with a $\mathbb{T}^2$-action having a discrete fixed point set such that $\Gamma$ describes $M$ in the strong sense for the action of the first $S^1$-factor of $\mathbb{T}^2$, and $M$ satisfies Property A. Let $v_j=(v_{j,1},v_{j,2})$ and $a_j$ be the elements of $\mathbb{Z}^2$ and $\mathbb{Z}$ that are used to construct $M$ in Theorem \[t31\], $1 \leq j \leq k$, where $k$ is the number of fixed points of $M$. Moreover, since $(p_2,p_1)$ is $g$-edge, $(p_2,p_3)$ is $h$-edge, $(p_3,p_4)$ is $g$-edge, and $M$ satisfies the condition (2) of Property A, we have $v_3=-v_1$. Similarly, since $(p_2,p_3)$ is $h$-edge, $(p_3,p_4)$ is $g$-edge, $(p_5,p_4)$ is $h$-edge, and $M$ satisfies Condition (3) of Property A, we have $v_4=-v_2$.
By using Theorem \[t31\], construct a 4-dimensional compact connected almost complex manifold $M'$ equipped with a $\mathbb{T}^2$-action having $k-1$ fixed points, by taking the following values for $v_j'$ and $a_j'$.
1. $v_1'=v_1$, $v_2'=v_2+v_3$, $v_3'=v_4$, $\cdots$, $v_{k-1}'=v_k$, $v_k'=v_1'$.
2. $a_1'=a_1+1$, $a_2'=1$, $a_3'=a_4+1$, $a_4'=a_5$, $\cdots$, $a_{k-1}'=a_k$, $a_{k}'=a_1+1$.
Since $v_1,v_2,v_3,v_4$ are in counterclockwise order, $v_3=-v_1$, and $v_4=-v_2$, it follows that $v_1'=v_1$, $v_2'=v_2+v_3$, $v_3'=v_4$ are also in counterclockwise order. Moreover, since $v_j$ and $a_j$ satisfy Conditions (1) and (2) in Theorem \[t31\], with $v_3=-v_1$ and $v_4=-v_2$ one can check that $v_j'$ and $a_j'$ also satisfy Conditions (1) and (2) in Theorem \[t31\]. Therefore, $v_j'$ and $a_j'$ satisfy all of the conditions in Theorem \[t31\]. Then by the construction, the fixed points $p_j'$ of $M'$ have the following $\mathbb{T}^2$-weights.
1. The fixed point $p_1'$ has $\mathbb{T}^2$-weights $\{v_1,-v_k\}$ same as $p_1$.
2. The fixed point $p_2'$ has $\mathbb{T}^2$-weights $\{v_2+v_3,-v_1\}$.
3. The fixed point $p_3'$ has $\mathbb{T}^2$-weights $\{v_4,-v_2-v_3\}$.
4. If $j \neq 1,2,3$, $\mathbb{T}^2$-weights of $p_j'$ and $p_{j-1}$ are equal.
Since $v_{2,1}=h$ and $v_{3,1}=g$, we have $v_{2,1}'=v_{2,1}+v_{3,1}=h+g$. Therefore, by the last claim of Theorem \[t31\], $p_2'$ and $p_{3}'$ lie in the same 2-sphere $S^2$, where the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M$ restricts to act on this $S^2$ with fixed points $p_2'$ and $p_{3}'$ that have $S^1$-weights $h+g$ and $-h-g$, respectively. Therefore, if we let $\Gamma'$ be a multigraph which is obtained from $\Gamma$ by replacing $(p_2,p_3)$-edge and $(p_3,p_4)$-edge with $(p_2,p_4)$-edge that has label $(g+h)$, removing $p_3$, then $\Gamma'$ describes $M'$ in the strong sense.
We check that $M'$ satisfies Property A.
First, we check Condition (1) of Property A. The fixed points $p_2$ and $p_3$ have $S^1$-weights $\{g,h\}$ and $\{g,-h\}$, respectively. Since $M$ satisfies Condition (1) of Property A, $g$ and $h$ are relatively prime. Since $v_{2,1}=h>0$ and $v_{3,1}=g>0$, we have $v_{2,1}'=v_{2,1}+v_{3,1}=h+g>0$. It follows that $p_2'$ and $p_3'$ have $S^1$-weights $\{g,g+h\}$ and $\{-g-h,-h\}$. Since $p_{i+1}'$ have the same $S^1$-weights as $p_i$ for $i \neq 1,2$ and $M$ satisfies Condition (1) of Property A, it follows that $M'$ satisfies (1) of Property A.
Second, we check Condition (2) of Property A. We check (2) of Property A for the edge $(p_2',p_3')$, and we only need to check if $(p_{1}',p_{k-1}')$ is $(g+h)$-edge, since then $(p_{1}',p_{k-1}')$ is $(g+h)$-edge, $(p_2',p_1')$ is $g$-edge, and $(p_2',p_3')$ is $(g+h)$-edge so that these edges are in the situation of Condition (2) of Property A. Suppose that $(p_{1}',p_{k-1}')$ is $(g+h)$-edge. This means that $(p_{1},p_{k-1})$ is $(g+h)$-edge. We show that $v_2'=-v_{k-1}'$. We have $v_{k-1,1}=-g-h$, $v_{1,1}=-g$, and $v_{2,1}=h$. Since $v_2=-a_1 \cdot v_1-v_{k-1}$, it follows that $v_{2,1}=-a_1 \cdot v_{1,1}-v_{k-1,1}$ and this implies that $a_1=-1$. Since $a_1'=a_1+1$, we have $a_1'=0$ and hence $v_2'=-a_1' \cdot v_1'-v_{k-1}'=-v_{k-1}'$. Therefore, Condition (2) of Property A holds for all edges in $\Gamma'$, since $v_1'=v_1$ and other $v_i'$ are the same as $v_{i+1}$ for $i \neq 1,2$, and $M$ satisfies Condition (2) of Property A.
Third, we check Condition (3) of Property A. This means that we check (3) of Property A for the edge $(p_2',p_3')$, and we only need to check if $(p_5',p_4')$ is $(g+h)$-edge, since then $(p_{2}',p_3')$ is $(g+h)$-edge, $(p_4',p_3')$ is $h$-edge, and $(p_5',p_4')$ is $(g+h)$-edge so that these edges are in the situation of Condition (3) of Property A. Suppose that $(p_5',p_4')$ is $(g+h)$-edge. This means that $(p_6,p_5)$ is $(g+h)$-edge. We show that $v_4'=-v_2'$. We have $v_{5,1}=-g-h$, $v_{4,1}=-h$, and $v_{3,1}=g$. Since $v_5=-a_4 \cdot v_4-v_3$, it follows that $v_{5,1}=-a_4 \cdot v_{4,1}-v_{3,1}$, and this implies that $a_4=-1$. Since $a_3'=a_4+1$, we have $a_3'=0$ and hence $v_4'=-a_3' \cdot v_3'-v_2'=-v_2'$. Therefore, Condition (3) of Property A holds for all edges in $\Gamma'$, since $v_1'=v_1$ and other $v_i'$ are the same as $v_{i+1}$ for $i \neq 1,2$, and $M$ satisfies Condition (3) of Property A.
Therefore, $M'$ satisfies Property A.
For general $v_1$, $v_2$, $v_3$, $v_4$ in counterclockwise order, $v_1$, $v_2+v_3$, $v_4$ need not be in counterclockwise order. This is why we need Conditions (2) and (3) of Property A so that in Lemma \[l36\] we have $v_1=-v_3$ and $v_2=-v_4$, and hence $v_1,v_2+v_3,v_4$ are in counterclockwise order.
Note that for Lemma \[l33\], Lemma \[l34\], Lemma \[l35\], and Lemma \[l36\], the converse also holds. That is, we can perform reverse operations of the operations in the lemmas.
With Theorem \[t31\], Lemma \[l32\], Lemma \[l33\], Lemma \[l34\], Lemma \[l35\], and Lemma \[l36\], we are ready to prove our main theorem, Theorem \[t11\].
Let the circle act effectively on a 4-dimensional compact almost complex manifold $M$ with a discrete fixed point set. By Lemma \[l26\], there exists a labeled directed multigraph $\Gamma''$ describing $M$ such that, for each edge $e$, the index of $i(e)$ in the isotropy submanifold $M^{\mathbb{Z}/(w(e))}$ is one less than the index of $t(e)$ in $M^{\mathbb{Z}/(w(e))}$. In particular, $\Gamma''$ has no self-loops; see Lemma \[l26\] and the explanation after it how we draw the multigraph $\Gamma''$.
Suppose that for each connected component $\Gamma_j''$ of $\Gamma''$, we can construct a 4-dimensional compact connected almost complex manifold $M_j$ equipped with a $\mathbb{T}^2$-action having a discrete fixed point set such that the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M_i$ is described by $\Gamma_j''$. Then by Lemma \[l27\], by taking the equivariant connected sum along free orbits of $M_j$’s, we can construct a 4-dimensional compact connected almost complex manifold $M_0$ equipped with a $\mathbb{T}^2$-action having a discrete fixed point set such that the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M_0$ is described by the disjoint union of $\Gamma_j''$’s, which is $\Gamma''$. Hence, it suffices to assume that $M$ is described by a single connected multigraph.
Therefore, from now on, suppose that $M$ is described by a connected labeled directed multigraph $\Gamma''$ such that for each edge $e$, the index of $i(e)$ in the isotropy submanifold $M^{\mathbb{Z}/(w(e))}$ is one less than the index of $t(e)$ in $M^{\mathbb{Z}/(w(e))}$.
We will show that by Theorem \[t31\] we can construct a 4-dimensional compact connected almost complex manifold $M''$ equipped with a $\mathbb{T}^2$-action having a discrete fixed point set such that the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M''$ is described by $\Gamma''$ in the strong sense and $M''$ satisfies Property A.
The proof is based on the induction on the largest label among the labels of all of the edges of $\Gamma''$. Let $l$ be the largest label among the labels of all of the edges in $\Gamma''$. If the largest label is 1, this means that every weight at each fixed point is either $+1$ or $-1$ and hence the action is semi-free. In other words, $\Gamma''$ is a semi-free multigraph. Then the claim follows by Lemma \[l32\].
Therefore, from now on, suppose that $l>1$. Suppose that $(p,q)$ is $l$-edge for some vertices (fixed points) $p$ and $q$. This means that $p$ has weight $+l$, $q$ has weight $-l$, and $p$ and $q$ lie in the same connected component of $M^{\mathbb{Z}_l}$. Then exactly one of the following holds:
1. $n_p=1$ and $n_q=1$.
2. $n_p=0$ and $n_q=1$.
3. $n_p=1$ and $n_1=2$.
4. $n_p=0$ and $n_q=2$.
Suppose that Case (1) holds. Then the $S^1$-weights at $p$ and $q$ are $\{-x_1,l\}$ and $\{-l,x_2\}$ respectively, for some positive integers $x_1$ and $x_2$. Since the action is effective and $l$ is the largest weight, $x_1<l$ and $x_2<l$. Since $p$ and $q$ lie in the same connected component of $M^{\mathbb{Z}_l}$, by Lemma \[l21\], the weights at $p$ and the weights at $q$ are equal modulo $l$, and this implies that $-x_1 \equiv x_2 \mod l$, i.e., $x_1+x_2=l$. This is precisely the case as in Figure \[fig2-2\] and Lemma \[l33\]. If we can construct a 4-dimensional compact connected almost complex manifold $M'$ equipped with a $\mathbb{T}^2$-action where the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M'$ is described by $\Gamma'$ in the strong sense and $M'$ satisfies Property A, where $\Gamma'$ is obtained from $\Gamma''$ by shrinking the $l$-edge $(p,q)$ to a vertex, then by Lemma \[l33\], we can construct a 4-dimensional compact connected almost complex manifold $M''$ equipped with a $\mathbb{T}^2$-action such that the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M''$ is described by $\Gamma''$ in the strong sense (which describes $M$) and $M''$ satisfies Property A. This step corresponds to Operation 1 in Theorem \[t11\].
Suppose that Case (2) holds. Then the $S^1$-weights at $p$ and $q$ are $\{x_1,l\}$ and $\{-l,x_2\}$ respectively, for some positive integers $x_1$ and $x_2$. Since the action is effective and $l$ is the largest weight, $x_1<l$ and $x_2<l$. Since $p$ and $q$ lie in the same connected component of $M^{\mathbb{Z}_l}$, by Lemma \[l21\], the weights at $p$ and the weights at $q$ are equal modulo $l$. This implies that $x_1 \equiv x_2 \mod l$, i.e., $x_1=x_2$. This is precisely the case as in Figure \[fig3-2\] and Lemma \[l34\]. If we can construct a 4-dimensional compact connected almost complex manifold $M'$ equipped with a $\mathbb{T}^2$-action where the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M'$ is described by $\Gamma'$ in the strong sense and $M'$ satisfies Property A, where $\Gamma'$ is obtained from $\Gamma''$ by changing the label $l$ of the edge $(p,q)$ by $l-x_1$, then by Lemma \[l34\], we can construct a 4-dimensional compact connected almost complex manifold $M''$ equipped with a $\mathbb{T}^2$-action such that the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M''$ is described by $\Gamma''$ in the strong sense (which describes $M$) and $M''$ satisfies Property A. This step corresponds to Operation 2 in Theorem \[t11\].
Suppose that Case (3) holds. Then the $S^1$-weights at $p$ and $q$ are $\{-x_1,l\}$ and $\{-l,-x_2\}$ respectively, for some positive integers $x_1$ and $x_2$. Since the action is effective and $l$ is the largest weight, $x_1<l$ and $x_2<l$. Since $p$ and $q$ lie in the same connected component of $M^{\mathbb{Z}_l}$, by Lemma \[l21\], the weights at $p$ and the weights at $q$ are equal modulo $l$, and this implies that $-x_1 \equiv -x_2 \mod l$, i.e., $x_1=x_2$. This is precisely the case as in Figure \[fig4-2\] and Lemma \[l35\]. If we can construct a 4-dimensional compact connected almost complex manifold $M'$ equipped with a $\mathbb{T}^2$-action where the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M'$ is described by $\Gamma'$ in the strong sense and $M'$ satisfies Property A, where $\Gamma'$ is obtained from $\Gamma''$ by changing the label $l$ of the edge $(p,q)$ by $l-x_1$, then by Lemma \[l35\], we can construct a 4-dimensional compact connected almost complex manifold $M$ equipped with a $\mathbb{T}^2$-action such that the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M''$ is described by $\Gamma''$ in the strong sense (which describes $M$) and $M''$ satisfies Property A. This step corresponds to Operation 3 in Theorem \[t11\].
Suppose that Case (4) holds. Then the $S^1$-weights at $p$ and $q$ are $\{x_1,l\}$ and $\{-l,-x_2\}$ respectively, for some positive integers $x_1$ and $x_2$. Since the action is effective and $l$ is the largest weight, $x_1<l$ and $x_2<l$. Since $p$ and $q$ lie in the same connected component of $M^{\mathbb{Z}_l}$, by Lemma \[l21\], the weights at $p$ and the weights at $q$ are equal modulo $l$, and this implies that $x_1 \equiv -x_2 \mod l$, i.e., $x_1+x_2=l$. This is precisely the case as in Figure \[fig5-2\] and Lemma \[l36\]. Let $p'$ be the terminal point of the $x_1$-edge whose initial point is $p$, and let $q'$ be the initial point of the $x_2$-edge whose terminal point is $q$, i.e., $(p,p')$ is $x_1$-edge and $(q',q)$ is $x_2$-edge. In other words, $p',p,q,q'$ correspond to $p_{i-2},p_{i-1},p_{i+1},p_{i+2}$ in Figure \[fig5-2\]. If we can construct a 4-dimensional compact connected almost complex manifold $M'$ equipped with a $\mathbb{T}^2$-action where the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M'$ is described by $\Gamma'$ in the strong sense and $M'$ satisfies Property A, where $\Gamma'$ is obtained from $\Gamma''$ by replacing the edge $(p,q)$ with two edges $(p,r)$ and $(r,q)$ with labels $x_2$ and $x_1$ respectively with adding a vertex $r$ (which corresponds to $p_i$ in Figure \[fig5-1\]), then by Lemma \[l36\], we can construct a 4-dimensional compact connected almost complex manifold $M''$ equipped with a $\mathbb{T}^2$-action such that the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M''$ is described by $\Gamma''$ in the strong sense (which describes $M$) and $M''$ satisfies Property A. This step corresponds to Operation 4 in Theorem \[t11\].
Whenever $(p,q)$ is $l$-edge, by the four steps above, the existence of the manifold $M''$ described by $\Gamma''$ in the strong sense (which describes $M$) and satisfying Property A is reduced to the existence of another manifold $M'$ described by $\Gamma'$ in the strong sense and satisfying Property A, where $\Gamma'$ has one less $l$-edge than $\Gamma''$. By repeating the arguments, the existence of the manifold $M''$ that is described by $\Gamma''$ in the strong sense and satisfies Property A reduces to the existence of a 4-dimensional compact connected almost complex manifold $M'''$ equipped with a $\mathbb{T}^2$-action having a discrete fixed point set constructed by Theorem \[t31\] satisfying Property A such that the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M'''$ is described by a connected 2-regular labeled directed multigraph $\Gamma'''$ in the strong sense such that the label of every edge is equal to 1, and for each edge $e$, the index of $i(e)$ in the isotropy submanifold $M^{\mathbb{Z}/(w(e))}$ ($=M$) is one less than the index of $t(e)$ in $M^{\mathbb{Z}/(w(e))}$ ($=M$). This implies that $\Gamma'''$ is a connected 2-regular semi-free multigraph. By Lemma \[l32\], such a manifold $M'''$ exists. This proves the first claim.
By Theorem \[t21\], the Todd genus of $M$ is equal to the number of fixed points of index 0. Since the 4 operations in Theorem \[t11\] do not reduce the number of fixed points (vertices) of index 0, it follows that $\textrm{Todd}(M)=\sum_{j=1}^k T(\Gamma_j)$.
The proof above has also proved the last claim of Theorem \[t11\] that given any 2-regular labeled directed multigraph $\Gamma$ constructed from a 2-regular semi-free multigraph followed by the 4 operations in Theorem \[t11\], there exists a 4-dimensional compact connected almost complex manifold $M$ equipped with a circle action having a discrete fixed point set that is described by $\Gamma$.
In fact, in this section we have proved the following theorem.
Let the circle act on a 4-dimensional compact almost complex manifold $M$ with a discrete fixed point set. Then we can construct a 4-dimensional compact connected almost complex manifold $M'$ equipped with a $\mathbb{T}^2$-action such that the action of the first $S^1$-factor of $\mathbb{T}^2$ on $M'$ has the same fixed point data as $M$.
This follows from Theorem \[t31\], Lemma \[l32\], Lemma \[l33\], Lemma \[l34\], Lemma \[l35\], Lemma \[l36\], and the proof of Theorem \[t11\].
\[r34\] Theorem \[t11\] does not imply that if a multigraph describes a 4-dimensional almost complex $S^1$-manifold, the manifold is unique. That is, a multigraph can describe two different $S^1$-manifolds. For this, let $M'$ be an $S^1$-manifold with 8 fixed points constructed from Theorem \[t31\] by taking $v_1=(1,0)$, $v_2=(1,1)$, $v_3=(-1,0)$, $v_4=(-1,-1)$, $v_5=(1,0)$, $v_6=(1,1)$, $v_7=(-1,0)$, $v_8=(-1,-1)$ as in Lemma \[l32\]. Next, for $i=1,2$, let $M_i$ be a $\mathbb{T}^2$-manifold with 4 fixed points constructed from Theorem \[t31\] by taking $v_1=(1,0)$, $v_2=(1,1)$, $v_3=(-1,0)$, $v_4=(-1,-1)$ also as in Lemma \[l32\]. By using Lemma \[l27\], take the equivariant connected sum of $M_1$ and $M_2$ along free orbits to construct a $\mathbb{T}^2$-manifold $M''$. From the constructions, $M'$ and $M''$ are not diffeomorphic; $M'$ has one chain of eight 2-spheres connecting 8 fixed points, while $M'$ has two chains of four 2-spheres, each chain connecting 4 fixed points. Both of $M'$ and $M''$ have 2 fixed points having weights $\{1,1\}$, 4 fixed points having weights $\{-1,1\}$, and 2 fixed points having weights $\{-1,-1\}$, and hence they have the same fixed point data. Therefore, Figure \[fig1\] describes both of $M'$ and $M''$. On the other hand, a disjoint uions of two copies of Figure \[fig6\] also describes both of $M'$ and $M''$. However, from the constructions, it is natural to take Figure \[fig1\] to describe $M'$, and to take a disjoint union $\Gamma'$ of two copies of Figure \[fig6\] to describe $M''$, as Figure \[fig1\] describes $M'$ in the strong sense and $\Gamma'$ describes $M'$ in the strong sense.
Comparison with complex manifolds and symplectic manifolds {#s4}
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In this section, we compare circle actions on almost complex manifolds, complex manifolds, and symplectic manifolds. In dimension 2, there is no difference; if a 2-dimensional compact connected oriented manifold $M$ admits a circle action, either $M$ is the 2-sphere $S^2$ or the 2-torus $\mathbb{T}^2$, and both of them are Kähler; both of them are almost complex, complex, and symplectic.
In dimension 4, however, the class of almost complex $S^1$-manifolds is bigger than that of complex $S^1$-manifolds and that of symplectic $S^1$-manifolds. If the circle acts on a 4-dimensional compact connected almost complex manifold $M$ with a non-empty discrete fixed point set, the Todd genus of $M$ takes any positive integer. On the other hand, if in addition $M$ is complex or symplectic and the circle action preserves the given structure, the Todd genus of $M$ must be equal to 1. Therefore, there are almost complex manifolds equipped with circle actions which cannot be complex or symplectic.
We prove Theorem \[t12\] that if $M$ is complex or symplectic, a multigraph describing $M$ can be constructed from a single 2-regular semi-free multigraph $\Gamma$ with $T(\Gamma)=1$ (Figure \[fig6\]) followed by the 4 operations in Theorem \[t11\].
If $M$ is a complex manifold, the Todd genus of $M$ is equal to 1; see [@CHK]. Suppose that $M$ is a symplectic manifold. Since there is a fixed point and $\dim M=4$, the action must be Hamiltonian; see [@MD]. Since the action is Hamiltonian, there is a unique fixed point of index 0. The Todd genus of $M$ is equal to the number of fixed points of index 0. It follows that the Todd genus of $M$ is equal to 1. Therefore, in either case the Todd genus of $M$ is equal to 1.
We apply Theorem \[t11\]. By Theorem \[t11\], a labeled directed multigraph $\Gamma'$ describing $M$ can be achieved from a 2-regular semi-free multigraph followed by the 4 operations in Theorem \[t11\]. Since $\textrm{Todd}(M)=1$, $\Gamma'$ has exactly 1 vertex of index 0. Since the 4 operations in Theorem \[t11\] do not reduce the number of vertices (fixed points) of index 0, to achieve $\Gamma'$ from a semi-free multigraph by the 4 operations in Theorem \[t11\], we must begin with a single 2-regular semi-free multigraph $\Gamma$ with $T(\Gamma)=1$ (see Figure \[fig6\]), i.e., $\Gamma$ has precisely one vertex of index 0.
Let the circle act on a 4-dimensional compact connected manifold, complex or symplectic, preserving the complex or symplectic structure. Assume that the fixed point set is non-empty and discrete. Carrell, Howard, and Kosniowski for a complex manifold [@CHK] and Karshon [@Ka] for a symplectic manifold proved that such a manifold can be obtained from $\mathbb{CP}^2$ or a Hirzebruch surface followed by blow ups, where any blow up only occurs at a fixed point of index 1. Let $p$ be a fixed point of index 1. Let $\{-a,b\}$ be the weights at $p$ for some positive integers $a$ and $b$. Blowing up $p$ equivariantly (either in complex or symplectic sense), we get two fixed points $p'$ and $p''$ that have weights $\{-a,a+b\}$ and $\{-a-b,b\}$ respectively, instead of $p$. For a multigraph describing a manifold, blowing up at a fixed point of index 1 corresponds to Operation 1 of Theorem \[t11\]; see Figure \[fig2\]. To compare our minimal model (Figure \[fig6\]) and their minimal models (Figures \[fig7-1\], \[fig7-2\], \[fig7-3\]), we first describe circle actions on $\mathbb{CP}^2$ and Hirzebruch surfaces.
Let $S^1$ act on $\mathbb{CP}^2$ by
$g \cdot [z_0:z_1:z_2]=[z_0:g^a z_1:g^{a+b} z_2]$
for all $g \in S^1 \subset \mathbb{C}$, for some positive integers $a$ and $b$ that are relatively prime. The action has three fixed points $p_1=[1:0:0]$, $p_2=[0:1:0]$, and $p_3=[0:0:1]$, and the weights at the fixed points are $\{a+b,a\}$, $\{-a,b\}$, and $\{-b,-a-b\}$, respectively. Figure \[fig7-1\] is the multigraph describing $\mathbb{CP}^2$.
\[e210\] Let $n$ be an integer. A Hirzebruch surface is defined as
$\{([z_0:z_1:z_2],[w_1:w_2])\in\mathbb{CP}^2\times\mathbb{CP}^1|z_1 w_2^n=z_2 w_1^n\}$.
For all $g \in S^1 \subset \mathbb{C}$, let $g$ act on the Hirzebruch surface by
$g \cdot ([z_0:z_1:z_2],[w_1:w_2]) = ([g^c z_0:z_1:g^{nd} z_2],[w_1:g^d w_2])$
for some positive integers $c$ and $d$ that are relatively prime. The action has 4 fixed points, $p_1=([1:0:0],[1:0]), p_2=([1:0:0],[0:1]), p_3=([0:1:0],[1:0])$, and $p_4=([0:0:1],[0:1])$, and the weights at the fixed points are $\{-c,d\}$, $\{nd-c,-d\}$, $\{c,d\}$, and $\{c-nd,-d\}$, respectively. If $c-nd$ is positive, Figure \[fig7-2\] with $e=c-nd$ describes the manifold. Suppose that $c-nd$ is negative. In this case, if $d=1$, Figure \[fig7-2\] with $e=nd-c$ describes the manifold, and if $d>1$, Figure \[fig7-3\] with $e=nd-c$ describes the manifold. If $d=1$, Figure \[fig7-3\] also describes the manifold, but we choose Figure \[fig7-2\] to describe the manifold so that the multigraph can be achieved from our minimal model. This results from the fact that any multigraph that we associate in this paper satisfies the conditions in Lemma \[l26\]; since $d=1$ is the smallest positive weight, for any edge $e$ with label 1, a multigraph we associate satisfies $n_{t(e)}=n_{i(e)}+1$ where $n_v$ is the index of a vertex $v$; see Definition \[d11\]. Note that when Figure \[fig7-2\] describes the manifold, $c \equiv e \mod d$ and when Figure \[fig7-3\] describes the manifold, $c \equiv -e \mod d$.
[.25]{}
\(a) [$p_1$]{}; (b) \[above right=of a\] [$p_2$]{}; (c) \[above left=of b\] [$p_3$]{}; (a) \[->\] edge node\[right\] [$a$]{} (b); (b) \[->\] edge node\[right\] [$b$]{} (c); (a) \[->\]edge node \[right\] [$a+b$]{} (c);
[.25]{}
\(A) [$p_1$]{}; (B) \[above left=of A\] [$p_2$]{}; (C) \[above right=of A\] [$p_3$]{}; (D) \[above right=of B\] [$p_4$]{}; (A) \[->\] edge node\[right\] [$d$]{} (B); (A) \[->\] edge node \[right\] [$c$]{} (C); (B) \[->\] edge node \[right\] [$e$]{} (D); (C) \[->\] edge node \[right\] [$d$]{} (D);
[.25]{}
\(A) [$p_1$]{}; (B) \[above left=of A\] [$p_3$]{}; (C) \[above =of B\] [$p_4$]{}; (D) \[above right=of C\] [$p_2$]{}; (A) \[->\] edge node\[right\] [$c$]{} (B); (A) \[->\] edge node \[right\] [$d$]{} (D); (B) \[->\] edge node \[right\] [$d$]{} (C); (C) \[->\] edge node \[left\] [$e$]{} (D);
On the other hand, as in Theorem \[t11\], our minimal model is described by a semi-free multigraph. Therefore, we shall discuss how to go from our minimal multigraph Figure \[fig6\] to their minimal multigraphs Figure \[fig7-1\], Figure \[fig7-2\], and Figure \[fig7-3\] with $d>1$ by our operations in Theorem \[t11\].
For a multigraph Figure \[fig7-1\] for $\mathbb{CP}^2$, perform the reverse of Operation 4 in Theorem \[t11\]. Then the resulting multigraph is Figure \[fig7-2\] if we set $a=c$, $b=d$, and $e=c$. This means that if we can go from our minimal multigraph Figure \[fig6\] to Figure \[fig7-2\] with $a=c$, $b=d$, and $e=c$ by our operations, then we can perform Operation 4 in Theorem \[t11\] to Figure \[fig7-2\] with $a=c$, $b=d$, and $e=c$ to reach Figure \[fig7-1\]. Therefore, for a multigraph for $\mathbb{CP}^2$, our task is to show how to go from Figure \[fig6\] to Figure \[fig7-2\].
From now on, we will show how to go from their minimal multigraphs to our minimal multigraph by performing reverses of our operations. Then one can chase the reversed operations and go from our minimal one to their minimal ones by performing our operations. Therefore, our task is to show how to go from Figure \[fig7-2\] and Figure \[fig7-3\] with $d>1$ to Figure \[fig6\].
We show how to go from Figure \[fig7-3\] with $d>1$ to Figure \[fig6\]. In Figure \[fig7-3\], by performing the reverse of Operation 2 $j_1$-times, replace the label $c$ of $(p_1,p_3)$-edge by $c-j_1d$ where $j_1$ is the biggest non-negative integer such that $c-j_1d$ is positive and call it $c_1$. Then $c_1<d$. Similarly, by performing the reverse of Operation 3 $j_2$-times, replace the label $e=nd-c$ of $(p_4,p_2)$-edge by $nd-c-j_2d$ where $j_2$ is the biggest non-negative integer such that $nd-c-j_2d$ is positive and call it $c_2$. Then $c_2<d$. Since $c_1=c-j_1d$, $c_2=nd-c-j_2d$, $0<c_1<d$, $0<c_2<d$, and $c_1+c_2$ is a multiple of $d$, we have $c_1+c_2=d$. The resulting multigraph is Figure \[fig8-1\]. Next, do the reverse of Operation 4 for $(p_1,p_2)$-edge. The resulting multigraph is Figure \[fig8-2\]. Next, do the reverse of Operation 1 for $(p_3,p_4)$-edge. Then the resulting multigraph is Figure \[fig8-3\].
Both of Figure \[fig7-2\] and Figure \[fig8-3\] with $d>1$ are now included in Figure \[fig9\] with $f \equiv h \mod g$ if we relabel the labels suitably, where $f$, $g$, and $h$ are pairwise prime. Therefore, it remains to show how to go from Figure \[fig9\] to Figure \[fig6\].
Now, by performing the reverse of Operation 2 for $(p_1,p_2)$-edge $k_1$-times, replace the label $f$ of $(p_1,p_2)$-edge by $f-k_1g$ where $k_1$ is the biggest non-negative integer such that $f-k_1g$ is positive and call it $f_1$. Similarly, by performing the reverse of Operation 3 for $(p_3,p_4)$-edge $k_2$ times, replace the label $h$ of $(p_3,p_4)$-edge by $h-k_2g$ where $k_2$ is the biggest non-negative integer such that $h-k_2g$ is positive and call it $h_1$. Since $f \equiv h \mod g$, this means that $f_1=h_1$ and $f_1<g$. Next, do the reverse of Operation 3 for $(p_2,p_4)$-edge $k_3$ times to replace the label $g$ of the $(p_2,p_4)$-edge by $g-k_3f_1$ where $k_3$ is the biggest non-negative integer such that $g-k_3f_1$ is positive and call it $g_1$. Do the reverse of Operation 2 for $(p_1,p_3)$-edge $k_4$ times to replace the label $g$ of the $(p_1,p_3)$-edge by $g-k_4f_1$ where $k_4$ is the biggest non-negative integer such that $g-k_4f_1$ is positive and call it $g_2$. Then we have that $g_1=g_2$ and $g_1<f_1$. Then we are still in the case of Figure \[fig9\], but now the labels of all edges are reduced. Therefore, if we repeat this process, we can reduce the label of every edge to 1. Hence we can reach our multigraph Figure \[fig6\]. This means that from Figure \[fig6\] we can reach Figure \[fig7-2\] and Figure \[fig8-3\] (with $d>1$) and hence Figure \[fig7-1\] by doing our Operations 1, 2, 3 and 4 in Theorem \[t11\].
[.25]{}
\(A) [$p_1$]{}; (B) \[above left=of A\] [$p_3$]{}; (C) \[above =of B\] [$p_4$]{}; (D) \[above right=of C\] [$p_2$]{}; (A) \[->\] edge node\[right\] [$c_1$]{} (B); (A) \[->\] edge node \[right\] [$d$]{} (D); (B) \[->\] edge node \[right\] [$d$]{} (C); (C) \[->\] edge node \[left\] [$d-c_1$]{} (D);
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\(A) [$p_1$]{}; (B) \[above left=of A\] [$p_3$]{}; (C) \[above =of B\] [$p_4$]{}; (D) \[above right=of C\] [$p_2$]{}; (E) \[above right=of A\] [$p_5$]{}; (A) \[->\] edge node\[right\] [$c_1$]{} (B); (A) \[->\] edge node \[right\] [$d-c_1$]{} (E); (B) \[->\] edge node \[right\] [$d$]{} (C); (C) \[->\] edge node \[left\] [$d-c_1$]{} (D); (E) \[->\] edge node \[right\] [$c_1$]{} (D);
[.25]{}
\(A) [$p_1$]{}; (B) \[above left=of A\] [$p_{3'}$]{}; (C) \[above right=of A\] [$p_{5'}$]{}; (D) \[above right=of B\] [$p_2$]{}; (A) \[->\] edge node\[right\] [$c_1$]{} (B); (A) \[->\] edge node \[right\] [$d-c_1$]{} (C); (B) \[->\] edge node \[right\] [$d-c_1$]{} (D); (C) \[->\] edge node \[right\] [$c_1$]{} (D);
[.25]{}
\(A) [$p_1$]{}; (B) \[above left=of A\] [$p_2$]{}; (C) \[above right=of A\] [$p_3$]{}; (D) \[above right=of B\] [$p_4$]{}; (A) \[->\] edge node\[right\] [$f$]{} (B); (A) \[->\] edge node \[right\] [$g$]{} (C); (B) \[->\] edge node \[right\] [$g$]{} (D); (C) \[->\] edge node \[right\] [$h$]{} (D);
Chern numbers and lower bound on the number of fixed points {#s5}
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As another consequence of Theorem \[t11\], we obtain a lower bound on the number of fixed points of a 4-dimensional compact almost complex $S^1$-manifold. Moreover, our bound is sharp.
The corollary holds if there is a fixed surface. Therefore, from now on, assume that the fixed point set is discrete. Let $N_i$ be the number of fixed points of index $i$. By Theorem \[t21\], the Todd genus of $M$ is equal to $N_0$, and $N_0=N_2$. By Corollary \[c23\], there exists $0 \leq i \leq 1$ such that $N_i>0$ and $N_{i+1}>0$. These imply that $N_0>0$, $N_1>0$, and $N_2>0$. In particular, $\textrm{Todd}(M)=N_0>0$. Since total number of fixed points is $N_0+N_1+N_2=2N_0+N_1=2 \cdot \textrm{Todd}(M)+N_1$, it follows that there are at least $2 \cdot \textrm{Todd}(M)+1$ fixed points. This proves the first part.
From now on, for any positive integers $N_0$ and $N_1$, we construct a 4-dimensional compact connected almost complex manifold $M$ equipped with a circle action having $2N_0+N_1$ fixed points, $N_0$ fixed points of index 0, $N_1$ fixed points of index 1, and $N_0$ fixed points of index 2.
First, suppose that $2N_0 \leq N_1$. In Theorem \[t11\], begin with a single connected semi-free multigraph $\Gamma$ with $T(\Gamma)=N_0$. The multigraph $\Gamma$ has $2N_0$ vertices of index 1. Next, Perform Operation 1 to any vertex of index 1 $N_1-2N_0$ times; the resulting manifold has $N_1$ vertices of index 1. This proves the corollary when $2N_0 \leq N_1$.
Second, suppose that $N_1<2N_0$. In Theorem \[t11\], begin with a single connected semi-free multigraph $\Gamma$ with $T(\Gamma)=N_0$. We label the fixed points as in Figure \[fig9’\]; for each $0 \leq j \leq N_0-1$, $p_{4j+1}$ has weights $\{1,1\}$, $p_{4j+2}$ has weights $\{-1,1\}$, $p_{4j+3}$ has weights $\{-1,-1\}$, and $p_{4j+4}$ has weights $\{-1,1\}$; see Figure \[fig13\].
Next, for each $j$, we do the following. We have $(p_{4j+1},p_{4j+2})$ is 1-edge, $(p_{4j+2},p_{4j+3})$ is 1-edge, and $(p_{4j+4},p_{4j+3})$ is 1-edge and hence these edges are in the situation of Figure \[fig3\] for any $0 \leq j \leq N_0-1$. If $j=0$, Perform Operation 3 in Theorem \[t11\] to $(p_{2},p_{3})$-edge to replace the label 1 of the $(p_{2},p_{3})$-edge with 2. If $1 \leq j \leq N_0-1$, perform Operation 3 in Theorem \[t11\] to $(p_{4j+2},p_{4j+3})$-edge $2j$ times to replace the label of the $(p_{4j+2},p_{4j+3})$-edge with $2j+1$.
Similarly, for each $0 \leq j \leq N_0-1$, we do the following. We have $(p_{4j+4},p_{4j+3})$ is 1-edge, $(p_{4j+5},p_{4j+4})$ is 1-edge, and $(p_{4j+5},p_{4j+6})$ is 1-edge and hence these edges are in the situation of Figure \[fig2\]. Perform Operation 2 in Theorem \[t11\] to $(p_{4j+5},p_{4j+4})$-edge $2j-1$ times to replace the label of the $(p_{4j+5},p_{4j+4})$-edge with $2j$.
This means that if we move from the vertex $p_1$ to $p_2$ to $p_3$ and so on, the labels of the edges are 1, 2, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, $\cdots$, 1, $2N_0-1$, 1, $2N_0$. See Figure \[fig14\] for the multigraph obtained.
We will perform Operation 4 $2N_0-1$ times to have exactly 1 fixed point of index 1. Now, $(p_2,p_3)$ is 2-edge, $(p_4,p_3)$ is 1-edge, $(p_5,p_4)$ is 2-edge, and $(p_5,p_6)$ is 1-edge; perform Operation 4 of Theorem \[t11\] to replace $(p_4,p_3)$-edge and $(p_5,p_4)$-edge with $(p_5,p_3)$-edge with label 3, removing $p_4$; see Figure \[fig15\]. Next, $(p_5,p_3)$ is 3-edge, $(p_5,p_6)$ is 1-edge, $(p_6,p_7)$ is 3-edge, and $(p_8,p_7)$ is 1-edge; perform Operation 4 of Theorem \[t11\] to replace $(p_5,p_6)$-edge and $(p_6,p_7)$-edge with $(p_5,p_7)$-edge with label 4, removing $p_6$; see Figure \[fig16\]. Next, $(p_5,p_7)$ is 4-edge, $(p_8,p_7)$ is 1-edge, $(p_9,p_8)$ is 4-edge, and $(p_9,p_{10})$ is 1-edge; perform Operation 4 of Theorem \[t11\] to replace $(p_8,p_7)$-edge and $(p_9,p_8)$-edge with $(p_9,p_7)$-edge with label 5, removing $p_8$. If we continue this, in the last step we have that $(p_{4N_0-3},p_{4N_0-1})$ is $2N_0$-edge, $(p_{4N_0},p_{4N_0-1})$ is 1-edge, $(p_{1},p_{4N_0})$ is $2N_0$-edge, and $(p_1,p_2)$ is 1-edge; perform Operation 4 of Theorem \[t11\] to replace $(p_{4N_0-1},p_{4N_0})$-edge and $(p_{1},p_{4N_0})$-edge with $(p_1,p_{4N_0-1})$-edge with label $2N_0+1$, removing $p_{4N_0}$. We have performed Operation 4 $(2N_0-1)$-times. The resulting manifold has 1 fixed point of index 1, namely $p_2$. Theorem \[t11\] guarantees the existence of such a manifold $M$.
If instead in the above we perform Operation 4 ($2N_0-N_1$)-times, then the resulting manifold has $N_0$ fixed points of index 0, $N_1$ fixed points of index 1, and $N_0$ fixed points of index 2. Theorem \[t11\] guarantees the existence of such a manifold $M$. This completes the proof.
[1,0]{}
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\(A) [$p_1$]{}; (B) \[above=of A\] [$p_2$]{}; (C) \[above right=of B\] [$p_3$]{}; (E) \[below right=of D\] [$p_5$]{}; (F) \[above=of E\] [$p_6$]{}; (G) \[above right=of F\] [$p_7$]{}; (H) \[below=of G\] [$p_8$]{}; (N) \[below right=of H\] [$p_{9}$]{}; (I) \[right=of N\] [$p_{4N_0-3}$]{}; (J) \[above=of I\] [$p_{4N_0-2}$]{}; (K) \[above right=of J\] [$p_{4N_0-1}$]{}; (L) \[below=of K\] [$p_{4N_0}$]{}; (M) \[below right=of L\] [$p_{1}$]{}; (A) \[->\] edge node\[right\] [$1$]{} (B); (B) \[->\] edge node \[right\] [$2$]{} (C); (E) \[->\] edge node \[right\] [$3$]{} (C); (E) \[->\] edge node \[right\] [$1$]{} (F); (F) \[->\] edge node \[right\] [$3$]{} (G); (H) \[->\] edge node \[right\] [$1$]{} (G); (I) \[->\] edge node \[right\] [$1$]{} (J); (J) \[->\] edge node \[left\] [$2N_0-1$]{} (K); (L) \[->\] edge node \[right\] [$1$]{} (K); (M) \[->\] edge node \[right\] [$2N_0$]{} (L); (N) \[->\] edge node \[right\] [$4$]{} (H);
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\(A) [$p_1$]{}; (B) \[above=of A\] [$p_2$]{}; (C) \[above right=of B\] [$p_3$]{}; (E) \[below right=of D\] [$p_5$]{}; (G) \[above right=of F\] [$p_7$]{}; (H) \[below=of G\] [$p_8$]{}; (N) \[below right=of H\] [$p_{9}$]{}; (I) \[right=of N\] [$p_{4N_0-3}$]{}; (J) \[above=of I\] [$p_{4N_0-2}$]{}; (K) \[above right=of J\] [$p_{4N_0-1}$]{}; (L) \[below=of K\] [$p_{4N_0}$]{}; (M) \[below right=of L\] [$p_{1}$]{}; (A) \[->\] edge node\[right\] [$1$]{} (B); (B) \[->\] edge node \[right\] [$2$]{} (C); (E) \[->\] edge node \[right\] [$3$]{} (C); (E) \[->\] edge node \[right\] [$4$]{} (G); (H) \[->\] edge node \[right\] [$1$]{} (G); (I) \[->\] edge node \[right\] [$1$]{} (J); (J) \[->\] edge node \[left\] [$2N_0-1$]{} (K); (L) \[->\] edge node \[right\] [$1$]{} (K); (M) \[->\] edge node \[right\] [$2N_0$]{} (L); (N) \[->\] edge node \[right\] [$4$]{} (H);
Note that three fixed points can only occur in dimension 4; see [@J2] whose proof adapts the proof for symplectic manifolds in [@J1]. If we chase what we have done to $v_i$ in the proof of Corollary \[c19\] via Lemmas \[l32\] through \[l36\], by using the following values for $v_i$ in Theorem \[t31\], we can construct a 4-dimensional compact connected almost complex $S^1$-manifold with $2 \cdot \textrm{Todd}(M)+1$ fixed points; $v_1=(1,0)$, $v_2=(2,1)$, $v_3=(-3,-1)$, $v_4=(4,1)$, $\cdots$, $v_{k-2}=(-k+2,1)$, $v_{k-1}=(k-1,1)$, $v_{k}=(-k,-1)$, where $k=2 \cdot \textrm{Todd}(M)+1$. The weights at $p_1$, $p_2$, $\cdots$, $p_k$ are $\{k,1\}$, $\{-1,2\}$, $\{-2,-3\}$, $\{3,4\}$, $\{-4,-5\}$, $\cdots$, $\{k-2,k-1\}$, and $\{-k-1,-k\}$, respectively. For any positive integer $N_0$, one can produce an example with $N_0$ fixed points of index 0 and 1 fixed point of index 1 in various ways. Note that the $v_i$ are barely in counterclockwise direction as $k$ increases.
The first claim follows since $c_1^2[M]=10 N_0- N_1$ and $c_2[M]=2N_0+N_1$, where $N_i$ is the number of fixed points of index $i$; see [@S]. The second claim follows from Corollary \[c19\].
Classification of semi-free actions with discrete fixed point sets {#s6}
==================================================================
In this section, we review the result on the fixed point data of a semi-free circle action on a compact almost complex manifold with a discrete fixed point set, and for any possible fixed point data for a semi-free action, we construct an almost complex $S^1$-manifold that has the given fixed point data. As mentioned in the introduction, Tolman and Weitsman [@TW] proved that a semi-free symplectic circle action with a discrete fixed point set is Hamiltonian if and only if the fixed point set is non-empty. Feldman [@Fe] and Li [@L] reproved this.
\[t51\] [@Fe], [@L], [@TW] Let the circle act symplectically and semi-freely on a compact connected symplectic manifold $M$ with a discrete fixed set. If the fixed point set is non-empty, the action must be Hamiltonian. Moreover, there are precisely $2^n$ fixed points, where $2n=\dim M$.
To prove Theorem \[t51\], Tolman and Weitsman proved the following lemma by using equivariant cohomology and the Atiyah-Bott-Berline-Vergne localization formula. This is also proved in [@J3] by using Theorem \[t21\].
\[l23\] [@J3], [@TW] Let the circle act semi-freely on a $2n$-dimensional compact almost complex manifold with a discrete fixed point set. Then for $0 \leq i \leq n$, $N_i=N_0 \cdot {n \choose i}$, where $N_i$ is the number of fixed points of index $i$.
For a symplectic action there are exactly $2^n$ fixed points, since the fixed point set is discrete and there is a unique fixed point of index 0, i.e., $N_0=1$. As another application of Theorem \[t11\], we prove an analogous result for almost complex manifolds.
\[t52\] Let $n \geq 2$. Given any positive integer $k$, there exists a $2n$-dimensional compact connected almost complex manifold $M$ equipped with a semi-free circle action having a discrete fixed point set, whose Todd genus is $k$. Moreover, for $0 \leq i \leq n$, $N_i=k \cdot {n \choose i}$, where $N_i$ is the number of fixed points of index $i$. In particular, the total number of fixed points is equal to $k \cdot 2^n$.
Let $\Gamma$ be a connected 2-regular semi-free multigraph with $T(\Gamma)=k$. By Lemma \[l32\], there exists a 4-dimensional compact connected almost complex manifold $M'$ equipped with a semi-free circle action having a discrete fixed point set which is described by $\Gamma$. The manifold $M'$ has $4k$ fixed points, $k$ fixed points of index 0, $2k$ fixed points of index 1, and $k$ fixed points of index 2. Let $M''$ be the product $S^2 \times \cdots \times S^2$ of $(n-2)$ copies of the 2-spheres, where on each $S^2$ the circle acts by rotating once. Since the rotation on each 2-sphere has 2 fixed points, the north pole and the south pole that have weights $-1$ and $+1$ respectively, $M''$ has $2^{n-2}$ fixed points and the number of fixed points with index $i$ is equal to ${n-2 \choose i}$ for each $0 \leq i \leq n-2$. Taking the product of $M'$ and $M''$ with a diagonal action, the desired manifold $M$ is constructed. Since $M'$ has $k$ fixed points of index 0 and $M''$ has 1 fixed point of index 0, it follows that $M$ has $k$ fixed points of index 0. Since the number of fixed points of index 0 is equal to the Todd genus of $M$ by Theorem \[t21\], the theorem follows.
The first claim follows from Lemma \[l23\]. The second claim follows from Theorem \[t52\].
Multigraphs as manifolds {#s7}
========================
By Lemma \[l26\], we know that for a circle action on a compact almost complex $S^1$-manifold $M$ with a discrete fixed point set, there exists a labeled directed multigraph describing $M$. In this final section, we ask the converse; that is, when a labeled directed multigraph can be realized as a multigraph describing an almost complex $S^1$-manifold. For a vertex $v$ of a labeled directed multigraph $\Gamma$, let $\{w(e) \ | \ i(e)=v\} \cup \{-w(e) \ | \ t(e)=v\}$ be the **weights** at $v$.
Let $\Gamma$ be an $n$-regular labeled directed multigraph.
(1) The multigraph $\Gamma$ is **symmetric**, if $N_i=N_{n-i}$ for all $i$, where $N_i$ is the number of vertices of index $i$.
(2) The multigraph $\Gamma$ is **effective**, if for each vertex $v$, the greatest common divisor of the labels of the edges of $v$ is equal to 1.
(3) The multigraph $\Gamma$ satisfies the **minimal property**, if an edge $e$ has label 1, then $n_{i(e)}+1=n_{t(e)}$, where $n_v$ denotes the index of a vertex $v$. In particular,
$|\{e \in E \ | \ w(e)=1, i(e)=p, n_p=j\}|=|\{e \in E \ | \ w(e)=1, t(e)=p, n_p=j+1\}|$ for $j \in \{0,1,\cdots,n-1\}$.
(4) The multigraph $\Gamma$ satisfies the **equal modulo property**, if $(p,q)$ is $w$-edge, the weights at $p$ and the weights at $q$ are equal modulo $w$.
For a multigraph that is 2-regular, directed, labelled, and loop-free, we give a sufficient condition for the multigraph to describe a 4-dimensional compact almost manifold equipped with an effective circle action having a discrete fixed point set.
\[t71\] Let $\Gamma$ be a 2-regular effective symmetric directed labeled multigraph without any loop. Suppose that $\Gamma$ satisfies the equal modulo property and the minimal property. Then there exists a 4-dimensional compact connected almost complex manifold $M$ equipped with an effective circle action having a discrete fixed point set that is described by $\Gamma$.
The proof is similar to that of Theorem \[t11\] in Section 3. The proof is based on induction on the largest label. Since $\Gamma$ is effective, the greatest common divisor of labels of edges at each vertex is equal to 1. Let $l$ be the largest label among all labels of the edges of $\Gamma$.
Suppose that $l=1$. Then every edge has label 1. By the minimal property, $\Gamma$ is semi-free. By Theorem \[t11\], there exists a required manifold $M$ which is described by $\Gamma$.
Therefore, from now on, assume that $l>1$. Let $(p,q)$ be the edge whose label is $l$ for some vertices $p$ and $q$. Since $l>1$ is the largest label and $\Gamma$ is effective, the labels of other edges at $p$ and $q$ are strictly smaller than $l$. By the minimal property and the symmetry property, $\Gamma$ has at least one vertex of index $i$ for $0 \leq i \leq 2$. Then we have 4 cases:
1. $n_p=1$ and $n_q=1$.
2. $n_p=0$ and $n_q=1$.
3. $n_p=1$ and $n_q=2$.
4. $n_p=0$ and $n_q=2$.
Suppose that Case (1) holds. Let $e_1$ be another edge at $p$ and let $e_2$ be another edge at $q$. Since $l$ is the largest label and $\Gamma$ is effective, the label $w(e_1)$ of $e_1$ is smaller than $l$. Similarly, $w(e_2)<l$. By the equal modulo property, since $(p,q)$ is $l$-edge, the weights $\{-w(e_1),l\}$ at $p$ and the weights $\{w(e_2),-l\}$ at $q$ are equal modulo $l$. This implies that $-w(e_1) \equiv w(e_2) \mod l$, i.e., $w(e_1)+w(e_2)=l$. This is the case of Figure \[fig2-2\]. Therefore, if we can construct a manifold $M'$ described by a multigraph $\Gamma'$ (Figure \[fig2-1\]) where $\Gamma'$ is obtained from $\Gamma$ by shrinking the $l$-edge $(p,q)$ to a vertex, by Theorem \[t11\] we can construct a manifold $M$ which is described by $\Gamma$.
In Cases (2), (3), and (4) of Theorem \[t71\], we proceed as in Cases (2), (3), and (4) of the proof of Theorem \[t11\] in Section \[s3\].
In each case, the existence of a manifold $M$ described by $\Gamma$ reduces to the existence of a manifold $M'$ described by $\Gamma'$, where $\Gamma$ is obtained from $\Gamma'$ by applying one of the 4 operations in Theorem \[t11\]. The multigraph $\Gamma'$ has one less $l$-edge than $\Gamma$. Continuing this, the existence of $M$ reduces to the existence of a manifold $M''$ which is described by a semi-free multigraph. By Theorem \[t11\], such a manifold $M''$ exists and this completes the proof.
[1]{}
K. Ahara and A. Hattori: *4 dimensional symplectic $S^1$-manifolds admitting moment map.* J. Fac. Sci. Univ. Tokyo Sect. IA, Math. **38** (1991), 251-298.
M. Audin: *Hamiltoniens périodiques sur les variétés symplectiques compactes de dimension 4*. Géométrie symplectique et mécanique, Proceedings 1988, C. Albert ed., Springer Lecture Notes in Math. 1416 (1990).
J. Carrell, A. Howard, and C. Kosniowski: *Holomorphic vector fields on complex surfaces.* Math. Ann. **204** (1973) 73-81.
K. E. Feldman: *Hirzebruch genus of a manifold supporting a Hamiltonian circle action*. Russ. Math. Surv. **56** (2001) 978-979.
R. Fintushel: *Classification of circle actions on 4-manifolds*. Trans. Amer. Math. Soc. **232** (1978) 377-390.
L. Godinho and S. Sabatini: *New tools for classifying Hamiltonian circle actions with isolated fixed points*, Found. Comput. Math. **14** (2014), 791-860.
F. Hirzebruch, T. Berger, and R. Jung: *Manifolds and modular forms*. Aspects of Mathematics, vol. E20, Vieweg, Braunschweig, (1992).
A. Hattori: *$S^1$-actions on unitary manifolds and quasi-ample line bundles*, J. Fac. Sci. Univ. Tokyo Sect. IA Math. **31** (1985), 433-486.
D. Jang and S. Tolman: *Hamiltonian circle actions on eight dimensional manifolds with minimal fixed sets.* Transform. Groups., **22** (2017), 353-359.
D. Jang: *Symplectic periodic flows with exactly three equilibrium points*. Ergod. Theor. Dyn. Syst., **34** (2014), 1930-1963.
D. Jang: *Circle actions on almost complex manifolds with isolated fixed points.* J. Geom. Phys., **119** (2017), 187-192.
D. Jang: *Symplectic circle actions with isolated fixed points.* J. Symplect. Geom. **15** (2017), 1071-1087.
D. Jang: *Circle actions on oriented manifolds with discrete fixed point sets and classification in dimension 4*. J. Geom. Phys., **133** (2018), 181-194.
D. Jang: *Circle actions on almost complex manifolds with 4 fixed points*. Math. Z. (2019), doi:10.1007/s00209-019-02267-z.
Y. Karshon: *Periodic Hamiltonian flows on four dimensional manifolds.* Mem. Am. Math. Soc. **672** (1999).
P. Li: *The rigidity of Dolbeault-type operators and symplectic circle actions.* Proc. Amer. Math. Soc. **140** (2012) 1987-1995.
M. Masuda: *Unitary toric manifolds, multi-fans and equivariant index*. Tohoku. Math. J. **51** (1999) 237-265.
D. McDuff: *The moment map for circle actions on symplectic manifolds.* J. Geom. Phys. **5** (1988), no. 2, 149-160.
P. S. Pao: *Non-linear circle actions on the 4-sphere and twisting spun knots.* Topology **17** (1978) no. 3, 291-296.
S. Sabatini: *On the Chern numbers and the Hilbert polynomial of an almost complex manifold with a circle action.* Commun. Contemp. Math., **19**, 1750043 (2017).
S. Tolman and J. Weitsman: *On semifree symplectic circle actions with isolated fixed points.* Topology **39** (2000), no. 2, 299-309.
S. Tolman: *On a symplectic generalization of Petrie’s conjecture.* Trans. Amer. Math. Soc. **362** (2010), no. 8, 3963-3996.
[^1]: MSC 2010: 58C30, 37C25, 58J20
|
---
abstract: 'It is crucial for physical realization of quantum information networks to first establish entanglement among multiple space-separated quantum memories and then at a user-controlled moment to transfer the stored entanglement to quantum channels for distribution and conveyance of information. Here we present an experimental demonstration on generation, storage and transfer of deterministic quantum entanglement among three spatially separated atomic ensembles. The off-line prepared multipartite entanglement of optical modes is mapped into three distant atomic ensembles to establish entanglement of atomic spin waves via electromagnetically-induced-transparency light-matter interaction. Then the stored atomic entanglement is transferred into a tripartite quadrature entangled state of light, which is space-separated and can be dynamically allocated to three quantum channels for conveying quantum information. The existence of entanglement among released three optical modes verifies that the system has capacity of preserving multipartite entanglement. The presented protocol can be directly extended to larger quantum networks with more nodes.'
author:
- 'Zhihui Yan$^{1,2\dagger}$'
- 'Liang Wu$^{1\dagger}$'
- 'Xiaojun Jia$^{1,2}$'
- 'Yanhong Liu$^{1}$'
- 'Ruijie Deng$^{1}$'
- 'Shujing Li$^{1,2}$'
- 'Hai Wang$^{1,2}$'
- 'Changde Xie$^{1,2}$'
- 'Kunchi Peng$^{1,2} $'
title: Establishing and storing of deterministic quantum entanglement among three distant atomic ensembles
---
Flying photons or bright optical beams are the best natural quantum channels, while usually matter systems are employed for memories at quantum nodes[@Kimble1; @Polzik0]. Single atoms[@Rempe1; @Haroche1], atomic ensembles[@Vuletic1; @Lett1; @Lam1; @Peng2; @Yu1; @Laurat1], trapped ions[Wineland1,Blatt1,Monroe2]{}, optomechanics[Wang1,Vahala1,Aspelmeyer1,Kiese]{}, superconductors[@Huard1], solid-state systems[@Gisin1; @Tittel1; @Sellars1; @Imamoglu1] and so on have been applied as quantum nodes. Especially, atomic ensembles are among the best candidates for quantum nodes to store and process quantum information due to the advantage of the collective enhancement of light-atom interaction[@Vuletic1; @Lett1; @Lam1; @Peng2; @Yu1; @Laurat1].
The entanglement of discrete quantum variables between two atomic ensembles has been experimentally achieved by means of Raman scattering approach[Kimble3,Pan1]{} or transferring quantum states of entangled photons into two atomic systems[@Kuzmich1; @Kimble4; @Guo1]. In 2010, Kimble’s group demonstrated measurement-induced entanglement stored in four atomic memories and coherent transfer of the atomic entanglement to four photonic channels[@Kimble2]. For the first time, their experiment proved that a multipartite entangled W state of atomic ensembles can be transferred into a photonic mode-entangled W state with the heralded entanglement and thus showed an advance in the distribution of multipartite entanglement across quantum networks. Besides above-mentioned schemes based on applying discrete quantum variables of single photons and atoms, the continuous-variable (CV) regime provides another avenue toward the realization of quantum information. To develop CV quantum information networks, CV entanglement between two macroscopic objects, i.e. atomic ensembles, has been investigated[@Polzik1; @Polzik2]. CV entanglement of spin-wave variances between two atomic ensembles has been experimentally realized via quantum non-demolition (QND) measurement[@Polzik1] and dissipation mechanism of atomic systems[@Polzik2], respectively. For implementing quantum computation[Monroe1]{} and quantum communication[@Gisin2], the established entanglement has to be stored in atomic memories and then to be controllably released on demand. Quantum memories for squeezing and entanglement of light have been theoretically investigated[@Yadsan; @Tikh], and the storage of CV entanglement between two atomic ensembles has been experimentally completed[@Polzik3]. So far, all experiments on generation and storage of CV entanglement of atomic systems are concentrated between two ensembles[@Polzik1; @Polzik2; @Polzik3]. In Ref.\[35\], a displaced entangled state of two sideband modes of an optical beam is used for the initial quantum resource to create entanglement between two atomic ensembles. To extend this method to multipartite entanglement more sidebands with different frequency shifts have to be prepared and the number of entangled sideband modes must be strictly restricted by the bandwidth of optical parametric amplifier, which is the device for generating optical entangled state in their system. On the other hand, it is difficult to space-separate these entangled optical submodes with small frequency intervals. Although a narrow-band optical cavity can be utilized for separating these optical modes, entanglement among them will be significantly reduced**[@Bachor1]**. These limitations make difficult to extend the experimental method of Ref.\[35\] to entangle more atomic ensembles. Up to now, it still remains a challenge to entangle more than two remote quantum memories in CV regime.
Here, we present an experimental demonstration on deterministically establishing, storing and releasing of CV entanglement among three atomic ensembles. At first, a tripartite optical entangled state is off-line prepared, and then the entanglement is transferred into three atomic ensembles located 2.6 meters apart from each other via electromagnetically-induced-transparency (EIT) interaction. After a given storage time the preserved atomic entanglement is controllably released into three separated quantum channels consisting of three entangled optical submodes. The criterion inequalities and dependence of entanglement among three released optical submodes on systematic parameters are theoretically deduced and multipartite entanglement transfer as well as storage are experimentally proved. Since the tripartite optical entangled state is generated by linearly optical transformation of three squeezed states of light, its three submodes are naturally space-separated[@Peng6; @Jia1]. The presented scheme can be directly extended to generate optical entangled states with more submodes if more squeezed states of light are available. In this way, entanglement of more atomic ensembles can be established.
Results {#results .unnumbered}
=======
**Experimental Setup.** Fig. 1a describes the experimental setup for generation, storage and transfer of tripartite entanglement. Three space-separated submodes $\hat{a}(0)_{\text{S1}}$, $\hat{a}(0)_{\text{S2}}$ and $\hat{a}(0)_{\text{S3}}$ of an optical entangled state off-line prepared in Part I interact respectively with three atomic memories A$_{1}$, A$_{2}$, A$_{3}$ located at three distant nodes to generate entanglement of spin waves among three atomic ensembles. The entanglement is stored within the three ensembles. Then the preserved entanglement is transferred back into an optical entangled state with three submodes $\hat{a}(t)_{\text{S1}}$, $\hat{a%
}(t)_{\text{S2}}$ and $\hat{a}(t)_{\text{S3}}$ after a storage time $t$ (Part II). At last, entanglement among three released optical submodes is measured by three balanced homodyne detectors BHD$_{1-3}$ (Part III).
Three narrow band entangled optical beams tuned to the $\left\vert
5S_{1/2},F=1\right\rangle $ $\leftrightarrow \left\vert 5P_{1/2},F^{\prime
}=1\right\rangle $ transition of Rubidium around 795 nm are obtained via linearly optical transformation of three squeezed states of light, which are generated by three degenerate optical parametric amplifiers (DOPA$_{1-3}$). The DOPA$_{1}$ and DOPA$_{2(3)}$ operating in parametric amplification and deamplification produce phase and amplitude quadrature squeezed states $\hat{a}_{\text{S1}}$ and $\hat{a}_{\text{S2(3)}}$, respectively[@Peng1; @Peng5]. The squeezing parameters ($r$) for the three squeezed states are assumed to be identical for simplicity. In fact, three DOPAs used in our experiment have totally identical configuration and thus their squeezing parameters are almost the same. The quadrature amplitudes $%
\hat{X}_{\text{S}j}=(\hat{a}_{\text{S}j}+\hat{a}_{\text{S}j}^{+})/\sqrt{2}$ and phases $\hat{P}_{\text{S}j}=(\hat{a}_{\text{S}j}-\hat{a}_{\text{S}%
j}^{+})/\sqrt{2}i$ ($j=1$, $2$, $3$) of output optical beams of three DOPAs are expressed as $\hat{X}_{\text{S}1}=$e$^{r}\hat{X}_{\text{S}1}^{(0)}$, $\hat{P}%
_{\text{S}1}=$e$^{-r}\hat{P}_{\text{S}1}^{(0)}$, $\hat{X}_{\text{S}2(3)}=$e$%
^{-r}\hat{X}_{\text{S}2(3)}^{(0)}$ and $\hat{P}_{\text{S}2(3)}=$e$^{r}\hat{P}%
_{\text{S}2(3)}^{(0)}$[@Peng5], where $\hat{X}(\hat{P})_{\text{S}%
1}^{(0)} $, $\hat{X}(\hat{P})_{\text{S}2}^{(0)}$ and $\hat{X}(\hat{P})_{%
\text{S}3}^{(0)}$ are amplitude (phase) quadratures of input optical beams $%
\hat{a}_{\text{S}1}^{(0)}$, $\hat{a}_{\text{S}2}^{(0)}$ and $\hat{a}_{\text{S%
}3}^{(0)}$ for DOPAs, respectively. By interfering three squeezed states of light on BS$%
_{1} $ and BS$_{2}$, we obtain a tripartite optical entangled state with quantum correlations of both amplitude quadratures $\hat{X}(0)_{\text{L}j}=(%
\hat{a}(0)_{\text{S}j}+\hat{a}(0)_{\text{S}j}^{+})/\sqrt{2}$ and phase quadratures $\hat{P}(0)_{\text{L}j}=(\hat{a}(0)_{\text{S}j}-\hat{a}(0)_{%
\text{S}j}^{+})/\sqrt{2}i$[@Peng5]. Then the three entangled optical beams are chopped into 500 ns pulses $\hat{a}(0)_{\text{S}1}$, $\hat{a}(0)_{\text{S}2}$ and $\hat{a}(0)_{\text{S}3}$, with three acoustical-optical modulators AOM$%
_{4-6}$, which are used for the input signals of three atomic ensembles, i.e. $^{87}$Rb vapour cells A$_{1}$, A$_{2}$ and A$_{3}$.
{width="8.6cm"}
. .
**Quantum state transfer between atomic spin waves and optical submodes.** The physical mechanism of light-matter interaction used for the experiment is EIT, which is a transparency phenomenon induced by optical field in an opaque medium by means of quantum interference[@Marangos1]. M. Fleischhauer and M. D. Lukin have theoretically demonstrated that when quantum fields propagate in EIT media there are form-stable quantum excitations associated with such propagation, named dark-state polaritons, and in this process the quantum state of light can be ideally transferred to collective atomic excitations and vice versa. Therefore, EIT effects can be applied to generate nonclassical states of atomic ensembles, to store optical quantum states, and reversibly to release stored quantum states into optical channels, respectively[@Lukin2; @Giacobino1; @Lvovsky1; @Furusawa1]. An atomic ensemble is represented by total angular momentum operator of collective atomic spins $\hat{J}=\sum_{i}\left\vert g\right\rangle
\left\langle m\right\vert $, and y, z-components of the collective atomic angular momentum play the role of canonical variables, i.e. $\hat{X}_{\text{A}}=(\hat{J%
}+\hat{J}^{+})/\sqrt{2}=\hat{J}_{y}/\sqrt{\langle \hat{J}_{x}\rangle }$, $%
\hat{P}_{\text{A}}=(\hat{J}-\hat{J}^{+})/\sqrt{2}i=\hat{J}_{z}/\sqrt{\langle
\hat{J}_{x}\rangle }$[@Polzik3]. When control optical beams are adiabatically switched off, the mapping relations of amplitude (phase) quadratures from input optical submodes $\hat{X}(\hat{P})(0)_{\text{L}j}$ to atomic spin waves $\hat{X}(\hat{P})(t)_{\text{A}j}$ after a storage time $t$ are expressed by[@Ou1]:
$$\begin{aligned}
\hat{X}(t)_{\text{A}j} &=&\sqrt{\eta _{_{\text{M}}}}\hat{X}(0)_{\text{L}j}+%
\sqrt{1-\eta _{_{\text{M}}}}\hat{X}_{\text{A}j}^{\text{vac}}\text{,} \\
\hat{P}(t)_{\text{A}j} &=&\sqrt{\eta _{_{\text{M}}}}\hat{P}(0)_{\text{L}j}+%
\sqrt{1-\eta _{_{\text{M}}}}\hat{P}_{\text{A}j}^{\text{vac}}\text{,} \notag\end{aligned}$$
where the mapping efficiency from input optical submodes to atomic spin waves is $\eta _{_{\text{M}}}=\eta _{_{\text{T}}}\eta _{_{\text{W}}}$e$%
^{-t/\tau _{s}}$, $\eta _{_{\text{T}}}$ is the optical transmission efficiency, $\eta _{_{\text{W}}}$ is the storage efficiency of light in atomic ensemble, and $\tau _{\text{s}}$ is the storage lifetime limited by atomic decoherence. The vacuum quadrature noises of atomic ensembles $\hat{X}%
(\hat{P})_{\text{A}j}^{\text{vac}}$ are introduced by limited mapping efficiency $\eta _{_{\text{M}}}$. Since canonical quadrature operators of atomic spin waves obey the same commutation relation with that of Gaussian optical states, i.e. $[\hat{X}_{\text{A}},\hat{P}_{\text{A}}]=i$, using similar procedure of deducing full tripartite inseparability criteria provided by Loock *et al.*, we can obtain a set of analogous criterion inequalities for atomic spin waves (see also the Method section).
The stored atomic entanglement can be transferred to tripartite entanglement among output three optical submodes $\hat{a}(t)_{\text{P}1}$, $\hat{a}(t)_{%
\text{P}2}$ and $\hat{a}(t)_{\text{P}3}$ by turning on control optical beams. The quadrature amplitudes and phases of released submodes, $\hat{X}(t)_{\text{L}%
j}=(\hat{a}(t)_{\text{P}j}+\hat{a}(t)_{\text{P}j}^{+})/\sqrt{2}$ and $\hat{P}%
(t)_{\text{L}j}=(\hat{a}(t)_{\text{P}j}-\hat{a}(t)_{\text{P}j}^{+})/\sqrt{2}%
i $ in terms of quadratures for atomic spin waves $\hat{X}(\hat{P})(t)_{%
\text{A}j}$ after a storage time $t$ are expressed by[@Ou1]:
$$\begin{aligned}
\hat{X}(t)_{\text{L}j} &=&-\sqrt{\eta _{_{\text{M}}}^{\prime }}\hat{X}(t)_{%
\text{A}j}+\sqrt{1-\eta _{_{\text{M}}}^{\prime }}\hat{X}_{\text{L}j}^{\text{%
vac}}\text{,} \\
\hat{P}(t)_{\text{L}j} &=&-\sqrt{\eta _{_{\text{M}}}^{\prime }}\hat{P}(t)_{%
\text{A}j}+\sqrt{1-\eta _{_{\text{M}}}^{\prime }}\hat{P}_{\text{L}j}^{\text{%
vac}}\text{,} \notag\end{aligned}$$
where the mapping efficiency from atomic spin waves to optical submodes $%
\eta _{_{\text{M}}}^{\prime }$ is the retrieval efficiency from atomic ensembles to light. The vacuum quadrature noises of optical submodes $\hat{X}%
(\hat{P})_{\text{L}j}^{\text{vac}}$ are introduced by the read process.
The full tripartite inseparability criteria for released optical modes are given by[@Loock1]:
$$\begin{aligned}
I(t)_{\text{L}1} &=&\langle \delta ^{2}(\hat{X}(t)_{\text{L}2}-\hat{X}(t)_{%
\text{L}3})\rangle /2+\langle \delta ^{2}(g_{\text{L}1}^{\prime }\hat{P}(t)_{%
\text{L}1} \notag \\
&&+\hat{P}(t)_{\text{L}2}+\hat{P}(t)_{\text{L}3})\rangle /2\geqslant 1\text{,%
} \\
I(t)_{\text{L}2} &=&\langle \delta ^{2}(\hat{X}(t)_{\text{L}1}-\hat{X}(t)_{%
\text{L}3})\rangle /2+\langle \delta ^{2}(\hat{P}(t)_{\text{L}1} \notag \\
&&+g_{\text{L}2}^{\prime }\hat{P}(t)_{\text{L}2}+\hat{P}(t)_{\text{L}%
3})\rangle /2\geqslant 1\text{,} \notag \\
I(t)_{\text{L}3} &=&\langle \delta ^{2}(\hat{X}(t)_{\text{L}1}-\hat{X}(t)_{%
\text{L}2})\rangle /2+\langle \delta ^{2}(\hat{P}(t)_{\text{L}1} \notag \\
&&+\hat{P}(t)_{\text{L}2}+g_{\text{L}3}^{\prime }\hat{P}(t)_{\text{L}%
3})\rangle /2\geqslant 1\text{.} \notag\end{aligned}$$
If any two in the three inequalities are simultaneously violated, the three submodes form a tripartite Greenberger-Horne-Zeilinger-like (GHZ-like) entangled state of light, where $g_{\text{L1}}^{\prime }$, $g_{\text{L2}%
}^{\prime }$ and $g_{\text{L3}}^{\prime }$ are the gain factors for optimizing the correlation variances for released submodes. From Eqs. (1) and (2), it can be seen that entanglement is limited by total mapping efficiencies $\eta $ ($\eta =\eta _{_{\text{M}}}\eta _{_{\text{M}}}^{\prime
} $) as well as squeezing parameter $r$. When the squeezing parameters $r$ for three DOPAs and the total mapping efficiencies $\eta $ for three atomic memories are the same, the values of left sides of three inequalities in Eq. (3) are identical, $I(t)_{\text{L1}}=I(t)_{\text{L2}}=I(t)_{\text{L3}}=I(t)_{%
\text{L}}$. The smaller $I(t)_{\text{L}}$ is, the higher the entanglement is. Fig. 2 shows the dependence of correlation variance combinations for three released submodes on the squeezing parameter $r$ of initial squeezed states and the total mapping efficiency $\eta $ with the storage time of 1000 ns. We can see that the combinations are reduced with the increase of squeezing parameter $r$ and total mapping efficiency $\eta $ (see also the Method section).
{width="8.6cm"}
When the control optical beams $\hat{a}_{\text{C1}}$, $\hat{a}_{\text{C2}}$ and $\hat{a}_{\text{C3}}$, which are tuned to $\left\vert 5S_{1/2},F=2\right\rangle $ $\leftrightarrow \left\vert
5P_{1/2},F^{\prime }=1\right\rangle $ transition of Rubidium, are adiabatically switched off, optical entanglement among three input submodes $\hat{a}(0)_{%
\text{S1}}$, $\hat{a}(0)_{\text{S2}}$ and $\hat{a}(0)_{\text{S3}}$ is transferred to atomic entanglement of spin waves according to Eq. (1) via EIT interaction. After a storage time of 1 $\mu s$, control optical beams $\hat{a}_{%
\text{C1}}$, $\hat{a}_{\text{C2}}$ and $\hat{a}_{\text{C3}}$ are switched on again, three optical submodes $\hat{a}(t)_{\text{S1}}$, $\hat{a}(t)_{\text{S2%
}}$ and $\hat{a}(t)_{\text{S3}}$ are released. The combinations of correlation variances in Eq. (3) are measured with three time domain BHD$_{1-3}$. The intensive coherent light $\hat{a}_{\text{L1}}$, $\hat{a}_{\text{L2}}$ and $\hat{a}_{\text{L3}}$ are utilized as local oscillators of BHD$_{1-3}$. The control and signal optical beams with orthogonally linear polarizations are combined on Glan-Thompson Polarizers (P$_{1-3}$) before atomic cells, and control optical beams are filtered out from the signal optical beams by Glan-Thompson Polarizers (P$_{4-6}$) and etalon filters (F$_{1-3}$). In storage and retrieval procedures 10000 traces of BHD output signals with 20 G samples/s are digitally filtered with a bandpass filter of 2.5 MHz and averaged to obtain the optimal entangled degree. In this case the low-frequency sideband noises resulting from pumping laser of DOPAs and atomic ensembles, as well as high frequency noises coming from parametric conversion in DOPAs and EIT process in atomic ensembles have been are filtered out. When both control and signal optical beams are blocked and only the local oscillators are remained, outputs of BHDs stand for the corresponding vacuum noise level[@Ou0].
------------------------------------------------------------------------- ---------------------- ------------------------ --------------------------
Correlation variances Values for input Values for atomic Values for released
for different combinations submodes (dB) spin waves (dB) submodes (dB)
$\langle \delta ^{2}(\hat{X}_{2}-\hat{X}_{3})\rangle $ -3.30$\pm $0.05 -0.56$\pm $0.03 -0.37$\pm $0.03
$\langle \delta ^{2}(g_{1}\hat{P}_{1}+\hat{P}_{2}+\hat{P}_{3})\rangle $ -2.93$\pm $0.05 -0.15$\pm $0.02 -0.10$\pm $0.02
$\langle \delta ^{2}(\hat{X}_{1}-\hat{X}_{3})\rangle $ -3.25$\pm $0.05 -0.53$\pm $0.03 -0.35$\pm $0.03
$\langle \delta ^{2}(\hat{P}_{1}+g_{2}\hat{P}_{2}+\hat{P}_{3})\rangle $ -2.91$\pm $0.05 -0.15$\pm $0.02 -0.10$\pm $0.02
$\langle \delta ^{2}(\hat{X}_{1}-\hat{X}_{2})\rangle $ -3.25$\pm $0.05 -0.52$\pm $0.03 -0.34$\pm $0.03
$\langle \delta ^{2}(\hat{P}_{1}+\hat{P}_{2}+g_{3}\hat{P}_{3})\rangle $ -2.90$\pm $0.05 -0.14$\pm $0.02 -0.09$\pm $0.02
------------------------------------------------------------------------- ---------------------- ------------------------ --------------------------
: The values of normalized correlation variances for different combinations.
{width="8.6cm"}
**Experimental Results.** The normalized correlation variances for different combinations of quadrature components are given in Table. 1, where gain factors ($g_{1}$, $g_{2}$ and $g_{3}$) are chosen as the optimal gains for minimizing the corresponding correlation variances. The correlation variances of input and released submodes are directly measured with three sets of BHDs. The normalized correlation variances among three atomic ensembles are inferred from Eq. (2), where the mapping efficiency $\eta _{_{%
\text{M}}}^{\prime }$ is about $68\%$ for our experimental system. The measured normalized quantum correlation variances are shown in Fig. 3. The squeezing parameter $r$ is $0.38$ and total mapping efficiency $\eta $ is about $16\%$. The combination of correlation variances for three released submodes is $I(t)_{\text{L}}=0.96\pm 0.01$, which is less than 1, thus according to criterion inequalities Eq. (3) the entanglement among released submodes is verified. The value is in agreement with the theoretically calculated result, which is marked with a black dot in Fig. 2. Because of the limitation of total mapping efficiency, the correlation variances of released entangled state are much higher than that of input state. However, correlation variances below the corresponding vacuum noise level in Fig. 3 and the violation of criterion inequalities in Eq. (3) certainly prove the existence of tripartite entanglement among three optical submodes released from atomic spin waves of three atomic ensembles. Thus the tripartite GHZ-like entanglement among three atomic ensembles is experimentally demonstrated.
Discussion {#discussion .unnumbered}
==========
In summary, deterministic quantum entanglement among three spatially separated quantum nodes is experimentally generated, stored and transferred. Within storage lifetime the multipartite entanglement is preserved in three space-separated atomic ensembles, and then at a desirable time the stored atomic entanglement can be controllably converted into three optical submodes to be quantum channels. Our work shows that multipartite CV entanglement can be established among remote macroscopic objects by transferring off-line prepared entanglement of optical beams into atomic ensembles via EIT interaction. Since unconditional CV entanglement among multipartite optical modes has been experimentally accomplished[Peng6,Furusawa2,Treps1]{}, mature quantum optical technology can be used for realizing entanglement of more nodes in a quantum networks. The obtained entanglement among atomic ensembles depends on mapping efficiency $\eta _{_{%
\text{M}}}$ from light to atoms and initial squeezing parameter $r$. The higher the squeezing and the mapping efficiency are, the better the entanglement among atomic ensembles is. In the presented experimental system, the total mapping efficiency is mainly limited by the optical transmission loss and memory (storage and retrieval) efficiency. The transmission loss (about 18%) comes from the optical losses of atomic cells, etalon filters, Glan-Thompson Polarizers and other optical components, which can be further reduced if better optical elements with lower losses are available.
The increased excess noises in released submodes from memories originate from fluorescence and coherent emission as well as spurious fluctuations in signal channels induced by the control optical beam[Lvovsky1]{}. On the other hand, since both EIT and four-wave mixing (FWM) effects are simultaneously generated in an ensemble of hot atoms[Phillips1]{}, the control optical beam acts as a far-detuned field on the signal transition in the undesired FWM process and spontaneously generates an idler field, which also can form excess noises. These mechanisms resulting in excess noises always exist in EIT light-matter interaction for any input states, whatever vacuum state, squeezed vacuum state, entangled state or others. Therefore some schemes used for improving EIT memory efficiency of classical signals[Lobino1,Phillips1,Lauk1]{}, such as decreasing detuning of probe and control optical beams, increasing the power of control optical beam and enhancing the temperature of atomic vapour, will unavoidably introduce more excess noises into atomic media and reduce quantum correlations among atomic ensembles. However, by optimizing experimental parameters, excess noises can be minimized for a given EIT experimental system[Lvovsky1,Giacobino1,Phillips1]{}.
It has been demonstrated that mapping efficiency can be significantly improved by means of the technology of optical cavity enhancement without introducing excess noise[@Pan3; @Walmsley1] and the storage lifetime can be dramatically increased if thermal atomic ensembles are replaced by cold atoms confined in three dimensional optical lattice[@Pan3]. The generation systems of optical squeezed states with high squeezing up to 15 dB has been available today[@Schnabel1], which can provide initial high quality quantum resource for establishing better multipartite entanglement among atomic memories. The presented scheme opens up a new possibility for constructing future quantum internet[@Kimble1] and implementing distribution quantum computation based on the use of deterministic CV entanglement resources of light and atomic memories[@Stace1].
Methods {#methods .unnumbered}
=======
**Experimental realization of tripartite entangled state of light.** A Ti:sapphire laser (Coherent MBR-110) pumped by green laser (Yuguang DPSS FG-VIIIB) outputs 3 W laser at about 795 nm wavelength, which is used for pumping light of second harmonic generator (SHG) and seed lights of DOPAs. The configuration of optical cavities for SHG and three DOPAs is identical bow tie type ring cavity with a $1\times 2\times 10$ mm periodically poled KTiOPO4 (PPKTP) crystal. Three DOPAs are pumped by the second harmonic fields at about 398 nm from SHG and the fundamental waves from SHG are utilized as three injected seed fields ($\hat{a}_{\text{S1}}^{(0)}$, $\hat{a}_{\text{%
S2}}^{(0)}$ and $\hat{a}_{\text{S3}}^{(0)}$). DOPA$_{1}$ and DOPA$_{2(3)}$ are operated at parametric amplification and deamplification to produce quadrature phase and amplitude squeezed state of light, respectively[@Peng2].
The three squeezed optical beams are interfered on two optical beam splitters. The quadrature phase squeezed field from DOPA$_{1}$ ($\hat{a}_{%
\text{S1}}$) and the quadrature amplitude squeezed field from DOPA$_{2}$ ($%
\hat{a}_{\text{S2}}$) are firstly interfered on a beam splitter (BS1) with the ratio of R:T=1:2 (R: reflectivity and T: transmissivity). Then, one of two output optical beams from BS1 and the quadrature amplitude squeezed light from DOPA$%
_{3}$ ($\hat{a}_{\text{S3}}$) are interfered on BS2 with the ratio of R:T=1:1. The relative phase between the two input optical beams on BS1(2) is kept at $2k\pi $ ($k$ is integer). Finally three entangled optical beams are chopped into three optical pulses $\hat{a}(0)_{\text{S1}}$, $\hat{a}(0)_{%
\text{S2}}$ and $\hat{a}(0)_{\text{S3}}$ by three AOMs. The three optical pulses are respectively injected into three atomic ensembles to be the input optical submodes. The quadratures amplitudes $\hat{X}(0)_{\text{L}j}=(\hat{a}(0)_{\text{S}j}+\hat{a%
}(0)_{\text{S}j}^{+})/\sqrt{2}$ and phases $\hat{P}(0)_{\text{L}j}=(\hat{a}%
(0)_{\text{S}j}-\hat{a}(0)_{\text{S}j}^{+})/\sqrt{2}i$ ($j=1,2,3$) of input optical submodes are expressed by[@Furusawa2; @Peng2]:
$$\hat{X}(0)_{\text{L}1}=\sqrt{\frac{1}{3}}\text{e}^{r}\hat{X}_{\text{S}%
1}^{(0)}+\sqrt{\frac{2}{3}}\text{e}^{-r}\hat{X}_{\text{S}2}^{(0)}\text{,}$$
$$\begin{aligned}
\hat{P}(0)_{\text{L}1} &=&\sqrt{\frac{1}{3}}\text{e}^{-r}\hat{P}_{\text{S}%
1}^{(0)}+\sqrt{\frac{2}{3}}\text{e}^{r}\hat{P}_{\text{S}2}^{(0)}\text{,} \\
\hat{X}(0)_{\text{L}2} &=&\sqrt{\frac{1}{3}}\text{e}^{r}\hat{X}_{\text{S}%
1}^{(0)}-\sqrt{\frac{1}{6}}\text{e}^{-r}\hat{X}_{\text{S}2}^{(0)}+\sqrt{%
\frac{1}{2}}\text{e}^{-r}\hat{X}_{\text{S}3}^{(0)}\text{,} \\
\hat{P}(0)_{\text{L}2} &=&\sqrt{\frac{1}{3}}\text{e}^{-r}\hat{P}_{\text{S}%
1}^{(0)}-\sqrt{\frac{1}{6}}\text{e}^{r}\hat{P}_{\text{S}2}^{(0)}+\sqrt{\frac{%
1}{2}}\text{e}^{r}\hat{P}_{\text{S}3}^{(0)}\text{,} \\
\hat{X}(0)_{\text{L}3} &=&\sqrt{\frac{1}{3}}\text{e}^{r}\hat{X}_{\text{S}%
1}^{(0)}-\sqrt{\frac{1}{6}}\text{e}^{-r}\hat{X}_{\text{S}2}^{(0)}-\sqrt{%
\frac{1}{2}}\text{e}^{-r}\hat{X}_{\text{S}3}^{(0)}\text{,} \\
\hat{P}(0)_{\text{L}3} &=&\sqrt{\frac{1}{3}}\text{e}^{-r}\hat{P}_{\text{S}%
1}^{(0)}-\sqrt{\frac{1}{6}}\text{e}^{r}\hat{P}_{\text{S}2}^{(0)}-\sqrt{\frac{%
1}{2}}\text{e}^{r}\hat{P}_{\text{S}3}^{(0)}\text{,}\end{aligned}$$
respectively.
The inequalities of full inseparability criteria for input tripartite entangled states of light are[@Loock1]:
$$\begin{aligned}
I(0)_{\text{L}1} &=&\langle \delta ^{2}(\hat{X}(0)_{\text{L}2}-\hat{X}(0)_{%
\text{L}3})\rangle /2+\langle \delta ^{2}(g_{\text{L}1}\hat{P}(0)_{\text{L}%
1}+\hat{P}(0)_{\text{L}2}+\hat{P}(0)_{\text{L}3})\rangle /2\geqslant 1\text{,%
} \\
I(0)_{\text{L}2} &=&\langle \delta ^{2}(\hat{X}(0)_{\text{L}1}-\hat{X}(0)_{%
\text{L}3})\rangle /2+\langle \delta ^{2}(\hat{P}(0)_{\text{L}1}+g_{\text{L}%
2}\hat{P}(0)_{\text{L}2}+\hat{P}(0)_{\text{L}3})\rangle /2\geqslant 1\text{,}
\notag \\
I(0)_{\text{L}3} &=&\langle \delta ^{2}(\hat{X}(0)_{\text{L}1}-\hat{X}(0)_{%
\text{L}2})\rangle /2+\langle \delta ^{2}(\hat{P}(0)_{\text{L}1}+\hat{P}(0)_{%
\text{L}2}+g_{\text{L}3}\hat{P}(0)_{\text{L}3})\rangle /2\geqslant 1\text{.}
\notag\end{aligned}$$
If any two in the three inequalities are simultaneously violated, the three submodes form a tripartite GHZ-like entangled state, where $g_{\text{L}%
1}$, $g_{\text{L}2}$ and $g_{\text{L}3}$ are gain factors for minimizing correlation variances of the input tripartite entangled state of light.
When the squeezing parameter $r$ ($r\geqslant 0$) for three DOPAs is the same, the gain factors in Eq. (5) should be the same, i.e. $g_{\text{L}1}=g_{%
\text{L}2}=g_{\text{L}3}=g_{\text{L}}$, and the values of left sides of three inequalities are identical, i.e. $I(0)_{\text{L}1}=I(0)_{\text{L}2}=I(0)_{\text{L}3}=I(0)_{%
\text{L}}$. Using Eqs. (4) and (5), the combination of normalized quantum correlation variances for input optical beams is obtained:
$$I(0)_{\text{L}}=\frac{12\text{e}^{-2r}+2(g_{\text{L}}+2)^{2}\text{e}%
^{-2r}+4(g_{\text{L}}-1)^{2}\text{e}^{2r}}{24}\text{.}$$
Calculating the minimum value of Eq. (6), we get the optimal gain factor $g_{%
\text{L}}^{\text{opt}}$:
$$g_{\text{L}}^{\text{opt}}=\frac{2\text{e}^{4r}-2}{2\text{e}^{4r}+1}\text{.}$$
If squeezing parameter $r$ is larger than 0, the combination of correlation variances will be less than 1 with the optimal gain factor and the input optical submodes of atomic ensembles are in a tripartite GHZ-like entangled state.
**Establishing and storing of tripartite atomic-ensemble entanglement.** In EIT memory medium, quantum state can be mapped from input optical submode $\hat{a}_{\text{S}}$ into atomic spin wave $\hat{J}$ and vice versus under the interaction with a the strong control optical beam $\hat{a}_{\text{C}}$[Fleischhauer1,Lukin2]{}. The control optical beam is treated as classical optical beam $A_{\text{C}}$ because it is much more intensive than the signal optical modes. In EIT process, the effective interaction Hamiltonian $\hat{H}_{\text{%
EIT}}$ between signal optical mode $\hat{a}_{\text{S}}$ and atomic spin wave $\hat{J}$ is given by[@Polzik1; @Ou1]:
$$\hat{H}_{\text{EIT}}=i\hbar \kappa A_{\text{C}}\hat{a}_{\text{P}}\hat{J}%
^{+}-i\hbar \kappa A_{\text{C}}\hat{a}_{\text{P}}^{+}\hat{J}\text{,}$$
which is similar to a beam-splitter interaction, where $\kappa $ stands for the interaction constant between light and atoms.
By solving Heisenberg motion equations with the Hamiltonian $\hat{H}_{%
\text{EIT}}$ (Eq. (8)), we obtain the expressions of quantum storage process and the mapping relations (Eq. (1)) of amplitude and phase quadratures from input optical submodes $\hat{X}(\hat{P})(0)_{\text{L}j}$ to atomic spin waves $\hat{X}(\hat{P})(t)_{\text{A}j}$ after a storage time $t$.
When the control optical beams are turned on, the input signal submodes are compressed in atomic ensembles due to the slow propagation under EIT interaction. On the moment of simultaneously shutting off three control optical beams, quantum entanglement among three pulse submodes will be mapped into atomic spin waves in the three ensembles, where the control optical beam plays the role of writing process. Using Eqs. (1) and (4), the amplitude (phase) quadratures of atomic spin waves $\hat{X}(\hat{P})(t)_{%
\text{A}j}$ after a storage time $t$ are obtained:
$$\begin{aligned}
\hat{X}(t)_{\text{A}1} &=&\sqrt{\frac{\eta _{_{\text{M}}}}{3}}\text{e}^{r}%
\hat{X}_{\text{S}1}^{(0)}+\sqrt{\frac{2\eta _{_{\text{M}}}}{3}}\text{e}^{-r}%
\hat{X}_{\text{S}2}^{(0)}+\sqrt{1-\eta _{_{\text{M}}}}\hat{X}_{\text{A}1}^{%
\text{vac}}\text{,} \\
\hat{P}(t)_{\text{A}1} &=&\sqrt{\frac{\eta _{_{\text{M}}}}{3}}\text{e}^{-r}%
\hat{P}_{\text{S}1}^{(0)}+\sqrt{\frac{2\eta _{_{\text{M}}}}{3}}\text{e}^{r}%
\hat{P}_{\text{S}2}^{(0)}+\sqrt{1-\eta _{_{\text{M}}}}\hat{P}_{\text{A}1}^{%
\text{vac}}\text{,} \notag \\
\hat{X}(t)_{\text{A}2} &=&\sqrt{\frac{\eta _{_{\text{M}}}}{3}}\text{e}^{r}%
\hat{X}_{\text{S}1}^{(0)}-\sqrt{\frac{\eta _{_{\text{M}}}}{6}}\text{e}^{-r}%
\hat{X}_{\text{S}2}^{(0)}+\sqrt{\frac{\eta _{_{\text{M}}}}{2}}\text{e}^{-r}%
\hat{X}_{\text{S}3}^{(0)}+\sqrt{1-\eta _{_{\text{M}}}}\hat{X}_{\text{A}2}^{%
\text{vac}}\text{,} \notag \\
\hat{P}(t)_{\text{A}2} &=&\sqrt{\frac{\eta _{_{\text{M}}}}{3}}\text{e}^{-r}%
\hat{P}_{\text{S}1}^{(0)}-\sqrt{\frac{\eta _{_{\text{M}}}}{6}}\text{e}^{r}%
\hat{P}_{\text{S}2}^{(0)}+\sqrt{\frac{\eta _{_{\text{M}}}}{2}}\text{e}^{r}%
\hat{P}_{\text{S}3}^{(0)}+\sqrt{1-\eta _{_{\text{M}}}}\hat{P}_{\text{A}2}^{%
\text{vac}}\text{,} \notag \\
\hat{X}(t)_{\text{A}3} &=&\sqrt{\frac{\eta _{_{\text{M}}}}{3}}\text{e}^{r}%
\hat{X}_{\text{S}1}^{(0)}-\sqrt{\frac{\eta _{_{\text{M}}}}{6}}\text{e}^{-r}%
\hat{X}_{\text{S}2}^{(0)}-\sqrt{\frac{\eta _{_{\text{M}}}}{2}}\text{e}^{-r}%
\hat{X}_{\text{S}3}^{(0)}+\sqrt{1-\eta _{_{\text{M}}}}\hat{X}_{\text{A}3}^{%
\text{vac}}\text{,} \notag \\
\hat{P}(t)_{\text{A}3} &=&\sqrt{\frac{\eta _{_{\text{M}}}}{3}}\text{e}^{-r}%
\hat{P}_{\text{S}1}^{(0)}-\sqrt{\frac{\eta _{_{\text{M}}}}{6}}\text{e}^{r}%
\hat{P}_{\text{S}2}^{(0)}-\sqrt{\frac{\eta _{_{\text{M}}}}{2}}\text{e}^{r}%
\hat{P}_{\text{S}3}^{(0)}+\sqrt{1-\eta _{_{\text{M}}}}\hat{P}_{\text{A}3}^{%
\text{vac}}\text{,} \notag\end{aligned}$$
respectively.
Since canonical quadrature operators of atomic spin waves obey the same commutation relation with that of Gaussian optical states, i.e. $[\hat{X}_{%
\text{A}},\hat{P}_{\text{A}}]=i$, using similar procedure of deducing full tripartite inseparability criteria provided by Loock *et al.*[@Loock1], we obtain a set of analogous criterion inequalities for atomic spin waves:
$$\begin{aligned}
I(t)_{\text{A1}} &=&\langle \delta ^{2}(\hat{X}(t)_{\text{A2}}-\hat{X}(t)_{%
\text{A3}})\rangle /2+\langle \delta ^{2}(g_{\text{A1}}\hat{P}(t)_{\text{A1}%
}+\hat{P}(t)_{\text{A2}}+\hat{P}(t)_{\text{A3}})\rangle /2\geqslant 1\text{,}
\\
I(t)_{\text{A2}} &=&\langle \delta ^{2}(\hat{X}(t)_{\text{A1}}-\hat{X}(t)_{%
\text{A3}})\rangle /2+\langle \delta ^{2}(\hat{P}(t)_{\text{A1}}+g_{\text{A2}%
}\hat{P}(t)_{\text{A2}}+\hat{P}(t)_{\text{A3}})\rangle /2\geqslant 1\text{,}
\notag \\
I(t)_{\text{A3}} &=&\langle \delta ^{2}(\hat{X}(t)_{\text{A1}}-\hat{X}(t)_{%
\text{A2}})\rangle /2+\langle \delta ^{2}(\hat{P}(t)_{\text{A1}}+\hat{P}(t)_{%
\text{A2}}+g_{\text{A3}}\hat{P}(t)_{\text{A3}})\rangle /2\geqslant 1\text{.}
\notag\end{aligned}$$
When any two in three inequalities are violated, the three atomic ensembles are in an entangled state of GHZ-like type, where $g_{\text{A1}}$, $g_{\text{A2}}$ and $g_{\text{A3}}$ are gain factors for atomic ensembles.
If the mapping efficiencies $\eta _{_{\text{M}}}$ for three atomic memories are the same, the gain factors in Eq. (10) should be the same, i.e. $g_{%
\text{A1}}=g_{\text{A2}}=g_{\text{A3}}=g_{\text{A}}$, and the values of three inequalities are also identical, i.e. $I(t)_{\text{A1}}=I(t)_{\text{A2}%
}=I(t)_{\text{A3}}=I(t)_{\text{A}}$. According to Eqs. (9) and (10), the combination of normalized quantum correlation variances for atomic spin waves after a storage time $t$ is obtained:
$$I(t)_{\text{A}}=\frac{12\text{e}^{-2r}+2(g_{\text{A}}+2)^{2}\text{e}%
^{-2r}+4(g_{\text{A}}-1)^{2}\text{e}^{2r}}{24}\eta _{_{\text{M}}}+(1+\frac{1%
}{4}g_{\text{A}})(1-\eta _{_{\text{M}}})\text{.}$$
Similarly, by minimizing $I(t)_{\text{A}}$ we get the optimal gain factor $g_{\text{A}}^{\text{opt}}$:
$$g_{\text{A}}^{\text{opt}}=\frac{2\eta _{_{\text{M}}}\text{e}^{4r}-2\eta _{_{%
\text{M}}}}{3\text{e}^{2r}+\eta _{_{\text{M}}}-3\eta _{_{\text{M}}}\text{e}%
^{2r}+2\text{e}^{4r}\eta _{_{\text{M}}}}\text{.}$$
The smaller $I(t)_{\text{A}}$ is, the better the atomic entanglement is. Fig. 4 shows the dependence of correlation variance combination for three atomic ensembles on the squeezing parameter $r$ of initial squeezed states and the mapping efficiency $\eta _{_{\text{M}}}$ with the storage time of 1000 ns. We can see that the combination is reduced with the increase of squeezing parameter $r$ and mapping efficiency $\eta _{_{\text{M}}}$. In our experiment, the squeezing parameter $r$ and the mapping efficiency $\eta _{_{\text{M}}}$ are 0.38 and 23%, respectively.
{width="8.6cm"}
**Quantum state transfer from stored entangled state of atomic spin waves to released optical submodes.** The control optical beams of read process enable to map quantum state from the atomic spin waves to released optical submodes. Using Eqs. (2) and (9), the amplitude (phase) quadratures $\hat{X}(\hat{P})(t)_{\text{L}j}$ of released optical submodes after a storage time $t$ are calculated. We have
$$\begin{aligned}
\hat{X}(t)_{\text{L}1} &=&\sqrt{\frac{\eta }{3}}\text{e}^{r}\hat{X}_{\text{S}%
1}^{(0)}+\sqrt{\frac{2\eta }{3}}\text{e}^{-r}\hat{X}_{\text{S}2}^{(0)}+\sqrt{%
1-\eta }\hat{X}_{\text{L}1}^{\text{vac}}\text{,} \\
\hat{P}(t)_{\text{L}1} &=&\sqrt{\frac{\eta }{3}}\text{e}^{-r}\hat{P}_{\text{S%
}1}^{(0)}+\sqrt{\frac{2\eta }{3}}\text{e}^{r}\hat{P}_{\text{S}2}^{(0)}+\sqrt{%
1-\eta }\hat{P}_{\text{L}1}^{\text{vac}}\text{,} \notag \\
\hat{X}(t)_{\text{L}2} &=&\sqrt{\frac{\eta }{3}}\text{e}^{r}\hat{X}_{\text{S}%
1}^{(0)}-\sqrt{\frac{\eta }{6}}\text{e}^{-r}\hat{X}_{\text{S}2}^{(0)}+\sqrt{%
\frac{\eta }{2}}\text{e}^{-r}\hat{X}_{\text{S}3}^{(0)}+\sqrt{1-\eta }\hat{X}%
_{\text{L}2}^{\text{vac}}\text{,} \notag \\
\hat{P}(t)_{\text{L}2} &=&\sqrt{\frac{\eta }{3}}\text{e}^{-r}\hat{P}_{\text{S%
}1}^{(0)}-\sqrt{\frac{\eta }{6}}\text{e}^{r}\hat{P}_{\text{S}2}^{(0)}+\sqrt{%
\frac{\eta }{2}}\text{e}^{r}\hat{P}_{\text{S}3}^{(0)}+\sqrt{1-\eta }\hat{P}_{%
\text{L}2}^{\text{vac}}\text{,} \notag \\
\hat{X}(t)_{\text{L}3} &=&\sqrt{\frac{\eta }{3}}\text{e}^{r}\hat{X}_{\text{S}%
1}^{(0)}-\sqrt{\frac{\eta }{6}}\text{e}^{-r}\hat{X}_{\text{S}2}^{(0)}-\sqrt{%
\frac{\eta }{2}}\text{e}^{-r}\hat{X}_{\text{S}3}^{(0)}+\sqrt{1-\eta }\hat{X}%
_{\text{L}3}^{\text{vac}}\text{,} \notag \\
\hat{P}(t)_{\text{L}3} &=&\sqrt{\frac{\eta }{3}}\text{e}^{-r}\hat{P}_{\text{S%
}1}^{(0)}-\sqrt{\frac{\eta }{6}}\text{e}^{r}\hat{P}_{\text{S}2}^{(0)}-\sqrt{%
\frac{\eta }{2}}\text{e}^{r}\hat{P}_{\text{S}3}^{(0)}+\sqrt{1-\eta }\hat{P}_{%
\text{L}3}^{\text{vac}}\text{.} \notag\end{aligned}$$
In this case of $g_{L1}^{\prime }=g_{L2}^{\prime }=g_{L3}^{\prime
}=g_{L}^{\prime }$, $I(t)_{L}$ in Eq. (3) can be calculated from Eq. (13):
$$I(t)_{\text{L}}=\frac{12\text{e}^{-2r}+2(g_{\text{L}}^{\prime }+2)^{2}\text{e%
}^{-2r}+4(g_{\text{L}}^{\prime }-1)^{2}\text{e}^{2r}}{24}\eta +(1+\frac{1}{4}%
g_{\text{L}}^{\prime })(1-\eta )\text{.}$$
The optimal gain factor ($g_{\text{L}}^{\prime \text{opt}}$) for read out process equals to:
$$g_{\text{L}}^{\prime \text{opt}}=\frac{2\eta \text{e}^{4r}-2\eta _{_{\text{M}%
}}^{\prime }}{3\text{e}^{2r}+\eta -3\eta \text{e}^{2r}+2\text{e}^{4r}\eta }%
\text{.}$$
Fig. 2 is obtained from Eq. (14), where the optimal gain in Eq. (15) is applied.
**Atomic ensemble.** The atomic energy levels of Rubidium used for quantum memory medium is illustrated in Fig. 3b. The collective coherence of the ground state $\left\vert 5S_{1/2},F=1\right\rangle $ and meta state $\left\vert 5S_{1/2},F=2\right\rangle $ is used to store nonclassical state of light. For balancing the storage efficiency and excess noises from atoms, the frequencies of both signal and control optical beams are detuned by $\Delta _{\text{s}}=700$ MHz (the detuning of signal optical beam from the transition between energy levels g and e) and $\Delta _{\text{c%
}}=700.5$ MHz (the detuning of control optical beam from the transition between energy levels m and e), respectively. The detuning is realized by two sets of double-pass 1.7 GHz acousto-optical modulators (AOM$_{1-2}$). Three rubidium vapour cells of 7.5 cm-long with 10 torr of neon buffer gas in a $\mu $ metal magnetic shielding are used as atomic media, where the neon buffer gas prevents thermal diffusion to increase the atomic coherence. The three rubidium atomic cells are heated to around 65 **$%
^{\circ }$C** in our experiment.
**Experimental time sequence.** In the beginning of each period, the laser is turned on for 2 ms by AOM$_{3}$ and split into three optical beams $\hat{a}_{\text{S1}}^{(0)}$, $\hat{a}_{\text{S2}%
}^{(0)}$ and $\hat{a}_{\text{S3}}^{(0)} $, which are used for input signals of DOPAs respectively for persistently locking the length of optical cavities. Then the input signals of DOPAs are turned off for 20000 ns by AOMs$_{3}$ to generate three squeezed vacuum states of light. Within the locking period the three entangled optical submodes are chopped into 500 ns pulses with AOMs$_{4-6}$, which are used for input submodes $\hat{a}(0)_{\text{S1}}$, $\hat{a}(0)_{\text{S2}}$ and $\hat{a}(0)_{%
\text{S3}}$ of three atomic ensembles. Once the signal pulses enter into the atomic cell, the control optical beams $\hat{a}_{\text{C1}}$, $\hat{a}_{\text{C2}}$ and $\hat{a}_{\text{C3}}$ are switched off by AOM$_{7}$ to complete the quantum storage. At an user-controlled moment (1000 ns for our experiment) within storage lifetime, the control optical beams $\hat{a}%
_{\text{C1}}$, $\hat{a}_{\text{C2}}$ and $\hat{a}_{\text{C3}}$ are turned on again by AOM$_{7}$ at to obtain three released optical pulses $\hat{a}(t)_{\text{S1}}$, $\hat{a}(t)_{%
\text{S2}}$ and $\hat{a}(t)_{\text{S3}}$.
Data Availability {#data-availability .unnumbered}
=================
The data that support the findings of this study are available from the corresponding author on request.
[99]{} Kimble, H. J. The quantum internet. *Nature* **453**, 1023-1030 (2008).
Hammerer, K., Sorensen, A. S. & Polzik, E. S. Quantum interface between light and atomic ensembles. *Rev. Mod. Phys.* **82**, 1041-1093 (2008).
Specht, H. *et al.* A single-atom quantum memory. *Nature* **473**, 190–193 (2011).
Facon, A. *et al.* A sensitive electrometer based on a Rydberg atom in a Schrodinger-cat state. *Nature* **535**, 262-265 (2016).
Simon, J., Tanji, H. , Ghosh, S. & Vuletic, V. Single-photon bus connecting spin-wave quantum memories. *Nature Phys*. **3**, 765-769 (2007).
Marino, A. M., Pooser, R. C., Boyer, V. & Lett, P. D. Tunable delay of Einstein–Podolsky–Rosen entanglement. *Nature* **457**, 859-862 (2009).
Hosseini, M., Sparkes, B. M., Campbell, G., Lam, P. K. & Buchler, B. C. High efficiency coherent optical memory with warm rubidium vapour. *Nature Commun.* **2**, 174 (2011).
Xu, Z. X. *et al.* Long Lifetime and High-Fidelity Quantum of photonic polarization qubit by lifting Zeeman degeneracy. *Phys. Rev. Lett.* **111**, 240503 (2013).
Chen, Y. H. *et al.* Coherent optical memory with high Storage efficiency and large fractional delay. *Phys. Rev. Lett.* **110**, 083601 (2013).
Parigi, V. *et al.* Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory. *Nature Commun.* **6**, 7706 (2015).
Langer, C. *et al.* Long-lived qubit memory using atomic ions. *Phys. Rev. Lett.* **95**, 060502 (2005).
Stute, A. *et al.* Tunable ion-photon entanglement in an optical cavity. *Nature* **485**, 482-485 (2012).
Hucul, D. *et al.* Modular entanglement of atomic qubits using photons and phonons. *Nature Phys.* **11**, 37-42 (2014).
Fiore, V. *et al.* Storing Optical information as a mechanical excitation in a silica optomechanical resonator. *Phys. Rev. Lett.* **107**, 133601 (2011).
Lee, H. *et al.* Spiral resonators for on-chip laser frequency stabilization. *Nature Commun.* **4**, 2468 (2013).
Riedinger, R. *et al.* Nonclassical correlations between single photons and phonons from a mechanical oscillator. *Nature* **530**, 313-316 (2016).
Kiesewetter, S., Teh, R. Y., Reid, M. D. & Drummond, P. D. Pulsed entanglement of two optomechanical oscillators. Preprint at http://arXiv.org/abs/1611.09903 (2016).
Flurin, E., Roch, N., Pillet, J., Mallet, F. & Huard, B*.* Superconducting quantum node for entanglement and storage of microwave radiation. *Phys. Rev. Lett.* **114**, 090503 (2015).
Clausen, C. *et al.* Quantum storage of photonic entanglement in a crystal. *Nature* **469**, 508-511 (2011).
Saglamyurek, E. *et al.* Broadband waveguide quantum memory for entangled photons. *Nature* **469**, 512-515 (2011).
Zhong, M. *et al.* Optically addressable nuclear spins in a solid with a six-hour coherence time. *Nature* **517**, 177-180 (2015).
Gao, W. B. *et al.* Observation of entanglement between a quantum dot spin and a single photon. *Nature* **491**, 426-430 (2012).
Chou, C. W. *et al.* Measurement-induced entanglement for excitation stored in remote atomic ensembles. *Nature* **438**, 828-832 (2005).
Yuan, Z-S. *et al.* Experimental demonstration of a BDCZ quantum repeater node. *Nature* **454**, 1098-1101 (2008).
Matsukevich, D. N. *et al.* Entanglement of remote atomic qubits. *Phys. Rev. Lett.* **96**, 030405 (2006).
Choi, K. S., Deng, H., Laurat, J. & Kimble, H. J. Mapping photonic entanglement into and out of a quantum memory. *Nature* **452**, 67-71 (2008).
Zhang, W. *et al.* Experimental realization of entanglement in multiple degrees of freedom between two quantum memories. *Nature Commun.* **7**, 13514 (2016).
Choi, K. S., Goban, A., Papp, S. B., Van Enk, S. J. & Kimble, H. J. Entanglement of spin waves among four quantum memories. *Nature* **468**, 412-416 (2010).
Julsgaard, B., Kozhekin, A. E. & Polzik, E. S. Experimental long-lived entanglement of two macroscopic objects. *Nature* **413**, 400-403 (2001).
Krauter, H. *et al.* Entanglement generated by dissipation and steady state entanglement of two macroscopic objects. *Phys. Rev. Lett.* **107**, 080503 (2011).
Ladd, T. D. *et al.* Quantum computers. *Nature* **464**, 45-53 (2010).
Sangouard, N., Simon, C., Riedmatten, H. D. & Gisin, N. Quantum repeaters based on atomic ensembles and linear optics. *Rev. Mod. Phys.* **83**, 33-80 (2011).
Yadsanappleby, H. & Serafini, A. Would one rather store squeezing or entanglement in continuous variable quantum memories. *Phys. Lett. A* **375**, 1864-1869 (2011).
Tikhonov, K. S., Golubeva, T. Y. & Golubev, Y. M. Entangled states of signal pulses in multimode quantum memory. *Opt. Spectrosc.* **118**, 773-780 (2015).
Jensen, K. *et al.* Quantum memory for entangled continuous-variable states. *Nature Phys.* **7**, 13-16 (2011).
Bachor, H. A. & Ralph, T. C. A Guide to Experiments in Quantum Optics 2nd edn (WILEY-VCH, 2004).
Su, X. L. *et al.* Experimental preparation of eight-partite cluster state for photonic qumodes. *Opt. Lett.* **37**, 5178-5180 (2012).
Jia, X. J. *et al.* Superactivation of Multipartite Unlockable Bound Entanglement. *Phys. Rev. Lett.* **108**, 190501 (2012).
Fleischhauer, M., Imamoglu, A. & Marangos, J. P. Electromagnetically induced transparency: Optics in coherent media. *Rev. Mod. Phys.* **77**, 633-673 (2005).
Fleischhauer, M. & Lukin, M. D. Dark-state polaritons in electromagnetically induced transparency. *Phys. Rev. Lett.* **84**, 5094-5097 (2000).
Cviklinski, J. *et al.* Reversible quantum interface for tunable single-sideband modulation. *Phys. Rev. Lett.* **101**, 133601 (2008).
Appel, J., Figueroa, E., Korystov, D., Lobino, M. & Lvovsky, A. I. Quantum memory for squeezed light. *Phys. Rev. Lett.* **100**, 093602 (2008).
Honda, K. *et al.* Storage and retrieval of a squeezed vacuum. *Phys. Rev. Lett.* **100**, 093601 (2008).
Jia, X. *et al.* Experimental realization of three-color entanglement at optical fiber communication and atomic storage wavelengths. *Phys. Rev. Lett.* **109**, 253604 (2012).
Wu, L. *et al.* Experimental generation of tripartite polarization entangled states of bright optical beams. *Appl. Phys. Lett.* **108**, 161102 (2016).
Ou, Z. Y. Efficient conversion between photons and between photon and atom by stimulated emission. *Phys. Rev. A* **78**, 023819 (2008).
Van Loock, P. & Furusawa, A. Detecting genuine multipartite continuous-variable entanglement. *Phys. Rev. A* **67**, 052315 (2003).
Ou, Z. Y, Pereira, S. F., Kimble, H. J. & Peng, K. C. Realization of the Einstein-Podolsky-Rosen Paradox for Continuous Variables. *Phys. Rev. Lett.* **68**, 3663 (1992).
Yokoyama, S. *et al.* Ultra-large-scale continuous-variable cluster states multiplexed in the time domain. *Nature Photon.* **7**, 982-986 (2013).
Roslund, J., Medeiros, R., Jiang, S., Fabre, C. & Treps, N. Wavelength-multiplexed quantum networks with ultrafast frequency combs. *Nature Photon.* **8**, 109-112 (2014).
Lobino, M., Kupchak, C., Figueroa, E. & Lvovsky. A. I. Memory for Light as a Quantum Process. *Phys. Rev. Lett.* **102**, 203601 (2009).
Phillips, N. B., Gorshkov, A. V. & Novikova, I. Light storage in an optically thick atomic ensemble under conditions of electromagnetically induced transparency and four-wave mixing. *Phys. Rev. A* **83**, 063823 (2011).
Lauk, N., O’Brien, C. & Fleischhauer M. Fidelity of photon propagation in electromagnetically induced transparency in the presence of four-wave mixing. *Phys. Rev. A* **88**, 013823 (2013).
Yang, S.-J., Wang, X.-J., Bao, X.-H. & Pan, J.-W. An efficient quantum light–matter interface with sub-second lifetime. *Nature Photon*. **10,** 381-384 (2016).
Saunders, D. J. *et al.* Cavity-enhanced room-temperature broadband Raman memory. *Phys. Rev. Lett.* **116**, 090501 (2016).
Vahlbruch, H., Mehmet, M., Danzmann, K. & Schnabel, R. Detection of 15 dB squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency. *Phys. Rev. Lett.* **117**, 110801 (2016).
Barrett, S. D., Rohde, P. P. & Stace, T. M. Scalable quantum computing with atomic ensembles. *New J. Phys.* **12**, 093032 (2010).
Fleischhauer, M., Imamoglu, A. & Marangos, J. P. Electromagnetically induced transparency: optics in coherent media. *Rev. Mod. Phys.* **77**, 633-673 (2005).
He, Q. Y., Reid, M. D., Giacobino, E., Cviklinski, J. & Drummond, P. D. Dynamical oscillator-cavity model for quantum memories. *Phys. Rev. A* **79**, 022310 (2009).
**Acknowledgments**
The authors acknowledge financial support from the Key Project of the Ministry of Science and Technology of China (Grant No. 2016YFA0301402), the Natural Science Foundation of China (Grants Nos. 11322440, 11474190, 11304190, 11654002), FOK YING TUNG Education Foundation, the Program for Sanjin Scholars of Shanxi Province.
**Author contributions**
X.J., H.W. and C.X. conceived the original idea. Z.Y., L.W., X.J. and K.P. designed the experiment. Z.Y., L.W., Y.L. and R.D. constructed and performed the experiment. Z.Y., X.J. and S.L. accomplished theoretical calculation and the data analysis. Z.Y., X.J., C.X. and K.P. wrote the paper. All the authors reviewed the manuscript.
**Additional information**
**Competing financial interests**: The authors declare no competing financial interests.
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---
abstract: 'In a recent article by Dickey, J. O., Marcus, S.L. and O. de Viron, 2011, the authors show evidences for correlations in the multi-decadal variability of Earth’s rotation rate \[i.e., length of day (LOD)\], the angular momentum of the core (CAM), and natural surface air temperature (SAT). Previous investigators have already found that the LOD fluctuations are largely attributed to core-mantle interactions and that the SAT appears strongly anti-correlated with the decadal LOD. As the above authors note, the cause of this common variability needs to be further investigated and studied. In fact, “since temperature cannot affect the CAM or LOD to a sufficient extent, the results favor either a direct effect of Earth’s core-generated magnetic field (e.g., through the modulation of charged-particle fluxes, which may impact cloud formation) or a more indirect effect of some other core process on the climate-or yet another process that affects both”. The main aim of the present research note is to further support the above results, using a proper continuous wavelet analysis, and to evidence some early detected time-period relations existing between LOD and SAT data.'
address: 'Mathematical and Physical Models, Enel Research, Pisa - Italy'
author:
- Stefano Sello
title: 'On the correlation between air temperature and the core Earth processes: Further investigations using a continuous wavelet analysis'
---
Introduction
============
We recall here some brief introductive considerations of the argument under study, as well reported in the reference article by Dickey, J. O., Marcus, S.L. and O. de Viron, 2011. The length of day (i.e., the time needed by Earth to make one full rotation) fluctuates around its mean 24-h value, over a broad range of periods. Earth rotation is observed with a very high accuracy from precise geodetic techniques such as very-long-baseline interferometry, which allows for the determination of the length of day (LOD) at the 0.02-ms level (less than 1 cm at the equator). Earth rotation data collected before 1975 are the results of traditional optical observations; thus uncertainties during this period are significantly larger than for current determinations. The observed fluctuation of the LOD can be interpreted, at seasonal time scales, in terms of angular momentum exchange between the solid Earth and the superficial fluid layers (i.e. the atmosphere and the ocean), without the need to invoke influences from the core. For the atmosphere, variations in global angular momentum are dominated by the effect of the winds (the wind term), with the moment-of-inertia changes (the pressure term) being an order of magnitude smaller (Hide and Dickey 1991; Rosen 1993). The ocean contributions, being yet smaller than the pressure term, are significant in the closure of the angular momentum balance (Marcus et al. 1998). On the other hand, at decadal and longer time scales (multi-decadal), the main source of rotational variation is the interaction between the mantle and core, as substantiated by the significant correlation between the low-degree zonal components of the magnetic field and the LOD (Jault and Le Mouel 1991). The decadal LOD variations (e.g., $\sim 4 $ ms around 1900) are too large in amplitude to be explained by the atmosphere ($\sim 1 $ ms in amplitude). The long-term global mean Earth surface air temperature (SAT) is significantly anti-correlated with decadal and longer LOD (e.g., Lambeck and Cazenave 1976). Variability at periods of 60-80 yr has been well established in LOD (e.g., Jault and Le Mouel 1991; Roberts et al. 2007) and in the core angular momentum (CAM), using both observational data (Zatman and Bloxham 1997; Dickey and de Viron 2009) and theoretical studies (Mound and Buffett 2007).
It is well known that there are no direct observations of the fluid core motions. However, geomagnetic fields, generated by the dynamo inside the fluid core, can be used to infer the motion field and its velocity by utilizing several basic assumptions and the induction equation. In fact, if the solid Earth-core system is assumed to be isolated, the changes of the solid-Earth angular momentum, and thus its rotation speed and the associated LOD, can be deduced from changes of the CAM. The main result of the above authors is that LOD, CAM and corrected (i.e. for anthropogenic effects) global mean air temperature show strong evidence of multi-decadal variability, with a period in the broad range: 60-80-yr. Oscillations in global temperatures with periods in the 65-70-yr range were originally reported by Schlesinger and Ramankutty (1994). Subsequent observational studies and simulations with coupled atmosphere-ocean models have found similar multi-decadal climatic modes, typically originating in the North Atlantic Ocean; however, the excitation source or sources of these oscillations have not been unambiguously identified (Knight 2009). The authors work suggests that the same core processes that are known to affect Earth’s rotation and magnetic field (Roberts et al. 2007) may also contribute to the excitation of such modes, possibly through geomagnetic modulation on regional as well as global scales (Usoskin et al. 2008). For more details see the full article by Dickey, J. O., Marcus, S.L. and O. de Viron, 2011.
Results
=======
Following the final comment of the authors: “Further work remains to be done, especially in linking common modes of variability between Earth’s subsystems and better describing the physical connections between them”, here we used, as starting point, the curves of Fig. 1 in the original article (see the reported Figure 1), considering the -LOD behavior and the corrected temperature series from GISS beginning in 1880 and ending 2002.9.
The authors main results are: “...the detection of a broadband variability centered at 78 yr (common variability ranges from 67 to 86 yr from SSA method). Oscillations in global temperatures with periods in the 65-70 yr are well known. Our work suggests that the same core processes that are known to affect Earth’s rotation and magnetic field may also contribute to the excitation of such modes, possibly through geomagnetic modulation of near-Earth charged particle fluxes that may influence cloud nucleation processes, and hence the planetary albedo, on regional as well as global scales.”
The main goal of the following wavelet analysis is to confirm the above global correlation between -LOD and Corrected Temperature (i.e. avoiding the anthropogenic component) at multi-decadal scales and furthermore to better investigate time-frequency correlations between the series in order to strictly correlate the processes involved, observing their modes temporal behaviors. For mathematical details on this wavelet analysis, see: G. Ranucci and S. Sello, 2007.
The following Figure 2 shows the wavelet analysis performed on the first series: -LOD data Note that the y-axis is shown in logarithmic scale in order to better see all the frequency range of interest involved (here selected from: 1 yr to: 121 yr). Black contour lines are boundaries of confidence regions at 95%.
Main results: we confirm a dominant and time-persistent frequency band in the periodicity range: 50-92 yrs, with a central peak at 65 yr, clearly visible up to 1900. Another persistent frequency band is near 34 yr. Further, other less persistent frequencies are: 33.8 - 13.8 - 8.41 - 3.7 - 3.0 and 1.7 yr. Some of these frequencies are well time localized and may be the object of further future investigations.
The following Figure 3 shows the wavelet analysis performed on the second series: Corrected T data. Note that the scales of the wavelet are the same as Figure 1 but the contour power levels here are rescaled to the different intensity energy level of the series. Black contour lines are boundaries of confidence regions at 95%.
Main results: we confirm again a dominant and time-persistent frequency band in the range: 60-120 yrs, with a central peak at 78.25 yr, clearly visible in the full time range considered. Another persistent frequency band is near 30.8 yr. Further, other less persistent frequencies are now: 19.8 - 9.0 - 4.7 yr. Some of these frequencies are again well time localized but with interesting differences, both in relative intensity and/or time-frequency position from those of the -LOD case. These may be the object of further future investigations. A qualitative analysis of the main common periodicities detected both in -LOD and Temp series, leads to the preliminary conclusion that there is a general shift towards longer periodicities (from -LOD to Temp data) together with a phase shift of few years, probably related to the time response of atmosphere to changes of LOD. The excitation of similar modes shifted in time seems to exclude some external processes that affect Earth’s core and climate simultaneously. In fact, the authors already note “an approximately 8-yr lag between the (negative) LOD and the corrected temperature; this lag agrees with the 8-yr lag between changes in Earth’s rotational speed and surface geomagnetic field perturbations found by Roberts et al. (2007)”.
The following Figure 4 shows the wavelet analysis performed on the third series: -CAM data Note that the scales of the wavelet are the same as Figure 1. Black contour lines are boundaries of confidence regions at 95%.
Main results: we confirm again a dominant and time-persistent frequency band in the long multi-decadal range. Here, due to the smoother and shorter CAM data, we were able to resolve, as maximum period, only the value: 55.7 yr, clearly visible in the full time range considered. Other significant periodicities are: 33.1 - 22.2 - 13.2 - 8.16 and 2.6 yr. Some of these frequencies are again well time localized but with visible differences, mainly corresponding to different time positions from those of the -LOD case. However, there are clear common periodicities, as expected, with -LOD data. These localized periodicities may be the object of further future investigations.
Conclusions
===========
The above wavelet analysis results support the idea, stated in the article by Jean O. Dickey, Steven L. Marcus and Oliver de Viron, that some sources of the (natural) global temperatures oscillations could be identified with the core processes that affect Earth’s rotation and magnetic field. In particular, the excitation of similar modes shifted in time seems to exclude the possibility of external (e.g. solar) processes that affects the core and climate simultaneously. It seems more likely that some core processes, that induce mode excitations of rotation and magnetic fields, drive global air temperatures oscillations too. Through the multi-resolution wavelet analysis we can see that not only multi-decadal oscillations but also some high energetic decadal and sub-decadal well time-localized periodicities appear somewhat correlated and thus we need further investigations in order to give a correct interpretation. The time-frequency approach, available e.g. with a proper wavelet analysis, could help further detailed investigations and interpretations on this interesting suggested relation.
References
==========
Dickey, J. O., Marcus, S.L. and O. de Viron:, 2011 Air Temperature and Anthropogenic Forcing: Insights from the Solid Eart, J. Climate, 24, 569-574.
Dickey, J. O., and O. de Viron, 2009: Leading modes of torsional oscillation within the earth’s core. Geophys. Res. Lett., 36, L15302, doi:10.1029/2009GL038386.
Hansen, J., and S. Lebedeff, 1987: Global trends of measured surface air temperature. J. Geophys. Res., 92, 13 345-13 372.
Hide, R., and J. O. Dickey, 1991: Earth’s variable rotation. Science, 253, 629-637.
Jault, D., and J. L. Le Mouel, 1991: Exchange of angular momentum between the core and the mantle. J. Geomag. Geoelectr., 43, 111-119.
Lambeck, K., and A. Cazenave, 1976: Long-term variations in the length of day and climatic change. Geophys. J. Roy. Astron. Soc., 46, 555-573.
Marcus, S. L., Y. Chao, J. O. Dickey, and P. Gegout, 1998: Detection and modeling of nontidal oceanic effects on Earth’s rotation rate. Science, 281, 1656-1659. Mound, J., and B. Buffett, 2007: Viscosity of the earth’s fluid core and torsional oscillations. J. Geophys. Res., 112, B05402, doi:10.1029/ 2006JB004426.
Ranucci, G. and Sello, S., 2007: Search for periodicities in the experimental solar neutrino data: A wavelet approach, Phys. Rev. D, 75, 7.
Roberts, P. H., Z. J.Yu, and C. T. Russell, 2007: On the 60-year signal from the core. Geophys. Astrophys. Fluid Dyn., 101, 11-35.
Rosen, R. D., 1993: The axial momentum balance of Earth and its fluid envelope. Surv. Geophys., 14, 1-29.
Schlesinger,M. E., and N. Ramankutty, 1994: An oscillation in the global climate system of period 65-70 years. Nature, 367, 723-726.
Knight, J. R., 2009: The Atlantic multidecadal oscillation inferred from the forced climate response in coupled general circulation models. J. Climate, 22, 1610-1625.
Usoskin, I. G., M. Korte, and G. A. Kovaltsov, 2008: Role of centennial geomagnetic changes in local atmospheric ionization. Geophys. Res. Lett., 35, L05811, doi:10.1029/2007GL033040.
Zatman, S., and J. Bloxham, 1997: Torsional oscillations and the magnetic field within the earth’s core. Nature, 388, 760-763.
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abstract: 'We continue the work by Aschbacher, Kinyon and Phillips \[AKP\] as well as of Glauberman \[Glaub1,2\] by describing the structure of the finite Bruck loops. We show that a finite Bruck loop $X$ is the direct product of a Bruck loop of odd order with either a soluble Bruck loop of $2$-power order or a product of loops related to the groups $PSL_2(q)$, $q= 9$ or $q \geq 5$ a Fermat prime. The latter possibility does occur as is shown in \[Nag1, BS\]. As corollaries we obtain versions of Sylow’s, Lagrange’s and Hall’s Theorems for loops.'
author:
- 'B. Baumeister, A. Stein'
title: 'The finite Bruck Loops[^1]'
---
Introduction
============
Let $(X,\circ)$ be a finite loop; that is a finite set together with a binary operation $\circ$ on $X$, such that there exists an element $1 \in X$ with $1 \circ x = x \circ 1 =x$ for all $x \in X$ and such that the left and right translations $$\lambda_x : X \to X ,~ y \mapsto x \circ y,\quad \rho_y : X \to X,~ x \mapsto x \circ y$$ are bijections. Loops can be thought of as groups without associativity law.
Given a loop $X$, let $G:= \langle \rho_x: x \in X \rangle \le {~{\rm Sym}}(X)$, the so called [*enveloping group of $X$*]{}. The set $K:=\{ \rho_x: x \in X \}$ is a transversal to $H:={{\rm Stab}}_G(1)$ and $(G,H,K)$ is called the [*Baer envelope of $X$*]{}. This connection between loops and transversals in groups goes back to Baer [@B], see Section 2. In \[Asch\] Aschbacher started the study of loops using group theory \[Asch\], which turned out to be a very powerful tool \[Asch, AKP, Nag1, BS\].
Though loops are a generalisation of groups, general loops can be very wild due to the missing associativity: Left-and right inverses may not be identical, powers of elements may not be definable in the usual way and many loops without proper subloops besides the cyclic groups of prime order exist.
In search for natural restrictions on loops, Bol discovered in [@Bol] the following identity, which today is known as the [*(right) Bol identity*]{}: $$\quad ((x \circ y) \circ z) \circ y = x \circ ((y \circ z) \circ y) \mbox{ for all }~x,y,z \in X.$$
A loop is called a (right) [*Bol loop*]{}, if it satisfies the above identity. Bol himself showed, that this generalisation of associativity is quite natural, but that groups are not the only examples of Bol loops.
One consequence of the identity is, that the subloop generated by one element is a (cyclic) group. Therefore powers and inverses of elements are well defined.
Examples of Nagy \[Nag2\] however imply, that general Bol loops may still be quite wild: While groups of odd order are soluble due to work of Feit and Thompson, simple Bol loops of odd non-prime order exist. Furthermore there are noncyclic simple Bol loops, which occur from transversals in soluble groups.
A natural further restriction is the following identity known as the [*automorphic inverse property*]{} AIP: $$\quad (x \circ y)^{-1} = x^{-1} \circ y^{-1} \mbox{ for all } x,y \in X.$$
This identity implies, that the inverse map $\iota: X \to X, x \mapsto x^{-1}$ is an automorphism of the loop. Bol loops with AIP are called [*Bruck loops*]{} and generalise abelian groups. In the literature Bruck loops occur also under other names, such as $K$-loops \[Kreuz, Kiech\] or gyrocommutative gyrogroups \[Ung\].
Glauberman showed in [@GG1] and [@GG2], that Bruck loops of odd order behave very well: These loops are soluble and allow generalisations of many theorems of group theory. His famous $Z^\ast$-theorem was originally a byproduct of this work.
Then, forty years later, Aschbacher, Kinyon and Phillips showed that the following holds in finite Bruck loops [@AKP]:
- Elements of 2-power order and elements of odd order commute in a more general sense, see Theorem 2 of \[AKP\].
- Bruck loops are a central product of a subloop of odd order and a subloop generated by elements of 2-power order.
- Simple Bruck loops are of 2-power exponent.
- The structure of minimal simple Bruck loops is very restricted.
This leaves the Bruck loops of 2-power exponent to be studied. Notice, that in the simplest case of Bruck loops of exponent 2, the automorphic inverse property is already a consequence of the Bol identity and the exponent 2 assumption.
In [@A] Aschbacher gave powerful restrictions on the structure of minimal simple Bol loops of exponent 2. Using the restrictions given in Aschbacher’s paper, Nagy and independently Baumeister and Stein found a simple Bol loop of exponent 2 and size 96 in April 2007 \[Nag1, BS\]. Furthermore Nagy produced an infinite sequence of simple Bol loops of exponent 2 \[Nag1\]. A bit later a simple Bruck loop of exponent 4 and size 96 was found by Baumeister and Stein \[BS\].
Thus the weakening of the associativity law produces lots of generalised elementary abelian groups, which are simple Bruck loops and live in non-soluble groups. Recall, that Aschbachers paper [@A] and its generalisations in [@AKP] restrict only the structure of minimal simple loops. Suddenly the question arose, whether the class of Bruck loops is maybe as wild as the class of general Bol loops.
In this paper we determine the structure of a finite Bruck loop showing that the structure of a finite Bruck loop is not as wild as suspected. The definition of a loop envelope and a twisted subgroup is given in the next section. Recall that for $G$ a group $O(G)$ is the biggest normal subgroup of $G$ of odd order and that $O^{2^\prime}(G)$ is the smallest normal subgroup of $G$ such that $G/O^{2^\prime}(G)$ is a group of odd order.
\[direct product\] Let $X$ be a finite Bruck loop. Then the following holds.
- $X = Y \times Z$ where $Y$ is a subloop with $|Y|$ odd and $Z$ a subloop of 2-power exponent.
- Let $G$ be the enveloping group of $X$. Then $G = O(G) \times O^{2^\prime}(G)$.
- A loop envelope $(G,H,K)$ of $Z$ where $H$ acts on $K$ and $K$ is a twisted subgroup consisting of $2$-power elements satisfies the following.
- $\overline{G} =G/O_2(G) \cong D_1 \times D_2 \times \cdots \times D_e$ with $D_i \cong {{\rm PGL}}_2(q_i)$ for $q_i\ge 5$ a Fermat prime or $q_i=9$ and $e$ a non-negative integer,
- $D_i \cap \overline{H}$ is a Borel subgroup in $D_i$,
- $F^\ast(G) = O_2(G)$,
- $ \overline{K}$ is the set of involutions in $\overline{G}\setminus{\overline{G}^\prime}$.
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- Notice, that it may be $e = 0$ in which case $G$ is a $2$-group and $X$ a soluble loop!
- If $(G,H,K)$ is the Baer envelope of $Z$, then it satisfies the assumptions required in the theorem.
To prove Theorem \[direct product\] we also show that if $X$ is a finite Bruck loop with loop envelope $(G,H,K)$, then $O_2(G)$ is the group to a subloop of $X$.
A direct consequence of the theorem is the following.
\[characterisation soluble\] Let $X$ be a finite Bruck loop with enveloping group $G$. Then $X$ is soluble if and only if $G= O(G) \times O_2(G)$.
Now not only the class of Bruck loops is understood better, but also the more general class of Bol-$A_r$-loops.
\[Ar-loops\] Let $X$ be a Bol-$A_r$-loop. Then there is a normal subloop $Y$ of $X$ which is a group such that $X/Y$ is as described in Theorem \[direct product\].
As a group theoretic corollary we obtain:
Let $G$ be a finite group and $H \le G$, such that there is a transversal $K$ to $H$ in $G$ which is the union of $1 \in G$ and $G$-conjugacy classes of involutions. If $G= \langle K \rangle$, then $(G,H,K)$ is a loop envelope to a Bruck loop of exponent $2$ with $H$ acting on $K$. Therefore, Theorem \[direct product\](c) describes $G$, $H$ and $K$.
Moreover, we show Sylow’s Theorem for the prime $2$.
\[Sylow2\] \[Sylow’s Theorem\] Let $X$ be a finite Bruck loop.
- There is a subloop $P$ of $X$ such that $|P|=|X|_2$.
- All subloops of $X$ of size $|X|_2$ are conjugate under $H$, the group of inner automorphisms of $X$.
- If $Y\le X$ with $|Y|$ a power of $2$, then there is an $h \in H$ such that $Y \le P^h$.
In fact, Theorems 12 and 14 of \[Glaub2\] as well as Theorem \[direct product\] yield if $p$ is an odd prime which divides the order of $X$ and which does not divide $q+1$ for any Fermat prime $q$ or $q = 9$, then there is a subloop $P$ of $X$ such that $|P|=|X|_p$.
We also get Lagrange’s Theorem for Bruck loops.
\[Lagrange\] \[Lagrange’s Theorem\] Let $X$ be a finite Bruck loop and $Y \le X$ a subloop. Then $|Y|$ divides $|X|$.
Finally, the Theorem of Hall holds as well:
\[Hall\] \[Hall’s Theorem\] Let $X$ be a finite Bruck loop and let $\Pi$ be the set of primes dividing the order of $X$. Then $X$ is soluble if and only if there is a Hall $\pi$-subloop in $X$ for every subset $\pi$ of $\Pi$.
There is even a stronger version of that theorem:
\[Hall2\] Let $X$ be a finite Bruck loop. Then $X$ is soluble if and only if there is a Sylow subloop in $X$ for every prime dividing $|X|$.
The organisation of the paper is as follows. In the next section we recall the relevant definitions and the notation. Then in the third section we collect our previous results which will be needed in the proofs of the theorems. In Section 4 we prepare the proof of Sylow’s Theorem for the special case of Bruck loops of $2$-power exponent by calculating the number of elements in the intersection of $K$ with a Sylow $p$-subgroup. Moreover, we prove Theorems \[Sylow2\] and \[Hall\] in the special case of Bruck loops of $2$-power exponent in that section. Finally Theorems \[direct product\], ..., \[Hall2\] will be shown in full generality in the last section.
Definitions and Notation
========================
We follow the notation of Aschbacher [@A] and [@AKP]. Please keep in mind that in this paper we study exclusively finite loops.
Baer observed that loops can be translated into the language of group theory [@B]. This translation is as follows. Let $X$ be a loop and let $$\rho: X \to Sym(X),~~~x \to \rho_x.$$ Define $$G := {~{\rm RMult}}(X) := \langle \rho_x~|~ x \in X\rangle \le Sym(X)$$ $$H:={{\rm Stab}}_G(1),~\mbox{where}~1 \in X,~\mbox{and}~K:=\{ \rho_x~|~ x \in X \}.$$
Then
1. $1 \in K$ and $K$ is a transversal to all conjugates of $H$ in $G$.
2. $H$ is core free.
3. $G = \langle K \rangle $.
The group $G$ is called the [*enveloping group of $X$*]{} (or right multiplication group) and the triple $(G,H,K)$ the [*Baer envelope of the loop*]{}. Baer also observed that whenever $(G,H,K)$ is a triple with $G$ a group, $H \le G$ and $K \subseteq G$ satisfying condition (1), then we get a loop on $K$ by setting $x \circ y = z$, $x,y \in K$ whenever $z$ is the element in $K$ such that $Hxy = Hz$. This loop is called the [*loop related to $(G,H,K)$*]{}.
The triple $(G,H,K)$ with $G$ a group, $H \le G$ and $K \subseteq G$ is called a [*loop folder*]{}, [*faithful loop folder*]{} or [*loop envelope*]{} if (1), (1) and (2) or (1) and (3) hold, respectively. In general there are many different loop folders to a given loop.
If $X$ is a Bol loop and $(G,H,K)$ the Baer envelope of $X$, then $K$ is a [*twisted subgroup*]{}, that is $1 \in K$ and whenever $x,y \in K$, then $x^{-1}$ and $xyx$ is in $K$. If, moreover, $X$ is a Bruck loop, then $H$ acts on $K$ by conjugation, \[Asch, 4.1\]. A [*Bruck folder*]{} is a loop folder $(G,H,K)$, if the following holds
- $K$ is a twisted subgroup
- $H$ acts on $K$ by conjugation.
We say that a Bruck folder is a [*BX2P-folder*]{}, if also
- The elements in $K$ are of $2$-power order
If $(G,H,K)$ is a [*BX2P-folder*]{}, then the loop related to it is a Bruck loop of $2$-power exponent \[BSS\]. Moreover, notice that the subgroup $H$ of the Baer envelope induces automorphisms on $X$ in a Bruck loop. These are called the [*inner automorphisms of $X$*]{}.
Subloops, homomorphisms, normal subloops, factor loops and simple loops are defined as usual in universal algebra: A [*subloop*]{} is a nonempty subset which is closed under loop multiplication.
[*Homomorphisms*]{} are maps between loops which commute with loop multiplication. The map defines an equivalence relation on the loop, such that the product of equivalence classes is again an equivalence class. [*Normal subloops*]{} are preimages of 1 under a homomorphism and therefore subloops. A normal subloop defines a partition of the loop into blocks (cosets), such that the set of products of elements from two blocks is again a block. Such a construction gives factor loops as homomorphic images with the block containing 1 as the kernel. [*Simple loops*]{} have only the full loop and the 1-loop as normal subloops.
For instance if $(G,H,K)$ is a loop folder defining a loop $X$ and $G_0$ a normal subgroup of $G$ which contains $H$, then $(G_0, H, G_0 \cap K)$ is a loop folder to a normal subloop $X_0$ of $X$.
A [*subfolder*]{} $(U,V,W)$ is a loop folder with $U \le G$, $V \le U \cap H$ and $W \subseteq U \cap K$. The loop to a subfolder of a loop folder is a subloop of the loop to the loop folder.
Finally we recall the definition of a soluble loop given in \[Asch\]. A loop $X$ is [*soluble*]{} if there exists a series $1 = X_0 \leq \cdots \leq X_n= X$ of subloops with $X_i$ normal in $X_{i+1}$ and $X_{i+1}/X_i$ an abelian group.
Let $\pi$ be a set of primes. A natural number $n$ is a [*$\pi$-number*]{} if $n =1$ or $n$ is the product of powers of primes in $\pi$. Assume that $X$ is a loop such that every element of the loop generates a group. We say that $X$ is a $\pi$-loop, if the order of $X$ is a $\pi$-number. Notice that this definition is different from the one given in \[Glaub1\]. For loops of odd order these two concepts coincides (see \[Glaub1, p. 394, Corollary 2\]), but not for loops of even order (see the Aschbacher loop in \[BS\]).
In order to distinguish the two concepts we propose to use the following notations: A [*local $\pi$-loop*]{} is a loop such that the orders of the elements are all $\pi$-numbers and a [*global $\pi$-loop*]{} is a loop such that the order of the loop is a $\pi$-number.
Then there are local $2$-loops which are not global $2$-loops, see \[BS\].
We say that a subloop $Y$ of $X$ is a [*$\pi$-Hall subloop*]{}, if $|Y|_\pi = |X|_\pi$.
Previous results
================
In the following let $(G,H,K)$ be a loop folder. We can see from the structure of a subgroup $U$ of $G$, if it defines a subloop.
\[subfolders\] \[BSS, 2.1, 2.2\] A subgroup $U \le G$ gives rise to a subfolder $(U,V,W)$, if and only if $U = (U \cap H)(U \cap K)$. Then $V = U \cap H$ and $W = U \cap K$. In particular, subgroups of $G$ which contain either $H$ or $\langle K \rangle$ give rise to subfolders.
\[BruckFolder\]\[BSS, 2.15 (2) - (4)\] Let $(G,H,K)$ be a Bruck folder. Then the following holds.
1. There exists a unique $\tau \in {{\rm Aut}}(G)$ with $[H,\tau]=1$ and $k^\tau = k^{-1}$ for all $k \in K$.
2. The set $\Lambda= \tau K \subseteq {{\rm Aut}}(G)$ is $G$-invariant.
3. Subfolders and homomorphic images are Bruck folder.
Notice, that \[BruckFolder\](3) implies that subloops of Bruck loops are again Bruck loops. In order to prove Sylow’s Theorem we will work in a group slightly bigger than $G$.
\[Lambda\]
Let $(G,H,K)$ be a Bruck folder and $\tau \in {{\rm Aut}}(G)$ the automorphism introduced in \[BruckFolder\](1). Then let $${\bf G^+} := G\langle \tau \rangle,$$ the semidirect product of $G$ with $\tau$, $${\bf H^+}:= H \langle \tau \rangle \le G^+~\mbox{and}~
{\bf \Lambda }:= \tau K \subseteq G^+.$$ By \[BruckFolder\](1) and (2) $\Lambda$ is a $G^+$-invariant set of involutions.
The following are powerful facts.
\[noHinvert\]\[BSS, 2.17, 3.3\] Let $(G,H,K)$ be a BX2P-folder and let $\overline{G}=G/O_2(G)$. Then
- $k^2 \in O_2(G)$ for all $k$ in $K$.
- $1 \in \overline{K}$ and $\overline{K}$ is a union of $\overline{G}$-conjugacy classes.
- Let $g \in G$ and $h \in H$. If $(h^g)^k = (h^g)^{-1}$ for some $k$ in $K$, then $h^2 = 1$.
We already have some information on soluble Bruck loops, see also \[AKP, Corollary 4\].
\[soluble\_loops\] \[BSS, 3.8, 3.9, 3.10\] Let $(G,H,K)$ be a BX2P-envelope to a Bruck loop $X$ of $2$-power exponent. Then the following holds
- $X$ is soluble if and only if $|X|$ is a power of $2$.
- If $X$ is soluble, then $G$ is a $2$-group.
- If $G = O_2(G)H$, then $X$ is soluble.
In \[BSS\] we introduced the concept of passive groups.\
[**Definition**]{} A finite nonabelian simple group $S$ is called [*passive*]{}, if whenever $(G,H,K)$ is a BX2P-folder with $$F^\ast(G/O_2(G)) \cong S,$$ then $G = O_2(G) H$.\
In that case consequently the loop to $(G,H,K)$ is of 2-power size and soluble by \[soluble\_loops\](3). The main theorems of \[BSS\], respectively \[S\] are the following.
\[EnvelopeGroups\] \[BSS, Theorem 1\] Let $(G,H,K)$ be a loop envelope to a Bruck loop of exponent $2$ such that $K$ is a twisted subgroup, such that $H$ acts on $K$ and such that $K$ consists of elements of $2$-power order. Moreover, assume that every non-abelian simple section of $G$ is either passive or isomorphic to ${{\rm PSL}}_2(q)$ for $q=9$ or $q \ge 5$ is a Fermat prime. Then the following holds
- $\overline{G}:= G/O_2(G) \cong D_1 \times D_2 \times \cdots \times D_e$ with $D_i \cong {{\rm PGL}}_2(q_i)$ for $q_i\ge 5$ a Fermat prime or $q_i=9$ and $e$ a non-negative integer,
- $D_i \cap \overline{H}$ is a Borel subgroup in $D_i$,
- $F^\ast(G) = O_2(G)$,
\[MainClassificationTheorem\] \[S, Theorem 1\] Let $S$ be a non-abelian simple group, then $S$ is passive or isomorphic to ${{\rm PSL}}_2(q)$ for $q=9$ or $q \ge 5$ a Fermat prime.
It follows that the order of a Bruck loop of exponent $2$ is very restricted:
\[order\] Let $X$ be a Bruck loop of $2$-power exponent. Then $$|X| = 2^{a}\prod_{i=1}^e (q_i+1)$$ for some $e \in {\mathbb{N}}\cup \{0\}$ and $q_i = 9$ or a Fermat prime. Moreover, $|X|_2 = 2^{a+e}$. If $(G,H,K)$ is the Baer envelope of $X$, then $2^a = |O_2(G):O_2(G) \cap H|$.
Bruck loops of 2-power exponent
===============================
In this section $(G,H,K)$ will always be a BX2P-envelope to a non-soluble Bruck loop.
By the definition of a BX2P-envelope it follows that $(G,H,K)$ is a loop envelope to a Bruck loop of exponent $2$ such that $H$ acts on $K$, such that $K$ is a twisted subgroup and such that the elements in $K$ are of $2$-power order. As by Theorem \[MainClassificationTheorem\] every non-abelian simple non-passive group is isomorphic to ${{\rm PSL}}_2(q)$ for $q=9$ or $q \ge 5$ a Fermat prime, Theorem \[EnvelopeGroups\] yields
1. $\overline{G}:= G/O_2(G) \cong D_1 \times D_2 \times ... \times D_e$ $~\mbox{with}~D_i \cong {{\rm PGL}}_2(q_i)$ and $q_i=9$ or a Fermat prime $q_i \ge 5$.
2. $D_i \cap \overline{H}=:B_i$ is a Borel subgroup of $D_i$.
Let $\pi_i$ be the projection of $\overline{G}$ onto $D_i$. We first study $\overline{H}$ and $\overline{K}$ in more detail.
\[HisBorel\]
1. $\overline{H}= \prod\limits_{i=1}^e B_i$.
2. If $\overline{k} \in \overline{K}$ and $1 \le i \le e$, then $\pi_i(\overline{k})$ is either $1$ or an involution in $D_i\setminus{D_i^\prime}$.
By Theorem \[EnvelopeGroups\] $$B:=\prod\limits_{i=1}^e B_i \le \overline{H}.$$ The involutions in $D_i^\prime$ invert elements of odd order which are conjugate to elements in $B_i$. Therefore, \[noHinvert\](2) and (3) imply (2).
Next we aim to show $\overline{H}= B$. As $B_i$ is a maximal subgroup of $D_i$, it follows $\pi_i(\overline{H}) = B_i$ or $D_i$, for $1 \le i \le e$.
Assume that $B < \overline{H}$. Then $\pi_j(\overline{H}) = D_j$ for some $1 \le j \le e$. This implies $\langle B_j^{\overline{H}} \rangle = D_j$ is a subgroup of $\overline{H}$. Let $\overline{k}$ be an element in $\overline{K}$ and consider $\pi_j(\overline{k})$. If $\pi_j(\overline{k}) \ne 1$, then $\pi_j(\overline{k})$ inverts some element of odd prime order in $D_j$ by Baer-Suzuki. Then $\overline{k}$ inverts an element in $\overline{H}$ and therefore $k$ inverts an element in $H$ \[Asch, (8.1)(1)\]. This yields a contradiction to \[noHinvert\]. Therefore, $\pi_j(\overline{k})=1$ for all $\overline{k} \in \overline{K}$, which contradicts $\overline{G}= \langle \overline{K} \rangle$.
This shows $B = \overline{H}$.
Some subloops of Bruck loops of $2$-power exponent
--------------------------------------------------
Now we can prove, that $O_2(G)$ is a group to a subloop:
\[O2subloop\] $O_2(G) H \cap K = O_2(G) \cap K$ and $O_2(G)= (O_2(G) \cap H)(O_2(G) \cap K)$.
By \[HisBorel\], $O_2(\overline{H})=1$. By \[subfolders\] the subgroup $O_2(G)H$ defines a subloop, which is soluble by \[soluble\_loops\] (3). Therefore $O_2(G)H = H(O_2(G)H \cap K)$ and $\langle O_2(G)H \cap K \rangle$ is a $2$-group by \[soluble\_loops\] (2), which yields $\langle O_2(G)H \cap K \rangle \le O_2(O_2(G)H) = O_2(G)$. Hence $O_2(G)H \cap K = O_2(G)\cap K$ and the Dedekind identity implies the statement.
Then application of Lemma \[subfolders\] shows that $O_2(G)$ is a group to a subloop. There are lots of other subloops: Let $I:= \{1,2,...,e\}$ and let $G_J$ be the full preimage of $\prod\limits_{j \in J} D_j$ for $J \subseteq I$.
\[manysubloops\] $G_J =(G_J \cap H)(G_J \cap K)$ for every $J \subseteq I$.
For $J=\emptyset$ this is \[O2subloop\] and for $J=I$ this is the loop folder property.
Let $x \in G_J$ and $x=h k$ with $h \in H$, $k \in K$. Let $l \in I-J$. As $\pi_l(x)=1$, we cannot have $\pi_l(\overline{k}) \ne 1$: Else by \[HisBorel\](2), $\pi_l(\overline{k})$ is some involution of ${{\rm PGL}}_2(q_l)$ outside ${{\rm PSL}}_2(q_l)$. But $\pi_l(\overline{H})=B_l$ and $B_l$ contains only involutions from ${{\rm PSL}}_2(q_l)$. So $\pi_l(\overline{k})=1$, thus $\pi_l(\overline{h})=1$ too. This implies the statement.
Preparations for Sylow’s Theorem
--------------------------------
Our next goal is to produce subloops to certain Sylow-2-subgroups $P$ of $G$. Therefore we calculate $|P^+ \cap \Lambda|$.
For every $J \subseteq I$, $\overline{G}$ has a unique conjugacy class ${\cal C}_J$ of elements such that whenever $t \in {\cal C}_J$, then $\pi_i(t)=1$ for $i \not\in J$ and $\pi_i(t)$ is some involution in $D_i\setminus{D_i^\prime}$ for $i \in J$. Moreover $$|{\cal C}_J|= \prod_{j \in J} q_j \frac{q_j-1}{2}.$$
This is immediate from the structure of $\overline{G}$. Recall, that for $q$ odd, the centraliser of an involution in ${{\rm PGL}}_2(q)$ is the normaliser of a torus of size either $q-1$ or $q+1$. In our case $q-1$ is divisible by 4, so inner involutions of ${{\rm PSL}}_2(q)$ have a centraliser of size $2(q-1)$ while outer involutions have centraliser size $2(q+1)$.
For $J \subseteq I$ let $t \in {\cal C}_J$. We denote by $O_2(G^+)t$ the full preimage of $t$ in $G^+$. The number $n_J:= |O_2(G^+)t \cap \Lambda|$ is well defined and independent of the choice of $t\in {\cal C}_J$. Then $$n_\emptyset = |O_2(G^+) \cap \Lambda| = |O_2(G) \cap K|= |O_2(G):O_2(G) \cap H|$$ by \[manysubloops\].
$$n_J = \frac{n_\emptyset \cdot 2^{|J|}}{ \prod\limits_{j \in J} (q_j-1) }$$
By \[manysubloops\] $G_J$ defines a subloop, so $|G_J: G_J \cap H|=|G_J \cap K| = |G_J^+ \cap \Lambda|$. As $$|G_J:G_J \cap H| = |\overline{G_J}:\overline{G_J} \cap \overline{H}| |O_2(G) :O_2(G) \cap H|,$$ we have $$|G_J:G_J \cap H|= n_\emptyset \prod_{j \in J} (q_j+1).$$
On the other hand $$|G_J^+ \cap \Lambda|= \sum_{L \subseteq J} n_L |{\cal C}_L|.$$ We therefore get a system of equations for the $n_J$.
Now the statement can be shown by induction on $|J|$. For example for $|J|=1$ we get the equation $n_\emptyset (q_j+1) = n_\emptyset + n_{\{j\}} \cdot q_j \frac{q_j-1}{2}$, which gives $n_{\{j\}} = \frac{2 n_\emptyset} {q_j-1} $. In general we have: $$n_\emptyset \prod_{j \in J} (q_j+1) = \sum_{L \subseteq J} n_L \prod_{j \in L} q_j \frac{q_j-1}{2}.$$ For $L \subseteq J$, $L \ne J$ we have the formula for $n_L$ by induction. On the other hand for any numbers $q_j, j \in J$ the equation $$\prod_{j \in J} (q_j +1) = \sum_{L \subseteq J} \prod_{j \in L} q_j$$ holds. After some calculation this gives exactly the formula for $n_J$.
\[SylowLoopExists\] Let $P \in {{\rm Syl}}_2(G)$. Then $$|P \cap K|=|P^+ \cap \Lambda|= 2^e n_\emptyset = |G:H|_2=|X|_2.$$ If $P \cap O_2(G)H \in {{\rm Syl}}_2(O_2(G)H)$, then $P= (P \cap H)(P \cap K)$.
Notice that $|P^+ \cap \Lambda|$ is independent of the choice of $P$, as $\Lambda$ is $G^+$-invariant.
Furthermore, $|P \cap K| = |P^+ \cap \Lambda|$ as $\tau \in O_2(G^+) \le P^+$.
We choose $P \in {{\rm Syl}}_2(G)$ with $P \cap O_2(G)H \in {{\rm Syl}}_2(O_2(G)H)$. Then also $P^+ \cap O_2(G^+)H^+ \in {{\rm Syl}}_2(O_2(G^+)H^+)$.
Let $i \in I$ and consider $P_i=\pi_i(\overline{P}) \in {{\rm Syl}}_2(D_i)$. Then $P_i$ is a dihedral group, $P_i \cap \overline{H}$ is a cyclic group of size $q_i-1$. The other coset of $P_i \cap \overline{H}$ in $P_i$ consists entirely of involutions, half of them involutions in $D_i^\prime$ and half of them in $D_i\setminus{D_i^\prime}$. As all involutions in $D_i\setminus{D_i^\prime}$ are conjugate in $\overline{G}$, it follows $$\pi_i(\overline{P^+}) \cap \overline{\Lambda} = 1 + \frac{q_i-1}{2},$$ where $1$ is a summand as $1 \in \overline{\Lambda}$. This shows for $J \subseteq I$: $$|\overline{P^+} \cap {\cal C}_J| = \prod_{j \in J} \frac{q_j-1}{2}.$$ As $$|P^+ \cap \Lambda| = \sum_{J \subseteq I} n_J |\overline{P^+} \cap {\cal C}_J|,$$ it follows that $$|P^+ \cap \Lambda| = \sum_{J \subseteq I} \frac{n_\emptyset 2^{|J|}}{\prod\limits_{j \in J} (q_j-1)} \prod_{j \in J} \frac{q_j-1}{2} = 2^{|I|} n_\emptyset = 2^e n_\emptyset.$$ By the Dedekind identity we have $O_2(G^+) (P^+ \cap H^+)= P^+ \cap O_2(G^+)H^+$. This gives $$\frac{|O_2(G^+)||P^+\cap H^+|}{|O_2(G^+) \cap P^+ \cap H^+|}= |P^+\cap O_2(G^+)H^+|.$$ Theorem \[EnvelopeGroups\] yields $$|P^+ \cap O_2(G^+)H^+| = \frac{|G^+|_2}{|G^+:O_2(G^+)H^+|_2} = \frac{|G^+|_2}{2^e}.$$ As $O_2(G) \leq P$ and $$|O_2(G)|/|O_2(G) \cap P \cap H|
= |O_2(G):O_2(G) \cap H| =n_\emptyset,$$ it follows that $$|P^+ \cap H^+|=|P \cap H|=\frac{|G|_2}{2^e n_\emptyset}$$ and therefore, $$|P:P \cap H|= 2^e n_\emptyset.$$ Hence $|P:P \cap H| = |P \cap K|$, which yields $P= (P \cap H)(P \cap K)$.
Moreover, $$|X|_2=|K|_2 = |G:H|_2 = |G:O_2(G)H|_2 |O_2(G)H:H|_2$$ $$= 2^e |O_2(G):O_2(G) \cap H| = 2^e n_\emptyset.$$
Sylow’s theorem for Bruck loops of $2$-power exponent
-----------------------------------------------------
Next we show, that the subloops of size $|X|_2$ have some nice properties. Therefore, we need to recall some facts about ${{\rm PGL}}_2(q)$.
\[PGLlemma\] Let $Z \cong {{\rm PGL}}_2(q)$ with $q=9$ or $q\ge 5$ a Fermat prime. Let $B$ be a Borel subgroup of $G$ and ${\cal C}$ the class of involutions in $Z\setminus{Z^\prime}$.
1. $B$ has two orbits on ${{\rm Syl}}_2(Z)$: one orbit of size $q$ and one of size $\frac{|B|}{2}$.
2. If $P \in {{\rm Syl}}_2(Z)$, then either $P \cap B \in {{\rm Syl}}_2(B)$ or $|P \cap B|=2$.
3. Let $A \subseteq \{1 \} \cup {\cal C}$ and suppose that $D= \langle A \rangle$ is a $2$-group such that $D=(D \cap B)A$. Then there is a $Q \in {{\rm Syl}}_2(Z)$ such that $D \le Q$ and $Q \cap B \in {{\rm Syl}}_2(B)$.
Let $\Omega$ be the set of points of the projective line related to $Z$. Then $|\Omega| = q+1$ and $Z$ acts triply transitive on $\Omega$. Moreover, $B$ is the stabiliser in $Z$ of a point $a$ of $\Omega$ and every Sylow $2$-subgroup of $G$ is the setwise stabiliser of two points of $\Omega$.
It follows that $B$ has two orbits on the set of pairs: one consisting of the pairs containing $a$ and the other one consisting of those not containing $a$. Their length are $q$ and $q(q-1)/2$, respectively. This shows (1).
If $P \in {{\rm Syl}}_2(Z)$ fixes a pair in the first orbit, then $P \cap B \in {{\rm Syl}}_2(B)$. If $P$ fixes a pair in the second orbit, then $P \cap B$ fixes a point and a pair of points setwise and is therefore just an involution, which is (2).
As $D=(D \cap B)A$, it follows that $|D:D\cap B| = 2$. Thus $D\cap B$ fixes the point $a$ and $a^D$ is of length $2$. Let $Q$ be the stabiliser of $a^D$ in $G$. Then $Q \in {{\rm Syl}}_2(Z)$, $D \le Q$ and $Q \cap B \in {{\rm Syl}}_2(B)$, which is (3).
The following is fundamental for the proof of the $2$-Sylow Theorem.
\[goodSylow\] Let $(G,H,K)$ be a faithful BX2P-envelope and $U$ a $2$-subgroup of $G$ such that
- $U = \langle U \cap K \rangle$
- $U=(U \cap H)(U \cap K)$.
Then there is a Sylow-2-subgroup $Q$ of $G$ such that $U \le Q$ and $Q \cap O_2(G) H \in {{\rm Syl}}_2(O_2(G)H)$.
For fixed $1 \leq i \leq e$ let $$Z:= \pi_i(\overline{G}), D:=\pi_i(\overline{U}), B:=\pi_i(\overline{H})~\mbox{and}~A:=\pi_i(\overline{U} \cap \overline{K}).$$ By \[HisBorel\] $${\cal C}:=\pi_i(\overline{K}) -\{1 \}$$ is the class of involutions in $Z\setminus{Z^\prime}$. Moreover by the homomorphism property of $\pi_i$ it follows that $$\pi_i(\overline{U}) = \pi_i(\overline{U} \cap \overline{H}) \pi_i(\overline{U} \cap \overline{K}).$$ This yields, as $$\pi_i(\overline{U} \cap \overline{H}) \le \pi_i(\overline{U}) \cap \pi_i(\overline{H})= D \cap B,$$ that $D = (B\cap B)A$. Hence, $Z,B,A,D$ satisfy the assumptions of \[PGLlemma\](3). Therefore, \[PGLlemma\](3) implies that $\pi_i(U)$ is contained in a Sylow $2$-subgroup $Q_i$ of $\pi_i(\overline{G})$ and that $Q_i \cap \overline{H}$ is a Sylow $2$-subgroup of $\pi_i(\overline{H})$.
Let $Q$ be the preimage of $\prod\limits_{i \in I}Q_i$. Then $U \leq Q$ and $$Q \cap O_2(G) H \in {{\rm Syl}}_2( O_2(G) H)$$ as asserted.
\[SylowLoops\] Let $X$ be a finite Bruck loop of $2$-power exponent and $Y$ a soluble subloop of $X$.
1. Then there is a subloop $Z$ of $X$ such that $Y \le Z$ and $|Z|=|X|_2$.
2. All subloops of $X$ of size $|X|_2$ are conjugate under $H$.
If $Y$ is soluble, then $Y$ is a $2$-loop by \[soluble\_loops\](1). Let $(G,H,K)$ be a faithful BX2P-envelope to $X$. Then there is a subgroup $U$ of $G$ such that
- $U = \langle U \cap K \rangle$
- $U = (U \cap H)(U \cap K)$ by \[subfolders\]
- $U$ is a $2$-group by \[soluble\_loops\](2)
Hence by \[goodSylow\] there is a Sylow $2$-subgroup $Q$ of $G$ such that $U$ is contained in $Q$ and such that $Q \cap O_2(G)H \in Syl_2(O_2(G)H)$. Now \[SylowLoopExists\] and \[subfolders\] imply that $(Q,Q \cap H, Q \cap K)$ is a subfolder of our chosen folder. Let $Z$ the subloop of $X$ related to that subfolder. As $Q \cap O_2(G)H$ is a Sylow $2$-subgroup of $O_2(G)H$, the intersection $Q \cap H$ is a Sylow $2$-subgroup of $H$. As $Q = (Q \cap H)(Q \cap K)$ it follows that $|Q \cap K| = |X|_2$, which proves (1).
Let $Y_2$ be a subloop of $X$ of size $|X|_2$. Then $Y_2$ is soluble and therefore by (1) there is Sylow $2$-subgroup $P$ of $G$ such that $(P, P \cap H, P\cap K)$ is a subfolder to a subloop $Z_2$ which contains $Y_2$ and which is of order $|X|_2$. This shows that $Y_2 = Z_2$. Recall that $G/O_2(G) = D_1 \times \cdots \times D_e$ with $D_i \cong {{\rm PGL}}_2(q_i)$ and $e \geq 0$. Then, as $P \cap H$ is a Sylow $2$-subgroup of $H$ (see also \[PGLlemma\](2)), according to \[PGLlemma\](1) there is an element $h$ in $H$ which maps $Q$ onto $P$.
Now we can easily prove Theorem \[Sylow2\].\
[**Proof of Theorem \[Sylow2\] for Bruck loops of $2$-power exponent.**]{} As the set consisting of the $1$-element of $X$ is a soluble subloop of $X$, Corollary \[SylowLoops\] yields (1). (2) is the second statement of the corollary.
If $Y$ is a subloop of $X$ of $2$-power order, then $Y$ is soluble by \[soluble\_loops\](1). Therefore (3) follows from Corollary \[SylowLoops\] as well.
Lagrange’s theorem
------------------
Now we can prove Lagrange’s theorem for Bruck loops of $2$-power exponent.\
[**Proof of Theorem \[Lagrange\] for $X$ a Bruck loop of $2$-power exponent.**]{} First of all notice, that by \[BruckFolder\](3) $Y$ is a Bruck loop of $2$-power exponent. By \[SylowLoopExists\], we have $|Y|_2 \le |X|_2$: There is a subloop of $Y$ of size $|Y|_2$ by Theorem \[Sylow2\] in case of Bruck loops of $2$-power exponent, which is soluble by \[soluble\_loops\]. Let $U\le G$ be the $2$-group related to this subloop; so $|U \cap K| = |Y|_2$. As by \[SylowLoopExists\] $|P \cap K| = |X|_2$ for any Sylow-2-subgroup of $G$, $|Y|_2$ is a divisor of $|X|_2$.
Suppose $Y$ is nonsoluble. Then $|Y|_{2'} \ne 1$. There is a subgroup $U\le G$ such that $U =(U \cap H)(U \cap K)$, $U = \langle U \cap K \rangle$ and $|Y|=|U:U \cap H| = |U \cap K|$. By \[soluble\_loops\] $U$ is not a soluble group. In the following we apply Theorem \[EnvelopeGroups\] on $U$ and on $G$ and use the notation introduced there. The map $\theta: U \to G: u \mapsto O_2(G) u$ gives a homomorphism from $U$ into $\overline{G}$ and an injection from $U/(O_2(U)\cap O_2(G))$ into $\overline{G}$.
Assume there is $D_i \leq \overline{G}$ such that $\pi_i(\overline{U})$ is non-soluble and properly contained in $D_i$. Then by Theorem \[EnvelopeGroups\] $\pi_i(\overline{U}) \cong {{\rm PGL}}_2(5)$ and $D_i \cong {{\rm PGL}}_2(9)$. Elements of odd order from $U \cap H$ map to elements of odd order in $\overline{H}$, which yields a contradiction as $\pi_i(\overline{U\cap H}) \cong 5:4$ and $\overline{H}$ is a $\{2,3\}$-group. Hence components of $U/O_2(U)$ project surjectively onto components of $G/O_2(G)$. This implies together with \[order\] that $$|Y| = 2^a\prod_{j\in J}(q_j+1)~\mbox{where}~J~\mbox{ is a subset of }~\{1, \ldots , e\}.$$ Thus the odd part of $|Y|$ divides $|X|$.
As we already saw that $|Y|_2 \le |X|_2$, it follows that the order of $Y$ divides the order of $X$.
The finite Bruck loops
======================
In this section we prove the main theorems.\
[**Proof of Theorem \[direct product\].**]{} Let $X$ be a finite Bruck loop. Then according to \[AKP, Theorem 1\] $$X = O^{2^\prime}(X) * O(X),~\mbox{where}~Z:= O^{2^\prime}(X)$$ is the subloop generated by all $2$-elements of $X$ and $$Y:= O(X)$$ the largest normal subloop of $X$ of odd order. Notice, that the definition of the subloop $Z$ is different from that one in \[AKP\]. Let $(G,H,K)$ be the Baer-envelope of $X$ and set $$G_2:= O^{2^\prime}(G).$$ Then $G_2 = {\langle}\rho_x ~|~ x \in Z {\rangle}\leq {~{\rm Sym}}(X)$, see \[AKP, 6.1\]. Moreover, $G_1:= O^{2}(G)$ is the enveloping group of $O(X)$ and $G = G_1 *G_2$, see \[AKP, Proof of 6.1\]. Set $U:= G_1 \cap G_2$ and $T = Z \cap Y$. Then $U$ is a subgroup of $Z(G)$ and therefore, it acts semiregularly on $Z$.
Moreover, $Z/T$ is a Bruck loop of $2$-power exponent with enveloping group $G_2/V$ where $V \leq U$ is the enveloping group of $T$, see \[AKP\] Theorem 1 and \[Asch\] 2.6. In particular, $V$ is a subgroup of $Z(G_2)$ of odd order.
Set $\tilde{G}_2=G_2/V$. Then $\tilde{G}_2/O_2(\tilde{G}_2) \cong D_1 \times \cdots \cong D_e$ with $D_i \cong {{\rm PGL}}_2(q_i)$, where $q_i$ is a Fermat prime or $9$ by Theorems \[EnvelopeGroups\] and \[MainClassificationTheorem\]
We claim that $G_2$ splits over $V$. Clearly, $O_2(G_2)V$ splits over $V$. Therefore, $G_2$ splits over $V$ if and only if $\overline{G}_2:=G_2/O_2(G_2)$ splits over $\overline{V}$.
Moreover, $\overline{G}_2$ splits over $\overline{V}$ if and only if the full preimage $L_i$ of $D_i$ in $\overline{G}_2$ splits over $\overline{V}$, for $1 \leq i \leq e$. As $L_i/\overline{V} \cong {{\rm PGL}}_2(q_i)$ and as $\overline{V}$ is of odd order it follows that the extension splits or that $q_i = 9$ and $|L_i^\infty \cap \overline{V}| = 3$, see \[Atlas, p. XVI, Table 5\]. Assume the latter. Then every involution in $L_i\setminus{L_i^\prime}$ inverts $L_i^\infty \cap \overline{V}$, see the action of the automorphism $2_3$ of ${{\rm PSL}}_3(4)$ on the Schur-multiplier of order $3$ of ${{\rm PSL}}_3(4)$ in \[Atlas, p. 23 \], which contradicts $V \leq Z(G_2)$. This shows that $G_2$ splits over $V$.
Hence, as $G_2 = O^{2^\prime}(G)$, we get $V = 1$ as well as $T = 1$. Thus (a) and (b) follow. The fact that $T =1$, also implies that $Z$ is a Bruck loop of $2$-power exponent. Part (c) is a direct consequence of Theorem \[EnvelopeGroups\].\
[**Proof of Theorem \[Sylow2\].**]{} The assertion follows from Theorem \[direct product\] and the proof of Theorem \[Sylow2\] in the case of Bruck loops of $2$-power exponent.\
[**Proof of Theorem \[Lagrange\].**]{} Following Bruck, it is enough to show, that this condition holds already in simple Bruck loops, see \[Bruck, Chapter V, p. 93, Lemma 2.1\]. As simple Bruck loops are either of prime order or of $2$-power exponent, we get the result from the proof of Theorem \[Lagrange\] in the case of Bruck loops of $2$-power exponent.
A different proof of Theorem \[Lagrange\] without quoting Bruck is to apply Theorem \[direct product\], Corollary 4, p. 395, in \[Glaub1\] and the proof of Theorem \[Lagrange\] in the case of Bruck loops of $2$-power exponent.\
[**Proof of Theorem \[Hall\].**]{} Every finite Bruck loop of odd order is soluble, see Theorem 14 of \[Glaub2\], and contains therefore Hall $\pi$-subloops, Theorem 12 \[Glaub2\]. If $X$ is a soluble Bruck loop, then by Theorem \[direct product\] $X$ is the direct product of a Bruck loop of odd order and a Bruck loop of $2$-power order. Hence there is a $\pi$-subloop for every subset $\pi$ of $\Pi$.
Now assume that $X$ is a Bruck loop such that there is a $\pi$-subloop for every subset $\pi$ of $\Pi$. If $X$ is of odd order, then it is soluble \[Glaub2\]. Assume now that $O^{2^\prime}(X) \neq 1$. If $O^{2^\prime}(X)$ is not of $2$-power order, then there is $q_i$, $q_i = 9$ or $q_i \geq 5$ a Fermat prime such that a prime divisor $r\neq 2$ of $q_i+1$ divides $|O^{2^\prime}(X)|$, see Corollary \[order\]. As there is no Bruck loop of $r$-power order, see Theorem \[direct product\], it follows that $O^{2^\prime}(X)$ is of $2$-power order and therefore soluble by \[soluble\_loops\].\
[**Proof of Theorem \[Hall2\].**]{} The previous proof also shows Theorem \[Hall2\].
Open questions
--------------
There are still some open questions on Bruck loops:\
- For which $q$ exist $M$-loops and/or $N$-loops as defined in [@AKP] and [@A]? There are only known examples for $q=5$.
- Are there infinitely many Fermat primes ? This is number theory...
- What is the structure of simple Bruck loops in detail? There are known examples with two composition factors of type ${~{\rm Alt}}_5$. Are the nonabelian composition factors of $G$ pair wise isomorphic in a simple Bruck loop?
- Is there a way to get the structure of $O_2(G)$ under control in $M$-loops, $N$-loops and/or Bruck loops of 2-power exponent (For the definition of an $N$ and an $M$-loop see \[BSS\])?
[99]{}
M.Aschbacher, On Bol loops of exponent 2, [*J. Algebra*]{} [**288**]{} (2005), 99-136
M.Aschbacher, M.Kinyon, J.D.Phillips, Finite Bruck loops, [*Transactions of the AMS*]{} [**358**]{}, No.7 (2006), 3061-3075 J.H.Conway, R.T.Curtis, S.P.Norton, R.A.Parker, R.A.Wilson [*An ATLAS of finite groups*]{} Oxford University Press, 1985 R.Baer, Nets and groups, [*Trans. Amer. Math. Soc*]{} [**47**]{} (1939), 110-141
B.Baumeister, A.Stein, Self-invariant $1$-Factorizations of Complete Graphs and Finite Bol Loops of Exponent $2$, to appear in [*Beiträge zur Algebra und Geometrie*]{}
B.Baumeister, A.Stein, G.Stroth, On Bruck Loops of $2$-power Exponent, submitted
G.Bol [*Gewebe und Gruppen*]{} Math. Ann. [**114**]{} (1937), 414-431 R.H.Bruck, [*A survey of Binary Systems*]{}, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, Heft 20. Reihe: Gruppentheorie, Springer Verlag, Berlin etc 1958 G.Glauberman, On Loops of odd order, [*J. Algebra*]{} [**1**]{} (1964), 374-396 G.Glauberman, On Loops of odd order II, [*J. Algebra*]{} [**8**]{} (1968), 393-414 M.W.Liebeck, The classification of finite Moufang loops, [*Math. Proc. Cambridge Philos. Soc*]{} [**102**]{} (1987),33-47 H. Kiechle, [*The Theory of $K$-Loops*]{}, Lecture Notes in Mathematics 1778, Springer, Berlin, Heidelberg, New-York, 2002. A. Kreuzer, Inner mappings of Bruck loops, [*Math. Proc. Cambridge Philos. Soc.*]{} [**123**]{} (1998), 53-57 G.Nagy, [*A class of simple proper Bol loops*]{}, [*Preprint*]{} G.Nagy [*Finite simple left Bol loops*]{}
http://www.math.u-szeged.hu/~nagyg/pub/simple_bol_loops.html
A.Stein, On Bruck Loops of 2-power Exponent, II, Preprint 2009 A.A.Ungar, Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovectors Spaces Kluwer Academic Publishers, Doldrechts-Boston-London 2001
[^1]: This research is part of the project “Transversals in Groups with an application to loops” GZ: BA 2200/2-2 funded by the DFG
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abstract: 'Stars are powerful sources for weakly interacting particles that are produced by nuclear or plasma processes in their hot interior. These fluxes can be used for direct measurements (e.g. solar or supernova neutrinos) or the back-reaction on the star can be used to derive limits on new particles. We discuss two examples of current interest, the search for solar axions by the CAST experiment at CERN and stellar-evolution limits on the size of putative large extra dimensions.'
author:
- 'Georg G. Raffelt'
title: 'Stars and Fundamental Physics[^1]'
---
Introduction
============
Astrophysics and cosmology provide a natural testing ground for virtually any new idea in the area of elementary particle physics. Usually one may first think of the early universe or perhaps high-energy cosmic rays when searching for astrophysical arguments in favor or against a new particle-physics model. However, there are a number of interesting cases where the low energies available in stars are quite sufficient for rather useful and restrictive tests of high-energy physics phenomena.
The basic idea is very simple. Stars are powerful sources for weakly interacting particles such as neutrinos, gravitons, hypothetical axions, and other new particles that can be produced by nuclear reactions or by thermal processes in the hot stellar interior. The solar neutrino flux is now routinely measured with such precision that compelling evidence for neutrino oscillations has accumulated. The measured neutrino burst from supernova (SN) 1987A has been used to derive many useful limits. Even when the particle flux can not be measured directly, the absence of visible decay products, notably x- or $\gamma$-rays, can provide important information. The properties of stars themselves would change if they lost too much energy into a new channel. This “energy-loss argument” has been widely used to constrain a long list of particles and particle properties. All of this has been extensively reviewed [@Raffelt1996; @Raffelt:1999tx] and is now widely appreciated among particle physicists [@Groom:in].
Therefore, instead of reviewing once more the general ideas I will rather focus on two topical examples of current interest that nicely illustrate the overall methods. One is the search for solar axions by the CAST experiment at CERN (Sec. 2). The other is the possibility that space-time has large extra dimensions. This hypothesis predicts a “tower” of graviton modes that can be produced in stars, notably in SN cores or neutron stars. The most restrictive limits on the size of the extra dimensions arises from the astrophysical arguments presented in Sec. 3. A brief summary and outlook is given in Sec. 4.
Axion-Like Particles
====================
Axions are hypothetical particles that are predicted in the context of a theoretical scheme to solve the CP problem of strong interactions [@Kim:ax; @Cheng:1987gp]. This is the problem that quantum chromodynamics (QCD) ought to violate the CP symmetry in that the neutron should have a large electric dipole moment, contrary to experimental evidence. This observation can be explained by a new symmetry, the Peccei-Quinn symmetry, that is spontaneously broken at some large energy scale $f_a$, the Peccei-Quinn scale or axion decay constant. Axions are the “almost” Nambu-Goldstone bosons of this new symmetry and as such nearly massless.
Phenomenologically one should think of axions as the neutral pion’s little brother. Model-dependent details aside, the axion’s mass and couplings are given by those of the $\pi^0$, scaled with $f_\pi/f_a$ where $f_\pi=93~{\rm MeV}$ is the pion decay constant. The axion decay constant $f_a$ is a free parameter and thus can be very large. Therefore, axions can be very light and very weakly interacting even though they are fundamentally a QCD phenomenon.
There are other plausible solutions of the strong CP problem. However, the Peccei-Quinn approach is particularly elegant and predicts something new—in the guise of axions it provides a handle for a possible experimental verification. Moreover, axions can play the role of the cosmic cold dark matter [@Kolb:vq]. Therefore, two fundamental problems would be solved by the existence of one new particle.
The experimental search for axions has focused on their predicted interaction with the electromagnetic field that would be of the form $${\cal L}_{a\gamma\gamma}=
{\textstyle\frac{1}{4}} g_{a\gamma\gamma}
F_{\mu\nu}\widetilde F^{\mu\nu}\,a=
-g_{a\gamma\gamma}{\bf E}\cdot{\bf B}\,a\,,$$ where $F$ is the electromagnetic field-strength tensor, $\widetilde F$ its dual, and ${\bf E}$ and ${\bf B}$ the electric and magnetic fields, respectively. The coupling strength is $$g_{a\gamma\gamma}
=\frac{\alpha}{2\pi f_a}\,C_\gamma,
\qquad C_\gamma=\frac{E}{N}-1.92\pm0.08\,,$$ where $E/N$ is the ratio of the electromagnetic and color anomalies, a model-dependent ratio of small integers. One popular case is the DFSZ model where $E/N=8/3$, another the KSVS model where $E/N=0$, but there are more general examples [@Kim:1998va].
Assuming $m_a=0.60~{\rm eV}\times 10^7~{\rm GeV}/f_a$ for the axion mass, Fig. \[fig:axgam\] shows $g_{a\gamma\gamma}$ as a function of $m_a$. The diagonal band marked “Axion Models” is somewhat arbitrarily delimited by the DFSZ and KSVZ models. The role of axions or axion-like particles is frequently assessed in the full two-dimensional $g_{a\gamma\gamma}$-$m_a$-space rather than the narrow band defined by conventional axion models, although this band remains the best-motivated location in this parameter space.
The electromagnetic interaction allows for the two-photon decay $a\to\gamma\gamma$ with a rate $\Gamma_{\rm decay}=g_{a\gamma\gamma}^2
m_a^3/64\pi$. This process is very slow if the axion mass is small and the coupling strength is weak. Therefore, it is more promising to consider the analogous process where one of the photons is virtual, i.e. an external electric or magnetic field. The $a\leftrightarrow\gamma$ conversion in the presence of an external $E$ or $B$ field is known as the Primakoff process; it was first considered for neutral pions half a century ago [@Primakoff1951].
If axions are the galactic dark matter, they can be detected in the laboratory by the “haloscope” technique [@Sikivie:ip]. One places a tunable high-Q microwave cavity in a strong magnetic field and measures the power output. If the resonance frequency matches $m_a$, the Primakoff-conversion can produce a measurable signal. Two pilot experiments [@Wuensch:1989sa; @Hagmann:tj] and a first full-scale search [@Asztalos:2001jk; @Asztalos:2001tf] exclude a range of coupling strength shown in Fig. \[fig:axgam\] that is marked “Haloscope.” The new generation of experiments in Livermore [@Asztalos:2001tf] and Kyoto [@Yamamoto:2000si] should cover the dashed area in Fig. \[fig:axgam\], perhaps leading to the discovery of axion dark matter.
![Limits on the axion-photon coupling $g_{a\gamma\gamma}$ as a function of axion mass $m_a$. The limits apply to any axion-like particle except for the “haloscope” search which assumes that axions are the galactic dark matter; the dotted region marks the projected sensitivity range of the ongoing full-scale searches. Limits for higher masses than shown here are reviewed in Ref. [@Masso:1997ru]. The light-grey region marks the foreseen CAST sensitivity.[]{data-label="fig:axgam"}](raffeltF1.eps){width="100.00000%"}
In a different region of masses and couplings axions are detectable with a related technique called the “helioscope” [@Sikivie:ip; @vanBibber:1988ge]. Thermal photons in the solar interior convert to axions by the Primakoff process in the microscopic electric fields of charged particles, producing a solar axion flux which peaks at energies of a few keV. If one views the Sun through a long dipole magnet, the axions partially back-convert into photons and become visible as x-rays at the far end of the magnet. A dedicated search for this effect by the Tokyo Axion Helioscope [@Inoue:2002qy] excludes the dark-grey region in Fig. \[fig:axgam\].
The conversion rate in the helioscope scales quadratically with the length $L$ and field-strength $B$ of the conversion region. Therefore, one can do much better in the new CAST project at CERN where a de-commissioned LHC test magnet is used as a “magnetic telescope” to search for solar axions [@Zioutas:1998cc; @CAST]. Mounted on a movable platform (Fig. \[fig:cast\]) allowing $\pm40^\circ$ horizontal and $\pm5^\circ$ vertical tracking, this instrument can achieve about 33 full days of alignment with the Sun per year. If we express the coupling strength as $g_{a\gamma\gamma}=g_{10}\,10^{-10}~{\rm GeV}^{-1}$, the solar axion flux at Earth is $g_{10}^2\,3.5\times10^{11}~{\rm cm^{-2}~s^{-1}}$. The conversion probability in the magnet is $g_{10}^2\,1.8\times10^{-17} (B/8.4\,{\rm T})^2(L/10\,{\rm m})^2$. For the two magnet bores with a cross section of $2\times 14~{\rm cm}^2$ we thus expect an x-ray event rate of $15\,g_{10}^4$ per day of exposure time.
In order to reach the sensitivity shown as a light-grey area in Fig. \[fig:axgam\] one needs to make great efforts to suppress background counts. One way is to focus the x-rays to a small detector region. Specifically, an engineering model for the seven x-ray telescopes of the Abrixas satellite has become available for this purpose and has been tested to be in good working condition. CAST should be able to take first data shortly, i.e. in the summer or fall of 2002.
![Schematic view of the CAST experiment at CERN.[]{data-label="fig:cast"}](raffeltF2.eps){width="100.00000%"}
Figure \[fig:axgam\] shows a loss of sensitivity for about $m_a>10~{\rm meV}$. The axion-photon conversion should be pictured as a phenomenon similar to neutrino oscillations [@Raffelt:1987im]. For larger $m_a$ the oscillation length becomes shorter than the magnet and the effective mixing angle is suppressed. This “momentum mismatch” between axions and photons can be overcome by giving the photons a refractive mass by virtue of a low-$Z$ gas such as helium. This approach was successfully employed in the Tokyo Helioscope [@Inoue:2002qy] and will be used in CAST as well. We may extend the sensitivity range to larger masses as shown in Fig. \[fig:axgam\] and in the neighborhood of $m_a\sim 1~{\rm eV}$ actually bite into the parameter range of conventional axion models.
At somewhat larger masses axions are already ruled out by a telescope search for spectral lines from $a\to\gamma\gamma$ decay in galaxy clusters [@Bershady:1990sw]. In the few-eV mass range axions would have been in thermal equilibrium in the early universe and contribute a small hot-dark matter component.
If we use the Sun as an axion source we must be sure that our sensitivity range is not excluded by an excessive modification of stellar properties by the axionic energy loss. An observable modification of the solar p-mode frequencies excludes $g_{a\gamma\gamma}$ values above the horizontal line in Fig. \[fig:axgam\] marked “Sun” [@Schlattl:1998fz]. Significantly smaller couplings are excluded because the energy-loss of horizontal-branch (HB) stars would shorten their helium-burning lifetime, reducing the relative number of HB stars observed in globular clusters [@Raffelt1996; @Raffelt:1999tx]; see the horizontal line in Fig. \[fig:axgam\] marked “HB Stars.” The CAST experiment advances into uncharted territory.
For very small axion masses, however, the CAST sensitivity range is already excluded by an argument involving SN 1987A. Axions would have been produced in the hot SN core by the Primakoff effect, and then would have back-converted into $\gamma$-rays in the galactic magnetic field. The non-observation of a $\gamma$-ray burst in the SMM instrument in coincidence with the observed SN 1987A neutrinos excludes $g_{a\gamma\gamma}$ values above the line marked SN 1987A [@Brockway:1996yr; @Grifols:1996id]. This limit applies only for about $m_a<10^{-9}~{\rm eV}$; for larger masses the conversion is suppressed by the mass difference relative to photons.
The magnetically induced transition from photons to axion-like particles in intergalactic space has been proposed as a mechanism that would make distant photon sources look dimmer, with important consequences for the interpretation of the SN Ia Hubble diagram [@Csaki:2001yk; @Csaki:2001jk; @Erlich:2001iq; @Deffayet:2001pc; @Mortsell:2002dd]. The relevant masses are very small, again to avoid suppressing the transition by a large axion-photon mass difference. Therefore, the relevant $g_{a\gamma\gamma}$ range is limited by the SN 1987A argument and thus falls outside the CAST sensitivity range.
Large Extra Dimensions
======================
The Planck scale of about $10^{19}$ GeV, relevant for gravitation, is very much larger than the electroweak scale of about 1 TeV of the particle-physics standard model. A radical new approach to solving this notorious hierarchy problem holds that there could be large extra dimensions, the main idea being that the standard-model fields are confined to a 3+1 dimensional brane embedded in a higher dimensional bulk where only gravity is allowed to propagate [@add98; @Antoniadis:1998ig; @add99; @Han:1999sg; @grw99]. This concept immediately puts stringent constraints on the size of the extra dimensions because Newton’s law holds at any scale which has thus far been observed, i.e. down to about 1 mm. Extra dimensions can only appear at a smaller scale.
Following common practice the new dimensions are taken to form an $n$-torus of the same radius $R$ in each direction. The Planck scale of the full higher dimensional space, $M_{{\rm P},n+4}$, can be related to the normal Planck scale,[^2] $M_{{\rm P},4}=1.22\times10^{19}~{\rm
GeV}$, by Gauss’ law [@add98] $$M_{{\rm P},4}^2 = R^n M_{{\rm P},n+4}^{n+2}.$$ Therefore, if $R$ is large then $M_{{\rm P},n+4}$ can be much smaller than $M_{{\rm P},4}$. If this scenario is to solve the hierarchy problem then $M_{{\rm P},n+4}$ must be close to the electroweak scale, i.e. $M_{{\rm P},n+4} < 10$–100 TeV. This requirement already excludes $n=1$ because $M_{{\rm P},n+4} \simeq 100$ TeV corresponds to $R \simeq 10^{8}$ cm. However, $n \geq 2$ remains possible, and particularly for $n=2$ there is the intriguing perspective that the extra dimensions could be accessible to experiments probing gravity at scales below 1 mm.
The most restrictive limits on $M\equiv M_{{\rm P},n+4}$ obtain from supernovae and neutron stars. The first example is the SN 1987A energy-loss argument. If large extra dimensions exist, the usual 4D graviton is complemented by a tower of Kaluza-Klein (KK) states, corresponding to new phase space in the bulk. These KK gravitons would be emitted from the SN core after collapse by nucleon bremsstrahlung $N+N\to N+N+{\rm KK}$. The KK gravitons interact with the strength of ordinary gravitons and thus are not trapped in the SN core. However, this energy-loss channel can compete with neutrino cooling because of the large multiplicity of KK modes and shorten the observable signal [@Cullen:1999hc; @SN1987A; @Hanhart:2001er; @Hanhart:2001fx]. This argument has led to the tight bound $R < 0.66~\mu$m ($M >
31$ TeV) for $n=2$ and $R < 0.8$ nm ($M > 2.75$ TeV) for $n=3$ [@Hanhart:2001fx].
The KK gravitons emitted by all core-collapse SNe over the age of the universe produce a cosmological background of these particles. Later they decay into all standard-model particles which are kinematically allowed; for the relatively low-mass modes produced by a SN the only channels are ${\rm KK} \to 2\gamma$, $e^+e^-$ and $\nu \bar\nu$. The relevant decay rates are $\tau_{2\gamma} = \frac{1}{2}\tau_{e^+e^-} =
\tau_{\nu \bar\nu} \simeq 6 \times 10^{9}~{\rm yr} \, (m/100~{\rm
MeV})^{-3}$ [@Han:1999sg]. Therefore, over the age of the universe a significant fraction of the produced KK modes has decayed into photons, contributing to the diffuse cosmic $\gamma$-ray background observed by EGRET. This argument implies that if the number of extra dimensions $n=2$ or 3, their radius $R$ must be about a factor of 10 smaller than implied by the SN 1987A cooling limit, i.e. for $n=2$ one finds $R < 0.9 \times 10^{-4}$ mm or $M \geq
84$ TeV. For $n=3$ the new limit is $R < 0.19 \times 10^{-6}$ mm or $M > 7$ TeV [@Hannestad:2001jv].
This, however, is not the end of the story. We later realized that the KK gravitons emitted by the SN core will stay gravitationally trapped because most of them are produced near their kinematical threshold, i.e. with barely relativistic velocities [@Hannestad:2001xi]. Therefore, every neutron star is surrounded by a halo of KK gravitons which is dark except for the decays into $\simeq 100~{\rm MeV}$ neutrinos, $e^+e^-$ pairs and $\gamma$-rays. In principle, this radiation can be directly observed. Conversely, the non-observation allows one to set stringent limits. In addition, the radiation impinges on the neutron star, keeping it hot, above the observed temperature in some cases such as the pulsar PSR J0953+0755. One obtains the limit $M>1680~{\rm TeV}$ for $n=2$ and $M>60~{\rm TeV}$ for $n=3$. In view of these limits one expects that if large extra dimensions solve the hierarchy problem, their number $n$ should probably exceed 4.
Similar arguments can be applied to other particles than gravitons that may exist and may be able to propagate in the bulk of the larger-dimensional space. The hypothetical majorons are one case in point [@Hannestad:2002ff].
Of course, there are loop holes to such limits. The size of the extra dimensions need not be equal, or there can be other than toroidal compactifications. The KK gravitons may be able to decay fast into invisible channels, and so forth. However, our main point is that straightforward astrophysical arguments lead to non-trivial and restrictive limits on the structure of this new theory.
Summary and Outlook
===================
Stars continue to provide some of the most restrictive limits on new particle-physics ideas. The much-discussed hypothesis that our space-time has extra dimensions that are compactified on the sub-millimeter scale is a recent case in point. In addition to deriving limits, there are opportunities for new discoveries. The CAST experiment at CERN searching for solar axions will have a sensitivity range that for the first time pushes beyond stellar-evolution limits and thus has a realistic chance of actually finding axion-like particles emitted by the Sun.
In future the observation of solar neutrinos will continue to provide valuable information. The ongoing efforts in neutrino physics virtually guarantee that large detectors will operate for many years to come; even a megatonne detector may be built to search for proton decay and to perform precision measurements at laboratory neutrino beams. Therefore, chances are that one will measure the neutrino burst from a galactic supernova, providing high-statistics information both on the SN event and a host of information of particle physics interest.
The recent excitement about the possible discovery of strange-matter stars [@Seife2002], even though not conclusive, illustrates that compact stars offer one of the few opportunities to discover the true ground state of nuclear matter.
Astroparticle physics is now an established research activity at the interface between inner space and outer space. The physics and observationd of stellar objects continue to offer a number of intriguing opportunities in this multi-faceted and interdisciplinary field.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was partly supported by the Deutsche Forschungsgemeinschaft under grant No. SFB 375 and the ESF network Neutrino Astrophysics.
[99]{}
G.G. Raffelt: [*Stars as Laboratories for Fundamental Physics*]{} (Chicago University Press, Chicago, 1996)
G.G. Raffelt: “Particle physics from stars,” Ann. Rev. Nucl. Part. Sci. [**49**]{}, 163 (1999). D.E. Groom et al. (Particle Data Group): “Review of particle physics,” Eur. Phys. J. C [**15**]{}, 1 (2000). J.E. Kim: “Light pseudoscalars, particle physics and cosmology,” Phys. Rept. [**150**]{}, 1 (1987). H.Y. Cheng: “The strong CP problem revisited,” Phys. Rept. [**158**]{}, 1 (1988). E.W. Kolb and M.S. Turner: [*The Early Universe*]{} (Addison-Wesley, Redwood City, 1990).
J.E. Kim: “Constraints on very light axions from cavity experiments,” Phys. Rev. D [**58**]{}, 055006 (1998). H. Primakoff: “Photo-production of neutral mesons in nuclear electric fields and the mean life of the neutral meson,” Phys. Rev. [**81**]{}, 899 (1951).
P. Sikivie: “Experimental tests of the ‘invisible’ axion,” Phys. Rev. Lett. [**51**]{}, 1415 (1983), Erratum ibid. [**52**]{}, 695 (1984). W.U. Wuensch et al.: “Results of a laboratory search for cosmic axions and other weakly coupled light particles,” Phys. Rev. D [**40**]{}, 3153 (1989). C. Hagmann, P. Sikivie, N.S. Sullivan and D.B. Tanner: “Results from a search for cosmic axions,” Phys. Rev. D [**42**]{}, 1297 (1990). S.J. Asztalos et al.: “Experimental constraints on the axion dark matter halo density,” Astrophys. J. [**571**]{}, L27 (2002). S. Asztalos et al.: “Large-scale microwave cavity search for dark-matter axions,” Phys. Rev. D [**64**]{}, 092003 (2001). K. Yamamoto et al.: “The Rydberg-atom-cavity axion search,” hep-ph/0101200. E. Masso and R. Toldra: “New constraints on a light spinless particle coupled to photons,” Phys. Rev. D [**55**]{}, 7967 (1997). K. van Bibber, P.M. McIntyre, D.E. Morris and G.G. Raffelt: “A practical laboratory detector for solar axions,” Phys. Rev. D [**39**]{}, 2089 (1989). Y. Inoue et al.: “Search for sub-electronvolt solar axions using coherent conversion of axions into photons in magnetic field and gas helium,” Phys. Lett. B [**536**]{}, 18 (2002). K. Zioutas et al.: “A decommissioned LHC model magnet as an axion telescope,” Nucl. Instrum. Meth. A [**425**]{}, 482 (1999). CERN Axion Solar Telescope homepage at http://axnd02.cern.ch/CAST/
G. Raffelt and L. Stodolsky: “Mixing of the photon with low mass particles,” Phys. Rev. D [**37**]{}, 1237 (1988). M.A. Bershady, M.T. Ressell and M.S. Turner: “Telescope search for multi-eV axions,” Phys. Rev. Lett. [**66**]{}, 1398 (1991). H. Schlattl, A. Weiss and G. Raffelt: “Helioseismological constraint on solar axion emission,” Astropart. Phys. [**10**]{}, 353 (1999). J.W. Brockway, E.D. Carlson and G.G. Raffelt: “SN 1987A gamma-ray limits on the conversion of pseudoscalars,” Phys. Lett. B [**383**]{}, 439 (1996). J.A. Grifols, E. Masso and R. Toldra: “Gamma rays from SN 1987A due to pseudoscalar conversion,” Phys. Rev. Lett. [**77**]{}, 2372 (1996). C. Csaki, N. Kaloper and J. Terning: “Dimming supernovae without cosmic acceleration,” Phys. Rev. Lett. [**88**]{}, 161302 (2002). C. Csaki, N. Kaloper and J. Terning: “Effects of the intergalactic plasma on supernova dimming via photon axion oscillations,” Phys. Lett. B [**535**]{}, 33 (2002). J. Erlich and C. Grojean, “Supernovae as a probe of particle physics and cosmology,” Phys. Rev. D [**65**]{} (2002) 123510. C. Deffayet, D. Harari, J.P. Uzan and M. Zaldarriaga: “Dimming of supernovae by photon—pseudoscalar conversion and the intergalactic plasma,” hep-ph/0112118. E. Mörtsell, L. Bergström and A. Goobar: “Photon axion oscillations and type Ia supernovae,” astro-ph/0202153. N. Arkani-Hamed, S. Dimopoulos and G. Dvali: “The hierarchy problem and new dimensions at a millimeter,” Phys. Lett. B [**429**]{}, 263 (1998). I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. Dvali: “New dimensions at a millimeter to a Fermi and superstrings at a TeV,” Phys. Lett. B [**436**]{}, 257 (1998). N. Arkani-Hamed, S. Dimopoulos and G. Dvali: “Phenomenology, astrophysics and cosmology of theories with sub-millimeter dimensions and TeV scale quantum gravity,” Phys. Rev. D [**59**]{}, 086004 (1999). T. Han, J.D. Lykken and R. Zhang: Phys. Rev. D [**59**]{}, 105006 (1999). G.F. Giudice, R. Rattazzi and J.D. Wells: “Quantum gravity and extra dimensions at high-energy colliders,” Nucl. Phys. B [**544**]{}, 3 (1999). Y. Uehara: “A mini-review of constraints on extra dimensions,” hep-ph/0203244. S. Cullen and M. Perelstein: “SN 1987A constraints on large compact dimensions,” Phys. Rev. Lett. [**83**]{}, 268 (1999). V. Barger, T. Han, C. Kao and R.J. Zhang: “Astrophysical constraints on large extra dimensions,” Phys. Lett. B [**461**]{}, 34 (1999). C. Hanhart, D.R. Phillips, S. Reddy and M.J. Savage: “Extra dimensions, SN 1987A, and nucleon nucleon scattering data,” Nucl. Phys. B [**595**]{}, 335 (2001). C. Hanhart, J.A. Pons, D.R. Phillips and S. Reddy: “The likelihood of GODs’ existence: Improving the SN 1987A constraint on the size of large compact dimensions,” Phys. Lett. B [**509**]{}, 1 (2001). S. Hannestad and G.G. Raffelt: “New supernova limit on large extra dimensions: Bounds on Kaluza-Klein graviton production,” Phys. Rev. Lett. [**87**]{}, 051301 (2001). S. Hannestad and G.G. Raffelt: “Stringent neutron-star limits on large extra dimensions,” Phys. Rev. Lett. [**88**]{}, 071301 (2002). S. Hannestad, P. Keranen and F. Sannino: “A supernova constraint on bulk majorons,” hep-ph/0204231. C. Seife: “If it quarks like a star, it must be … strange?,” Science [**296**]{}, 238 (2002).
[^1]: Prepared for the Proceedings of the ESO-CERN-ESA Symposium on Astronomy, Cosmology and Fundamental Physics (4–7 March 2002, Garching, Germany).
[^2]: Some authors define the Planck mass as $M_{\rm P,4}=1.22\times10^{19}~{\rm GeV}/
(8\pi)^{1/2}=2.4\times10^{18}~{\rm GeV}$. Limits on $M_{{\rm P},n+4}$ in this system of units have been reviewed in Ref. [@Uehara:2002yv].
|
---
abstract: 'In this publication I discuss the phase diagram of a frustrated spin-1/2 Heisenberg model suggested in A. A. Nersesyan and A. M. Tsvelik, Phys. Rev. B[**67**]{}, 024422 (2003). The phase diagram contains $(\pi,0)$ and $(\pi,\pi)$ antiferromagnetic phases separated by the Valence Bond Crystal (VBC) state. I argue that the point of the phase diagram with deconfined spinons, predicted in the aforementioned work, is situated in the middle of VBC state, at the point where the dimerization order parameter changes sign.'
author:
- 'A. M. Tsvelik'
title: 'Confinement and deconfinement of spinons in a frustrated spin-1/2 Heisenberg model. '
---
Introduction
============
In our previous paper we described a model of spin-1/2 antiferromagnet (the so-called Confederate Flag or CF model) where the fine tuning of interactions gives rise to a state with fractional quantum spin number excitations (spinons) [@nerstsv]. The subsequent studies ([@moukouri],[@sindz], [@balst] and especially [@batista]) have provided a support to our results. In the original publication we did not discuss what happens if one deviates from this special point. Such discussion is a subject of the present publication. Here I discuss the phase diagram of CF model concentrating primarily on the vicinity of the deconfinement (D) point. This gives us a better understaning of the physics involved and also helps to put the CF model in the broader context of studies of frustrated magnetism. Though the corresponding literature is enormous (see, for example, [@lhuillier] for review), the theoretical efforts are primarily concentrated on idealized models (such as models of dimers or gauge field theory models) whose relation to microscopic models with realistic interactions is not clear. The standard argument invokes universality: the belief is that low energy behavior of such systems will be independent on microscopical details following some universal patterns. It is always interesting to check general considerations against concrete models. In that sense, CF model, being simply a model of a Heisenberg magnet with short range interactions, presents an almost unique example.
The literature knows two scenario for realization of the D-point. In one of them deconfined spinons exist on the boundary between antiferromagnetic and VBC state in the hypothetical situation when these states touch each other at a Quantum Critical Point [@subir]. In the other scenario, based on the study of models of quantum dimers, spinons appear at the boundary between two VBC states (the Roksar-Kivelson critical point) [@fradkin], [@sent]. As I shall argue in this paper, neither situation is realized in CF model. In that model the D-point separates two VBC states, as in the second scenario, but the analogy does not go much further. First, according to [@fradkin],[@sent], the spectrum at the D-point in the dimer models consists of spinless particles (“photons”) with a quadratic $\omega \sim k^2$ spectrum. This is absolutely incompatible with CF model which is approximately (1+1) Lorentz invariant when the interchain coupling is weak. This symmetry dictates that the dispersion of the gapless excitations along the chains direction must be linear. Second, one of the VBC phases in the dimer models contains a ’devil’s staircase’ of commensurate and incommensurate phases which does not agree with the situation in CF model where both VBC phases are simple. So it appears that CF model is quite distinct and does not fit into the known categories.
The model
=========
The model suggested in our original paper [@nerstsv] is a spin-1/2 Heisenberg magnet with spatially anisotropic exchange interactions. The exchange in one direction is much stronger than in the others and therefore this model can be viewed as a model of weakly coupled chains. Recently Batista and Trugman [@batista] have found an isotropic version of the CF model and shown that it has the same ground state degeneracy and possesses spin-1/2 excitations. Therefore the requirement of the space anisotropy is only a matter of convenience. It allows us to use the continuum limit in one direction and to apply the field theory methods. The existence of the deconfined point also does not depend on the number of transverse directions, therefore for the sake of simplicity I will discuss the two-dimensional version of the model. In that case the interaction pattern reminds the Confederate Flag (see Fig. 1). The CF model Hamiltonian is given by H\_[CF]{} = \_[j,n]{}{J\_[**S**]{}\_[j,n]{} \_[j+1,n]{} + \_[= 1]{} \_[j,n + ]{} } + V\_[quarter]{},\[CF\] where ${\bf S}_{j,n}$ are spin-1/2 operators, and $J_{\parallel}>> J_r , J_d > 0$ . The term $V_{quarter}$ contains a four-(and possibly higher) spin exchange interaction with a small coupling constant $\lambda \sim J_r^2/J_{\parallel}$. This term was absent in the original publication, but, as was recently demonstrated by Balents and Starykh [@balst], one needs to introduce it to fine-tune the model to the state with deconfined spinons.
=0.5
Assuming that the interchain couplings ($J_r, J_d, \lambda$) are much smaller than the exchange along the chains ($J_{\parallel}$) it is legitimate to adopt a continuum description of individual chains. In this description, the local spin densities are represented as sums of the smooth and staggered parts: \_[j,n]{} /a\_0 \_n (x) = [**M**]{}\_n(x) + (-1)\^j[**N**]{}\_n(x), x = j a\_0, \[continS\] $a_0$ being the lattice spacing in the chain direction.
=0.65
The low-energy dynamics of the spin–1/2 Heisenberg antiferromagnet H\_[1D]{} = J\_\_j([**S**]{}\_j[**S**]{}\_[j + 1]{}) is described by the SU$_1$(2) Wess-Zumino-Novikov-Witten model. The latter Hamiltonian can be written in terms of the so-called chiral vector [*current*]{} operators, ${\bf J}$ and $\bar{\bf J}$, satisfying the level $k =1$ Kac-Moody algebra (this approach has been described in a vast number of publications; (for a review see [@affleck1] or [@Book],[@me]): H\_[1D]{} x+ , \[WZNW\] with $v = \pi a_0J_{\parallel}/2$.
It is remarkable that the smooth part of magnetization, = [**J**]{} + |[**J**]{}, \[M\] and the spin current, = v([**J**]{} - |[**J**]{}), \[j\] are locally expressed in terms of the chiral currents.
In the CF model, the exchange is frustrated in the direction perpendicular to the chains. In order to eliminate the coupling of the leading relevant operators (the staggered magnetizations ${\bf N}_n (x) \cdot {\bf N}_{n+1} (x)$ and the dimerizations $\epsilon_n(x)\epsilon_{n + \mu}(x)$) one has to fine tune the couplings $J_r - 2J_d$ and $\lambda$. In the leading order in $J_r, J_d$ the fine tuning is achieved when $J_r - 2J_d =0, \lambda =0$.
To discuss the phase diagram of CF model one needs to deviate from the fine-tuned point with deconfined spinons. For weak interchain interactions one still can employ the continuum Hamiltonian obtained using the continuum description of individual chains following the asymptotic representation (\[continS\]) of the spin operators: && H = H\_1 + H\_2,\
&& H\_1 = \_[n =1]{}\^[2N]{} .\[model\]\
&& H\_2 = J \_[n, ]{} [**N**]{}\_[n]{}[**N**]{}\_[n +]{}\[V\] where $H_{1D}$ is given by Eq.(\[WZNW\]). For the lattice Hamiltonian presented on Fig. 1 we have $\gamma = J_r + 2J_d$ and $\delta J = J_r -2J_d$. One very important property of model (\[model\]) is its (1+1)-dimensional Lorentz invariance which survives at $\delta J \neq 0$. This dictates the form of the excitation spectrum for all particles: E\^2 = v\^2k\^2 + \^2(k\_) where $\Delta(k_{\perp})$ is a periodic function of the wave vector component perpendicular to the chains. I emphasise that this result is valid for an arbitrary number of chains. As I have mentioned in Introduction, this property precludes the existence of $k^2$ gapless modes characteristic for dimer model critical points.
Model (\[model\],\[V\]) can be viewed as the critical system (the bunch of non-interacting spin-1/2 chains) perturbed by relevant interactions. Each of these interactions generate their own energy scale according to their scaling dimension. The current-current interaction, being only marginally relevant, generates the scale M \~J\_\[Mass\] and the interaction of the staggered spin components generates a scale $\sim \delta J$. When $|\delta J| >> M$ the system orders antiferromagnetically (though in two dimensions only at $T =0$). The corresponding Neel wave vectors are $(\pi,0)$ for $\delta J < 0$ and $(\pi,\pi)$ for $\delta J > 0$. These things are absolutely obvious; it is less obvious however what happens in the opposite limit $M >> |\delta J|$. For weak interchain interactions the best I can do in this case is to study the four chain model, where non-perturbative calculations can be carried out explicitely. I believe that the four chains are representative enough to give an insight into what happens for an infinite system.
The four chain model as both solvable and representative case.
==============================================================
So let us consider the case of four chains with periodic boundary conditions in the transverse direction. Let me briefly recall the results for $\delta J =0$ obtained in [@nerstsv],[@smirnov]. As was noticed that at $\delta J =0$ the model acquires an additional symmetry: the sectors with different parity decouple. This follows from the fact that the relevant interactions couple only currents of different chirality belonging to different chains. Thus the relevant current-current interaction is ([**J**]{}\_1 + [**J**]{}\_3)(|[**J**]{}\_2 + |[**J**]{}\_4) + ([**J**]{}\_2 + [**J**]{}\_4)(|[**J**]{}\_1 + |[**J**]{}\_3) and the Hamiltonian (\[model\]) decouples into two parts: H = H\_+ + H\_- The $+$ parity sector contains $J_{1,3}$ and $\bar J_{2,4}$ currents and the $-$ parity sector contains $\bar J_{1,3}$ and $J_{2,4}$ currents. Each of the models represented by the Hamiltonians $H_{\pm}$ is integrable, which can be demonstrated by rewriting them in terms of familiar integrable models. Let us recall how it was done in [@nerstsv]. A sum of two $k =1$ SU(2) currents is the $k=2$ current; moreover, according to [@FatZam] the sum of two SU$_1$(2) WZNW models (the central charge 2) can be represented as the SU$_2$(2) WZNW model with central charge 3/2 and plus one massless Majorana fermion (a critical Ising model) with central charge 1/2. Using the results of [@FatZam] we rewrite the entire Hamiltonian density (\[WZNW\]) as follows (here only the (+)-parity part is written): &&[H]{}\_+ = [H]{}\_[massless]{} + [H]{}\_[massive]{}\
&&[H]{}\_[massless]{} = -\_[0]{}\_x\_[0]{} + |\_[0]{}\_x|\_[0]{}\[Ising\]\
&&[H]{}\_[massive]{} = (:[**I**]{}: + :|[**I**]{}|[**I**]{}:) + |[**I**]{} = (- \^a\_x\^a + |\^a\_x|\^a) - (\^a|\^a)\^2 \[O3\] where $a = 1,2,3$ and $${\bf I} = {\bf J}_1 + {\bf J}_3, ~~ \bar{\bf I} = \bar{\bf J}_2 + \bar{\bf J}_4$$ \_[1,3]{} = {\_0+ \[\]}, |[**J**]{}\_[2,4]{} = {|\_0 + \[\]}\[curr2\] The fields $\chi, \bar\chi$ stand for real (Majorana) fermions.
Eq.(\[Ising\]) describes a critical Ising model; the corresponding excitations are gapless and non-magnetic; they appear in the sectors with both parities.
Let me say several words about the O(3) Gross-Neveau model (\[O3\]). Though Majorana fermion description presents some advantages, the staggered magnetization components as well as their product (\[V\]) are nonlocal with respect to these fermions. It turns out that the latter interaction can be expressed in terms of order and disorder parameter operators of the eight Ising models corresponding to each Majorana fermion species. Since we are interested in the case when $|\delta J| \leq M$, we have to recast the perturbation in terms of the nonchiral fields of models $H_{\pm}$. This can be done using Abelian bosonization representation for individual chains and the correspondence between $C=1$ theory and two critical Ising models (see [@Book], [@shelton], [@me]). The net result for the density of the perturbation Hamiltonain obtained after some algebra is
&&[H]{}\_2 = J([**N**]{}\_1 + [**N**]{}\_3)([**N**]{}\_2 + [**N**]{}\_4) =\
&&J(\_0\^+\_0\^-)\[(\_1\_2\_3)\^+(\_1\_2\_3)\^- - 3(\_1\_2\_3)\^+(\_1\_2\_3)\^- + ...\] +\
&&J(\_0\^+\_0\^-)\[3(\_1\_2\_3)\^+(\_1\_2\_3)\^- + (\_1\_2\_3)\^+(\_1\_2\_3)\^- + ...\] \[stagint\] where the dots stand for the terms which do not have finite averages at $\delta J$. $\s_a^{\pm}, \mu_a^{\pm}$ ($a = 0,1,2,3$ are order and disorder parameter operators of the Ising models associated with the Majorana fermions $\chi_a,\bar\chi_a$ from $+$ and $-$ sectors. As we shall demonstrate later, this interaction leads to confinement of spin-1/2 particles.
The dimerization order parameter is &&([**N**]{}\_1 - [**N**]{}\_3)([**N**]{}\_2 - [**N**]{}\_4) =\
&&(\_0\^+\_0\^-)\[(\_1\_2\_3)\^+(\_1\_2\_3)\^- + 3(\_1\_2\_3)\^+(\_1\_2\_3)\^- + ...\] +\
&&(\_0\^+\_0\^-)\[- 3(\_1\_2\_3)\^+(\_1\_2\_3)\^- + (\_1\_2\_3)\^+(\_1\_2\_3)\^- + ... \]\[dimer\] As I have said, the terms in the square brackets in Eq.(\[stagint\]) have nonzero vacuum averages even at $\delta J =0$. The unperturbed model has four ground states: with ${\langle}(\s_1\s_2\s_3)^{\pm}{\rangle}\neq 0$ (the $\s^{\pm}$ vacua) and ${\langle}(\mu_1\mu_2\mu_3)^{\pm}{\rangle}\neq 0$ (the $\mu^{\pm}$ vacua). Replacing in (\[stagint\]) the corresponding products by their vacuum expectation values $\sim M^{3/8}$ and identifying $(\mu_0^+\mu_0^-) = \cos(\sqrt\pi\Phi_0), (\s_0^+\s_0^-) = \sin(\sqrt\pi\Phi_0)$, we get &&[H]{}\_2 \~(J) M\^[3/4]{} (\^0)\
&&\~(J) M\^[3/4]{} (\^0) \[forms\] and 0 if for one parity ${\langle}\s{\rangle}\neq 0$ and ${\langle}\mu{\rangle}\neq 0$ for the other. Thus the ground state degeneracy is now reduced to two; the order (disorder) parameters in both sectors now condense simultaneously. The $\s \rightarrow \mu$ degeneracy is not lifted, since two forms of the potential (\[forms\]) are equivalent under the uniform shift $\sqrt\pi\Phi_0 \rightarrow \sqrt\pi\Phi_0 + \pi/2$ which does not affect the gradient term $({\partial}_{\mu}\Phi_0)^2$. Perturbation (\[V\]) couples the critical Ising model sectors with different parity; being projected on the state with ${\langle}\mu^{+}{\rangle}, {\langle}\mu^-{\rangle}\neq 0$ the resulting Hamiltonian becomes the sine-Gordon one: &&[H]{}\_[SG]{} = |\^+\_0\^3\_x\^+\_0 + |\^-\_0\^3\_x\^-\_0 + AM\^[3/4]{}(J)(\^+\_0\^-\_0 + 3\^+\_0\^-\_0) =\
&& \[\_0\^2 + (\_[x]{}\_0)\^2\] + A (J) M\^[3/4]{} (\_0 + \^[-1]{}3) where $\tilde A \sim 1$ is a numerical coefficient, $\tau^3$ is the Pauli matrix and $\Pi_0$ is the momentum density operator. The cosine term gives rise to the mass gap \~(J)\^[4/7]{}M\^[3/7]{} \[massd\] The average dimerization is &&([**N**]{}\_1 - [**N**]{}\_3)([**N**]{}\_2 - [**N**]{}\_4)=\
&& \~M\^[3/4]{} (0)\
&& \~- M\^[3/4]{} (0) This average does not depend on the choice of vacuum (that is whether $\s$ or $\mu$ are in the condensate): \~(J)\^[1/7]{}M\^[6/7]{} Thus the dimerization changes its sign when $\delta J$ goes through zero following a power law dependence. The exponent is quite small and it is possible that in the limit of infinite number of chains the transition is the 1st order, as in the isotropic CF model [@batista]. I believe that this is indicative of the physics behind the deconfinement: it occurs on the boundary between two Valence Bond crystalline orders.
=0.45
=0.65
Confinement of Spinons
======================
The problem of kinks confinement in the potential (\[stagint\]) is somewhat peculiar differing from the standard confinement problems studied by various authors [@mussardo], [@affleck], [@fonseca]. In our case the string tension is provided by the $\cos[\sqrt\pi\Phi_0], \sin[\sqrt\pi\Phi_0]$ terms and becomes energy dependent. Though it is not directly related to the main topic of the paper, it is interesting enough on its own right. So I will spent some time discussing this problem.
Let us consider a kink interpolating between the $\s$- and $\mu$-vacua in the $+$ sector ($[\mu\s]^{+}$ kink). It carries an isotopic (spin) index $\alpha = \pm 1/2$. According to (\[stagint\]), the creation of such a kink will lead to the rise of the total energy proportional to the system size. To prevent this, one has to create a similar kink in the $-$ sector. If we have two kinks $\mu\s$ - one centered at $x=x_1$ and the other one at $x=x_2$, the effective potential for the $\sqrt\pi\Phi = \sqrt\pi\Phi_0 + \tan^{-1}3$ field is A(J)M\^[3/4]{}{(x\_1 -x)+ (x - x\_2)} The kinks are heavy particles whose mass $M$ far exceeds the masses of excitations of the $\Phi_0$ field $\sim \Delta$. This justifies the approximation which takes the kink configuarations as step functions. At $x < x_1$ field $\Phi$ is locked at $\sqrt\pi$, at $x > x_2$ it is locked at $3\sqrt\pi/2$. The solution in the middle is $$\sqrt\pi\Phi = \frac{\pi}{2}\frac{x - x_1}{x_2 - x_1}$$ such that the total energy difference between the vacuum without kinks and the vacuum in the presence of two kinks is V = B\^2\^2|x\_[12]{}| + \[potential\] where $B \sim 1$ (though its numerical value can be calculated, it is not of an interest here) and B\^2\^2 = A (J)M\^[3/4]{}
=0.5
This energy difference provides an effective confining double-well potential in which the $[\mu\s]^{+}-[\mu\s]^{-}$ bound states are formed. Recall that the potential does not depend on the spin configuration (this will be no longer the case when $\Delta \sim M$). Therefore the bound states form 4-fold degenerate isotopic multiplets consisting of SU(2) triplets and singlets. The mass spectrum of kink bound states in such double-well potential is somewhat different from the purely linear confining potential studied in [@mussardo],[@affleck],[@fonseca]: && M\_n - 2M\
&& B{ + \^[1/2]{}(n + 1/2)\[1 (n)\] + ...}, n << (M/)\^[2/5]{} \[spec\] where $$\delta(n) \sim \exp\left[- (\pi/2)^{17/8}(M/B\Delta)^{3/4}\frac{1}{\sqrt{n + 1/2}}\right]$$ and M\_n - 2M B(n\^2B/M)\^[1/3]{}, n >> (M/)\^[2/5]{} \[spec1\] Not all these bound states are stable; the particles with masses greater than $2M + \Delta$ can decay into particles with smaller masses emitting excitations of the $\Phi_0$ field. Therefore particles with $n > \sqrt{M/\Delta}$ are unstable. Since the power 1/2 is quite close to 2/5, the mass sequence of Eq.(\[spec1\]) is never reached.
As far as kink-antikink states are concerned, for them the string potential is not repulsive, but attractive and they are expelled from the spectrum.
Conclusion
==========
The study of the four chain case gives reasons to believe that the infinite system has a phase diagram presented on Fig. 4. The deconfinement point appears in the middle of the VBC phase, as in the dimer models considered in [@fradkin],[@sent]. The excitation spectra are different, however. The spinless modes of the dimer models have a quadratic spectrum $\omega \sim k^2$; such spectrum cannot emerge in CF model due to the (1+1)-dimensional Lorentz invariance. Thus I conclude that the D-point of CF model belongs to a universality class different from the universality class of the Roksar-Kivelson critical point.
In conclusion to this paper I would like to point out a rather curious parallel between the problem of frustrated magnetism and another long standing problem of condensed matter physics, namely the problem of heavy fermion state formation in rare earth compounds. In these compounds magnetic moments of highly localized electrons belonging to rare earth ions $f$ shells interact with delocalized electrons from the broad conduction band. The problem is discussed in the literature in terms of competition between the Kondo screening and the induced interspin interaction (the so-called RKKY interaction). On the formal level this competiton looks exactly like the competition between the less relevant current-current interaction in model (\[model\]) and the more relevant interaction of staggered magnetisations (\[V\]). Indeed, the energy scale generated by the Kondo screening (Kondo temperature) is exponentially small in the coupling constant as in (\[Mass\]) and the energy scale generated by the RKKY interaction is proportional to the square of the spin-fermion coupling. Though one would expect that $\exp(-1/g)$ is always much smaller than $g^2$, there is a vast class of materials where the Kondo screening is manifest at temperatures much larger than the temperature of magnetic ordering. One possible explanation is that the RKKY interaction with its oscillatory behavior, is highly frustrated which leads to cancellations similar to the one considered in this paper.
I am grateful to O. Starykh, I. Zaliznyak, A. Nersesyan and F. H. L. Essler for discussions and interest to the work. I acknowledge the support from US DOE under contract number DE-AC02 -98 CH 10886.
[99]{} A. A. Nersesyan and A. M. Tsvelik, Phys. Rev. B[**67**]{}, 024422 (2003). S. Moukouri, cond-mat/0305608. P. Sindzingre, Phys. Rev. B[**69**]{}, 094418 (2004). O. A. Starykh and L. Balents, cond-mat/0402055. C. D. Batista and S. A. Trugman, cond-mat/0407632. G. Misguich and C. Lhuillier, in [*Frustrated Spin Systems*]{}, World Scientific (2003), ed. by H. T. Diep; cond-mat/0310405. T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, M. P. A. Fisher, Science, [**303**]{}, 1490 (2003). E. Fradkin. D. A. Huse, R. Moessner, V. Oganesyan and S. L. Sondhi, Phys. Rev. B[**69**]{}, 224415 (2004). A. Vishwanath, L. Balents and T. Senthil, Phys. Rev. B[**69**]{}, 224416 (2004). I. Affleck, [*Field Theory Methods and Quantum Critical Phenomena*]{}, Les Houches, session XLIX (Elsevier, New York, 1989). A. O. Gogolin, A. A. Nersesyan and A. M. Tsvelik, [*Bosonization in Strongly Correlated Systems*]{}, Cambridge University Press, 1999. A. M. Tsvelik, “Quantum Field Theory in Condensed Matter Physics”, 2nd edition, Cambridge University Press, 2003. F. A. Smirnov and A. M. Tsvelik, Phys. Rev. B[**68**]{}, 144412 (2003). A. B. Zamolodchikov and V. A. Fateev, Sov. J. Nucl. Phys. [**43**]{}, 657 (1986). D. G. Shelton, A. A. Nersesyan and A. M. Tsvelik, Phys. Rev. B[**53**]{}, 8521 (1996). G. Delfino and G. Mussardo, Nucl. Phys. B[**516**]{}, 675 (1998). I. Affleck, NATO ASI series E 349, ed. by A. Skjeltorp and D. Sherrington, Kluwer Academic (1998); condmat/9705127. P. Fonseca and A. Zamolodchikov, J. Stat. Phys. [**110**]{}, 527 (2003).
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---
abstract: 'In-band full-duplex systems can transmit and receive information simultaneously on the same frequency band. However, due to the strong self-interference caused by the transmitter to its own receiver, the use of non-linear digital self-interference cancellation is essential. In this work, we describe a hardware architecture for a neural network-based non-linear self-interference (SI) canceller and we compare it with our own hardware implementation of a conventional polynomial based SI canceller. In particular, we present implementation results for a shallow and a deep neural network SI canceller as well as for a polynomial SI canceller. Our results show that the deep neural network canceller achieves a hardware efficiency of up to $312.8$ Msamples/s/mm$^2$ and an energy efficiency of up to $0.9$ nJ/sample, which is $2.1\times$ and $2\times$ better than the polynomial SI canceller, respectively. These results show that NN-based methods applied to communications are not only useful from a performance perspective, but can also be a very effective means to reduce the implementation complexity.'
author:
- '[^1][^2][^3] [^4][^5]'
bibliography:
- 'IEEEabrv.bib'
- 'bibliography.bib'
title: 'Hardware Implementation of Neural Self-Interference Cancellation'
---
Introduction
============
In-band full-duplex (FD) communications have for long been considered to be impractical due to the strong self-interference (SI) caused by the transmitter to its own receiver. However, recent work on the topic (e.g., [@Jain2011; @Duarte2012; @Bharadia2013]) has demonstrated that it is, in fact, possible to achieve sufficient SI cancellation to make FD systems viable. Typically, SI cancellation is performed in both the radio frequency (RF) domain and the digital domain to cancel the SI signal down to the level of the receiver noise floor. There are several ways to achieve RF cancellation that can be broadly categorized into *passive RF cancellation* and *active RF cancellation*. Some form of RF cancellation is generally necessary to avoid saturating the analog front-end of the receiver. Passive RF cancellation can be obtained by using, e.g., circulators, directional antennas, beamforming, polarization, or shielding. Active RF cancellation is commonly implemented by transforming the transmitted RF signal appropriately to emulate the SI channel using analog components and subtracting the resulting SI cancellation signal from the received SI signal [@Jain2011; @Bharadia2013]. Alternatively, an additional transmitter can be used to generate the SI cancellation signal from the transmitted baseband samples [@Duarte2012].
However, a residual SI signal is typically still present at the receiver after RF cancellation has been performed. This residual SI signal can, in principle, be easily canceled in the digital domain, since it is caused by a known transmitted signal. Unfortunately, in practice, several transceiver non-linearities distort the SI signal. Some examples of non-linearities include baseband non-linearities (e.g., digital-to-analog converter (DAC) and analog-to-digital converter (ADC)) [@Balatsoukas2015], IQ imbalance [@Balatsoukas2015; @Korpi2014], phase-noise [@Sahai2013; @Syrjala2014], and power amplifier (PA) non-linearities [@Balatsoukas2015; @Korpi2014; @Anttila2014; @Korpi2017]. These effects need to be taken into account using intricate polynomial models to cancel the SI to the level of the receiver noise floor. These polynomial models perform well in practice, but their implementation complexity grows rapidly with the maximum considered non-linearity order. Principal component analysis (PCA) is an effective complexity reduction technique that can identify the most significant non-linearity terms in a parallel Hammerstein model [@Korpi2017]. However, with PCA-based methods, the transmitted digital baseband samples need to be multiplied with a transformation matrix to generate the SI cancellation signal, thus introducing additional complexity. Moreover, whenever the SI channel changes, the high-complexity PCA operation needs to be re-run. To the best of our knowledge, no hardware implementation of a polynomial SI canceller has been reported in the open literature to date. Only the work of [@Campo2018] has made a step in this direction, since the authors considered quantization aspects of polynomial SI cancellers.
In the past few years, there has been renewed interest in the use of neural networks (NNs) to augment or replace a range of signal processing tasks in communications systems [@OShea2017a; @Wang2017a; @Mao2018a; @Gunduz2019; @Qin2019a; @Balatsoukas2019sips]. NNs are particularly well-suited to tackle non-linear signal processing problems, where traditional model-based algorithms are unavailable or too complex for analytical treatment. However, NN-based solutions can also be used in cases where traditional model-based algorithms suffer from prohibitively high implementation complexity. For example, NNs have been used successfully to perform digital predistortion (DPD) in wireless systems [@Tarver2019b; @Hongyo2019], non-linear leakage cancellation in FDD transceivers [@Ploder2019], as well as optical fiber non-linearity compensation [@Hager2018a]. NNs have also been used for non-linear SI cancellation in full-duplex communications [@Balatsoukas2018; @Guo2018; @Kristensen2019] and it was shown in [@Balatsoukas2018] that they can achieve similar SI cancellation performance with a state-of-the-art polynomial SI cancellation model, but with much lower computational complexity.
Existing NN hardware accelerators, such as [@Zhang2015; @Chen2016], mainly target applications where both the size of the NN and the number of inputs is very large, and where producing a few tens of outputs per second is sufficient. Communications applications, on the other hand, use relatively small NNs with few inputs, but need to provide millions of outputs per second. As such, communications applications generally require different and more specialized NN hardware accelerator architectures. However, to date, only a small number of works have considered these hardware-related issues in the context of communications applications. Specifically, the works of [@Aoudia2019; @Wodiany2019] study NN quantization as a first step towards hardware implementation, while the authors of [@Kurzo2018; @Tarver2019a] describe actual hardware implementations of simple NNs for SI cancellation in full-duplex communications and DPD, respectively.
### Contribution {#contribution .unnumbered}
In this work, we present a hardware implementation of the SI cancellation method proposed in [@Balatsoukas2018] to quantify and translate the computational complexity gains over the state-of-the-art polynomial based model of [@Korpi2017] into real-world hardware resource utilization gains. Moreover, we also implement an instance of the deep NN canceller proposed in [@Kristensen2019] which leads to a significant additional complexity reduction compared with the shallow NN canceller of [@Balatsoukas2018]. Since, to the best of our knowledge, no polynomial SI canceller implementations have been reported in the literature, we also present a hardware architecture for a reference polynomial SI canceller. We note that this hardware architecture can also be used for other related applications such as digital predistortion and leakage cancelletion in FDD transceivers. We provide FPGA and ASIC implementation results that clearly demonstrate the significant gains with respect to the polynomial SI canceller that can be achieved by both the shallow and the deep NN-based SI cancellers in terms of resource utilization, throughput, and energy efficiency.
### Outline {#outline .unnumbered}
The remainder of this paper is organized as follows. Section \[sec:background\] provides background on full-duplex communications and digital SI cancellation using polynomial cancellers, while Section \[sec:nncanc\] describes how SI cancellation can be achieved using NNs. In Section \[sec:nnhardware\] we describe our proposed NN-based SI canceller hardware architecture and in Section \[sec:polyhardware\] we describe our proposed baseline polynomial-based SI canceller hardware architecture. In Section \[sec:results\], we compare the performance and the complexity of a conventional polynomial SI canceller with the NN-based SI cancellers. In Section \[sec:results\] we also provide FPGA and ASIC implementation results. Finally, Section \[sec:conclusion\] concludes this paper.
Conventional Digital Self-Interference Cancellation {#sec:background}
===================================================
Fig. \[fig:block\] shows a block diagram of a full-duplex transceiver. On the transmitter side, the digital baseband samples $x[n] \in \mathbb{C}$, where $n$ is the sample index, are converted to an analog signal using a digital-to-analog converter (DAC), up-converted to a carrier frequency $f_c$ using an IQ mixer, amplified using a power amplifier (PA), and filtered using a bandpass (BP) filter. The transmitted signal leaks to the receiver through an SI channel $h_{\text{SI}}$ and is then filtered using a BP filter, amplified using an LNA, downconverted using an IQ mixer, and digitized using an analog-to-digital-converter (ADC). The SI channel $h_{\text{SI}}$ also models the passive RF SI cancellation. An RF cancellation signal is subtracted from the received SI signal at some point before the LNA to avoid saturating the receiver. Since the transmitter and the receiver are co-located, they share a common local oscillator (LO) signal in order to minimize the effect of phase noise on the SI signal.
If we assume, for simplicity of exposition, that there is no signal-of-interest from a remote node and no thermal noise, then the received signal $y[n]$ in Fig. \[fig:block\] consists only of the residual SI signal after RF SI cancellation has been performed. We denote the received signal in this special case by $y_{\text{SI}}[n]$. The goal of digital SI cancellation is to reproduce an accurate copy of $y_{\text{SI}}[n]$, denoted by $\hat{y}_{\text{SI}}[n]$, based on samples of the transmitted baseband signal $x[n]$. This signal is then subtracted from $y[n]$ so that the residual SI signal is $y_{\text{SI}}[n] - \hat{y}_{\text{SI}}[n]$. If $\hat{y}_{\text{SI}}[n]$ is reconstructed perfectly, then the SI can be canceled entirely and $y_{\text{SI}}[n] - \hat{y}_{\text{SI}}[n] = 0$. In practice, as discussed previously, due to the presence of thermal noise and transceiver non-linearities, perfect SI cancellation is difficult to achieve. The SI cancellation performance $C_{\text{dB}}$ is typically evaluated as: $$\begin{aligned}
C_{\text{dB}} &= 10 \log_{10} \left( \frac{\sum_n |y_{\text{SI}}[n]|^2}{\sum_n |y_{\text{SI}}[n] - \hat{y}_{\text{SI}}[n]|^2} \right).\end{aligned}$$
Linear Self-Interference Cancellation {#sec:lincanc}
-------------------------------------
Linear SI cancellation is the simplest form of SI cancellation that ignores all non-linear effects of the various components in Fig. \[fig:block\]. The linear SI cancellation signal is constructed as: $$\begin{aligned}
\hat{y}_{\text{SI}}[n] & = \sum _{l=0}^{L-1}\hat{h}[l]x[n-l], \label{eq:lincanc}\end{aligned}$$ where $\hat{h}[l] \in \mathbb{C},~l \in \{0,\hdots,L-1\},$ models the SI channel $h_{\text{SI}}$ and any other memory effect in the transceiver chain. The parameters $\hat{h}[l]$ can be obtained from training samples either in a one-shot fashion using standard least-squares (LS) estimation or adaptively using an iterative version of the LS estimation algorithm, such as least mean squares (LMS) or recursive least squares (RLS).
Polynomial Non-Linear Self-Interference Cancellation {#sec:polycanc}
----------------------------------------------------
Each active component in the transceiver model shown in Fig. \[fig:block\] is generally a dynamic non-linear system. This means that linear cancellation alone is, in most cases, not accurate enough to cancel a sufficiently large fraction of the SI signal. It has been shown that the transmitter IQ imbalance and the PA non-linearities typically dominate all remaining non-linearities [@Anttila2014; @Korpi2017]. This is true in particular when the transmitter and receiver chains use the same local oscillator signal for upconversion, as shown in Fig. \[fig:block\], so that the effect of phase noise becomes negligible. As such, the SI cancellation signal $\hat{y}_{\text{SI}}[n]$ can be constructed as [@Anttila2014; @Korpi2017]: $$\begin{aligned}
\hat{y}_{\text{SI}}[n] & = \sum _{\substack{p=1,\\p \text{ odd}}}^{P} \sum_{l=0}^{L-1} \hat{h}_p[l]\left(K_1x[n-l] + K_2x^{*}[n-l]\right)^p.\end{aligned}$$ where the parameters $\hat{h}_p[l] \in \mathbb{C},~l \in \{0,\hdots,L-1\},~p \in \{1,3,\hdots,P\}$, model the joint effect of $\hat{h}^{\text{SI}}[l]$ and the memory effects introduced by the PA for the harmonic of order $p$, and $K_1$ and $K_2$ are parameters that model the IQ imbalance. Only odd values for $p$ are considered because even harmonics typically lie out-of-band and are filtered out by the transmitter and receiver BP filters. With some arithmetic manipulations $\hat{y}_{\text{SI}}[n]$ can be re-written as [@Anttila2014; @Korpi2017]: $$\begin{aligned}
\hat{y}_{\text{SI}}[n] & = \sum _{\substack{p=1,\\p \text{ odd}}}^P \sum_{q=0}^p\sum_{l=0}^{L-1}\hat{h}_{p,q}[l] x[n-l]^{q}x^*[n-l]^{p-q}, \label{eq:final}\end{aligned}$$ where the parameters $\hat{h}_{p,q}[l] \in \mathbb{C}$ capture the joint effect of $\hat{h}_{p}[l]$ and of the IQ imbalance parameters $K_1$ and $K_2$. The model in is linear with respect to the parameters $\hat{h}_{p,q}[l]$, and therefore, similarly to linear SI estimation, the parameters $\hat{h}_{p,q}[l]$ can be estimated based on training samples using some variant of the LS estimation algorithm. The *basis functions* of the polynomial model in are defined as: $$\begin{aligned}
\text{BF}_{p,q}(x) & = x^{q}(x^*)^{p-q}. \label{eq:basisfunction}\end{aligned}$$ The number of distinct basis functions in is [@Korpi2017]: $$\begin{aligned}
N_{\text{BF}} & = \frac{L}{4}\left(P+1\right)\left(P+3\right). \label{eq:nbf}\end{aligned}$$ Using , the expression for $\hat{y}[n]$ in can be re-written in a more compact form: $$\begin{aligned}
\hat{y}_{\text{SI}}[n] & = \sum _{\substack{p=1,\\p \text{ odd}}}^P \sum_{q=0}^p\sum_{l=0}^{L-1}\hat{h}_{p,q}[l] \text{BF}_{p,q}\left(x[n-l]\right). \label{eq:finalbf}\end{aligned}$$ We note that linear cancellation is a special case of the polynomial model in when only considering the single term for $p=1$ and $q=1$.
Computational Complexity
------------------------
Assuming that each complex multiplication is implemented using three real-valued multiplications and five real-valued additions and that each complex addition is implemented using two real-valued additions, it can directly be deduced from that the total number of real-valued multiplications and additions that are required by the linear SI canceller is: $$\begin{aligned}
N_{\text{ADD,lin}} & = 7L - 2,\\
N_{\text{MUL,lin}} & = 3L.\end{aligned}$$ Moreover, if we ignore the computation of the basis functions for simplicity,[^6] the total number of real-valued multiplications and additions that are required by the polynomial SI canceller (which also includes the linear cancellation term) is [@Balatsoukas2018]: $$\begin{aligned}
N_{\text{ADD,poly}} & = \frac{7}{4}L\left(P+1\right)\left(P+3\right)-2, \label{eq:addpoly}\\
N_{\text{MUL,poly}} & = \frac{3}{4}L\left(P+1\right)\left(P+3\right). \label{eq:multpoly}\end{aligned}$$ We note that the expression for $N_{\text{ADD,poly}}$ in our previous work of [@Balatsoukas2018] erroneously ignored the five real-valued additions that are required to implement each complex multiplication. As such, the actual complexity of the polynomial canceller is even higher than that reported in [@Balatsoukas2018].
Neural Network Non-Linear Digital Self-Interference Cancellation {#sec:nncanc}
================================================================
Polynomial SI cancellation models such as work well in practice but are often highly redundant in the sense that many of the $\hat{h}_{p,q}[l]$ parameters are very close to zero. NN-based SI cancellers, on the other hand, can extract the essence of the non-linear structure of the SI signal from training data, which often significantly reduces the complexity of the SI cancellation model [@Balatsoukas2018]. A challenge when using NN cancellers is that the NN training process is inherently noisy due to the use of mini-batches for gradient estimation, which makes it difficult to achieve a very accurate reconstruction of the SI signal [@Kristensen2019]. To overcome this problem, [@Balatsoukas2018] used a NN to reconstruct only a particular part of the SI signal, while using conventional linear cancellation for the remainder of the SI. Specifically, in [@Balatsoukas2018] the SI signal was conceptually decomposed into a linear component and a non-linear component: $$\begin{aligned}
y_{\text{SI}}[n] & = y_{\text{SI,linear}}[n] + y_{\text{SI,nl}}[n].\end{aligned}$$ The SI cancellation is carried out in two steps. First, linear cancellation is used in order to reconstruct $\hat{y}_{\text{SI,linear}}[n]$ as: $$\begin{aligned}
\hat{y}_{\text{SI,linear}}[n] & = \sum_{l=0}^{L-1}\hat{h}[l] x[n-l]. \label{eq:linear}\end{aligned}$$ The parameters $\hat{h}[l]$ are obtained using LS estimation while considering the substantially weaker signal $\hat{y}_{\text{SI,nl}}[n]$ as noise. The linear SI cancellation signal is then subtracted from the SI signal in order to obtain: $$\begin{aligned}
y_{\text{SI,nl}}[n] & \approx y_{\text{SI}}[n] - \hat{y}_{\text{SI,linear}}[n].\end{aligned}$$ The task of the NN is limited to reconstructing $y_{\text{SI,nl}}[n]$ based on the appropriate $x[n]$ samples.
As is common practice when training NNs, we normalize the input and output training samples so that $x[n]$ and $y_{\text{SI,nl}}[n]$ have unit variance (i.e., the variance of the real part and the variance of the imaginary part are both equal to $0.5$) and zero mean. To perform SI cancellation on the test data, the output of the NN is denormalized using the mean and variance estimated based on the training data.
Neural Network Structure
------------------------
Due to the universal approximation theorem [@Hornik1991], a feed-forward NN with one hidden layer, as depicted in Fig. \[fig:nn\], is sufficient to reconstruct the non-linear SI signal. While the work of [@Balatsoukas2018] only considered feedforward NNs with one hidden layer, it is possible to use any NN architecture to generate $\hat{y}_{\text{SI,nl}}[n]$. In particular, [@Kristensen2019] employed a deep feedforward NN and showed that using many layers with few neurons per layer has significant computational complexity advantages with respect to a shallow NN SI canceller that uses a single layer with more neurons. In all cases and as shown in Fig. \[fig:nn\], the cancellation NNs have $2L$ input nodes, which correspond to the real and imaginary parts of the $L$ delayed versions of $x[n]$, and two output nodes, which correspond to the real and imaginary parts of the target $\hat{y}_{\text{SI,nl}}[n]$ sample. In the following, we denote the number of hidden layers by $N_l$ and the number of hidden nodes per layer $N_h$.
Let the vector $\mathbf{l}_0$ contain the $2L$ inputs to the NN: $$\begin{aligned}
\mathbf{l}_0 & = \begin{bmatrix}
\Re{\left\{x[n]\right\}} & \Im{\left\{x[n]\right\}} & \hdots & \Re{\left\{x[n{-}L{+}1]\right\}} & \Im{\left\{x[n{-}L{+}1]\right\}}
\end{bmatrix}^{T}.\end{aligned}$$ The outputs of the first hidden layer neurons are given by: $$\begin{aligned}
\mathbf{l}_1 & = f_1\left(\mathbf{W}_{1}\mathbf{l}_{0} + \mathbf{b}_{1}\right), \label{eq:layer1}\end{aligned}$$ where $\mathbf{W}_{1}$ is an $N_h \times 2L$ matrix containing the hidden layer weights, $\mathbf{b}_{1}$ is an $N_h \times 1$ vector containing the hidden layer biases, and $f_1(\cdot)$ is the (vectorized) non-linear activation function used in the first hidden layer. The outputs of the neurons in the hidden layers $1 < l \leq N_l$ are: $$\begin{aligned}
\mathbf{l}_l & = f_l\left(\mathbf{W}_{l}\mathbf{l}_{l-1} + \mathbf{b}_{l}\right), \label{eq:layerl}\end{aligned}$$ where $\mathbf{W}_{l}$ is an $N_h \times N_h$ matrix containing the hidden layer weights, $\mathbf{b}_{l}$ is an $N_h \times 1$ vector containing the hidden layer biases, and $f_l(\cdot)$ is the (vectorized) non-linear activation function used in hidden layer $l$. Finally, the outputs of the output layer neurons are given by: $$\begin{aligned}
\mathbf{l}_{N_l+1} & = f_{N_l+1}\left(\mathbf{W}_{N_l+1}\mathbf{l}_{N_l} + \mathbf{b}_{N_l+1}\right), \label{eq:layero}\end{aligned}$$ where $\mathbf{W}_{N_l+1}$ is a $2 \times N_h$ matrix containing the output layer weights, $\mathbf{b}_{N_l+1}$ is a $2 \times 1$ vector containing the output layer biases, and $f_{N_l+1}$ is the activation function used in the output layer. As can be seen in Fig. \[fig:nn\], for $\mathbf{l}_{N_l+1}$ we have: $$\begin{aligned}
\mathbf{l}_{N_l+1} & =
\begin{bmatrix}
\Re{\left\{\hat{y}_{\text{nl}}[n]\right\}} &
\Im{\left\{\hat{y}_{\text{nl}}[n]\right\}}
\end{bmatrix}^T.\end{aligned}$$ The goal of the NN is to minimize the mean squared error between the expected NN output and the actual NN output: $$\begin{aligned}
\text{MSE} & = \frac{1}{N}\sum_{n=0}^{N-1}\left(\Re{\left\{y_{\text{SI,nl}}[n]\right\}}-\Re{\left\{\hat{y}_{\text{SI,nl}}[n]\right\}}\right)^2 \nonumber \\
& + \frac{1}{N}\sum_{n=0}^{N-1}\left(\Im{\left\{y_{\text{SI,nl}}[n]\right\}}-\Im{\left\{\hat{y}_{\text{SI,nl}}[n]\right\}}\right)^2, \label{eq:MSE}\end{aligned}$$ where $N$ is the total number of training samples. The MSE in is minimized by choosing appropriate values for $\mathbf{W}_{l},~\mathbf{b}_{l},~l \in \{1,\hdots,N_l+1\}$, using back-propagation [@Rumelhart1986].
Computational Complexity
------------------------
Let us assume that the NN uses the popular ReLU activation function in the hidden layers (which has similar complexity to a real-valued addition (i.e., $f_l = \text{ReLU}(\mathbf{x}) = \max(\mathbf{0},\mathbf{x})$, $l \in \{1,\hdots,N_l\}$) and a linear activation function in the output layer (i.e., $f_{N_l+1}(\mathbf{x}) = \mathbf{x}$). Then, the number of real-valued multiplications and additions that are required by a NN canceller with a single hidden layer and $N_h$ hidden neurons is [@Balatsoukas2018]: $$\begin{aligned}
N_{\text{ADD,NN}} & = (2L+3)N_h + 7L, \label{eq:addnnsingle}\\
N_{\text{MUL,NN}} & = (2L+2)N_h + 3L, \label{eq:multnnsingle}\end{aligned}$$ where the second term in both expressions comes from the linear SI canceller that is required for the NN SI canceller to work. Moreover, two additions are required to add the output of the linear SI canceller with the output of the NN canceller.[^7] For the more general NN described in [@Kristensen2019] with $N_l$ hidden layers with $N_h$ neurons each, - can be generalized to: $$\begin{aligned}
N_{\text{ADD,NN}} & = (2L+3 + (N_l{-}1)(N_h{+}1))N_h + 7L, \label{eq:addnn}\\
N_{\text{MUL,NN}} & = (2L + 2 + (N_l{-}1)N_h)N_h + 3L. \label{eq:multnn}\end{aligned}$$
![High-level architecture on the NN-based SI cancellation scheme [@Kurzo2018].[]{data-label="fig:si-canceller-nn"}](exportSchematics/SICancellerNN.pdf){width="1.0\columnwidth"}
Neural Network Canceller Hardware Architecture {#sec:nnhardware}
==============================================
In this section, we describe a generic hardware architecture that can be used to implement both the shallow NN-based SI canceller of [@Balatsoukas2018] and deeper NN-based SI cancellers such as the ones described in [@Kristensen2019]. We first provide an overview of the architecture, which is followed by a more detailed explanation of each component. In Fig. \[fig:si-canceller-nn\], we show the high-level architecture of a general NN-based canceller. The set of baseband samples $\left\{x[n],\hdots,x[n-L+1]\right\}$ is given as an input to a linear SI canceller and a NN-based SI canceller. These two SI cancellers operate in parallel in order to generate the linear and non-linear cancellation signals, respectively, which are then added (after the denormalization step for the NN) to produce the cancellation signal $\hat{y}_{\text{SI}}[n]$.
Macro-Pipeline Architecture
---------------------------
As shown in the example of Fig. \[fig:pipeline-archi\], in our architecture, the canceller NN layers are mapped to macro-pipeline stages. Each macro-pipeline stage requires several clock cycles to compute its outputs and it can start its computations as soon as valid outputs from the previous macro-pipeline stage become available. Due to the high throughput requirements of the SI cancellation task, we instantiate one macro-pipeline stage for each layer in the NN that is used for cancellation.
Let $\text{NE}_{l}$ denote the number of neurons in layer $l$. We note that $\text{NE}_0 = 2L$, $\text{NE}_{N_l+1} = 2$, and $\text{NE}_l = N_h$ for all hidden layers $l \in \{1,\hdots,\text{NE}_l\}$. The goal of a macro-pipeline stage is to compute $\mathbf{l}_{l}$ using expressions of the form -. Each element $j \in \{0,\hdots,{\text{NE}_l-1}\}$ of $\mathbf{l}_{l}$ can be computed as: $$\begin{aligned}
\mathbf{l}_{l}[j] = f_l \left( \mathbf{b}_l[j] + \sum_{i=0}^{\text{NE}_{l-1}-1} \mathbf{W}_l[i,j]\mathbf{l}_{l-1}[i] \right). \label{eq:dense-layer}\end{aligned}$$ The architecture of each macro-pipeline stage is shown in more detail in Fig. \[fig:dense-layer-archi\]. More specifically, each macro-pipeline stage contains an input interface, an array of $N_{\text{PE}}$ processing elements (PEs), a weights-and-biases memory, a control unit, and an output interface. We note that for simplicity, all weights, biases, and partial sums have a common bit-width of $Q$ bits and saturation is used in case of an overflow. More sophisticated quantization schemes are possible, but they are beyond the scope of this work.
The $N_{\text{PE}}$ PEs, whose internal structure is shown in Fig. \[fig:rpe-archi\], can be used to compute over multiple clock cycles using one of two possible schedules. In the neuron-by-neuron (NBN) schedule, neurons are processed sequentially and each of the $N_{\text{PE}}$ PEs computes a part of the sum in for a given neuron $j$. In the input-by-input (IBI) schedule, the inputs of layer $l$ (i.e., $\mathbf{l}_{l-1}$) are processed sequentially and the $N_{\text{PE}}$ PEs update the sum in with the term $\mathbf{W}[i,j]\mathbf{l}_{l-1}[i]$ for $N_{\text{PE}}$ distinct neurons in parallel. As an NBN macro-pipeline stage generates neuron output values sequentially, the optimal accelerator structure consists of an NBN macro-pipeline stage always being followed by an IBI macro-pipeline stage, allowing the IBI stage to start performing computations once the output of the first neuron of the preceding NBN stage has been computed. Once all inputs have been processed by the IBI stage, it immediately outputs multiple values to the NBN stage which follows it. Having an NBN stage after another NBN stage means that the second NBN stage would have to wait for all outputs of the previous stage to be generated before any processing can take place, and having an IBI stage followed by another IBI stage would mean that the second IBI stage cannot start processing before the first IBI stage has processed all its inputs. This structure of NBN and IBI stages connected in an alternating fashion masks a significant part of the latency and reduces the number of interconnects between two consecutive macro-pipeline stages. Since the exact architecture of each macro-pipeline stage depends on the processing schedule, we describe the details of the corresponding architectures separately in the next two sections.
![Example of a macro-pipeline architecture with two stages for a neural network with $N_l = 1$ hidden layers [@Kurzo2018]. More macro-pipeline stages can be added to the pipeline to implement neural networks of arbitrary depth $N_l$.[]{data-label="fig:pipeline-archi"}](exportSchematics/Layers.pdf){width="1.0\columnwidth"}
Neuron-by-Neuron Macro-Pipeline Architecture
--------------------------------------------
### Input Interface
The input interface consists of $N_{\text{PE}}$ multiplexers, which route each of the $\text{NE}_{l-1}$ elements of $\mathbf{l}_{l-1}$ to the correct PE.
### Processing Elements
In the NBN schedule, each PE is only associated with a single neuron, and therefore only a single partial sum needs to be stored in each PE. Thus, the PEs are simple multiply-and-accumulate (MAC) units and the memory shown in Fig. \[fig:rpe-archi\] is, in fact, a single $Q$-bit register.
### Control Unit
The main tasks of the control unit are to distribute the computations to the PEs and to stall the computations when no valid inputs are available or when the following macro-pipeline stage is not ready to accept new inputs. The computations are dispatched to the PEs as follows. When $N_{\text{PE}} \leq \text{NE}_{l-1}$, all $N_{\text{PE}}$ PEs are used to process a single neuron at a time and $\text{NE}_l\left\lceil\frac{\text{NE}_{l-1}}{N_{\text{PE}}}\right\rceil$ clock cycles are required to process all neurons. When $N_{\text{PE}} > \text{NE}_{l-1}$, we constrain $N_{\text{PE}}$ so that $N_{\text{PE}} = k \cdot \text{NE}_{l-1},~k \in \mathbb{N},$ and hence $k$ neurons are processed in parallel and $\left\lceil\frac{\text{NE}_l\text{NE}_{l-1}}{N_{\text{PE}}}\right\rceil$ clock cycles are required to process all neurons.
![Block diagram of the macro-pipeline stage architecture [@Kurzo2018].[]{data-label="fig:dense-layer-archi"}](exportSchematics/DenseLayer_TopLevel.pdf){width="1.025\columnwidth"}
### Weight and Bias Memories
The weight and bias memories for layer $l$ are used to store $\mathbf{W}_l$ and $\mathbf{b}_l$ and they can be written externally to re-configure the NN. The weights are organized in a memory that is $N_{\text{PE}}Q$ bits wide so that all PEs can be provided with data in parallel. A single word of the weight memory contains $N_{\text{PE}}$ weight values corresponding to $k$ different neurons. The bias memory, on the other hand, has a bit-width of $kQ$ bits.
### Output Interface
The output interface adds the partial sums from the $N_{\text{PE}}$ PEs using an adder tree, it adds the corresponding biases, and it applies the non-linear activation function $f_l$ for each of the $k$ neurons that are being processed in parallel. A register is added between the PEs and the output interface in order to reduce the critical path of the architecture. Moreover, the output interface forwards the outputs of the $k$ neurons that are processed in parallel to the next macro-pipeline stage.
### Latency
In the remainder of this work, we select $N_{\text{PE}}$ carefully so that both $\frac{\text{NE}_{l-1}}{N_{\text{PE}}}$ and $\frac{\text{NE}_l\text{NE}_{l-1}}{N_{\text{PE}}}$. With this setting, an NBN macro-pipeline stage requires $$\begin{aligned}
\mathcal{L}_l & = \frac{\text{NE}_l\text{NE}_{l-1}}{N_{\text{PE}}} + 1\end{aligned}$$ clock cycles to produce all outputs of NN layer $l$. However, one full set of outputs for a NN layer is actually produced every $\frac{\text{NE}_l\text{NE}_{l-1}}{N_{\text{PE}}}$ cycles, so that the throughput of the NBN macro-pipeline stage, measured in samples per clock cycle, is $$\begin{aligned}
\mathcal{T}_l & = \frac{N_{\text{PE}}}{\text{NE}_l\text{NE}_{l-1}}.\end{aligned}$$ Moreover, the first $k$ outputs of an NBN macro-pipeline stage become available after $$\begin{aligned}
\mathcal{L}_{l,\text{first}} & = \left\lfloor\frac{\text{NE}_{l-1}}{N_{\text{PE}}}\right\rfloor + 1\end{aligned}$$ clock cycles. Therefore, a potential IBI macro-pipeline stage that follows can already start its computations after the $\mathcal{L}_\text{first}$ clock cycles and that only $k \leq \text{NE}_{l}$ outputs need to be forwarded to the next stage at a time.
![Detailed view of the PE architecture that is used by both the NBN and the IBI macro-pipeline stages [@Kurzo2018].[]{data-label="fig:rpe-archi"}](exportSchematics/NN_PE.pdf){width="1.0\columnwidth"}
Input-by-Input Macro-Pipeline Architecture
------------------------------------------
### Input & Output Interfaces
The input and output interfaces of the IBI macro-pipeline stage are similar to that of the NBN macro-pipeline stage. The main difference is that the IBI output interface forwards the outputs of all $\text{NE}_l$ neurons that are processed in parallel to the next macro-pipeline stage.
### Processing Elements
In the IBI schedule, each PE can be associated with multiple neurons. Therefore, several partial sums potentially need to be stored in each PE. Thus, the PEs are MAC units and the memory shown in Fig. \[fig:rpe-archi\] has a $\left\lceil\frac{\text{NE}_l}{N_{\text{PE}}}\right\rceil Q$ bits.
### Control Unit
In the IBI schedule, when $N_{\text{PE}} \leq \text{NE}_l$, all $N_{\text{PE}}$ PEs are used to update the $\text{NE}_l$ neurons of layer $l$ sequentially with a new input value $\mathbf{l}[i]$ and $\text{NE}_{l-1}\left\lceil\frac{\text{NE}_l}{N_{\text{PE}}}\right\rceil$ clock cycles are required to process all neurons. When $N_{\text{PE}} > \text{NE}_l$, we constrain $N_{\text{PE}}$ so that $N_{\text{PE}} = k\text{NE}_l,~k \in \mathbb{N},$ and $k$ inputs are processed in parallel. Hence, $\left\lceil\frac{\text{NE}_l\text{NE}_{l-1}}{N_{\text{PE}}}\right\rceil$ clock cycles are required to process all neurons.
### Weight and Bias Memories
The weight and bias memories are similar to those of the NBN macro-pipeline stage. A single word of the weight memory contains $N_{\text{PE}}$ weights corresponding to $k$ different neurons. The bias memory has a bit-width of $\text{NE}_l Q$ bits in the IBI macro-pipeline stage.
### Latency
Similarly to the NBN schedule, we choose $N_{\text{PE}}$ carefully so that both $\frac{\text{NE}_l}{N_{\text{PE}}}$ and $\frac{\text{NE}_l\text{NE}_{l-1}}{N_\text{PE}}$ are always integers. Then, the latency and the throughput are $$\begin{aligned}
\mathcal{L}_l & = \frac{\text{NE}_l\text{NE}_{l-1}}{N_{\text{PE}}} + 1\end{aligned}$$ clock cycles and $$\begin{aligned}
\mathcal{T} & = \frac{N_{\text{PE}}}{\text{NE}_l\text{NE}_{l-1}},\end{aligned}$$ samples per clock cycle, respectively. Moreover, all $\text{NE}_l$ outputs of an IBI macro-pipeline stage become available simultaneously after: $$\begin{aligned}
\mathcal{L}_{l,\text{first}} & = \frac{\text{NE}_l\text{NE}_{l-1}}{N_{\text{PE}}} + 1 ,\end{aligned}$$ clock cycles.
Overall Neural Network Canceller Architecture
---------------------------------------------
The overall NN architecture consists of $N_l$ macro-pipeline stages with pipeline registers added between them. The first hidden layer uses an NBN macro-pipeline stage and the second hidden layer (or the output layer when $N_l=1$) uses an IBI macro-pipeline stage. Further layers use NBN and IBI macro-pipeline stages in an alternating fashion as previously discussed. The $\text{NE}_0 = 2L$ inputs $\mathbf{l}_0$ of the first NBN macro-pipeline stage that implements the computations of the first hidden layer are assumed to all be available in parallel. The number of PEs instantiated for layer $l$ is denoted by $N_{\text{PE},l}$. The computations for the linear canceller are done in parallel with the NN by instantiating a standard complex FIR filter with $N_{\text{PE},\text{linear}}$ complex-valued PEs. The latency of the linear canceller in clock cycles $$\begin{aligned}
\mathcal{L}_{\text{linear}} & = \left\lceil \frac{L}{N_{\text{PE},\text{linear}}} \right\rceil.\end{aligned}$$ Since the linear canceller is not pipelined, it holds that $\mathcal{T}_{\text{linear}}~=~ \frac{1}{\mathcal{L}_{\text{linear}}}$. The throughput of the overall NN canceller architecture is: $$\begin{aligned}
\mathcal{T} & = \min \left(\min_{l\in\{1,\hdots,N_l+1\}} \mathcal{T}_l, \; \mathcal{T}_{\text{linear}}\right).\end{aligned}$$ Since it is typically not very costly in terms of resources to ensure that $\mathcal{T}_{\text{linear}} \geq \mathcal{T}_l,~l\in\{1,\hdots,N_{l+1}\},$ in practice $\mathcal{T}$ is usually limited by $\min_{l} \mathcal{T}_l$. As opposed to the throughput, the latency of the overall NN canceller is more complicated to derive in general. However, in the special case where the number of PEs for each layer $l$ is chosen such that no stalling happens and $N_l+1$ is even, the latency can be calculated as: $$\begin{aligned}
\mathcal{L} & = \max \left(\sum _{l=1}^{(N_l+1)/2}\mathcal{L}_{2l-1,\text{first}}+\mathcal{L}_{2l}, \mathcal{L}_{\text{linear}}\right).\end{aligned}$$
Finally, we note that the denormalization step shown in Fig. \[fig:si-canceller-nn\] is constrained to scaling with powers of two, which can be implemented efficiently with simple shifting operations, both during training and during inference.
Polynomial Canceller Hardware Architecture {#sec:polyhardware}
==========================================
Since, to the best of our knowledge, there are no published implementations of polynomial SI cancellers in the literature, we provide our own optimized reference implementation. Our polynomial SI canceller architecture, which is shown in Fig. \[fig:poly-archi\], is largely based on the NN architecture since the main computational tasks of the two cancellers are very similar (i.e., computation of weighted sums). The main differences are that the input interface also computes the basis functions, that $N_{\text{CPE}}$ complex PEs (CPEs) are used to perform computations on complex values, and that there is only a single macro-pipeline stage. In the remainder of this section, we explain how the basis functions can be computed efficiently and we describe the polynomial SI canceller in more detail.
Basis Function Computation
--------------------------
The computation of the $N_{\text{BF}}$ basis functions in for each cancellation sample seems like a cumbersome task. Fortunately, we can show that the basis functions have a number of properties that enable their efficient computation. First, significant basis function re-use is possible. In particular, after $\hat{y}_{\text{SI}}[n-1]$ has been computed based on $\text{BF}_{p,q}(x[n{-}1{-}l]),~l\in \{0,\hdots,{L{-}1}\},~p\in \{1,3,\hdots,P\},~q \in \{0,\hdots,p\}$, the basis functions for $l\in \{0,\hdots,{L{-}2}\}$ can be stored and re-used for the computation of $\hat{y}_{\text{SI}}[n]$. As such, the only new basis functions that need to be computed for $\hat{y}_{\text{SI}}[n]$ are $\text{BF}_{p,q}(x[n]),~p\in \{1,3,\hdots,P\},~q \in \{0,\hdots,p\}$. This requires $\frac{L-1}{4}\left(P+1\right)\left(P+3\right)$ memory elements, but reduces the number of basis functions that need to be computed by a factor of $L$ from $\frac{L}{4}(P+1)(P+3)$ to $\frac{1}{4}(P+1)(P+3)$. Moreover, the following proposition shows two additional properties of the basis functions.
For the basis functions in , it holds that:
1. $\text{BF}_{p,q}(x) = \left(\text{BF}_{p,p-q}(x)\right)^*$
2. $\text{BF}_{p,q}(x) = x^2\text{BF}_{p-2,q-2}(x)$
Both properties follow from the definition of the basis function in . Specifically, for 1) we have: $$\begin{aligned}
\text{BF}_{p,q}(x) & = x^{q}(x^*)^{p-q} \nonumber \\
& = \left(x^{p-q}(x^*)^{p-(p-q)}\right)^* \nonumber \\
& = \left(\text{BF}_{p,p-q}(x)\right)^*,\end{aligned}$$ and for 2) we have: $$\begin{aligned}
\text{BF}_{p,q}(x) & = x^q(x^*)^{p-q} \nonumber\\
& = x^2x^{q-2}(x^*)^{p-2-(q-2)} \nonumber \\
& = x^2\text{BF}_{p-2,q-2}(x).\end{aligned}$$
Property 1) enables a computation reduction by a factor of two since for every $p \in \{1,3,\hdots,P\}$, it is sufficient to compute $\text{BF}_{p,q}(x)$ only for $q \in \left\{\frac{p+1}{2},\hdots,p\right\}$ and the remaining basis functions for $q \in \left\{0,\hdots,\frac{p-1}{2}\right\}$ can be obtained by simple conjugation. Moreover, property 2) reveals an efficient dynamic programming (DP) method to compute the basis functions for $x[n]$, which is shown in Algorithm \[alg:bf\]. Algorithm \[alg:bf\] requires one multiplication to pre-compute $(x[n])^2$ and $\frac{1}{8}(P+1)(P+3)-2$ multiplications for all executions of line \[alg:bf:mult\]. The conjugation in line \[alg:bf:conj\] does not require any multiplications as it is a simple sign change of the imaginary part of $\text{BF}_{p,p-q}(x)$. As such, the total number of multiplications to compute the basis functions for a baseband sample $x[n]$ is: $$\begin{aligned}
N_{\text{MUL},\text{BF}} & = \frac{1}{8}(P+1)(P+3)-1.\end{aligned}$$ One downside of the DP approach is that only the inner loop in Algorithm \[alg:bf\] can be parallelized. However, in most typical applications we have $P \leq 9$, so that the outer loop in Algorithm \[alg:bf\] is executed very few times. We note that, due to the efficiency of Algorithm \[alg:bf\], $N_{\text{MUL},\text{BF}}$ is significantly smaller than $N_{\text{MUL},\text{poly}}$, which justifies ignoring the multiplications of the basis function computations in for simplicity.
**Input:** $x[n]$ **Outputs:** $\text{BF}_{p,q}(x[n])$ for $p \in \{1,3,\hdots,P\},~q\in\{0,\hdots,p\}$ $\text{BF}_{1,0}(x[n]) \gets (x[n])^*$ $\text{BF}_{1,1}(x[n]) \gets x[n]$ $\text{BF}_{p,q}(x[n]) \gets \left(x[n]\right)^2\text{BF}_{p-2,q-2}(x[n])$ \[alg:bf:mult\] $\text{BF}_{p,p-q}(x[n]) \gets \text{BF}_{p,q}(x[n])^*$ \[alg:bf:conj\]
Polynomial Canceller Architecture
---------------------------------
We use a high-level structure that is similar to the NN-based cancellers in Fig. \[fig:si-canceller-nn\] in the sense that linear cancellation is done in parallel to non-linear cancellation and the polynomial SI canceller focuses only on the non-linear part of the SI signal. Since most of the SI signal is linear, removing the linear term separately significantly reduces the dynamic range of the values within the polynomial SI canceller, which in turn allows us to reduce the common quantization bit-width $Q$ for the real and the imaginary parts of the involved quantities.
### Input & Output Interfaces
The input interface consists of $N_{\text{CPE}}$ multiplexers, which route each of the $N_{\text{BF}}$ BFs to the correct CPE in order to compute parts of the sum in . As mentioned previously, the input interface also computes the BFs using $N_{\text{CPE},\text{BF}}$ CPEs. Since only the inner loop in Algorithm \[alg:bf\] can be parallelized, it is reasonable to constrain $N_{\text{PE},\text{BF}}$ so that $N_{\text{PE},\text{BF}} \leq \frac{P+1}{2}$. The number of clock cycles required to compute all BFs with $N_{\text{PE},\text{BF}}$ PEs is: $$\begin{aligned}
\mathcal{L}_{\text{BF}} & = 1+\sum _{\substack{p=3,\\p \text{ odd}}}^{P} \left\lceil\frac{p+1}{2N_{\text{CPE},\text{BF}}}\right\rceil, \label{eq:latencybf}\end{aligned}$$ where one clock cycle is used to pre-compute $x^2$ and the result (as well as $x^*$) are stored in two $2Q$-bit registers. The $\frac{L-1}{4}\left(P+1\right)\left(P+3\right)$ BFs that are re-used are stored in a circular buffer.
The output interface consists of an adder tree that adds up the partial sums stored in the $N_{\text{CPE}}$ CPEs in order to produce the final result.
![Block diagram of the polynomial canceller architecture.[]{data-label="fig:poly-archi"}](exportSchematics/PolyApprox_TopLevel.pdf){width="1.025\columnwidth"}
### Complex Processing Elements
The $N_{\text{CPE}}$ CPEs are complex MAC units with a $Q$-bit register to store partial sums. The complex MAC units are implemented using three real-valued multipliers and five real-valued adders.
### Control Unit
Similarly to the NN-based canceller, the main tasks of the control unit are to distribute the computations to the CPEs and to stall the computations when no valid inputs are available. The control unit schedules the operation so that the CPEs first compute the terms of that are based on BFs that are already available in the circular buffer. In the meantime, the input interface computes the $\frac{1}{4}(P+1)(P+3)$ BFs that depend on the new sample $x[n]$.
### Parameter Memory
The parameter memory is used to store the complex-valued $\hat{h}_{p,q}$ parameters of the polynomial canceller. The memory contains $\left\lceil\frac{N_{\text{BF}}}{N_{\text{CPE}}}\right\rceil$ words that are $2QN_{\text{CPE}}$ bits wide so that all $N_{\text{CPE}}$ CPEs can be provided with the parameters in parallel.
### Latency
In most practical cases, the latency of computing the new BFs is significantly smaller than the latency of computing the terms of , so that the latency of this operation is masked entirely and can be ignored. For example, for $P=7$ and $N_{\text{CPE},\text{BF}} = 1$, according to it takes only $10$ clock cycles to compute the new BFs. Setting $N_{\text{CPE},\text{BF}} = 2$ reduces the number of required cycles to $6$. As such, we can safely assume that the latency of the polynomial canceller is limited by the computation of . The latency of the polynomial SI canceller is given by: $$\begin{aligned}
\mathcal{L}_{\text{poly}} & = \left\lceil\frac{N_{\text{BF}}}{N_{\text{CPE}}} + 1,\right\rceil,\end{aligned}$$ where one clock cycle is required by the adder tree in the output interface to produce the final output. Since a pipeline register is inserted before the adder tree of the output interface, the throughput of the polynomial SI canceller, measured in samples per clock cycle, is given by $\mathcal{T}_{\text{poly}} = \frac{1}{\mathcal{L}_{\text{poly}}}$.
Numerical and Hardware Implementation Results {#sec:results}
=============================================
In this section, we compare the polynomial SI canceller with the NN-based SI cancellers in terms of their SI cancellation performance and their hardware implementation complexity. Specifically, we first provide a performance comparison of the polynomial SI canceller with the shallow NN SI canceller of [@Balatsoukas2018] as well as with an instance of a deep NN SI canceller of [@Kristensen2019]. We then present a comparison of FPGA and ASIC implementation results for the polynomial SI canceller and the NN-based SI cancellers in an equi-performance scenario.
Self-Interference Cancellation Performance Comparison
-----------------------------------------------------
### Comparison Setup {#sec:comparison}
The complexity expressions for the polynomial SI canceller in - and the NN SI cancellers in - can not be compared directly because they contain different sets of parameters. Thus, in order to perform a fair complexity comparison, we select values for $L$, $P$, $N_l$, and $N_h$ so that the compared polynomial and NN cancellers have as similar SI cancellation performance as possible.
For the performance evaluation, we use the dataset that was used in [@Balatsoukas2018], which consists of a $10$ MHz QPSK-modulated OFDM SI signal sampled at $20$ MHz that is generated using an actual full-duplex testbed with a transmit power of $10$ dBm.[^8] The dataset contains $20,000$ time-domain SI baseband samples, out of which 90% is used for training and 10% for the evaluation of the SI performance. For NN training, we use a mini-batch size of $B=32$ and the Adam optimizer [@Kingma2015] with a learning rate of $\lambda = 0.004$.
### Results
In Fig. \[fig:performance\], we show the power spectral density (PSD) of the received SI signal $y_{\text{SI}}[n]$ before any SI cancellation is performed, the PSD of the received signal when no transmission takes place (i.e., the *effective* noise floor of the receiver), as well as the PSDs of the SI signals after linear SI cancellation and after non-linear SI cancellation with the polynomial and NN-based cancellers. The results are obtained for $L=13$ for all SI cancellers and $P=5$ for the polynomial SI canceller. Moreover, the shallow NN has $N_l=1$ hidden layer with $N_h = 18$ neurons, while the deep NN has $N_l=5$ hidden layers with $N_{h}= 6$ neurons per layer. These parameter values are chosen as follows. First, we select $L$ by slowly increasing its value until no further *linear* cancellation gains are obtained. We then use the polynomial SI canceller and we increase $P$ until the gain in SI cancellation performance becomes very small. When going from $P=5$ to $P=7$, the SI cancellation only improves by $0.3$ dB while the computational complexity almost doubles, so that $P=5$ provides a sensible complexity-performance trade-off. We then use the same value of $L$ for the NN-based SI cancellers and we select $N_l$ and $N_h$ to match the performance of the polynomial SI canceller.
We observe that, with these parameter settings, all SI cancellers indeed achieve very similar performance and can cancel the SI very close to the receiver noise floor, with the polynomial canceller being $0.4$ dB and $0.5$ dB better than the shallow and the deep NN-based cancellers, respectively. We note that, in light of our recent results in [@Kristensen2019 Fig. 5], using $P=5$ for the polynomial SI canceller results in a more fair comparison than using $P=7$ as was done in [@Balatsoukas2018]. However, as we show in the sequel, even in this case there are very clear advantages in terms of the implementation complexity when using an NN-based canceller.
In Fig. \[fig:convergence\], we show the training convergence behavior of the shallow and deep NN SI cancellers. We observe that the shallow and deep NNs require only $7$ and $12$ training epochs to achieve more than $5$ dB of non-linear cancellation, respectively. Training for up to $20$ epochs further increases the cancellation performance to approximately $5.5$ dB, but there are clear diminishing returns. Moreover, we observe that the cancellers have similar performance on the training and test samples, meaning that there are no overfitting issues. We note, however, that during our experiments we observed that the deep NN is more sensitive to the weight initialization.
In Table \[tab:perf\], we show the complexity of the three SI cancellers in terms of the number of real-valued multiplications and additions given by - and -. We observe that the polynomial SI canceller requires the largest number of additions, while the shallow NN SI canceller requires the largest number of multiplications. The deep NN SI canceller, on the other hand, achieves a significant complexity reduction as it requires $25$% fewer multiplications and $60\%$ fewer additions than the polynomial SI canceller.
------------------- -------- -------- -----------
Cancellation (dB) $44.4$ $44.4$ $44.3$
$L$ $13$ $13$ $13$
$P$ $5$ n/a n/a
$N_l$ n/a $1$ $5$
$N_h$ n/a $18$ $6$
Real Add. $1090$ $613$ **$433$**
Real Mult. $468$ $543$ **$351$**
------------------- -------- -------- -----------
: Comparison of the SI cancellation performance and the complexity of polynomial and NN-based cancellers.[]{data-label="tab:perf"}
FPGA and ASIC Implementation Results {#sec:implresults}
------------------------------------
### Comparison Setup {#comparison-setup}
In Section \[sec:comparison\], we already showed that using $L=13$ for all cancellers, $P=5$ for the polynomial canceller, a single hidden layer with $N_h = 18$ neurons for the shallow NN canceller, and $N_l=5$ hidden layers with $N_{h}= 6$ for the deep NN canceller, leads to practically identical SI cancellation performance. However, in order to perform a meaningful comparison of FPGA and ASIC implementation results, the quantization bit-width $Q$ for the different cancellers also needs to be selected to individually minimize the implementation complexity while keeping the performance of the SI cancellers similar. In Fig. \[fig:quantization\], we show the cancellation performance for the polynomial SI canceller and the NN SI cancellers as a function of the quantization bit-width $Q$. We observe that both NN SI cancellers generally require a lower quantization bit-width $Q$ compared to the polynomial SI canceller to achieve the same SI cancellation performance. Moreover, for the hardware implementation results presented in this section, we can choose $Q=17$ for the shallow NN SI canceller, $Q=19$ for the deep NN SI canceller, and $Q=20$ for the polynomial SI canceller, as this choice leads to effectively identical SI cancellation performance and a very small loss with respect to the corresponding floating-point implementations for all cancellers. The deep NN SI canceller requires two additional integer bits compared to the shallow NN SI canceller, due to larger absolute output values in the hidden layers. This effectively shifts the bit-width versus SI cancellation performance curve of the deep NN SI canceller by two bits to the right compared to the shallow NN SI canceller.
For the shallow NN SI canceller, we set $N_{\text{PE},1} = 52$ and $N_{\text{PE},2} = 4$ so that $\mathcal{T}_1 = \mathcal{T}_2 = \nicefrac{1}{9}$. With this setting, the macro-pipeline is perfectly balanced and one cancellation sample is produced every $9$ clock cycles. Furthermore, $N_{\text{CPE},\text{linear}} = 2$ CPEs are instantiated for the NN SI canceller to ensure that the linear cancellation step can be completed in the same number of cycles. For the deep NN SI canceller, we set $N_{\text{PE},1} = 26$ for the first hidden layer, $N_{\text{PE},l} = 6$ for the remaining hidden layers, and $N_{\text{PE},6} = 2$ for the output layer so that $\mathcal{T}_l = \nicefrac{1}{6}$ for all $l \in \{1,\hdots,6\}$ and the macro-pipeline is balanced. The deep NN SI canceller uses $N_{\text{CPE},\text{linear}} = 3$ CPEs for the linear canceller due to the increased throughput requirements. The shallow NN SI canceller thus requires a total of $56$ PEs and the deep NN SI canceller requires a total $52$ PEs, but the deep NN SI canceller requires one more CPE than the shallow NN canceller. Finally, for the polynomial canceller, we use $N_{\text{CPE}} = 12$ complex PEs so that the $156$ complex multiplications required to compute $\eqref{eq:finalbf}$ for $L=13$ and $P=5$ can be carried out in $\mathcal{L}_{\text{poly}} = 13$ clock cycles. We also use $N_{\text{CPE},\text{BF}} = 1$ CPE for the BF computation, meaning that $\mathcal{L}_{\text{BF}} = 10$. Since $\mathcal{L}_{\text{poly}} > \mathcal{L}_{\text{BF}}$, one cancellation sample is produced every $\mathcal{L}_{\text{poly}} = 13$ clock cycles.
--------------------- --------------------------- --------------------------- ------------------
Slices $2099$ ($2.77$%) $\mathbf{1700}$ ($2.24$%) $2212$ ($2.91$%)
LUT (logic) $5693$ ($1.88$%) $\mathbf{2833}$ ($0.93$%) $3761$ ($1.24$%)
LUT (RAM) $\mathbf{884}$ ($0.29$%) $1678$ ($0.55$%) $1767$ ($0.58$%)
Registers $\mathbf{2462}$ ($0.41$%) ${2625}$ ($0.43$%) $3887$ ($0.64$%)
DSP Slices $\mathbf{42}$ ($1.50$%) $62$ ($2.21$%) ${61}$ ($2.07$%)
Frequency (MHz) $85.5$ $\mathbf{86.6}$ $69.7$
T/P (Msamples/s) $6.6$ $9.6$ $\mathbf{11.6}$
T/P (cycles/sample) $13$ $9$ $\mathbf{6}$
Latency (ns) $152$ $\mathbf{115}$ $304$
Latency (cycles) $13$ $\mathbf{10}$ $21$
--------------------- --------------------------- --------------------------- ------------------
: FPGA Implementation Results (Virtex-7 XC7VX485). The polynomial canceller uses $L=13$ and $P=5$, the shallow NN canceller uses $N_l=1$ and $N_h=18$, the deep NN canceller uses $N_l=5$ and $N_h=6$.[]{data-label="tab:fpga-res"}
Target
----------------------------------------- --------------------- --------------- ------------------ --------------------- --------------- -----------------
Poly. ($P=5$) Shallow NN Deep NN Poly. ($P=5$) Shallow NN Deep NN
Area (mm$^2$) $\mathbf{0.179}$ $0.255$ ${0.187}$ $\mathbf{0.155}$ $0.238$ ${0.162}$
Area (kGE) $\mathbf{366\,131}$ $521\,654$ ${380\,967}$ $\mathbf{317\,165}$ $485\,142$ ${330\,125}$
Frequency (MHz) $\mathbf{354}$ $326$ ${351}$ $270$ $180$ $120$
Throughput (Msamples/s) $27.2$ $36.2$ $\mathbf{58.5}$ $20.0$ $20.0$ $20.0$
Throughput (cycles/sample) $13$ $9$ $\mathbf{6}$ $13$ $9$ $\mathbf{6}$
Latency (ns) $37$ $\mathbf{31}$ $60$ $\mathbf{48}$ $56$ $175$
Latency (cycles) $13$ $\mathbf{10}$ $21$ $13$ $\mathbf{10}$ $21$
Total Power (mW) $\mathbf{49.8}$ $61.9$ ${54.5}$ $31.3$ $38.0$ $\mathbf{22.1}$
Internal Power (mW) $\mathbf{26.0}$ $35.7$ ${30.0}$ $16.9$ $21.1$ $\mathbf{11.7}$
Switching Power (mW) $\mathbf{22.4}$ $24.4$ ${23.2}$ $13.3$ $15.4$ $\mathbf{9.4}$
Leakage Power (mW) $1.4$ $1.8$ $\mathbf{1.3}$ $1.1$ $1.5$ $\mathbf{1.0}$
Hardware Efficiency (Msamples/s/mm$^2$) $152.0$ $142.0$ $\mathbf{312.8}$ $\mathbf{129.0}$ $84.0$ $123.5$
Energy Efficiency (nJ/sample) $1.8$ $1.7$ $\mathbf{0.9}$ $1.6$ $1.9$ $\mathbf{1.1}$
### FPGA Implementation Results {#sec:resultsfpga}
In Table \[tab:fpga-res\], we show place-and-route (PAR) results on a Xilinx Virtex-7 XC7VX485 (speed grade -2) FPGA, which contains a total of $75.9$k slices, $303.6$k LUTs, $607.2$k flip-flops, and $2.8$k DSP slices. A clock frequency target of $100$ MHz is used for all cancellers.
We observe that the shallow NN SI canceller has a lower slice and LUT as logic utilization than the polynomial SI canceller and a $45$% higher throughput. The higher throughput of the shallow NN SI canceller comes both from a lower number of cycles per sample and from a slightly higher operating frequency compared to the polynomial SI canceller. We also note that the polynomial SI canceller requires approximately $32$% fewer DSP slices than both the shallow and the deep NN SI canceller. The deep NN SI canceller uses more resources than the shallow NN SI canceller, but it has a $77$% higher throughput than the polynomial SI canceller. The main additional cost for the deep NN SI canceller compared to the shallow NN SI canceller comes from registers and LUTs used as logic. Even though the deep NN SI canceller is not able to achieve a clock frequency as high as the other cancellers, it still has the highest throughput. We also observe that the shallow NN SI canceller has the lowest latency, followed by the polynomial SI canceller and the deep NN SI canceller.
### ASIC Implementation Results {#sec:resultsasic}
In Table \[tab:asic\], we present ASIC implementation results for the polynomial SI canceller and the two NN SI cancellers using a 28 nm FD-SOI technology. We target two different points in terms of the operating frequency, namely, a maximum throughput point and a point where each SI canceller achieves a throughput of exactly $20$ MS/s, which is sufficient for the dataset that we consider. In both cases, we use slow-slow corners, an operating voltage of $0.7$ V, and an operating temperature of $125$ C. For the power results, post-PAR simulations are used both to verify the design and to accurately estimate the switching activity.
We observe that, for the maximum throughput operating point, the NN SI cancellers are both significantly faster and more energy efficient than the polynomial SI canceller. However, the polynomial SI canceller is generally smaller and has a lower power consumption. More specifically, the shallow NN SI canceller is $33$% faster but also $30$% larger than the polynomial SI canceller. As a result, its area efficiency is $7$% lower than that of the polynomial SI canceller. The deep NN SI canceller, on the other hand, is only $4$% larger than the polynomial SI canceller and at the same time $115$% faster and has a $106$% better area efficiency. The energy efficiency of the NN SI cancellers is also significantly better compared to the polynomial SI canceller. Specifically, the shallow and deep NN SI cancellers improve the energy efficiency by $6$% and $50$% compared to the polynomial SI canceller, respectively. Finally, we observe that the deep NN SI canceller has the worst latency ($60$ ns), followed by the polynomial SI canceller ($37$ ns) and the shallow NN canceller ($31$ ns).
As mentioned previously, for the dataset that we use in this work, a throughput of $20$ MS/s is sufficient for real-time operation. We observe that the relaxed timing requirements, in this case, reduce the area of the polynomial SI canceller by $13$%, the shallow NN SI canceller by $7$% and the deep NN SI canceller by $13$% compared to the results for the maximum throughput operating point. Interestingly, the energy efficiency per sample only improves for the polynomial canceller, whereas it becomes slightly worse for the two NN SI cancellers. Nevertheless, the deep NN SI canceller is still $30$% more energy efficient than the polynomial SI canceller and only $4$% less area efficient. Moreover, at this operating point, the polynomial canceller has the lowest latency ($48$ ns) and the deep NN canceller has the lowest power consumption.
Conclusion {#sec:conclusion}
==========
In this paper, we presented a high-throughput hardware architecture for a NN-based SI cancellation scheme for full-duplex radios. We also presented, to the best of our knowledge, the first efficient hardware architecture for polynomial SI cancellation in the literature, which we used as a comparison baseline for the NN-based SI cancellers. Our implementation results show that the NN SI cancellers have significantly lower computational complexity than a conventional polynomial SI canceller, which translates into substantial area and energy savings when the schemes are implemented in hardware. Specifically, an ASIC implementation of a deep NN-based SI canceller has up to $2.1\times$ and $2\times$ better hardware efficiency and energy effiency when compared to a conventional polynomial SI canceller, respectively.
[^1]: Y. Kurzo is with ON Semiconductor, 2074 Marin-Epagnier, Switzerland (e-mail: [email protected]).
[^2]: A. Kristensen and A. Burg are with the Telecommunications Circuits Laboratory, École polytechnique fédérale de Lausanne, 1015 Lausanne, Switzerland (e-mail: {andreas.kristensen,andreas.burg}@epfl.ch).
[^3]: A. Balatsoukas-Stimming is with the Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands (e-mail: [email protected]).
[^4]: Parts of this work were presented at the 2018 Asilomar Conference on Signals, Systems, and Computers [@Kurzo2018].
[^5]: This work was supported by the Swiss National Science Foundation under project \#200021\_182621.
[^6]: We note that this simplification is justified in Section \[sec:polyhardware\].
[^7]: We note that these two additions were not included in [@Balatsoukas2018], but we include them here for the sake of accuracy.
[^8]: The dataset is available at <https://github.com/abalatsoukas/fdnn>.
|
---
abstract: |
We determine non-perturbatively the normalisation parameter $\zm\zp/\za$ as well as the Symanzik coefficients $\bm$ and $\ba-\bp$, required in ${\rm O}(a)$ improved quark mass renormalisation with Wilson fermions. The strategy underlying their computation involves simulations in $\NF=3$ QCD with ${\rmO}(a)$ improved massless sea and non-degenerate valence quarks in the finite-volume Schrödinger functional scheme. Our results, which cover the typical gauge coupling range of large-volume $\NF=2+1$ QCD simulations with Wilson fermions at lattice spacings below $0.1\,\Fm$, are of particular use for the non-perturbative calculation of ${\rm O}(a)$ improved renormalised quark masses.\
address:
- |
Università di Roma “Tor Vergata”, Dipartimento di Fisica,\
Via della Ricerca Scientifica 1, 00133 Rome, Italy
- |
INFN, “Rome Tor Vergata” Division, c/o Dipartimento di Fisica,\
Via della Ricerca Scientifica 1, 00133 Rome, Italy
- 'Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland'
- |
Westfälische Wilhelms-Universität Münster, Institut für Theoretische Physik,\
Wilhelm-Klemm-Stra[ß]{}e 9, 48149 Münster, Germany
author:
- Giulia Maria de Divitiis
- Patrick Fritzsch
- Jochen Heitger
- Carl Christian Köster
- Simon Kuberski
- Anastassios Vladikas
title: 'Non-perturbative determination of improvement coefficients $\bm$ and $\ba-\bp$ and normalisation factor $\zm \zp/\za$ with $\NF = 3$ Wilson fermions '
---
Lattice QCD ,Non-perturbative Effects ,Symanzik Improvement
|
---
author:
- Kenji Mitsuhashi
- Joemini Poudel
- 'Thomas P. Matthews'
- 'Alejandro Garcia-Uribe'
- 'Lihong V. Wang [^1]'
- 'Mark A. Anastasio [^2]'
title: '[[A forward-adjoint operator pair based on the elastic wave equation for use in transcranial photoacoustic computed tomography]{}]{}'
---
[^1]: Department of Biomedical Engineering, Washington University in St.Louis, St.Louis, MO USA ().
[^2]: Department of Biomedical Engineering, Washington University in St.Louis, St.Louis, MO USA ().
|
---
author:
- |
The IceCube Collaboration [^1]\
[*<http://icecube.wisc.edu/collaboration/authors/icrc19_icecube>*]{}\
E-mail:
bibliography:
- 'bib.bib'
title: 'Measurement of the multi-TeV neutrino cross section with IceCube using Earth absorption'
---
First Measurement of Neutrino Cross Section with IceCube
========================================================
Neutrinos are weakly interacting particles, which interact through charge current and neutral current interactions. The cross section of neutrino interactions has only measured in neutrino beams at accelerator experiments up to $400\, \mathrm{GeV}$. IceCube is able to detect neutrinos at energies orders of magnitude higher than accelerator experiments. Measuring the absorption of neutrinos within the Earth at multi-TeV has allowed for the cross section measurement to be extended beyond the range of accelerator experiments. The early development of this idea was to use neutrinos from accelerators to measure the Earth’s density profile [@JAIN1999193]. With the development of neutrino telescopes this idea was adapted to use the atmospheric neutrino flux. In 2002, Hooper proposed using the neutrino flux to study the neutrino cross section instead of the Earth’s density [@Hooper2002]. He proposed comparing the ratio of upgoing to downgoing neutrinos, however this does not take into account the change in absorption with zenith angle.
![ Neutrino absorption is observed by measuring how the the neutrino energy spectrum changes with zenith angle. Neutrinos from the horizon have no absorption and thus provide a baseline. Neutrinos with a near vertical trajectory show the increase in absorption with energy. The right figure shows the transmission probability predicted by the Standard Model for neutrinos to transit the Earth as a function of energy and zenith angle. Both charge current and neutral current interactions are included. The dashed line indicates the core mantle boundary [@xsec]. \[anime\]](CrossSectionAnimation.png){width="90.00000%"}
The detection principle for this analysis, shown in Figure \[anime\], is that neutrinos which traverse the Earth are absorbed compared to those originating from near the horizon. The difference in the flux for different zenith angles and energy can be used to detect a change in the cross section from the Standard Model prediction. The transmission probability in Figure \[anime\] includes charge current and neutral current (NC) interactions which interact via a boson exchange, $W^{\pm}$ for charged current and $Z^{0}$ for NC. In neutral current interactions, the outgoing lepton will be a neutrino with a reduced energy. In charged current interactions, the effect is that the neutrino is lost in the interaction. This is the cause of neutrinos being absorbed within the Earth as more charged current interactions occur.
The IceCube neutrino detector has a sample of diffuse neutrinos coming through the Earth which are mostly produced in cosmic ray air showers, with a fraction coming from astrophysical sources [@6yrDiffuse]. IceCube uses Digital Optical Modules (DOMs) arrayed in a grid to detect Cherenkov light from the charged secondary particles [@IceCubeInst]. For this analysis IceCube’s upgoing muon sample was used to measure the muon neutrino flux.
This study used one year of data from when the detector was in partial configuration, consisting of 79 strings each containing 60 DOMs. The final sample contains 10,784 through going muon neutrino events that was used to produce the first multi-TeV cross section measurement $1.30_{-0.19}^{+0.21}(stat.)_{-0.43}^{+0.39}(syst.)$ times the Standard Model prediction [@xsec]. The result shown in Figure \[result\] is compared to the Standard Model prediction for the expected mixture of neutrinos and antineutrinos in IceCube. This result is consistent with the expected charged and neutral current interactions. The result is found from a maximum likelihood fit of the data, binned in zenith and energy. The Monte Carlo simulations include the prediction of the astrophysical and atmospheric neutrino contribution where the cross section is included as a parameter for the probability of a neutrino to be absorbed as it travels through the Earth. The transmission probability is calculated by propagating neutrinos through Earth assuming a specific model with cross section varied by a multiple of the Standard Model value. The propagation is done for a range of zenith angles where the neutrino travels through the Earth with an energy range of $10^2 -10^8 \mathrm{GeV}$. This analysis used the Preliminary Reference Earth Model for the Earth’s density [@prem].
The nuisance parameters for this fit include previous IceCube measurements of the astrophysical spectrum which are based on the assumption that the Standard Model cross section is correct. An increase in neutrino cross section would result in fewer neutrinos at the detector, which will affect the number of events in the fit. To insure this measurement conserves the total number of events this fit uses the product of each flux with the cross section in order to constrain to prior data. Thus, as the cross section increases the fluxes decrease to preserve the total number of events. This means that the fit is sensitive to neutrino absorption in the Earth and not the number of observed events.
The uncertainties in the cross section measurement are a combination of statistical and systematic uncertainties. The statistical uncertainty was obtained from performing the fit with fixed nuisance parameters. The systematic uncertainties are a combination of effects such as the optical properties of the ice, the angular acceptance of DOMs, and the density distribution of the Earth model. They are also impacted by variations in the atmospheric pressure at neutrino production as well as uncertainties on the measurements for prompt and astrophysical energy spectrum.
![Compilation of neutrino charged current cross section measurements divided by neutrino energy, from accelerator experiments [@particledata], theoretical predication from [@xsecCooper] and IceCube’s current result [@xsec]. Figure from [@xsec]. \[result\]](CrossSectionResults.png){width="90.00000%"}
Extended Neutrino Cross Section Measurement
===========================================
An extension to the cross section measurement is underway with improved statistics and systematics to investigate signs of possible beyond the Standard Model (BSM) interactions. This extended analysis will include 8 years of IceCube data, increasing the sample to over 300,000 events, which will result in the reduction of statistics uncertainty by 8-10%. As a result, systematics will become the more dominant effect. However the extended analysis will use the latest IceCube estimates of detector effects as well as measurements of prompt and astrophysical spectrum components. The extended analysis will also fit the multiple of cross section for discrete energy bins to observe any changes in the Standard Model prediction at high energies.
The effective area of the muon neutrino sample at different zenith angles for the extended analysis is shown in Figure \[area\]. The effective area of the sample has increased for all zenith angles compared to that used in the 1 year analysis, especially at higher energies above $10\, \mathrm{PeV}$ with near vertical zenith angles.
![The effective are of IceCube for the diffuse muon neutrino event selection [@DiffuseICRC] which will be used for the extended analysis. The effective area is improved as compared to the sample used in the first cross section measurement [@xsec]. The separate zenith bins show a high improvement for the near vertical events ($cos(\theta)\in (-1.00,-0.75)$). \[area\]](EffectiveAreaApproved.png){width="90.00000%"}
The extended analysis uses the newly developed software SQuIDs, which calculates the evolution of quantum mechanical ensembles, in this case neutrinos, through a given medium [@squid]. It can be used for calculating the propagation of a neutrino, accounting for interactions and oscillations. It also allows for separate variation in the charge and neutral current interaction. In the extended analysis each interaction type will have its own transmission probability for a given energy and zenith angles. The extended analysis will then measure the cross section multiple for charge and neutral current separately. In the Standard Model, the neutral current to charged current interaction cross section would scale proportionally, if new physics processes are occurring it would be indicated by a larger proportion of charged current interactions. BSM processes such as leptoquarks, sphalerons, and extra dimensions [@SpencerXsec] would cause interactions where the neutrino will disappear, this will be measured as an increase in the charged current cross section. Thus, the extended analysis may see an indication of these processes if the charged and neutral current cross section significantly deviate.
This analysis will further explore the possible effects of nuclear shadowing due to the altered parton distributions functions of nucleons in heavy nuclei, compared to isolated protons. Nuclear shadowing could have effects on the cross section measurement, which will be most noticeable at vertical trajectories where the neutrino travels through the Earth’s core.
The extended analysis will be more comparable with IceCube’s other analyses such as the high energy starting track cross section analysis which measures the cross section using neutrino high energy starting events (HESE) [@HESEScience] to fit the cross section of interaction over all zenith angles [@HESExsec]. This analysis is able to make the highest energy cross section measurement and will be the first all flavor neutrino cross section measurement. The HESE analysis also compares to the results found in [@MercXsec] which used a smaller sample of starting events. The extended muon neutrino analysis uses a sample with larger statistics sample it also incorporates energy bins which will compliment the HESE analysis. The extended sample is also more sensitive to higher energies than the 1 year thus will have more overlap with the HESE cross section measurement.
[^1]: For collaboration list, see PoS(ICRC2019) 1177.
|
---
abstract: 'Let $\rho$ be a two-dimensional even Galois representation which is induced from a character $\chi$ of odd order of the absolute Galois group of a real quadratic field. After imposing some additional conditions on $\chi$, we attach $\rho$ to a Hecke eigenclass in the cohomology of ${{\rm GL}}(2,{\mathbb Z})$ with coefficients in a certain infinite-dimensional vector space over a field of characteristic not equal to 2.'
address:
- 'Boston College, Chestnut Hill, MA 02467'
- 'Brigham Young University, Provo, UT 84602'
author:
- Avner Ash
- Darrin Doud
bibliography:
- 'biblio.bib'
date: 'February 23, 2017'
title: 'Even Galois representations and the cohomology of ${{\rm GL}}(2,{\mathbb Z})$'
---
Introduction
============
In this paper a Galois representation will be a continuous representation $\rho:G_{\mathbb Q}\to{{\rm GL}}(n,{\mathbb F})$ where ${\mathbb F}$ is either a topological field of characteristic 0 or a finite field. When the characteristic of ${\mathbb F}$ is not two, we say that $\rho$ is odd if the image of complex conjugation is conjugate to a diagonal matrix with alternating $1$’s and $-1$’s on the diagonal. If ${\mathbb F}$ has characteristic two, every Galois representation is considered to be odd. When $n=2$, $\rho$ is odd, and ${\mathbb F}$ is a finite field, Serre’s conjecture [@Serre-1987] (now a theorem of Khare and Wintenberger [@KW1; @KW2]) states that $\rho$ is attached to a modular form that is an eigenform of the Hecke operators. This means that the characteristic polynomial of the image of a Frobenius element at an unramified prime $\ell$ under $\rho$ equals a certain polynomial created from the eigenvalues of the Hecke operators at $\ell$. Other papers [@ADP; @AS; @Herzig] conjecture a similar attachment for $n\geq 2$, with modular forms replaced by elements of arithmetic cohomology groups. Work of Scholze [@Scholze] proves that any eigenclass of the Hecke operators in the cohomology of a congruence subgroup of ${{\rm SL}}(n,{\mathbb Z})$ with coefficients in a finite-dimensional admissible module $M$ over a field ${\mathbb F}$ has an attached Galois representation. For a field ${\mathbb F}$ of characteristic 0, this theorem was already proven in [@HLTT] by Harris, Lan, Taylor and Thorne. Caraiani and Le Hung [@Caraiani] showed that the representation guaranteed by Scholze’s theorem must be odd. (“Admissible" means that if ${\mathbb F}$ has characteristic 0 then $M$ is an algebraic representation, and if ${\mathbb F}$ has positive characteristic, then the matrices used to define the Hecke operators act on $M$ via reduction modulo some fixed integer.)
In this paper, we attach certain even Galois representations to eigenclasses in arithmetic cohomology groups. The details of our main result may be seen in Theorem \[T:main\], at the end of the paper. Following [@Caraiani] we know that we will need to use a non-admissible, infinite dimensional coefficient module for the cohomology. We also have to be careful with the exact definition of “attachment", which we now explain.
Let $f$ be a modular form of weight $k\geq 0$ on the upper half plane, with level $\Gamma_1(N)$ and nebentype $\theta$, and suppose that $f$ is an eigenform for the Hecke operators $T_\ell$ and $T_{\ell,\ell}$ for all $\ell\nmid N$. Denote the eigenvalue of $T_\ell$ by $a_\ell$, and the eigenvalue of $T_{\ell,\ell}$ by $A_\ell$. When $k\geq 2$, and $f$ is holomorphic, there is a Galois representation $\rho$ such that for all $\ell\nmid N$, $$\det(I-\rho(\operatorname{Frob}_\ell)X)=1-a_\ell X+\ell A_\ell X^2,$$ where (in this case), $A_\ell$ is easily seen to be equal to $\ell^{k-2}\theta(\ell)$.
The cases when $k=0$ or $1$ are different, because then $f$ is not cohomological. If $k=1$ and $f$ is holomorphic, or if $k=0$ and $f$ is a Maass form where the eigenvalue of the Laplacian is $1/4$, there is an attached Galois representation (this is only conjectural in the Maass form case) with finite image. In both cases, the motivic weight of $f$ is 0, and the characteristic polynomial of $\operatorname{Frob}_\ell$ equals $$1-a_\ell X + \theta(\ell) X^2.$$ These forms of the Hecke polynomials depend on the usual normalization of the Hecke operators.
In this paper, we will deal with the cohomology of ${{\rm GL}}(2,{\mathbb Z})$ with a nonadmissible, infinite dimensional coefficient module and an even Galois representation. Using the same normalization of the Hecke operators as in the finite dimensional coefficient setting, we define attachment of the Galois representation as follows (compare [@ADP Def. 2.1]). Note that this definition of “attachment" is for the purposes of this paper only, although we may guess that it will be the correct definition to use for all even two-dimensional Galois representations.
Let $V$ be a Hecke module over the field ${\mathbb F}$, and let $v\in V$ be an eigenvector for the Hecke operators $T_\ell$ and $T_{\ell,\ell}$ for almost all primes. Let $a_\ell$ be the eigenvalue of $T_\ell$ acting on $v$, and $A_\ell$ the eigenvalue of $T_{\ell,\ell}$ acting on $v$. Let $\rho:G_{\mathbb Q}\to{{\rm GL}}(2,{\mathbb F})$ be a Galois representation. We say that $\rho$ is attached to $v$ if, for almost all $\ell$, $$\det(I-\rho(\operatorname{Frob}_\ell)X)=1-a_\ell X+A_\ell X^2.$$
Our main theorem (Theorem \[T:main\]) then takes the following form.
Let $K$ be a real quadratic field of discriminant $d$, let ${\mathbb F}$ be a field with characteristic not equal to 2, and let $\chi:G_K\to{\mathbb F}^\times$ be a character with finite image, satisfying certain conditions (described in Theorem \[T:main\]). Then $\rho:G_{\mathbb Q}\to{{\rm GL}}(2,{\mathbb F})$ given by $\rho=\operatorname{Ind}_{G_K}^{G_{\mathbb Q}}\chi$ is an even Galois representation, and is attached to a Hecke eigenclass in $H^1({{\rm GL}}(2,{\mathbb Z}),{M_{S,q}}^*)$, where $q$ is a character related to $K$ (see Definition \[D:q\*\]), ${M_{S,q}}$ is defined in Definition \[D:MSQ\], and the asterisk denotes ${\mathbb F}$-dual.
The coefficient module ${M_{S,q}}$ that we use is naturally defined in terms of the field $K$. It is a non-algebraic infinite-dimensional module somewhat related to the kind that we we used in [@AD1] to study reducible cases of the Serre-type conjecture for $GL(3)/{\mathbb Q}$. Any Maass eigenform with eigenvalue $1/4$ is conjectured to have a Galois representation attached. Also, the Galois representations we work with in this paper are known to be attached to Maass forms. Our innovation is to prove attachment to something cohomological. Besides the intrinsic interest of this, we hope to be able to use our main theorem, combined with techniques similar to those of [@AD1], to prove a Serre type conjecture for the sum of $\rho$ and a character such that the three-dimensional representation as a whole is odd, in the context of $GL(3)/{\mathbb Q}$.
The idea of our proof is the following. We view $K$ as a two-dimensional ${\mathbb Q}$-vector space. We construct a $GL(2,{\mathbb Q})$ module $M$ consisting of formal sums of homothety classes of ${\mathbb Z}$-lattices in $K$, where the homotheties are given as multiplication by the elements of a carefully chosen subgroup ${K_{S,q}}$ of $K^\times$. We use homothety classes, rather than the lattices themselves, so that the stabilizer of a homothety class in ${{\rm GL}}(2,{\mathbb Z})$ will be an infinite cyclic group generated by the image $g$ of a unit in the ring of integers of $K$ under an embedding of $K$ into $GL(2)$ as a non-split torus. This matrix $g$ also stabilizes a closed geodesic in the quotient of the upper half plane modulo ${{\rm GL}}(2,{\mathbb Z})$. Our initial idea was to work with the fundamental classes of these closed geodesics, but of course we don’t want to view them in the homology of ${{\rm GL}}(2,{\mathbb Z})$ with admissible coefficients for the reasons stated above. Instead we use the more algebraic approach of this paper.
We focus on the submodule ${M_{S,q}}$ of $M$ which consists of formal sums that have finite support modulo the center and which have central character $q$. As we just said, the stabilizer of a homothety class of lattices is an infinite cyclic group. This allows us to use Shapiro’s lemma to write the homology $H_1$ of ${{\rm GL}}(2,{\mathbb Z})$ with coefficients in ${M_{S,q}}$ in terms of the $H_1$ of these cyclic stabilizers, which is an algebraic version of the fundamental classes of the corresponding closed geodesics. We then have to understand how the Hecke operators act.
We use the method of partial Hecke operators described in [@AD1] to get a tractable formula for the action of a Hecke operator on $H_1({{\rm GL}}(2,{\mathbb Z}),{M_{S,q}})$. Now a class in that homology group has finite support (modulo the center) on chains, and the Hecke operators always expand the support. So there will not be any Hecke eigenvectors in the homology group $H_1({{\rm GL}}(2,{\mathbb Z}),{M_{S,q}})$. We must seek for Hecke eigenvectors in the dual space $H^1({{\rm GL}}(2,{\mathbb Z}),{M_{S,q}}^*)$. We interpret elements of the dual space as functions on the space of lattices in $K$. In order for us to construct such functions in a way that makes it possible to compute the Hecke operators at $\ell$, we use the Bruhat-Tits graph $T_\ell$ for ${{\rm GL}}(2,{\mathbb Q}_\ell)$ or a double cover ${\mathcal T}^2_\ell$ of ${\mathcal T}_\ell$, depending on whether $\ell$ is split or inert in $K$. We then relate the Hecke operators at $\ell$ to a Laplacian on ${\mathcal T}_\ell$ (or ${\mathcal T}^2_\ell$) and to the action of the center. This allows us to construct lattice functions that have the desired Hecke eigenvalues.
These functions are restricted infinite products over the rational primes of the local functions we construct on the graphs. Lattices which are fractional ideals in $K$ play a special role in the study of ${M_{S,q}}$ and we call them “idealistic" lattices. The construction of the local functions depends on the crucial distinction between idealistic and non-idealistic lattices. To define a cohomology class, the infinite product has to satisfy a certain global invariance property (proved in Section \[S:Invariance\]), which is guaranteed by the fact that $\chi$ can be viewed as a global character on ideals.
In case ${\mathbb F}$ has characteristic 2, where the distinction between odd and even Galois representations breaks down, a simplified version of our construction works to attach $\rho$ to a Hecke eigenclass in the cohomology of ${{\rm GL}}(2,{\mathbb Z})$ with coefficients in the analog of ${M_{S,q}}$ (where now $q$ would be the identity central character.) In this case, we can work directly with the usual Bruhat-Tits graph for both split and inert primes. We do not cover this in this paper because when the characteristic of ${\mathbb F}$ equals 2, $\rho$ is deemed to be odd and is known to be attached to a homology class with admissible coefficients, by the work of Khare and Winteberger [@KW1; @KW2] referred to above.
We thank Dick Gross, David Hansen, Richard Taylor, and especially Kevin Buzzard for helpful comments and answers to questions that arose during the course of this research.
Lattices and Homotheties in $K$ {#S:Lattices}
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Fix a real quadratic field $K$, its ring of integers ${\mathfrak O}$ and an element $\omega\in{\mathfrak O}$ such that ${\mathfrak O}={\mathbb Z}[\omega]$. Let $d$ be the discriminant of $K/{\mathbb Q}$. Let $\epsilon$ be a fundamental unit, i.e. a unit whose image modulo $\pm1$ generates ${\mathfrak O}^\times/\{\pm1\}$.
Consider $K$ as a two-dimensional vector space over ${\mathbb Q}$. By a lattice in $K$, we will mean a free ${\mathbb Z}$-module of rank $2$ contained in $K$. Such a module has as a ${\mathbb Z}$-basis two ${\mathbb Q}$-linearly independent elements.
Let $Y$ be the set of all column vectors ${{}^t(a,b)}\in K^2$ with $b\neq 0$ and $a/b\notin {\mathbb Q}$. If we let ${{\bar\omega}}={{}^t(\omega,1)}\in Y$, then every element of $Y$ is of the form $\gamma {{\bar\omega}}$ for some $\gamma\in{{\rm GL}}(2,{\mathbb Q})$. In addition, given $y,y'\in Y$, there is a unique $\gamma\in{{\rm GL}}(2,{\mathbb Q})$ with $y=\gamma y'$. There is a natural action of $K^\times$ by scalar multiplication on $Y$, which we write as a right action.
Let $y={{}^t(a,b)}\in Y$. Define $\Lambda_y$ to be the ${\mathbb Z}$-lattice in $K$ generated by $a$ and $b$ (i.e. the set of all integer linear combinations of $a$ and $b$).
Note that for $\alpha\in K^\times$, we have $\Lambda_{y\alpha }=\alpha\Lambda_y$.
Let $H\subseteq K^\times$ be a multiplicative subgroup of $K^\times$. Two lattices $\Lambda_1$ and $\Lambda_2$ in $K$ will be said to be [*homothetic*]{} if there is some $\alpha\in K^\times$ such that $\Lambda_1=\alpha\Lambda_2$. If $\alpha\in H$, we will say that the lattices are [*$H$-homothetic*]{}.
Homothety and $H$-homothety of lattices are equivalence relations on the set of all lattices in $K$.
Let $H$ be a multiplicative subgroup of $K^\times$. Define $Y/H$ to be the quotient of $Y$ with respect to the right action of scalar multiplication by $H$. The left action of ${{\rm GL}}(2,{\mathbb Q})$ on $Y$ then gives a left action of ${{\rm GL}}(2,{\mathbb Q})$ on $Y/H$.
There is a bijection between ${{\rm GL}}(2,{\mathbb Z})$-orbits of elements of $Y/H$ and $H$-homothety classes of lattices in $K$.
Denote the set of $H$-homothety classes of lattices in $K$ by ${\mathcal H}$. Define a map $$f:Y/H\to{\mathcal H}$$ by setting $f(x)$ equal to the $H$-homothety class of $\Lambda_y$ for any $y\in Y$ representing ${x}\in Y/H$. Since the various $y$ representing $x$ all differ by scalar multiples by some element of $H$, it is clear that $f$ is well defined. Since every lattice in $K$ is of the form $\Lambda_y$ for some $y\in Y$, the map $f$ is surjective. Finally, for $\gamma\in{{\rm GL}}(2,{\mathbb Z})$, we have that $\gamma x$ is represented by $\gamma y$, and that $\Lambda_y=\Lambda_{\gamma y}$. Hence, $f$ is constant on ${{\rm GL}}(2,{\mathbb Z})$-orbits, and so induces a surjective map $\hat f$ from the set of ${{\rm GL}}(2,{\mathbb Z})$-orbits in $Y/H$ to ${\mathcal H}$.
Now suppose that $x,x'\in Y/H$ are represented by $y,y'\in Y$, and $f(x)=f(x')$. Then $\Lambda_y=\alpha\Lambda_{y'}=\Lambda_{y'\alpha}$ for some $\alpha\in H$. Hence, the entries of both $y$ and $y'\alpha$ are a basis for $\Lambda_y$. Therefore, there is some $\gamma\in{{\rm GL}}(2,{\mathbb Z})$ such that $y=\gamma y'\alpha$. Then $x=\gamma x'$, so $x$ and $x'$ are in the same ${{\rm GL}}(2,{\mathbb Z})$-orbit. Hence, $\hat f$ is an injective map on ${{\rm GL}}(2,{\mathbb Z})$-orbits.
\[Lemma:mL\] Let $\Lambda$ be a lattice in $K$. Then there is a minimal positive integer $m$ such that $\epsilon^m\Lambda=\Lambda$.
Note that if $\Lambda$ and $\Lambda'$ are $K^\times$-homothetic, the lemma will be true for $\Lambda$ if and only if it is true for $\Lambda'$, with the same value of $m$ (since $K^\times$ is commutative.) Hence, we may, without loss of generality, assume that $\Lambda$ is contained in ${\mathfrak O}$. Since $\Lambda$ is a rank two ${\mathbb Z}$-submodule of ${\mathfrak O}$, it must have finite index in ${\mathfrak O}$. We may thus choose an $N\in{\mathbb Z}$ such that $N{\mathfrak O}\subseteq\Lambda\subseteq{\mathfrak O}$. Since ${\mathfrak O}/N{\mathfrak O}$ is finite and multiplication by $\epsilon$ permutes its elements, there is some positive $m\in{\mathbb Z}$ such that $\delta=\epsilon^m$ acts trivially on ${\mathfrak O}/N{\mathfrak O}$, and hence on $\Lambda/N{\mathfrak O}$. Then $\delta$ must take $\Lambda$ to itself, so $\delta\Lambda\subseteq\Lambda$. We must also have $\delta^{-1}\Lambda\subseteq\Lambda$, so $\Lambda\subseteq\delta\Lambda\subseteq\Lambda$, and therefore $\delta\Lambda=\Lambda$. The existence of a minimal positive $m$ satisfying the conditions of the theorem follows immediately from the existence of some positive $m$.
Given ${x}\in Y/H$, we define ${\Gamma_{{x}}}$ to be the stabilizer of ${x}$ in ${{\rm GL}}(2,{\mathbb Z})$, and ${\hat\Gamma_{{x}}}$ to be the quotient $$\frac{{\Gamma_{{x}}}}{{\Gamma_{{x}}}\cap\{\pm I\}}.$$
We will say that a subgroup $H$ of $K^\times$ is [*unit-cofinite*]{} if $H\cap{\mathfrak O}^\times$ has finite index in ${\mathfrak O}^\times$.
\[stabgen\] Let $H$ be a unit-cofinite subgroup of $K^\times$, and let ${x}\in Y/H$ be represented by $y\in Y$. Then ${\hat\Gamma_{{x}}}$ is a cyclic group, generated by the image of the unique element $g\in{\Gamma_{{x}}}$ satisfying $$gy=y\delta$$ where $\delta=\pm\epsilon^m$, and $m$ is smallest possible positive integer such that $\Lambda_y=\Lambda_{y\epsilon^m}$ and $\delta\in H$.
The notation in the theorem means that $\delta=\epsilon^m$ or $\delta=-\epsilon^m$, depending on which is in $H$. If both are in $H$, then $-1\in H$ and we set $\delta=\epsilon^m$. If $-1\notin H$, it is possible to have $-\epsilon^m\in H$ without having $\epsilon^m\in H$. Hence, it is necessary to choose $\delta=\pm\epsilon^m$ with $m$ minimal to get a generator of ${\Gamma_{x}}$.
Let $x$ be represented by $y={{}^t(a,b)}\in Y$. Choose the smallest positive $m$ such that $\epsilon^m\Lambda_y=\Lambda_y$ and one (or both) of $\epsilon^m$ and $-\epsilon^m\in H$. Let $\delta=\pm\epsilon^m\in H$. Then $\delta\Lambda_y=\Lambda_y$, so $y\delta$ is a basis of $\Lambda_y$. Hence, there is some $g\in{{\rm GL}}(2,{\mathbb Z})$ such that $gy=y\delta$. Since $\delta\in H$, we see that $g{x}={x}$.
We now show that every element in ${\Gamma_{{x}}}$ is (up to a sign) a power of $g$. Let $\eta\in{\Gamma_{{x}}}$. Then, since $\eta {x}={x}$, there is some $\alpha\in H$ such that $\eta y=y\alpha$. Now $\alpha$ is an eigenvalue of $\eta$, and $\eta\in{{\rm GL}}(2,{\mathbb Z})$, so $\alpha\in{\mathfrak O}^\times$. Hence, $\alpha=\pm\epsilon^r$. By the division algorithm and the minimality of $m$, we see that $\alpha=\pm\delta^k$ for some $k$. Hence, $\eta=\pm g^k$. If $\eta= g^k$ we are finished. If $\eta=- g^k$ then $-I\in\Gamma_x$ and $\eta\equiv g^k$ modulo $\Gamma_x\cap\{\pm I\}$.
Certain elements ${x}\in Y/H$ will be quite important to us; for these elements, the value of $m$ in the previous proof is determined solely by $H$.
Let $H$ be a multiplicative subgroup of $K^\times$. If ${x}\in Y/H$ can be represented by $y\in Y$ such that $\Lambda_y$ is a fractional ideal in $K$, then we say that ${x}$ is [*idealistic*]{}.
Note that determining whether ${x}$ is idealistic does not depend on the choice of $y\in Y$ representing ${x}$.
Let $H$ be a unit-cofinite subgroup of $K^\times$. If ${x}\in Y/H$ is idealistic, the value of $m$ in Theorem \[stabgen\] is equal to the smallest positive integer $k$ such that $\pm\epsilon^k\in H$.
\[D:generator\] Let $H$ be a unit-cofinite subgroup of $K^\times$. For ${x}\in Y/H$, denote the positive integer $m$ described in Theorem \[stabgen\] by $m_x$, and the element $g$ described in Theorem \[stabgen\] by $g_x$.
\[gammaxcor\] Let $H$ be a unit-cofinite subgroup of $K^\times$. If $x,x'\in Y/H$ are in the same ${{\rm GL}}(2,{\mathbb Z})$-orbit, then $m_x=m_{x'}$.
If $x=\gamma x'$ for $\gamma\in{{\rm GL}}(2,{\mathbb Z})$, then $\Gamma_{x'}=\gamma\Gamma_x\gamma^{-1}$, so $g_x$ is conjugate to $g_{x'}$. Then $g_x$ and $g_{x'}$ have the same eigenvalues, so $x$ and $x'$ have the same value of $m$.
Assume that $H$ does not contain $-1$. In this case, ${\Gamma_{x}}$ does not contain $-I$, so ${\hat\Gamma_{x}}={\Gamma_{x}}$ is cyclic, generated by $g_x$. Then, there is a canonical isomorphism $$I_x:H_1({\Gamma_{x}},{\mathbb F})\to{\Gamma_{x}}\otimes_{\mathbb Z}{\mathbb F}.$$
\[Def:zx\] If $-1\notin H$, and ${x}\in Y/H$, define $z_x$ to be the generator of $H_1({\Gamma_{x}},{\mathbb F})$ such that $I_x(z_x)=g_x\otimes 1$.
$(S,q)$-homotheties
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From now on, we fix a field ${\mathbb F}$ of characteristic not equal to $2$.
Let $y={{}^t(a,b)}\in Y$. We define an injective homomorphism $r_y:K^\times\to{{\rm GL}}(2,{\mathbb Q})$ by $$r_y(c){\begin{pmatrix}a\cr b\end{pmatrix}}={\begin{pmatrix}ac\cr bc\end{pmatrix}}$$ for $c\in K^\times$.
Let $M$ be any positive integer. Define $S_0(M)$ to be the largest subgroup of ${{\rm GL}}(2,{\mathbb Q})$ that can be mapped modulo $M$ to ${{\rm GL}}(2,{\mathbb Z}/M{\mathbb Z})$. Define $S(M)$ to be the kernel of reduction modulo $M$ from $S_0(M)$ to ${{\rm GL}}(2,{\mathbb Z}/M{\mathbb Z})$.
We note that for any $M$, $S_0(M)$ contains ${{\rm GL}}(2,{\mathbb Z})$.
\[D:S\_0\] Recall that $d=\operatorname{disc}(K)$. If the characteristic of ${\mathbb F}$ is nonzero, set $p$ equal to the characteristic of $F$; otherwise, set $p=1$. Fix positive integers $M,N$ with $M\geq 3$ such that $M\mid pdN$. Define $S_0=S_0(pdN)$ and $S=S(M)\cap S_0(pdN)$.
Since $S$ is the kernel of the composition $$S_0\to{{\rm GL}}(2,{\mathbb Z}/pdN{\mathbb Z})\to{{\rm GL}}(2,{\mathbb Z}/M{\mathbb Z}),$$ we see that $S$ has finite index in $S_0$. Further, since $M\geq3$, $-I\notin S$.
Recall that ${{\bar\omega}}={{}^t(\omega,1)}\in Y$. Define $$K_{S_0}=\{c\in K^\times:r_{{\bar\omega}}(c)\in S_0\}$$ and $$K_S=\{c\in K^\times:r_{{\bar\omega}}(c)\in S\}.$$ We note that $K_{S_0}$ and $K_S$ are multiplicative subgroups of $K^\times$.
Since $S\subseteq S_0$ has finite index, it is clear that $K_S\subseteq K_{S_0}$ with finite index.
If $\alpha\in K_{S_0}$, then $\alpha$ is relatively prime to $pdN$.
Since $\alpha\in K$, we may write $\alpha=\beta/n$ with $\beta\in{\mathfrak O}$ and $n\in{\mathbb Z}$. Since $\beta\in{\mathfrak O}$, we see that $r_{{\bar\omega}}(\beta)$ has integer entries. If a prime $b|dN$ and $b|n$, then, since the entries of $r_{{\bar\omega}}(\alpha)=\frac1nr_{{\bar\omega}}(\beta)$ can have no denominators divisible by $b$, it must be the case that $b$ divides every entry of $r_{{\bar\omega}}(\beta)$. This implies that $b|\beta$ in ${\mathfrak O}$. Canceling (repeatedly, if needed), we may take $n$ to be relatively prime to $pdN$.
Now $N^K_{\mathbb Q}(\alpha)=\det(r_{{\bar\omega}}(\alpha))=\det(r_{{\bar\omega}}(\beta))/n^2=N^K_{\mathbb Q}(\beta)/n^2$ must be relatively prime to $pdN$, so $N^K_{\mathbb Q}(\beta)$ is also relatively prime to $pdN$. Since $\beta\in{\mathfrak O}$, it must then be relatively prime to $pdN$, and so $\alpha$ is as well.
$K_S$ is a unit-cofinite subgroup of $K^\times$.
We note that $r_{{\bar\omega}}(\epsilon)\in{{\rm GL}}(2,{\mathbb Z})$. Hence, $r_{{\bar\omega}}(\epsilon)$ can be reduced modulo $M$ to give a matrix in ${{\rm GL}}(2,{\mathbb Z}/M{\mathbb Z})$. Since ${{\rm GL}}(2,{\mathbb Z}/M{\mathbb Z})$ is finite, there is some positive integer $k$ such that $$r_{{\bar\omega}}(\epsilon)^k=r_{{\bar\omega}}(\epsilon^k)\in S(M).$$ Since $r_{{\bar\omega}}(\epsilon)$ is also in $S_0$, we see that $\epsilon^k\in K_S$, so $K_S$ is unit-cofinite.
\[D:q\*\] Let $Z$ denote the set of scalar matrices $\zeta_r=rI$ where $r\in {\mathbb Q}^\times\cap K_{S_0}$. Let $\theta:{\mathbb Z}\to{\mathbb F}^\times$ be the quadratic Dirichlet character cutting out $K$, and define $q:Z\to{\mathbb F}^\times$ by $q(rI)=\theta(r)$ for $r\in{\mathbb Z}\cap K_{S_0}$, extended multiplicatively to $Z$.
We define $ q^*:K_{S_0}\to{\mathbb F}^\times$ to be the composition of the following multiplicative maps:
(1) The map taking $a\in K_{S_0}$ to the principal fractional ideal $(a)\subset K$,
(2) The map taking a fractional ideal to its prime factorization,
(3) The map taking a product of powers of prime ideals to the subproduct of powers of inert prime ideals,
(4) The map taking an inert prime ideal $(\ell)$ to $q(\ell I)$.
We note that $q^*$ is a homomorphism, and we define ${K_{S,q}}$ to be the kernel of $q^*|_{K_S}$. Then ${K_{S,q}}$ is a multiplicative subgroup of $K^\times$ and ${K_{S,q}}$ has index $2$ in $K_S$.
\[L:unit-cofinite\] ${K_{S,q}}$ is a unit-cofinite subgroup of $K^\times$.
This is true because $K_S$ is unit-cofinite, and any unit in $K_S$ is in the kernel of $q^*$, and so in ${K_{S,q}}$.
\[Lemma:q\*-q\] For $r\in{\mathbb Q}^\times\cap K_{S_0}$, $q(rI)=q^*(r)$.
Let $$r=\prod_{\ell\text{ prime}}\ell^{n_\ell}.$$ Then $$q(rI)=\theta(r)=\prod_{\ell\text{ prime}}\theta(\ell)^{n_\ell}=\prod_{\ell\text{ inert in $K$}}\theta(\ell)^{n_\ell}=\prod_{\ell\text{ inert in $K$}} q(\ell I)^{n_\ell}=q^*(r).$$ since $n_\ell=0$ for $\ell$ ramified in $K/{\mathbb Q}$, and $\theta(\ell)=1$ for $\ell$ split in $K$.
Let ${X_{S,q}}=Y/{K_{S,q}}$. Then ${\mathbb F}Y$ is an $(S_0,{K_{S,q}})$-bimodule, and we obtain an isomorphism $${\mathbb F}Y\otimes_{{K_{S,q}}}{\mathbb F}\cong {\mathbb F}{X_{S,q}}\otimes {\mathbb F}={\mathbb F}{X_{S,q}}$$ of $S_0$-modules.
\[L:X\^S-stab\] Let ${x}\in{X_{S,q}}$ be represented by $y\in Y$. Then
(a) ${\Gamma_{{x}}}=\{r_y(c):c\in{K_{S,q}}\}\cap{{\rm GL}}(2,{\mathbb Z}).$
(b) If $g\in{\Gamma_{{x}}}$, then $g=r_y(c)$ for some $c\in{\mathfrak O}^\times$.
(c) $-I\notin {\Gamma_{x}}$ and ${\Gamma_{{x}}}$ is infinite cyclic.
\(a) Suppose $g\in{\Gamma_{x}}$. Then we have that $gy=yc$ for some $c\in {K_{S,q}}$. Since $yc=r_y(c)y$, and the entries of $y$ are a ${\mathbb Q}$-basis of $K$, we see that $g=r_y(c)$. Hence, $g$ is in the given intersection, and any $g$ in the given intersection fixes ${x}$.
\(b) Let $g\in{\Gamma_{x}}$. Then $g=r_y(c)$, for some $c\in{K_{S,q}}$, and the characteristic polynomial of $g$ is the same as that of multiplication by $c$ on $K$. Since $g\in{{\rm GL}}(2,{\mathbb Z})$, we see that $c\in {\mathfrak O}^\times$.
\(c) This follows from Theorem \[stabgen\], Lemma \[L:unit-cofinite\], and the fact that $-1\notin K_{S,q}$.
Denote by ${{\tilde\omega}}$ the image in ${X_{S,q}}$ of ${{\bar\omega}}$.
Define $i^S=m_{{{\tilde\omega}}}$.
\[is\] Let ${x}\in{X_{S,q}}$.
(i) For any $x$, $i^S\mid m_x$.
(ii) If $x$ is idealistic, then $m_x=i^S$.
\(i) For any $\xi\in {X_{S,q}}$, let $\phi_\xi:\Gamma_\xi \to {K_{S,q}}$ be the injective homomorphism defined by $g=r_\xi(\phi_\xi(g))$ for $g\in\Gamma_\xi$. Then the image of $\phi_\xi$ is generated by $\epsilon^{m_\xi}$.
Now let ${x}\in{X_{S,q}}$ be represented by $y\in Y$. We have seen that any element of ${\Gamma_{{x}}}$ is of the form $r_y(c)$ for some $c\in{\mathfrak O}^\times$. From Lemma \[L:X\^S-stab\], we see that for $c\in{\mathfrak O}^\times$, $$r_y(c)\in{\Gamma_{{x}}}\implies c\in {K_{S,q}}\iff r_{{{\tilde\omega}}}(c)\in{\Gamma_{{{\tilde\omega}}}}.$$ Hence, the image of $\phi_x$ is contained in the image of $\phi_{{\tilde\omega}}$. Since both images are cyclic groups, we see that $i^S|m_x$.
\(ii) Since $x$ is idealistic, $\Lambda_y$ is a fractional ideal of $K$. Hence, $r_y(c)\in{{\rm GL}}(2,{\mathbb Z})$ for all $c\in{\mathfrak O}^\times$. Therefore, we see that for $c\in{\mathfrak O}^\times$, $$r_y(c)\in{\Gamma_{x}}\iff c\in K_{S,q}\iff r_{{{\tilde\omega}}}(c)\in{\Gamma_{{{\tilde\omega}}}}.$$ Clearly, then, $m_x=i^S$.
\[D:mx’\] For ${x}\in{X_{S,q}}$, set $m_x'=m_x/i^S$.
Finally, we prove the following lemma about the relationship between elements of $Z$ and elements of ${K_{S,q}}$.
\[Lemma:Z-KSQ\] Let $\zeta=rI\in{\mathbb Z}$ with $r\in{\mathbb Q}^\times\cap K_{S_0}$, let $\alpha\in{K_{S,q}}$, and let $y\in Y$. If $\zeta r_y(\alpha)\in{{\rm GL}}(2,{\mathbb Z})$, then $q(\zeta)=1$.
We have that $\zeta r_y(\alpha)=r_y(r\alpha)\in{{\rm GL}}(2,{\mathbb Z})$. Since the characteristic polynomial of $r\alpha$ is the same as the characteristic polynomial of $r_y(r\alpha)$, we see that $r\alpha$ is a unit in ${\mathfrak O}$. Hence, $q^*(r\alpha)=1$. Since $q^*(\alpha)=1$, by the multiplicativity of $q^*$ we see that $q^*(r)=1$. By Lemma \[Lemma:q\*-q\], it follows that $q(rI)=q(\zeta)=1$.
Defining the coefficient module ${M_{S,q}}$ {#MSQ}
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\[D:MSQ\] Define ${M_{S,q}}$ to be the ${\mathbb F}$-vector space of formal sums $$\sum_{{x}\in{X_{S,q}}} c_{{x}}{x}$$ with $c_{x}\in {\mathbb F}$, such that the sum is supported on a finite number of $Z$-orbits of ${X_{S,q}}$, and such that the coefficients satisfy the relation $$c_{\zeta{x}}=q^{-1}(\zeta)c_{{x}}$$ for all $\zeta\in Z$.
In this paper $q$ has order $2$, but we write $q^{-1}$ as a check on our computations and for possible generalizations to characters of higher orders.
Define $Z_{S,q}=\{\zeta_r\in Z:r\in {K_{S,q}}\}$.
Note that $Z_{S,q}$ is a subgroup of finite index in $Z$ (since ${K_{S,q}}$ has finite index in $K_{S_0}$). Let ${\mathcal B}$ be a collection of coset representatives of $Z_{S,q}$ inside $Z$.
Now $Z\,{{\rm GL}}(2,{\mathbb Z})$ is a group which acts on ${X_{S,q}}$. Hence, we may choose a collection of representatives of the $Z\,{{\rm GL}}(2,{\mathbb Z})$-orbits in ${X_{S,q}}$. We will denote such a collection by ${\mathcal A}$.
Note that given any $x\in{X_{S,q}}$, we may assume (possibly by changing ${\mathcal A}$) that $x\in {\mathcal A}$. For the remainder of this section, we fix a set ${\mathcal A}$ of $Z\,{{\rm GL}}(2,{\mathbb Z})$-orbit representatives.
Clearly each $Z$-orbit in ${X_{S,q}}$ contains at least one element of the form $gx$ with $g\in{{\rm GL}}(2,{\mathbb Z})$ and $x\in{\mathcal A}$.
\[D:Z-orbit-reps\] Let ${\mathcal C}$ be a collection of representatives of the $Z$-orbits in ${X_{S,q}}$, chosen so that each representative in ${\mathcal C}$ is of the form $gx$ for $g\in{{\rm GL}}(2,{\mathbb Z})$ and $x\in{\mathcal A}$.
\[Remark:z-orbits\] Note that ${\mathcal C}$ is not uniquely determined by ${\mathcal A}$. However, once a choice of ${\mathcal C}$ is fixed, any $Z$-orbit will contain a unique representative $gx\in{\mathcal C}$, and the element $x\in {\mathcal A}$ and the coset $g{\Gamma_{x}}$ of $g\in{{\rm GL}}(2,{\mathbb Z})$ are uniquely defined.
For the remainder of this section, we fix a choice of ${\mathcal C}$ corresponding to our choice of ${\mathcal A}$.
\[msqbasis\] The set $$\left\{\sum_{\zeta\in{\mathcal B}}q^{-1}(\zeta)\zeta x:x\in{\mathcal C}\right\}$$ is an ${\mathbb F}$-basis of ${M_{S,q}}$.
For any $x\in{X_{S,q}}$, the $Z$-orbit of $x$ is equal to the set $\{\zeta x:\zeta\in{\mathcal B}\}$. By the relation on the coefficients of an element in ${M_{S,q}}$, the coefficient of $\zeta x$ is equal to $q^{-1}(\zeta)$ times the coefficient of $x$. Since an element of ${M_{S,q}}$ is supported on finitely many $Z$-orbits, the lemma follows.
\[L:Z-action\] ${M_{S,q}}$ is an $S_0$-module. For $r\in{\mathbb Q}^\times\cap K_{S_0}$, the action of $\zeta_r$ on ${M_{S,q}}$ is via the scalar $q(\zeta_r)$.
For $s\in S_0$, we have $s\sum c_{x}{x}=\sum c_{{x}} s{x}\in{M_{S,q}}$, since $Z$ is in the center of $S_0$.
It suffices to prove the statement about the action of $\zeta_r$ on basis elements of the form $$\sum_{\zeta_\ell\in{\mathcal B}} q^{-1}(\zeta_\ell)\zeta_\ell{x},$$ with ${x}\in{\mathcal C}$. For $\zeta$ in $Z$, we will define $\bar\zeta$ to be the unique element of ${\mathcal B}$ such that $\zeta\in\bar\zeta {Z_{S,q}}$. The map from ${\mathcal B}$ to ${\mathcal B}$ given by $\zeta\mapsto\overline{\zeta\zeta_r}$ for a fixed $\zeta_r$ is a bijection. We also note that $q(\overline\zeta)=q(\zeta)$ for any $\zeta\in Z$.
Setting $u=r\ell$, we now have $$\begin{aligned}
\zeta_r\sum_{\zeta_\ell\in{\mathcal B}} q^{-1}(\zeta_\ell)\zeta_\ell{x}&=\sum_{\zeta_\ell\in{\mathcal B}} q^{-1}(\zeta_\ell)\zeta_\ell\zeta_r{x}\cr
&=\sum_{\zeta_\ell\in{\mathcal B}} q^{-1}(\zeta_\ell)\zeta_{\ell r}{x}\cr
&=\sum_{\overline\zeta_u\in{\mathcal B}} q^{-1}(\zeta_u\zeta_r^{-1})\zeta_u{x}\cr
&=q^{-1}(\zeta_r^{-1})\sum_{\overline\zeta_u\in{\mathcal B}} q^{-1}(\overline\zeta_u)\overline\zeta_u{x}\cr
&=q(\zeta_r)\sum_{\zeta_\ell\in{\mathcal B}} q^{-1}(\zeta_\ell)\zeta_\ell{x}.\qedhere\end{aligned}$$
\[C:unique-basis\] The basis described in Lemma \[msqbasis\] is independent of the choice of ${\mathcal C}$.
Let ${\mathcal C}$ and ${\mathcal C}'$ be two choices of $Z$-orbit representatives, as in Definition \[D:Z-orbit-reps\]. For a given $Z$-orbit, let $gx\in{\mathcal C}$ and $g'x\in{\mathcal C}'$ (with $g,g'\in{{\rm GL}}(2,{\mathbb Z})$) be the orbit representatives. Note that they will have the same representative $x\in{\mathcal A}$, since they are in the same $Z\,{{\rm GL}}(2,{\mathbb Z})$-orbit. Since $gx$ and $g'x$ are in the same $Z$-orbit, for some $\zeta\in Z$ we have $\zeta g'x= gx$, so that $\zeta g^{-1}g'x=x$, and (choosing a $y\in Y$ representing $x$) we see that $\zeta g^{-1}g'y\alpha=y$ for some $\alpha\in{K_{S,q}}$. Hence, $\zeta r_y(\alpha)\in{{\rm GL}}(2,{\mathbb Z})$, and by Lemma \[Lemma:Z-KSQ\] we see that $q(\zeta)=1$. The corresponding basis elements then differ by a factor of $\zeta$, so that by Lemma \[L:Z-action\] they differ by a scalar factor of $q(\zeta)=1$.
\[induced\] If we consider ${M_{S,q}}$ as a ${{\rm GL}}(2,{\mathbb Z})$-module, it is a sum of induced modules. In fact, we have isomorphisms of ${{\rm GL}}(2,{\mathbb Z})$-modules defined by $$e:\bigoplus_{{x}\in\mathcal A} {\mathbb F}[{{\rm GL}}(2,{\mathbb Z})]\otimes_{{\mathbb F}{\Gamma_{{x}}}}{\mathbb F}\to{M_{S,q}},$$ and $$f:{M_{S,q}}\to\bigoplus_{{x}\in\mathcal A} {\mathbb F}[{{\rm GL}}(2,{\mathbb Z})]\otimes_{{\mathbb F}{\Gamma_{{x}}}}{\mathbb F},$$ such that $e$ and $f$ are inverses of each other.
These isomorphisms depend on the choice of ${\mathcal A}$, which we suppress from the notation.
On basis elements of the form $g\otimes 1\in {\mathbb F}[{{\rm GL}}(2,{\mathbb Z})]\otimes_{{\mathbb F}{\Gamma_{{x}}}}{\mathbb F}$ for ${x}\in\mathcal A$, we define $e$ as $$e(g\otimes 1)=\sum_{\zeta_r\in{\mathcal B}}q(\zeta_r^{-1})\zeta_r g{x},$$ and extend linearly. Since any $g\in{\Gamma_{{x}}}$ acts trivially on ${x}$, this is well-defined, and it is clearly a homomorphism of ${{\rm GL}}(2,{\mathbb Z})$-modules.
We define $f$ on basis elements corresponding to $gx\in{\mathcal C}$ (with $x\in{\mathcal A}$ and $g\in{{\rm GL}}(2,{\mathbb Z})$) by $$f\left(\sum_{\zeta\in{\mathcal B}}q^{-1}(\zeta)\zeta gx\right)=g\otimes1\in{\mathbb F}\,{{\rm GL}}(2,{\mathbb Z})\otimes_{{\mathbb F}\,{\Gamma_{x}}}{\mathbb F}\subset\bigoplus_{x\in{\mathcal A}}{\mathbb F}\,{{\rm GL}}(2,{\mathbb Z})\otimes_{{\mathbb F}\,{\Gamma_{x}}}{\mathbb F},$$ and extending linearly. By Lemma \[msqbasis\], this gives a well defined ${\mathbb F}$-linear map from ${M_{S,q}}$ to $$\bigoplus_{x\in{\mathcal A}}{\mathbb F}\,{{\rm GL}}(2,{\mathbb Z})\otimes_{{\mathbb F}\,{\Gamma_{x}}}{\mathbb F}.$$
One sees that on basis elements, $e$ and $f$ are inverses. Hence, they are inverses of each other as ${\mathbb F}$-vector space maps. Since $e$ is a ${{\rm GL}}(2,{\mathbb Z})$-module map, so is $f$ and thus both are ${{\rm GL}}(2,{\mathbb Z})$-module isomorphisms.
Homology with coefficients in ${M_{S,q}}$ and Hecke operators {#coefficients}
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In this section, we will fix an element $x_0\in{X_{S,q}}$, and choose a set ${\mathcal A}$ of $Z\,{{\rm GL}}(2,{\mathbb Z})$-orbit representatives in ${X_{S,q}}$ that contains $x_0$.
As a consequence of Lemma \[induced\], we have that $${M_{S,q}}\cong \bigoplus_{{x}\in\mathcal A} \operatorname{Ind}_{{\Gamma_{{x}}}}^{{{\rm GL}}(2,{\mathbb Z})}{\mathbb F}.$$ Hence, by Shapiro’s lemma, we have $$H_1({{\rm GL}}(2,{\mathbb Z}),{M_{S,q}})\cong \bigoplus_{{x}\in\mathcal A}H_1(\Gamma_{{x}},{\mathbb F}).$$ Since ${\Gamma_{{x}}}$ is infinite cyclic, we have $$H_1({\Gamma_{{x}}},{\mathbb F})\cong H^0({\Gamma_{{x}}},{\mathbb F})\cong {\mathbb F}.$$ For each ${x}\in \mathcal A$, we choose a generator ${z_{{x}}}$ for $H_1({\Gamma_{{x}}},{\mathbb F})$, as in Definition \[Def:zx\].
We now examine an individual Hecke operator. Let $s\in S_0$, and let $E$ be a collection of single coset representatives for ${{\rm GL}}(2,{\mathbb Z})s{{\rm GL}}(2,{\mathbb Z})$, so that $${{\rm GL}}(2,{\mathbb Z})s{{\rm GL}}(2,{\mathbb Z})=\coprod_{s_\alpha\in E} {{\rm GL}}(2,{\mathbb Z})s_\alpha.$$ Then $E$ is a finite set. At this point, the $s_\alpha$ may be altered by left-multiplication by elements of ${{\rm GL}}(2,{\mathbb Z})$. We now adjust the elements of $E$ to make the computation of Hecke operators easier.
Because of our choice of ${\mathcal A}$, we have that $x_0\in{\mathcal A}$. For convenience in what follows, we will write ${\Gamma_{{x}_0}}={\Gamma_{0}}$. Recall that $\mathcal A$ is a collection of $Z\,{{\rm GL}}(2,{\mathbb Z})$-representatives of ${X_{S,q}}$. Hence, we may write any element of ${X_{S,q}}$ as $\zeta\gamma {x}$ for some ${x}\in{\mathcal A}$, $\zeta\in {\mathcal B}$ and $\gamma\in{{\rm GL}}(2,{\mathbb Z})$. Suppose that for $s_\alpha\in E$, we have $$s_\alpha{x}_0=\zeta\gamma{x}.$$ We will then adjust $s_\alpha$, replacing it by $\gamma^{-1}s_\alpha$, and denote the corresponding $\zeta$ by $\zeta_\alpha$, and the corresponding ${x}$ by ${x}_\alpha$, so that we have $$s_\alpha{x}_0=\zeta_\alpha{x}_\alpha.$$ We now fix this choice of $E$.
For a given $\zeta\in Z$ and ${x}\in{\mathcal A}$, we define $$E(\zeta{x})=\{s_\alpha\in E:s_\alpha{x}_0=\zeta{x}\}.$$ Since each $s_\alpha x_0$ is of the form $\zeta {x}$, we see easily that $E$ is a disjoint union of the $E(\zeta{x})$ as $\zeta$ runs through ${\mathcal B}$ and ${x}$ runs through ${\mathcal A}$ (and $E(\zeta{x})$ is empty for all but finitely many ${x}\in{\mathcal A}$).
Now, let $\zeta\in{\mathcal B}$, ${x}\in{\mathcal A}$, and (to avoid triviality) assume that for some $s_\alpha\in{\mathcal A}$, we have $s_\alpha{x}_0=\zeta{x}$. We will then define $W_{\zeta{x}}$ to be the set of all elements $g\in{{\rm GL}}(2,{\mathbb Z})s{{\rm GL}}(2,{\mathbb Z})$ such that $g{x}_0=\zeta{x}$. This set is nonempty, and stable under right multiplication by ${\Gamma_{0}}$ and under left multiplication by ${\Gamma_{{x}}}$. Hence, we may write it as a disjoint union of double cosets $$W_{\zeta{x}}=\coprod_{t\in T_{\zeta{x}}}{\Gamma_{{x}}}t{\Gamma_{0}},$$ for some subset $T_{\zeta{x}}\subseteq W_{\zeta{x}}$.
For each $t\in T_{\zeta{x}}$, we may choose a set $B_{\zeta x,t}$ of single coset representatives and write the double coset ${\Gamma_{{x}}}t{\Gamma_{0}}$ as a disjoint union of single cosets $${\Gamma_{{x}}}t{\Gamma_{0}}=\coprod_{t_\beta\in B_{\zeta{x},t}}{\Gamma_{{x}}}t_\beta.$$
\[L:partition\] With the notation described above,
1. $E$ is a disjoint union of the $E(\zeta{x})$ for $\zeta\in{\mathcal B}$ and ${x}\in{\mathcal A}$.
2. For each $t\in T_{\zeta{x}}$, we may choose $B_{\zeta{x},t}$ to be a subset of $E$.
3. With this choice, $E(\zeta{x})$ is the disjoint union of the $B_{\zeta{x},t}$ over all $t\in T_{\zeta{x}}$, and $B_{\zeta{x},t}={\Gamma_{{x}}}t{\Gamma_{0}}\cap E(\zeta{x})$.
First, note that because $\zeta$ is central, ${\Gamma_{\zeta{x}}}={\Gamma_{{x}}}$ for any ${x}\in {X_{S,q}}$.
We have seen (1) above.
For (2), choose $t\in T_{\zeta{x}}$, and let ${\Gamma_{{x}}}u$ be any single coset in ${\Gamma_{{x}}}t{\Gamma_{0}}$. Then $u{x}_0=\zeta{x}$, and $u$ is in some single coset ${{\rm GL}}(2,{\mathbb Z})s_\alpha$ for some $s_\alpha\in E$. Then $u=gs_\alpha$ for some $g\in{{\rm GL}}(2,{\mathbb Z})$. We then have that $$\zeta{x}=u{x}_0=gs_\alpha{x}_0=g\zeta_\alpha{x}_\alpha.$$ This implies that ${x}$ and ${x}_\alpha$ are in the same $Z\,{{\rm GL}}(2,{\mathbb Z})$-orbit, and hence equal (since both come from ${\mathcal A}$). Hence, $g\in{\Gamma_{{x}}}$, and consequently $\zeta_\alpha x=\zeta x$. Therefore $\zeta_\alpha\zeta^{-1}\in {Z_{S,q}}$ and since $\zeta_\alpha,\zeta\in{\mathcal B}$, it follows that $\zeta_\alpha=\zeta$. Hence, we have $s_\alpha\in E(\zeta{x})$, and ${\Gamma_{{x}}}u={\Gamma_{{x}}}s_\alpha$, so we see that we may take the coset representative of ${\Gamma_{{x}}}u$ to be $s_\alpha\in E$.
For (3), we first note that any coset ${\Gamma_{{x}}}t$ for $t\in T_{{x}}$ contains exactly one $s_\alpha$: part (2) shows that it contains at least one; if it contained two, say $s_\alpha$ and $s_\delta$, then they would differ by left multiplication by $\gamma\in{\Gamma_{{x}}}\subseteq{{\rm GL}}(2,{\mathbb Z})$, which would imply ${{\rm GL}}(2,{\mathbb Z})s_\alpha={{\rm GL}}(2,{\mathbb Z})s_\delta$ and therefore $s_\alpha=s_\delta$. Hence, it suffices to show that each $s_\alpha\in E(\zeta{x})$ is contained in $B_{\zeta{x},t}$ for some $t\in T_{\zeta{x}}$. This is, however, clear, since such an $s_\alpha$ is contained in $$W_{\zeta{x}}=\bigcup_{t\in T_{\zeta{x}}}{\Gamma_{{x}}}t{\Gamma_{0}}.$$ The last assertion is now clear.
From this point on, we will take $B_{\zeta{x},t}={\Gamma_{{x}}}t{\Gamma_{0}}\cap E(\zeta{x})$.
We continue to keep a fixed ${x}_0$. As $\zeta\in{\mathcal B}$, ${x}\in{\mathcal A}$, and $t\in T_{\zeta{x}}$ vary, finitely many $B_{\zeta{x},t}$ will be nonempty. We denote these sets by $B_1,\ldots,B_J$ for $J\in{\mathbb Z}$ positive, and for $j\in\{1,\ldots,J\}$ we write ${x}_j$, $\zeta_j$, and $t_j$ for the corresponding values of $\zeta$, ${x}$, and $t$, respectively. In addition, we will write ${\Gamma_{j}}$ for ${\Gamma_{{x}_j}}$. With this notation, we now have $$t_j{x}_0=\zeta_j{x}_j\quad\text{and}\quad s_\alpha{x}_0=\zeta_j{x}_j$$ for all $s_\alpha\in B_j$.
We note that for $s_\alpha\in B_j$, there exist $\eta_\alpha\in{\Gamma_{j}}$ and $\delta_\alpha\in{\Gamma_{0}}$ such that $s_\alpha=\eta_\alpha t_j\delta_\alpha$.
\[L:stab-reps\] Fix $j\in\{1,\ldots,J\}$, and for each $s_\alpha\in B_j$, write $s_\alpha=\eta_\alpha t_j \delta_\alpha$ with $\delta_\alpha\in{\Gamma_{0}}$ and $\eta_\alpha\in{\Gamma_{j}}$. Let $d_j=|B_j|$. Then $${\Gamma_{0}}=\coprod_{s_\alpha\in B_j}(t_j^{-1}{\Gamma_{j}}t_j\cap{\Gamma_{0}})\delta_\alpha$$ so that we have $$[{\Gamma_{0}}:t_j^{-1}{\Gamma_{j}}t_j\cap{\Gamma_{0}}]=d_j.$$
First we show that the displayed cosets are all different. Suppose $s_\alpha,s_\beta\in B_j$ with $\delta_\alpha \delta_\beta^{-1}=t_j^{-1}\gamma_jt_j\in t_j^{-1}{\Gamma_{j}}t_j\cap{\Gamma_{0}}$ for some $\gamma_j\in{\Gamma_{j}}$. Then $\delta_\alpha=t_j^{-1}\gamma_jt_j\delta_\beta$. Hence $$\begin{aligned}
{\Gamma_{j}}s_\alpha&={\Gamma_{j}} \eta_\alpha t_j\delta_\alpha\cr
&={\Gamma_{j}} \eta_\alpha t_j(t_j^{-1}\gamma_jt_j\delta_\beta)\cr
&={\Gamma_{j}} t_j\delta_\beta\cr
&={\Gamma_{j}} \eta_\beta t_j\delta_\beta\cr
&={\Gamma_{j}} s_\beta,\end{aligned}$$ where we have used that $\eta_\alpha,\eta_\beta,\gamma_j\in{\Gamma_{j}}$. Hence, $s_\alpha=s_\beta$, and we see that the cosets $(t_j^{-1}{\Gamma_{j}}t_j\cap{\Gamma_{0}})\delta_\alpha$ are pairwise disjoint for $s_\alpha\in B_j$.
It remains only to show that the union of the cosets is all of ${\Gamma_{0}}$. Let $g\in{\Gamma_{0}}$. Since ${\Gamma_{j}}t_j{\Gamma_{0}}$ is a disjoint union of cosets of the form ${\Gamma_{j}} s_\alpha$ for some $\alpha\in B_j$, we have that $t_jg=\gamma_js_\alpha$ for some $\gamma_j\in{\Gamma_{j}}$ and $s_\alpha\in B_j$. Then, since $s_\alpha=\eta_\alpha t_j\delta_\alpha$, for $\eta_\alpha\in{\Gamma_{j}}$ and $\delta_\alpha\in{\Gamma_{0}}$, we have $t_jg=\gamma_j \eta_\alpha t_j\delta_\alpha$. Hence $$g\delta_\alpha^{-1}=t_j^{-1}\gamma_j\eta_\alpha t_j\in t_j^{-1}{\Gamma_{j}}t_j\cap{\Gamma_{0}},$$ so $g\in (t_j^{-1}{\Gamma_{j}}t_j\cap{\Gamma_{0}})\delta_\alpha.$
The Hecke operator $T_s={{\rm GL}}(2,{\mathbb Z})s{{\rm GL}}(2,{\mathbb Z})$ acts in the usual way on the homology $H_1({{\rm GL}}(2,{\mathbb Z}),{M_{S,q}})$ (see below), and hence, via the Shapiro isomorphism on the group $$\bigoplus_{{x}\in\mathcal A} H_1({\Gamma_{{x}}},{\mathbb F}).$$ We now work out the details of the action on this latter group. To do this, we use the following lemma concerning transfers and corestrictions. This lemma is standard, and follows easily from [@Brown Sec. III.9].
\[transfer\] Let $A$ be an infinite cyclic group with generator $a$, and $B\subset A$ a subgroup of index $d$, and suppose that $A$ acts trivially on ${\mathbb F}$. For a group $G$ acting trivially on ${\mathbb F}$, we identify $H_1(G,{\mathbb F})$ canonically with $G^{ab}\otimes_{\mathbb Z}{\mathbb F}$.
(i) The transfer map ${{\rm tr}}:H_1(A,{\mathbb F})\to H_1(B,{\mathbb F})$ takes the generator $a\otimes 1$ to the generator $a^d\otimes 1$.
(ii) The corestriction map ${i}:H_1(B,{\mathbb F})\to H_1(A,{\mathbb F})$ (i.e. the map induced by the inclusion $B\subset A$) takes the generator $a^d\otimes 1$ to $d$ times the generator $a\otimes 1$.
With notation as above, we will use the techniques of [@AD1] to write the Hecke operator $T_s$ (given as a sum of actions of all the $s_\alpha$) acting on a generator $z_0$, as a sum of partial Hecke operators $U_{t_j}$ given as a sum of actions of the $s_{\beta,j}\in B_j$, so that $U_{t_j}$ maps $H_1({\Gamma_{0}},{\mathbb F})$ to $H_1({\Gamma_{j}},{\mathbb F})$.
More precisely, if $F_\bullet$ is a resolution of ${\mathbb F}$ by free ${\mathbb F}{{\rm GL}}(2,{\mathbb Q})$-modules, then $T_s$ on $H_1({{\rm GL}}(2,{\mathbb Z}),{M_{S,q}})$ sends the class of a cycle (using an obvious notation for elements of $F_\bullet$ and ${M_{S,q}}$) $\sum_f f\otimes_{{\mathbb F}{{\rm GL}}(2,{\mathbb Z})} m(f)$ to the class of $\sum_\alpha \sum_f s_\alpha f\otimes_{{\mathbb F}{{\rm GL}}(2,{\mathbb Z})} s_\alpha m(f)$. The partial Hecke operator $U_{t_j}$ sends the class of a cycle $\sum_f f\otimes_{{\Gamma_{0}}} \lambda(f)$ to the class of $\sum_{s_{\beta,j}\in B_j}\sum_f s_{\beta,j}f\otimes_{{\Gamma_{j}}} \lambda(f)$.
The following lemma follows immediately from Theorem 3.1 in [@AD1].
\[ad1\] $T_s$ composed with the Shapiro isomorphism equals $\sum_{j=1}^J U_{t_j}$.
From now on we will also use $T_s$ to stand for $\sum_{j=1}^J U_{t_j}$, depending on the context.
Next, we write the partial Hecke operator in terms of the transfer, corestriction, and an adjoint map.
\[Thm:partial-hecke\] Recall that $x_0\in\mathcal A$, and let $z_0$ be the generator of $H^1({\Gamma_{0}},{\mathbb F})$ chosen in Definition \[Def:zx\]. Then $U_{t_j}:H_1({\Gamma_{0}},{\mathbb F})\to H_1({\Gamma_{j}},{\mathbb F})$ is given as the composition of the three maps $$H_1({\Gamma_{0}},{\mathbb F})\to H_1({\Gamma_{0}}\cap s_j^{-1}{\Gamma_{j}}s_j,{\mathbb F})
\xrightarrow{\phi_j}
H_1(s_j{\Gamma_{0}}s_j^{-1}\cap {\Gamma_{j}},{\mathbb F})\to H_1({\Gamma_{j}},{\mathbb F}),$$ where the first map is the transfer, the second map is the map induced on homology by the pair of maps $(\operatorname{Ad}t_j,\tau_j)$, where $\operatorname{Ad}t_j$ is conjugation by $t_j$ on the group, and $\tau_j$ is multiplication by $q(\zeta_j)$ on the coefficient module, and the third map is corestriction.
We take a resolution $F_\bullet$ of ${\mathbb F}$ by free ${\mathbb F}[{{\rm GL}}(2,{\mathbb Q})]$-modules and let $Z_0$ be a cycle representing $z_0$. The map in Shapiro’s lemma taking $H_1({\Gamma_{0}},{\mathbb F})$ into $H_1({{\rm GL}}(2,{\mathbb Z}),{M_{S,q}})$ sends $Z_0$ to $Z_0\otimes e_*(I\otimes 1)$. Then, on the level of cycles, we have $$T_s(Z_0\otimes e_*(I\otimes 1))=\sum_{j=1}^J\sum_{s_{\beta}\in B_j}s_{\beta}Z_0\otimes s_{\beta}e_*(I\otimes 1).$$ If we let $w_j$ be the composition of the three maps in the statement of the theorem, then using Lemma \[ad1\] we see that what we need to show is that $$w_j(Z_0)\otimes e_*(I\otimes 1)=\sum_{s_{\beta}\in B_j}s_{\beta}Z_0\otimes s_{\beta}e_*(I\otimes 1).$$
On the level of cycles, using Lemma \[L:stab-reps\], the transfer of $Z_0$ is $$\sum_{s_{\beta}\in B_j}\delta_{\beta}Z_0,$$ where we recall that for $s_\beta\in B_j$, we have written $s_\beta=\eta_\beta t_j\delta_\beta$, with $\eta_\beta\in{\Gamma_{j}}$ and $\delta_\beta\in{\Gamma_{0}}$. Applying $\phi_j$ yields $$q(\zeta_j)\sum_{s_\beta\in B_j}(t_j\delta_\beta t_j^{-1})(t_jZ_0).$$ Finally, applying the corestriction gives $$q(\zeta_j)\sum_{s_\beta\in B_J}t_j\delta_\beta Z_0=q(\zeta_j)\sum_{s_\beta\in B_j}s_\beta Z_0.$$
We see that it suffices to show that $q(\zeta_j)e_*(I\otimes 1)=s_\beta e_*(I\otimes 1)$.
We have that $$\begin{aligned}
s_\beta e_*(I\otimes 1)&=s_\beta\sum_{\zeta_r\in{\mathcal B}}q(\zeta_r^{-1})\zeta_r{x}_0\cr
&=\sum_{\zeta_r\in{\mathcal B}}q(\zeta_r^{-1})\zeta_rs_\beta{x}_0\cr
&=\sum_{\zeta_r\in{\mathcal B}}q(\zeta_r^{-1})\zeta_r\zeta_j{x}_j\cr
&=\zeta_j\sum_{\zeta_r\in{\mathcal B}}q(\zeta_r^{-1})\zeta_r{x}_j\cr
&=q(\zeta_j)\sum_{\zeta_r\in{\mathcal B}}q(\zeta_r^{-1})\zeta_r{x}_j\cr
&=q(\zeta_j)e_*(I\otimes 1).\qedhere\end{aligned}$$
We now apply Theorem \[Thm:partial-hecke\] to compute $U_{t_j}(z_0)$.
\[C:partial-hecke\] The partial Hecke operator $U_{t_j}$ in Theorem \[Thm:partial-hecke\] satisfies $$U_{t_j}(z_0)=e_jq(\zeta_j)z_j$$ where $e_j=[{\Gamma_{j}}:t_j{\Gamma_{0}}t_j^{-1}\cap{\Gamma_{j}}]$.
By definition, $d_j=[{\Gamma_{0}}:t_j^{-1}{\Gamma_{j}}t_j\cap{\Gamma_{0}}]$ and $e_j=[{\Gamma_{j}}:t_j{\Gamma_{0}}t_j^{-1}\cap{\Gamma_{j}}]$. Hence $g_0^{d_j}$ is a generator of ${\Gamma_{0}}\cap t_j^{-1}{\Gamma_{j}}t_j$, and $g_j^{d_j}$ is a generator of $t_j{\Gamma_{0}}t_j^{-1}\cap{\Gamma_{j}}$. Considering $H_1({\Gamma_{0}},{\mathbb F})$ as ${\Gamma_{0}}\otimes_{\mathbb Z}F$ we have $z_0=g_0\otimes 1$. Hence, by Lemma \[transfer\](i), the transfer takes $z_0$ to $g_0^{d_j}\otimes 1$, which is then mapped to $g_j^{d_j}\otimes q(\zeta_j)$ by $\phi_j$. Finally, by Lemma \[transfer\](ii), the corestriction maps this to $e_j(g_j\otimes q(\zeta_j))=e_jq(\zeta_j)(g_j\otimes 1)=e_jq(\zeta_j)z_j$.
We now compute the value of $e_j$. Recall that $t_j{x}_0=\zeta_j{x}_j$. For $j=0,\dots,J$, choose $y_j\in Y$ such that ${x}_j$ is represented by $y_j$, and recall that ${\Gamma_{j}}$ is the stabilizer of ${x}_j$ in ${{\rm GL}}(2,{\mathbb Z})$ and $\epsilon$ is the fundamental unit of ${\mathfrak O}$ which we chose at the beginning of Section \[S:Lattices\]. Let $g_j\in\Gamma_j$ and $m_j\in{\mathbb Z}$ be defined as in Definition \[D:generator\] (with $H={K_{S,q}}$). Then $g_j$ is a generator of $\Gamma_j$ and $m_j>0$. Set $\delta_j=\pm\epsilon^{m_j}$, where the sign is chosen so that $g_jy_j=y_j\delta_j$ and $\delta_j\in{K_{S,q}}$.
\[H5\] With notation as above, $$e_j={{\rm LCM}}(m_0,m_j)/m_j,\qquad d_j={{\rm LCM}}(m_0,m_j)/m_0,$$ and $e_jm_j=d_jm_0$.
First, $g_0y_0=y_0\delta_0.$ Since $t_j{x}_0=\zeta_j{x}_j$, we have $t_jy_0=\alpha_j\zeta_jy_j$ for some $\alpha_j\in{K_{S,q}}$. Hence, $t_j^{-1}y_j=\alpha_j^{-1}\zeta_j^{-1}y_0$.
It follows that $t_jg_0t_j^{-1}y_j=y_j\delta_0$. In addition, $g_jy_j=y_j\delta_j$, $g_0$ generates $\Gamma_0$, and $g_j$ generates $\Gamma_j$.
We may choose a generator $h$ of $\Gamma_j\cap t_j\Gamma_0t_j^{-1}$, so that $h$ will be the smallest power of $t_jg_0t_j^{-1}$ that is contained in $\Gamma_j$. This power must be the smallest positive integer $k$ such that $\delta_0^k$ is a power of $\delta_j$. Since $\delta_0,\delta_j\in{K_{S,q}}$ and $-1\notin{K_{S,q}}$, we see that $k$ will be the smallest positive integer such that $km_0$ is a multiple of $m_j$. Hence, $km_0={{\rm LCM}}(m_0,m_j)$, and we see that $$h y_j=y_j\left(\pm\epsilon^{{{\rm LCM}}(m_0,m_j)}\right).$$ It follows that $$e_j={{\rm LCM}}(m_0,m_j)/m_j.$$ Reversing the roles of $\Gamma_0$ and $\Gamma_j$ and switching $t_j$ and $t_j^{-1}$, we obtain $$d_j={{\rm LCM}}(m_0,m_j)/m_0.\qedhere$$
Elements of $H^1({{\rm GL}}(2,{\mathbb Z}), {M_{S,q}}^*)$ interpreted as functions on lattices
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We now interpret the cohomology of the dual of ${M_{S,q}}$ as a collection of functions on a space of lattices.
Let $\Phi$ be a function from lattices in $K$ to ${\mathbb F}$. We will say that $\Phi$ is $q$-homogeneous if $\Phi(\alpha L)=q(\alpha I)\Phi(L)$ for all $\alpha\in {\mathbb Q}^\times\cap K_{S_0}$ and all lattices in $L$.
We will say that $\Phi$ is ${K_{S,q}}$-invariant if $\Phi(\alpha L)=\Phi(L)$ for all $\alpha\in{K_{S,q}}$ and all lattices $L$.
Note that since $q$ is trivial on ${K_{S,q}}$, a function $\Phi$ can be both $q$-homogeneous and ${K_{S,q}}$-invariant. In addition, we note that since $K$ is a real quadratic field, $q(-I)=1$. If this were not the case, the fact that $-L=L$ for any lattice $L$ in $K$ would force all $q$-homogeneous functions to be identically 0.
\[Lemma:homothety-to-cohomology\] There is an isomorphism between $H^1({{\rm GL}}(2,{\mathbb Z}),{M_{S,q}}^*)$ and the vector space of ${\mathbb F}$-valued functions on lattices in $K$ that are $q$-homogeneous and ${K_{S,q}}$-invariant.
Choose a set ${\mathcal A}$ of representatives of the $Z\,{{\rm GL}}(2,{\mathbb Z})$-orbits in ${X_{S,q}}$. This choice of ${\mathcal A}$ yields an isomorphism of ${{\rm GL}}(2,{\mathbb Z})$-modules $$f:{M_{S,q}}\to\bigoplus_{{x}\in\mathcal A}{\mathbb F}\,{{\rm GL}}(2,{\mathbb Z})\otimes_{{\mathbb F}{\Gamma_{{x}}}}{\mathbb F}.$$
This induces an isomorphism (via Shapiro’s Lemma) $$\begin{aligned}
H_1({{\rm GL}}(2,{\mathbb Z}),{M_{S,q}})&\cong\bigoplus_{{x}\in\mathcal A}H_1({{\rm GL}}(2,{\mathbb Z}),{\mathbb F}\,{{\rm GL}}(2,{\mathbb Z})\otimes_{{\mathbb F}{\Gamma_{{x}}}}{\mathbb F})\cr
&\cong \bigoplus_{{x}\in\mathcal A}H_1({\Gamma_{{x}}},{\mathbb F})\cr
&\cong\bigoplus_{{x}\in\mathcal A}{\mathbb F}.\end{aligned}$$ Using the natural duality between $H_1({{\rm GL}}(2,{\mathbb Z}),{M_{S,q}})$ and $H^1({{\rm GL}}(2,{\mathbb Z}),{M_{S,q}}^*)$, we see that determining an element of $H^1({{\rm GL}}(2,{\mathbb Z}),{M_{S,q}}^*)$ is the same as giving a function from $\mathcal A$ to ${\mathbb F}$.
We now show that there is an isomorphism between the vector space of functions from $\mathcal A$ to ${\mathbb F}$ and the vector space of ${K_{S,q}}$-invariant $q$-homogeneous functions on lattices in $K$.
Let $h$ be any $q$-homogeneous ${K_{S,q}}$-invariant function on lattices in $K$. Since every element in ${\mathcal A}$ can be lifted uniquely to a ${K_{S,q}}$-homothety class of lattices in $K$, $h$ defines a function $g$ on $\mathcal A$. Namely, given $x\in\mathcal A$, lift $x$ to $y\in Y$ and set $g(x)=h(\Lambda_y)$, where $\Lambda_y$ is the lattice spanned by the entries of $y$. Since $y$ is well-defined up to ${K_{S,q}}$-homotheties and $h$ is ${K_{S,q}}$-invariant, this gives a well-defined function $g$.
Given a function $g$ on $\mathcal A$ and a lattice $L$ in $K$, $L$ corresponds (by choosing a basis $y={{}^t(a,b)}\in Y$) to an element $x'\in {X_{S,q}}$, which lies in the $Z\,{{\rm GL}}(2,{\mathbb Z})$-orbit of a unique $x\in\mathcal A$. Let $x'=\zeta\gamma x$, with $\zeta\in Z$ and $\gamma\in{{\rm GL}}(2,{\mathbb Z})$. Define $h(L)=q(\zeta)g(x)$. Note that if $x'=\zeta'\gamma'x$ with $\zeta'\in Z$ and $\gamma'\in{{\rm GL}}(2,{\mathbb Z})$, then we have $(\zeta^{-1}\zeta')(\gamma^{-1}\gamma')x=x$. Hence, $(\zeta^{-1}\zeta')(\gamma^{-1}\gamma')=r_y(\alpha)$ for some $\alpha\in {K_{S,q}}$. By Lemma \[Lemma:Z-KSQ\], this implies that $q(\zeta)=q(\zeta')$, so $h$ is well defined. Then $h$ is a $q$-homogeneous ${K_{S,q}}$-invariant function on lattices.
These two maps (taking $h$ to $g$ and $g$ to $h$) are easily seen to be inverses, and preserve addition and scalar multiplication.
The Branched Bruhat-Tits graph and the Laplacian
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In order to construct functions on lattices that are eigenfunctions of the Hecke operators, we will use a modification of the Bruhat-Tits building [@Casselman; @Serre], in which we lift the Bruhat-Tits building to a finite branched cover.
For each prime $\ell$ unramified in $K$, let $K_\ell$ denote $K\otimes{\mathbb Q}_\ell$. Then $K_\ell$ is a two-dimensional vector space over ${\mathbb Q}_\ell$.
\[identify\] If $\ell$ is inert, then $K_\ell$ is a quadratic field extension of ${\mathbb Q}_\ell$. We fix the integral basis $\{1,\omega\}$ of $K$, and we identify $K_\ell$ with ${\mathbb Q}_\ell^2$ by identifying $1$ and $\omega$ with the standard basis elements $e_1,e_2\in{\mathbb Q}_\ell^2$.
If $\ell$ splits in $K$, then $(\ell)=\lambda\lambda'$ for prime ideals $\lambda,\lambda'$ in ${\mathfrak O}$ lying over $\ell$. Each of the completions $K_\lambda$ and $K_\lambda'$ is then isomorphic to ${\mathbb Q}_\ell$. Restricting these isomorphisms to $K$, we obtain two distinct Galois conjugate embeddings $i_\lambda,i_{\lambda'}:K\to{\mathbb Q}_\ell$. We then identify $K_\ell=K\otimes_{\mathbb Q}{\mathbb Q}_\ell$ with ${\mathbb Q}_\ell^2$ via the map taking $$t\otimes 1\mapsto (i_\lambda(t),i_{\lambda'}(t)).$$ We abbreviate the notation by writing $t\mapsto (t,t')$.
By a lattice in $K_\ell$, we will mean a rank two ${\mathbb Z}_\ell$-submodule of $K_\ell$.
If $L$ is a lattice in $K$, then $L_\ell=L\otimes_{\mathbb Z}{\mathbb Z}_\ell$ is a lattice in $K_\ell$.
Let $\ell$ be a prime, and $n$ a positive integer. Denote the elements of ${\mathbb Q}_\ell^\times$ with $\ell$-adic valuation divisible by $n$ by $V_n$. We note that $V_n$ is a subgroup of index $n$ of ${\mathbb Q}_\ell^\times$.
Let $L_1$ and $L_2$ be lattices in $K_\ell$. We say that $L_1$ and $L_2$ are [*$n$-homothetic*]{} if $L_1=\alpha L_2$ for some $\alpha\in V_n$. Then $n$-homothety is an equivalence relation, and we call an equivalence class an $n$-homothety class of lattices in $K_\ell$.
Let $n$ a positive integer, $K$ a real quadratic field, and $\ell$ a prime unramified in $K$ . The branched Bruhat-Tits graph ${\mathcal T}_\ell^n$ is the graph whose vertices are $n$-homothety classes of lattices in $K_\ell$. Two vertices are joined by an edge if there are representative lattices $L_1$ and $L_2$ of the vertices such that $L_2\subset L_1$ or $L_1\subset L_2$ with index $\ell$.
The Bruhat-Tits tree is a special case of the branched Bruhat-Tits graph in which $n=1$. When $n=1$, we may denote ${\mathcal T}_\ell^n$ by ${\mathcal T}_\ell$. When $n>1$, we will typically write vertices of ${\mathcal T}_\ell^n$ with a superscript $n$, i.e. $t^n\in{\mathcal T}_\ell^n$.
Let $L$ be a lattice in $K_\ell$. Denote the vertex of ${\mathcal T}_\ell^n$ represented by $L$ by $\varpi(L)$. Denote the vertex of ${\mathcal T}_\ell$ represented by $L$ by $\pi(L)$. Given a vertex $t^n\in{\mathcal T}_\ell^n$, there is a unique vertex $s\in{\mathcal T}_\ell$ containing $t^n$; we write $s=\pi(t^n)$.
Note that for any lattice $L$ with $\varpi(L)=t^n$, $\pi(t^n)=\pi(L)$. To keep our notation less cluttered, if $L$ is a lattice in $K_\ell$, we will often denote $\varpi(L)$ by $L$, as long as the context makes this usage clear.
We note that for any vertex $t\in{\mathcal T}_\ell$, there are exactly $n$ vertices $t^n\in {\mathcal T}_\ell^n$ with $\pi(t^n)=t$. If $L$ is a lattice in $K_\ell$ representing $t$, these $n$ vertices of ${\mathcal T}_\ell^n$ are represented by $$L,\ell L,\ldots,\ell^{n-1}L.$$
If $t$ is a vertex of ${\mathcal T}_\ell$, we will call the set $\{t^n\in{\mathcal T}_\ell^n: \pi(t^n)=t\}$ the [*fiber*]{} of $t$ and also the [*fiber*]{} of $t^n$ for any $t^n$ in that set .
\[D:idealistic\] A vertex $t^n$ is [*idealistic*]{} if $t^n$ is the $n$-homothety class of $I_\ell$ for some fractional ideal $I$ of $K$.
We now review some facts about completions $L_\ell$ of lattices in $K$. Let $\ell$ be a prime of ${\mathbb Q}$.
By [@Weil V.2, Corollary to Theorem 2], the operations of sum and intersection of lattices in $K$ commute with completion at $\ell$. In addition, by [@Weil V.3, Theorem 2], a lattice $L$ in $K$ is determined by its set of completions $L_w$ for all finite places $w$ of ${\mathbb Q}$. In fact $$L=\bigcap_w K\cap L_w.$$ Finally, completion at a finite place $w$ of finitely generated ${\mathbb Z}$-modules is an exact functor [@Eisenbud Theorem 7.2].
Applying these facts to fractional ideals of $K$, we note that if $I$ is an ideal of ${\mathfrak O}$ of norm prime to $\ell$, then $I_\ell$ is an ideal of ${\mathfrak O}_\ell$ of index prime to $\ell$, so $I_\ell={\mathfrak O_\ell}$. In addition, multiplication of relatively prime ideals (i.e. intersection) commutes with completion at $\ell$. Hence, for an ideal $I$, the completion $I_\ell$ depends only on the factors of $I$ of $\ell$-power norm.
Now suppose that $t^n\in{\mathcal T}_\ell^n$ is idealistic. Then we may assume that $t^n$ is represented by an ideal $I_\ell$, where $I$ is an ideal with $\ell$-power norm in ${\mathfrak O}$. If $\ell$ is inert in $K$, such an $I$ must be principal, so $I_\ell$ is ${\mathbb Q}_\ell$-homothetic to ${\mathfrak O_\ell}$. Hence, $t^n$ is idealistic if and only if $\pi(t^n)$ is represented by ${\mathfrak O_\ell}$.
On the other hand, if $\ell$ splits in $K$, then $\ell{\mathfrak O}=\lambda\lambda'$, where $\lambda,\lambda'$ are prime ideals of ${\mathfrak O}$ lying over $\ell$. We then see that $t^n\in{\mathcal T}_\ell^n$ is idealistic if and only if $\pi(t^n)$ is represented by an ideal of the form $\lambda^k$ or $(\lambda')^k$. In particular, if $\pi(t_1^n)=\pi(t_2^n)$, then $t_1^n$ and $t_2^n$ are either both idealistic, or both nonidealistic.
Let $L_1\supset L_2$ be lattices in $K_\ell$ with $[L_1:L_2]=\ell$. Let $t_1^n=\varpi(L_1)\in{\mathcal T}_\ell^n$. Then there are precisely two vertices $t_2^n,t_3^n\in{\mathcal T}_\ell^n$ with $\pi(t_2^n)=\pi(t_3^n)=\pi(L_2)$, such that there is an edge between the two pairs $(t_1^n,t_2^n)$ and $(t_1^n,t_3^n)$. If we let $t_2^n$ be represented by $L_2$, then $t_3^n$ is represented by $\ell^{-1} L_2$.
Clearly, if we take $t_2^n=\varpi(L_2)$ and $t_3^n=\varpi(\ell^{-1}L_2)$, we see that $t_2^n$ and $t_3^n$ are distinct and have the desired properties. It remains to show that there is no third vertex $t_4^n$, distinct from $t_2^n$ and $t_3^n$, with $\pi(t_4^n)=\pi(L_2)$, and such that there is an edge between $t_4^n$ and $t_1^n$.
Suppose that there is an edge between $t_1^n$ and $t_4^n$. Then either there is a lattice $L_3$ representing $t_4^n$ such that $L_1\supset L_3$ and $[L_1:L_3]=\ell$ or there is a lattice $L_3$ representing $t_4^n$ such that $L_1\subset L_3$ and $[L_3:L_1]=\ell$.
Now suppose $L_3$ is homothetic to $L_2$, say with $L_3=\alpha L_2$, where $\alpha\in{\mathbb Q}_\ell^\times$.
If $L_1\supset L_3$ has index $\ell$, then we have that $\ell^{-1}L_2\supset L_1$ has index $\ell$ and $L_1\supset\alpha L_2$ has index $\ell$. Hence, multiplying by $\ell$, we see that $L_2\subset \ell\alpha L_2$ with index $\ell^2$. This implies that $v_\ell(\alpha)=0$, so that $\alpha L_2=L_2$, so $t_4^n=t_2^n$.
On the other hand, if $L_1\subset L_3$ with index $\ell$, we have that $L_2\subset\alpha L_2$ has index $\ell^2$. Hence $v_\ell(\alpha)=-1$, and we see that $\alpha L_2=\ell^{-1}L_2$, so $t_4^n=t_3^n$.
Let $n\geq 1$, let $t^n$ be a vertex in ${\mathcal T}^n_\ell$, and let $t=\pi(t^n)\in{\mathcal T}_\ell$. Let $s\in{\mathcal T}_\ell$ be a neighbor of $t$. Then there are exactly two neighbors $s_1^n$ and $s_2^n$ of $t^n$ in ${\mathcal T}^n_\ell$ with $\pi(s_1^n)=\pi(s_2^n)=s$. If $L$ represents $t^n$, then exactly one of $s_1^n$ and $s_2^n$ is represented by a sublattice $L'$ of $L$ of index $\ell$; the other is represented by $\ell^{-1}L'$, which contains $L$ with index $\ell$.
Let $n\geq 1$, let $t^n\in{\mathcal T}_\ell^n$ be a vertex represented by a lattice $L$ in $K_\ell$, and let $s_1^n, s_2^n\in{\mathcal T}_\ell^n$ be two neighbors of $t_n$ with $\pi(s_1^n)=\pi(s_2^n)$. Call the neighbor represented by a sublattice of index $\ell$ in $L$ a [*downhill*]{} neighbor of $t^n$; call the other an [*uphill*]{} neighbor of $t$.
Let $t^n\in{\mathcal T}_\ell^n$. We define the [*tier*]{} of $t^n$ to be the distance between $\pi(t^n)$ and $\pi({\mathfrak O_\ell})$ in ${\mathcal T}_\ell$. A neighbor of $t^n$ of higher tier than $t^n$ will be called an [*outer neighbor*]{} of $t^n$: a neighbor of lower tier will be called an [*inner neighbor*]{}.
Each $t^n\in{\mathcal T}_\ell^n$ has precisely $\ell+1$ downhill neighbors and $\ell+1$ uphill neighbors. The use of uphill and downhill matches our intuition; if $s^n$ is a downhill neighbor of $t^n$, then $t^n$ is an uphill neighbor of $s^n$.
Each vertex of positive tier has precisely $\ell$ downhill outer neighbors, and $1$ downhill inner neighbor. It also has precisely $\ell$ uphill outer neighbors, and $1$ uphill inner neighbor.
A vertex of tier 0 has only outer neighbors; $\ell+1$ of them are uphill, and $\ell+1$ are downhill.
There is a natural action of the group ${{\rm GL}}(2,{\mathbb Q}_\ell)$ on ${\mathbb Q}_\ell^2$, namely matrix multiplication with elements of ${\mathbb Q}_\ell^2$ considered as column vectors. We transfer this action to $K_\ell$ via the identification that we have made between $K_\ell$ and ${\mathbb Q}_\ell^2$. The action of $g\in{{\rm GL}}(2,{\mathbb Q}_\ell)$ is invertible, and preserves ${\mathbb Q}_\ell$-linear combinations, so it maps bases of ${\mathbb Q}_l^2$ to bases, maps lattices to lattices, and preserves $n$-homothety of lattices. Hence, multiplication by $g$ defines a bijection from ${\mathcal T}_\ell^n$ to ${\mathcal T}_\ell^n$.
Let $F({\mathcal T}_\ell^n)$ be the set of ${\mathbb F}$-valued functions on the vertices of ${\mathcal T}_\ell^n$.
The [*Laplace operator*]{} $\Delta_\ell^n$ on $F({\mathcal T}_\ell^n)$ is defined by $$\Delta_\ell^n(f)(t^n)=\sum_{u^n}f(u^n),$$ where the sum runs over the $\ell+1$ downhill neighbors $u^n\in{\mathcal T}_\ell^n$ of $t^n\in {\mathcal T}_\ell^n$.
In the next lemma, we describe how the coset representatives for a Hecke operator act on lattices. Recall Lemmas \[L:partition\], \[L:stab-reps\] and \[ad1\] for the definition of the sets $B_j$, the coset representatives $s_{\beta,j}$, and the integers $d_j$. Note that these definitions depend on a choice of an element $x_0\in{X_{S,q}}$ and a choice of $Z\,{{\rm GL}}(2,{\mathbb Z})$-orbit representatives ${\mathcal A}$ containing $x_0$.
\[ell+1\] Let $x_0\in{X_{S,q}}$ and let ${\mathcal A}$ be a set of $Z\,{{\rm GL}}(2,{\mathbb Z})$-orbit representatives containing $x_0$. Let $x_0$ be represented by $y=(a_0,b_0)\in Y$ with $a_0,b_0\in{\mathfrak O}$, and let $L_0=L(a_0,b_0)$ be the lattice generated by $a_0$ and $b_0$. Let $s=\operatorname{diag}(\ell,1)$ and let $${{\rm GL}}(2,{\mathbb Z})s{{\rm GL}}(2,{\mathbb Z})=\coprod_{\alpha}{{\rm GL}}(2,{\mathbb Z})s_\alpha$$ with the $s_\alpha$ chosen and partitioned as described in Section \[coefficients\].
(i) $\mathcal L=\{s_\alpha L_0\}$ consists of the $\ell+1$ lattices of index $\ell$ contained in $L_0$.
(ii) $\mathcal L$ is partitioned into the subsets $$\mathcal L_j=\{s_{\alpha}L_0|s_\alpha\in B_j\},$$ and $|\mathcal L_j|=d_j$.
(iii) The same is true of the completions at $\ell$: $\mathcal L_\ell=\{s_\alpha( L_0)_\ell\}$ consists of the $\ell+1$ lattices of index $\ell$ contained in $(L_0)_\ell$, and these are partitioned into the subsets $$\mathcal L_{\ell,j}=\{s_{\alpha}(L_0)_\ell|\alpha\in B_j\}.$$ and $|\mathcal L_{\ell,j}|=d_j$
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(i) If $$s_\alpha\begin{pmatrix}a_0\cr b_0\end{pmatrix}=\begin{pmatrix}a_\alpha\cr b_\alpha\end{pmatrix},$$ then $s_\alpha L_0=L(a_\alpha,b_\alpha)$. Since $s_\alpha$ is an integral matrix of determinant $\ell$, it is clear that $L(a_\alpha,b_\alpha)$ has index $\ell$ in $L_0$, and all sublattices of $L_0$ of index $\ell$ arise this way.
(ii) Since $\{s_\alpha\}$ is partitioned by the sets $B_j$, it is clear that the lattices are partitioned as indicated.
(iii) If $L$ has index $\ell$ in $L_0$, then the completion $L_\ell$ has index $\ell$ in $(L_0)_\ell$, since taking completions of finitely generated modules is an exact functor. Given two lattices $L\neq M$, each having index $\ell$ in $L_0$, we note that for all places $w\neq\ell$, $L_w=M_w=(L_0)_w$. Since a lattice is determined by its completions at all finite places, we must have $L_\ell\neq M_\ell$.
Let $\phi_\ell^n\in F({\mathcal T}_\ell^n)$, let $x_0\in{X_{S,q}}$, and let ${\mathcal A}$ be any set of $Z\,{{\rm GL}}(2,{\mathbb Z})$-orbit representatives of ${X_{S,q}}$ containing $x_0$. Define the sets $B_j$ in terms of $x_0$ and ${\mathcal A}$ as in Lemma \[ell+1\]. If, for all choices of ${\mathcal A}$ and for all $y={{}^t(a_0,b_0)}\in Y$ with $a_0,b_0\in{\mathfrak O}$ representing $x_0$, we have that $\phi_\ell^n$ is constant on the set $$\{(s_{\beta,j}L(a_0,b_0))_\ell|\beta=1,\ldots,d_j\}$$ of vertices of ${\mathcal T}_\ell^n$, then we will say that $\phi_\ell^n$ is [*locally constant*]{} relative to $T_\ell$ and $x_0$.
If $\phi_\ell^n$ is locally constant relative to $T_\ell$ and all $x_0\in{X_{S,q}}$, then we say that $\phi_\ell^n$ is [*locally constant*]{}.
We remark that the condition $a_0,b_0\in{\mathfrak O}$ could be relaxed to $a_0,b_0\in K$ without effect. This is true because for any pair $a_0,b_0\in K$, there is an integer $m$ such that $m^na_0,m^nb_0\in{\mathfrak O}$; then $L(a_0,b_0)$ and $L(m^na_0,m^nb_0)$ are $n$-homothetic and hence define the same vertex of ${\mathcal T}_\ell^n$.
Let $t_0^n\in{\mathcal T}_\ell^n$ be the vertex represented by the lattice ${\mathfrak O_\ell}$.
\[L:gl2zl-invariant\] The action of ${{\rm GL}}(2,{\mathbb Z}_\ell)$ on ${\mathcal T}_\ell^n$ permutes the vertices of ${\mathcal T}_\ell^n$, fixes vertices of tier 0, and preserves edges (including whether the edge is uphill or downhill) and the tier of each vertex.
Since the action of ${{\rm GL}}(2,{\mathbb Z}_\ell)$ is invertible, it is clear that the map it induces on vertices is a bijection. In addition, if $\gamma\in{{\rm GL}}(2,{\mathbb Z}_\ell)$, and $L_1\subset L_2$ with index $\ell$, then $\gamma L_1\subset \gamma L_2$ with index $\ell$, so edges are preserved (including whether the edge is uphill or downhill).
Since the action of ${{\rm GL}}(2,{\mathbb Z}_\ell)$ fixes ${\mathbb Z}_\ell^2$, which is identified with ${\mathfrak O_\ell}$, it fixes vertices of tier 0. Since it preserves neighbors, a simple inductive argument shows that it maps each vertex to a vertex of the same tier.
\[mult-epsilon\] Multiplication by the fundamental unit $\epsilon\in K\subset K_\ell$ induces a permutation on the vertices of ${\mathcal T}_\ell^n$ given (on the level of ${\mathbb Z}_\ell$-lattices) by multiplication by a matrix in ${{\rm GL}}(2,{\mathbb Z}_\ell)$.
Suppose that $\ell$ is inert in $K$. In this case (see Definition \[identify\]), we have identified $K_\ell={\mathbb Q}_\ell\oplus{\mathbb Q}_\ell\omega$ with ${\mathbb Q}_\ell^2$. Since multiplication by $\epsilon$ is ${\mathbb Q}$-linear on $K$ it induces a ${\mathbb Q}_\ell$-linear map on $K_\ell$. Hence, multiplication by $\epsilon$ is represented by a matrix in ${{\rm GL}}(2,{\mathbb Q}_\ell)$. Since multiplication by $\epsilon$ is an automorphism of ${\mathfrak O_\ell}$, and ${\mathfrak O_\ell}$ is identified with ${\mathbb Z}_\ell^2\subset{\mathbb Q}_\ell^2$, this matrix has entries in ${\mathbb Z}_\ell$, and since $\epsilon$ has norm $\pm1$, the matrix must have determinant $\pm1$, and we see that the matrix is in ${{\rm GL}}(2,{\mathbb Z}_\ell)$.
Now suppose that $\ell$ is split. Referring to Definition \[identify\] again, we have identified $K_\ell$ with ${\mathbb Q}_\ell^2$, where $c\in K$ is identified with $(c,c')\in{\mathbb Q}_\ell^2$. Hence, multiplication by $\epsilon$ is represented by the matrix $\operatorname{diag}(\epsilon,\epsilon')$, which is in ${{\rm GL}}(2,{\mathbb Z}_\ell)$.
\[loc-const\] Let $\phi\in F({\mathcal T}^n_\ell)$ be a function on the vertices of ${\mathcal T}^n_\ell$. Assume that for every vertex $t^n\in{\mathcal T}^n_\ell$, $\phi$ is constant on the set of non-idealistic outer downhill neighbors $u^n$ of $t^n$. Then $\phi$ is locally constant relative to $T_\ell$ and any $x_0\in{X_{S,q}}$.
Assume that $\phi$ satisfies the conditions of the lemma. Let $x_0\in{X_{S,q}}$, choose any collection ${\mathcal A}$ of orbit representatives containing $x_0$, and choose any $y={{}^t(a_0,b_0)}$ with $a_0,b_0\in{\mathfrak O}$ representing $x_0$. Partition the set $\{s_\alpha\}$ of coset representatives for the Hecke operator $T_\ell$ as in Lemma \[ell+1\].
Let $t^n$ be the vertex of ${\mathcal T}_\ell^n$ represented by $L_0=L(a_0,b_0)$. For each set $B_j$, we wish to show that $\phi$ is constant on the set $\{(s_{\beta,j}L_0)_\ell|s_{\beta,j}\in B_j\}$. Choose any $s_{\beta,j}$ and let $u^n$ be the downhill neighbor of $t^n$ represented by $(L_1)_\ell$, where $L_1=s_{\beta,j}L_0$. Then $L_1$ is homothetic to a lattice with a basis representing $x_j$. We now divide the proof into 3 cases.
1. Suppose $u^n$ is idealistic. Then $L_1$ is a fractional ideal of $K$, and is homothetic to a fractional ideal with basis representing $x_j\in{\mathcal A}$. Hence, $m_j=i^S$, so $d_j=1$ by Lemmas \[is\] and \[H5\]. Hence, there is only one vertex on which $\phi$ must be constant.
2. Suppose that $u^n$ is the unique downhill inner neighbor of $t^n$. Recall from Theorem \[stabgen\] that ${\Gamma_{0}}=\Gamma{x_0}$ fixes $x_0$ and is generated by an element $g_0$ that acts on $L_0$ as multiplication by $\delta_0=\pm\epsilon^{m_x}$ with the sign chosen so that $\delta_0\in {K_{S,q}}$. From Theorem \[stabgen\], we see that $$g_0L_0=\delta_0L_0=L_0.$$ Since multiplication by $\delta_0$ fixes $L_0$, it also fixes $(L_0)_\ell=t^n$. Multiplication by $\delta_0$ also fixes each element of the fiber of $t_0^n=\varpi({\mathfrak O_\ell})$, so it must fix the unique downhill path from $t^n$ to the fiber of $t_0^n$. Hence, multiplication by $\delta_0$ must fix $u^n$.
Now, both $L_1$ and $\delta_0L_1$ are sublattices of $L_0$ of index $\ell$. Since both must represent $u^n$, we see that they are equal. Since $\delta_0L_1=L_1$, we see that $m_j|m_0$, so that $d_j=1$. Hence, again, there is only one vertex on which $\phi$ must be constant.
3. Suppose $u^n$ is a nonidealistic outer downhill neighbor of $t^n$. By cases 1 and 2, no vertex in $\{(s_{\beta,j}L_0)_\ell|\beta\in B_j\}$ can be idealistic or a downhill inner neighbor of $t_n$. Hence, $\phi$ is constant (by hypothesis) on all the vertices in the desired set.
Construction of functions on lattices; comparison between the Laplacian and a Hecke operator
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\[D:q-homogeneous\] Let $q:Z\to{\mathbb F}^\times$ be the character defined in Definition \[D:q\*\], and let $\ell$ be a prime of ${\mathbb Z}$ that is unramified in $K$. We say that a function $f\in F({\mathcal T}_\ell^n)$ is $q$-homogeneous (or just homogeneous, if $q$ is understood) if, for all lattices $L$ in $K$, $$f(\ell L_\ell)=q(\ell I)f(L_\ell).$$
\[Def:glob\] For all finite places $w$ of ${\mathbb Q}$ unramified in $K$, let $n_w=1$ if $w$ is inert in $K$, let $n_w=2$ if $w$ splits in $K$. Fix a prime $\ell$ of ${\mathbb Q}$ not dividing $pdN$, and let $W$ be the set of all finite places of ${\mathbb Q}$ not dividing $\ell pdN$. For $w\in W$, let $\phi_w\in F({\mathcal T}_w^{n_w})$ denote a homogeneous function such that $\phi_w({\mathfrak O_w})=1$. We view the functions $\phi_w$ as fixed by the context, and do not include them in the following notation for $\Phi$. For any homogeneous $\phi_\ell\in F({\mathcal T}_\ell^{n_\ell})$, define the function $\Phi(\phi_\ell)$ on lattices $L$ in $K$ by the formula $$\Phi(\phi_\ell)(L)=\phi_\ell(L_\ell)\prod_{w\in W}\phi_w(L_w).$$
\[new\] The infinite product in the definition makes sense and $\Phi(\phi_\ell)$ is $q$-homogeneous. The map $\phi_\ell\mapsto \Phi(\phi_\ell)$ is ${\mathbb F}$-linear.
For any given $L$, we have that $L_w={\mathfrak O_w}$ for almost all $w$, so the product is actually finite. The linearity of the map $\phi_\ell\mapsto \Phi(\phi_\ell)$ is clear. Now suppose $\alpha\in {\mathbb Q}^\times\cap K_{S_0}$ and $L$ is a lattice. Then $\alpha$ is prime to $pdN$ and factors as $$\alpha=\ell^{f_\ell}\prod_{w\in W} w^{f_w}.$$ Then $$\Phi(\phi_\ell)(\alpha L)=\phi_\ell(\alpha L_\ell)\prod_{w\in W}\phi_w(\alpha L_w)=
\phi_\ell(\ell^{f_\ell} L_\ell)\prod_{w\in W}\phi_w(w^{f_w} L_w)
.$$ Since $\phi_\ell$ and all the $\phi_w$ are homogeneous, this equals $$q(\ell^{f_\ell}I)\phi_\ell( L_\ell)\left(\prod_{w\in W}q(w^{f_w}I)\right)\left(\prod_{w\in W}\phi_w(L_w)\right)=
q(\alpha I)\Phi(\phi_\ell)(L).\qedhere$$
We now proceed to the main theorem of this section: the comparison between the Hecke operator and the Laplace operator.
By Lemma \[Lemma:mL\] and the fact that ${K_{S,q}}$ is unit-cofinite, we see that for any lattice $L\subseteq K$, there is a minimal positive integer $m_L$ such that $\epsilon^{m_L}L=L$ and one of $\pm\epsilon^{m_L}\in{K_{S,q}}$. If $L_1$ and $L_2$ are $K^\times$-homothetic lattices in $K$, it is clear that $m_{L_1}=m_{L_2}$. Set $m'_L=m_L/i^S$. By Theorem \[stabgen\], if $L=L(a,b)$ and $x$ is the image in ${X_{S,q}}$ of $y={}^t(a,b)$ then $m_L=m_x$. Therefore, by Lemma \[is\], $i^S|m_L$ and $m'_L$ is a positive integer.
Let $\psi_\ell\in F({\mathcal T}_\ell^{n_\ell})$. We define the [*transform*]{} of $\psi_\ell$ to be the function $\hat\psi_\ell\in F({\mathcal T}_\ell^n)$ given by the formula $$\hat\psi_\ell(t^n)=m'_L\psi_\ell(t^n),$$ where $L$ is any lattice in ${\mathfrak O}$ of $\ell$-power index, such that $L_\ell$ represents $t^n$.
Given $\psi_\ell\in F({\mathcal T}_\ell^n)$, the transform $\hat\psi_\ell$ is well defined.
We need to show that for $t^n\in{\mathcal T}_\ell^n$, the value of $m_L$ does not depend on the lattice $L$ chosen to represent $t^n$. Note that up to homothety by powers of $\ell^n$, there is a unique lattice $\Lambda\subseteq K_\ell$ representing $t^n$. By [@Weil V.2, Theorem 2] there is a unique lattice $L\subseteq {\mathfrak O}$ of $\ell$-power index such that $L_\ell=\Lambda$. Since $\Lambda$ is uniquely defined up to homothety by powers of $\ell^n$, so too is $L$. Finally, since homothety does not change the value of $m_L$, we see that $m_L$ does not depend on the choice of $L$, so $m'_L$ does not.
If $\psi_\ell({\mathfrak O_\ell})=1$, then $\hat\psi_\ell({\mathfrak O_\ell})=1$, since $m'_{{\mathfrak O}}=1$. In addition, if ${\mathbb F}$ has characteristic $0$, then a function $\psi_\ell$ is determined by its transform; this fails if any $m'_L$ is divisible by the characteristic of ${\mathbb F}$.
Let $\ell\nmid pdN$ be prime. If $\psi_\ell\in F({\mathcal T}_\ell^n)$ is homogeneous, then $\hat\psi_\ell$ is also homogeneous.
If $t^n\in{\mathcal T}_\ell^n$ is represented by $L_\ell$, with $L$ a lattice of $\ell$-power index in ${\mathfrak O}$, then $\ell t^n$ is represented by $\ell L_\ell$. Since $m_{L}'=m_{\ell L}'$, we have $$\hat\psi_\ell(\ell t^n)=m_{\ell L}'\psi_\ell(\ell t^n)=m_L'q(\ell I)\psi_\ell(t^n)=q(\ell I)\hat\psi_\ell(t^n).\qedhere$$
We now fix a set ${\mathcal A}_0$ of representatives of the $Z\,{{\rm GL}}(2,{\mathbb Z})$-orbits in ${X_{S,q}}$. Recall from Lemma \[Lemma:homothety-to-cohomology\] that this choice fixes an isomorphism between the cohomology group $$H^1({{\rm GL}}(2,{\mathbb Z}),{M_{S,q}}^*)$$ and $q$-homogenous, ${K_{S,q}}$-invariant functions on lattices.
\[Theorem:hecke-transform\] Let $\ell\nmid pdN$ be prime. For each finite place $w\in W$, fix a homogeneous function $\phi_w\in F({\mathcal T}_\ell^{n_w})$, as in Definition \[Def:glob\]. Let $n=n_\ell$, and let $\psi_\ell\in F({\mathcal T}_\ell^n)$ be homogeneous. Assume that $\Phi(\hat\psi_\ell)$ is ${K_{S,q}}$-homothety invariant. (It will be $q$-homogeneous by Lemma \[new\].)
As in Lemma \[Lemma:homothety-to-cohomology\] and its proof, view $\Phi(\hat\psi_\ell)$ as an element of $$H^1({{\rm GL}}(2,{\mathbb Z}),{M_{S,q}}^*)\cong H_1({{\rm GL}}(2,{\mathbb Z}),{M_{S,q}})^*.$$ That is to say, view $\Phi(\hat\psi_\ell)$ as an ${\mathbb F}$-valued functional on $H_1({{\rm GL}}(2,{\mathbb Z}),{M_{S,q}})$, via the pairing $$\langle\Phi(\hat\psi_\ell),\bullet\rangle:H_1({{\rm GL}}(2,{\mathbb Z}),{M_{S,q}})\to{\mathbb F}.$$
If $\psi_\ell$ is locally constant relative to $T_\ell$ and $x_0$, then $$\langle\Phi(\hat\psi_\ell)T_\ell,z_0\rangle=m_0\langle\Phi(\Delta_\ell\psi_\ell),z_0\rangle.$$
By Lemma \[ad1\], $T_\ell=\sum_{j=1}^J U_{t_j}$. By Corollary \[C:partial-hecke\], for $1\leq j\leq J$, we have $$U_{t_j}(z_0)=e_jq(\zeta_j)z_j.$$ Then $$\begin{aligned}
\langle\Phi(\hat\psi_\ell)T_\ell,z_0\rangle&=\langle\Phi(\hat\psi_\ell),T_\ell z_0\rangle\cr
&=\sum_{j=1}^Jq(\zeta_j)e_j\langle\Phi(\hat\psi_\ell),z_j\rangle\cr
&=\sum_{j=1}^Jq(\zeta_j)e_jm_j'\langle\Phi(\psi_\ell),z_j\rangle.$$ We have $e_jm_j'=m_0'd_j$ by Lemma \[H5\], so $$\langle\Phi(\hat\psi_\ell)T_\ell,z_0\rangle=\sum_{j=1}^Jq(\zeta_j)m_0'd_j\langle\Phi(\psi_\ell),z_j\rangle.$$
Now, for a fixed $j$, we will analyze the term $\langle\Phi(\psi_\ell),z_j\rangle$. Recall the definition of the partial Hecke operators $U_{t,j}$ and of the matrices $s_{\beta,j}$ from the paragraphs before Lemma \[ad1\]. Also recall the matrices $t_j$ and the fact that $t_j{x}_0=\zeta_j{x}_j$ from the paragraphs before Lemma \[L:stab-reps\] .
Because $\Phi$ is invariant under ${K_{S,q}}$-homothety, we may choose $a_j,b_j\in K$ so that ${{}^t(a_j,b_j)}\in Y$ represents ${x}_j\in {X_{S,q}}$. In fact, we choose $a_0,b_0\in K$ so that ${{}^t(a_0,b_0)}\in Y$ represents ${x}_0$, and then set $${\begin{pmatrix}a_j\cr b_j\end{pmatrix}}=\zeta_j^{-1}t_j{\begin{pmatrix}a_0\cr b_0\end{pmatrix}}.$$ We then obtain $$\begin{aligned}
\langle\Phi(\psi_\ell),z_j\rangle&=\Phi(\psi_\ell)(L(a_j,b_j))\cr
&=\psi_\ell(L(a_j,b_j)_\ell)\prod_{w \in W}\phi_w(L(a_j,b_j)_w).\end{aligned}$$ Since $t_j$ is an integral matrix with determinant $\ell$, we know that $t_j\in{{\rm GL}}(2,{\mathfrak O}_w)$ for all $w \in W$. Factor $$\zeta_j^{-1}=\left(\ell^{f_{\ell,j}}\prod_{w \in W}w^{f_{w,j}}\right)I.$$ (By Definition \[D:q\*\], primes not in $W\cup\{\ell\}$ cannot divide the numerators or denominators of the diagonal entries of the matrix $\zeta_j\in Z$.)
Then $L(a_j,b_j)_w$ is the same as the lattice $w^{f_{w,j}}L(a_0,b_0)_w$. Set $$c=\prod_{w\in W}\phi_w(L(a_0,b_0)_w).$$ Since each $\phi_w$ is homogeneous, we obtain $$\langle\Phi(\psi_\ell),z_j\rangle=\psi_\ell(L(a_j,b_j)_\ell)\ c\prod_{w \in W}q(w^{f_w,j}I).$$ Hence, we see that $$\begin{aligned}
\langle\Phi(\hat\psi_\ell)T_\ell,z_0\rangle&=\sum_{j=1}^Jq(\zeta_j)d_jm_0'\langle\Phi(\psi_\ell),z_j\rangle\cr
&=cm_0'\sum_{j=1}^Jq(\zeta_j)d_j\psi_\ell(L(a_j,b_j)_\ell)\prod_{w \in W}q(w^{f_{w,j}}I)\cr
&=cm_0'\sum_{j=1}^Jq(\ell^{f_{\ell,j}}I)d_j\psi_\ell(L(a_j,b_j)_\ell),\cr\end{aligned}$$ where we have used the factorization of $\zeta_j^{-1}$.
On the other hand, since $\psi_\ell$ is assumed to be locally constant with respect to $T_\ell$ and $x_0$, and any $s_{\beta,j}$ takes any vertex to a downhill neighbor, we have that for each $s_{\beta,j}$, $$\begin{aligned}
\psi_\ell(s_{\beta,j}L(a_0,b_0)_\ell)&=\psi_\ell(t_jL(a_0,b_0)\ell)\cr
&=\psi_\ell(\zeta_jL(a_j,b_j)\ell)\cr
&=\psi_\ell((\ell^{f_{\ell,j}})^{-1}L(a_j,b_j)_\ell)\cr
&=q((\ell^{f_{\ell,j}})^{-1}I)\psi_\ell(L(a_j,b_j)_\ell),\end{aligned}$$ since $\psi_\ell$ is homogeneous.
Hence, using the fact that $d_j=|B_j|$, we have that $$\begin{aligned}
\langle\Phi(\Delta_\ell^n\psi_\ell),z_0\rangle&=\Phi(\Delta_\ell^n\psi_\ell)(L(a_0,b_0))\cr
&=(\Delta_\ell^n(\psi_\ell)(L(a_0,b_0)_\ell)\prod_{w \in W}\phi_w(L(a_0,b_0)_w)\cr
&=c(\Delta_\ell^n(\psi_\ell)(L(a_0,b_0)_\ell)\cr
&=c\sum_{j=1}^J\sum_{s_{\beta,j}\in B_j}\psi_\ell(s_{\beta,j}L(a_0,b_0)_\ell)\cr
&=c\sum_{j=1}^Jd_jq((\ell^{f_{\ell,j}})^{-1}I)\psi_\ell(L(a_j,b_j)_\ell),\end{aligned}$$ where we have used Lemma \[ell+1\]. Multiplying both sides of the last equality by $m_0'$ yields the assertion of the theorem, because $q$ has order $2$.
\[laplace-eigenvector\] For any ${x}_0\in {\mathcal A}_0$, let $z_0\in H_1({\Gamma_{{x}_0}},{\mathbb F})$ be the corresponding homology generator, and let $L_0$ be a lattice corresponding to ${x}_0$. Assume that $\psi_\ell$ is locally constant relative to $T_\ell$ and $x_0$. Further, assume that $\Phi(\hat\psi_\ell)$ is $q$-homogeneous and ${K_{S,q}}$-invariant, and that $(\Delta_\ell^n\psi_\ell)((L_0)_\ell)=\mu\psi_\ell((L_0)_\ell)$. Then $$\langle\Phi(\hat\psi_\ell)T_\ell,z_0\rangle=\mu\langle\Phi(\hat\psi_\ell),z_0\rangle.$$
From Theorem \[Theorem:hecke-transform\] and linearity we have $$\begin{aligned}
\langle\Phi(\hat\psi_\ell)T_\ell,z_0\rangle&=m_0'\langle\Phi(\Delta_\ell\psi_\ell),z_0\rangle\cr
&=m_0'\langle\Phi(\mu\psi_\ell),z_0\rangle\cr
&=\mu\langle\Phi(m_0'\psi_\ell),z_0\rangle\cr
&=\mu\langle\Phi(\hat\psi_\ell)T_\ell,z_0\rangle.\qedhere\end{aligned}$$
\[eigenclass\] Assume that $\psi_\ell$ is locally constant relative to $T_\ell$ and every $x\in {X_{S,q}}$, that $\Phi(\hat\psi_\ell)$ is $q$-homogeneous and ${K_{S,q}}$-invariant, and that $\Delta_\ell\psi_\ell=\mu\psi_\ell$.
Then $\Phi(\hat\psi_\ell)$ is an eigenclass for $T_\ell$ with eigenvector $\mu$ and it is an eigenclass for $T_{\ell,\ell}$ with eigenvector $\theta(\ell)$.
First, we show that $\Phi(\hat\psi_\ell)\neq 0$. By definition, $$\Phi(\hat\psi_\ell)(L)=\hat\psi_\ell(L_\ell)\prod_{w\in W}\phi_w(L_w).$$ By construction, $\phi_w({\mathfrak O}_w)=1$ for every $w\in W$ and $\psi_\ell({\mathfrak O_\ell})=1$. Since $m'_{{\mathfrak O_\ell}}=1$, also $\hat\psi_\ell({\mathfrak O}_\ell)=1$. Therefore, $\Phi(\hat\psi_\ell)({\mathfrak O})=1$.
For any ${x}\in{\mathcal A}_0$, write $z_{{x}}\in H^1({\Gamma_{{x}}},{\mathbb F})$ for the homology generator corresponding to ${x}$. By Corollary \[laplace-eigenvector\] and our hypothesis, for each $x\in{\mathcal A}_0$, we have $$\langle\Phi(\hat\psi_\ell)T_\ell,z_x\rangle=\langle\mu\Phi(\hat\psi_\ell),z_x\rangle.$$ Since $\Phi(\hat\psi_\ell)$ is in the dual space to $H_1({{\rm GL}}(2,{\mathbb Z}),{M_{S,q}})$, and $\{z_{x}:{x}\in{\mathcal A}_0\}$ spans $H_1({{\rm GL}}(2,{\mathbb Z}),{M_{S,q}})$, we are finished with $T_\ell$.
As for $T_{\ell,\ell}$, its action is given by the double coset of the central element $\ell I$. So this is just a single coset, and its action on homology is given by the central character $q$ on the coefficient module ${M_{S,q}}$. Since $q(\ell I)=\theta(\ell)$, $$\langle\Phi(\hat\psi_\ell)T_{\ell,\ell},z_x\rangle=\langle\Phi(\hat\psi_\ell),T_{\ell,\ell}z_x\rangle=\langle\Phi(\hat\psi_\ell),\theta(\ell)z_x\rangle=\langle\theta(\ell)\Phi(\hat\psi_\ell),z_x\rangle.$$ Hence, $\Phi(\hat\psi_\ell)T_{\ell,\ell}=\theta(\ell)\Phi(\hat\psi_\ell)$.
Constructing locally constant eigenfunctions
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Recall that $\theta$ is the quadratic Dirichlet character associated to the real quadratic field $K/{\mathbb Q}$, and $q$ is the character on $Z$ defined by setting $q(r I)=\theta(r)$ for $r\in {\mathbb Z}\cap K_{S_0}$ and extending multiplicatively to $Z$. Since $K$ is real quadratic, $q(-I)=\theta(-1)=1$. Fix an ${\mathbb F}$-valued character $\chi$ on the group of ideals of $K$ relatively prime to $N$ for some positive integer $N$.
In this section, we will construct locally constant $q$-homogeneous functions $\psi_\ell^0$ on ${\mathcal T}_\ell^{n_\ell}$ that are eigenfunctions of the Laplace operator with eigenvalues related to $\chi$. We do this first for inert primes $\ell$.
\[T:inert-psi\] Let $\ell$ be a prime of ${\mathbb Q}$ that is inert in $K/{\mathbb Q}$ and does not equal the characteristic of ${\mathbb F}$. Then there is a locally constant $q$-homogeneous function $\psi_\ell^0\in F({\mathcal T}_\ell^2)$ that is an eigenvalue of the Laplace operator with eigenvalue 0 and satisfies $\psi_\ell^0({\mathfrak O_\ell})=1$.
We define $\psi_\ell^0$ inductively.
For vertices of tier 0, we define $\psi_\ell^0({\mathfrak O_\ell})=1$ and $\psi_\ell^0(\ell{\mathfrak O_\ell})=\theta(\ell)=-1$. We see easily that $\psi_\ell^0$ is homogeneous on the vertices of tier $0$.
On vertices $t^2\in{\mathcal T}_\ell^2$ of tier 1, we define $\psi_\ell^0(t^2)=0$. Clearly $\psi_\ell^0$ is $\theta$-homogeneous on vertices of tier 1. In addition, since all downhill neighbors of a vertex of tier 0 have tier 1, we can now compute $\Delta_\ell^2(\psi_\ell^0)$ on vertices of tier 0; we find that its value is $0$, as desired. Finally, $\psi_\ell^0$ is constant on all downhill neighbors of vertices of tier 0.
On each vertex $t^2\in {\mathcal T}_\ell^2$ of tier 2, let $u^2\in {\mathcal T}_\ell^2$ be the unique uphill neighbor of $t^2$ of tier 1, and we let $v^2$ be the unique downhill neighbor of $u^2$ of tier 0. We define $\psi_\ell^0(t^2)=-\psi_\ell^0(v^2)/\ell$. Because the unique uphill neighbor of $\ell t^2$ of tier 1 is $\ell u^2$, which has a unique downhill neighbor of tier 0 equal to $\ell v^2$, we see that with this definition, $\psi_\ell^0$ is homogeneous on vertices of tier 1. In addition, for any vertex $u^2$ of tier 1, $\psi_\ell^0$ is constant on the downhill neighbors of $u^2$ of higher tier, since its value on such vertices depends only on its value on the unique downhill inner neighbor of $u^2$. Finally, we have constructed $\psi_\ell^0$ so that $$\Delta_\ell^2(\psi_\ell^0)(u^2)=0$$ for each vertex $u^2$ of tier 1.
We continue; for vertices $t^2\in{\mathcal T}_\ell^2$ of odd tier, we define $\psi_\ell^0(t^2)=0$. This guarantees that for vertices $u^2$ of even tier, $\Delta_\ell^2(\psi_\ell^0)(u^2)=0$, and that $\psi_\ell^0$ is constant on all downhill neighbors of $u^2$ of higher tier. Further, with this definition, $\psi_\ell^0(\ell t^2)=0=\theta(\ell)\psi_\ell^0(t^2)$ so that $\psi_\ell^0$ is homogeneous on vertices of odd tier.
For a vertex $t^2\in{\mathcal T}_\ell^2$ of positive even tier, let $u^2$ be the unique uphill inner neighbor of $t^2$, and let $v^2$ be the unique downhill inner neighbor of $u^2$. We define $\psi_\ell^0(t^2)=-\psi_\ell^0(v^2)/\ell$. Clearly $\psi_\ell^0$ is constant on all downhill outer neighbors of $u^2$ (since its value on such neighbors depends only on its value on $v^2$). As in the case of tier 2, we see that $\psi_\ell^0(\ell t^2)=\theta(\ell) \psi_\ell^0(t^2)$, and $\Delta_\ell^2(\psi_\ell^0)(u^2)=\psi_\ell^0(v^2)+\ell(-\psi_\ell^0(v^2)/\ell)=0$.
With this construction, we see that $\psi_\ell^0$ is homogeneous, locally constant, and is an eigenfunction of $\Delta_\ell^2$ with eigenvalue 0.
\[L:inert-invariant\] For an inert prime $\ell$, the function $\psi_\ell^0$ defined above is ${{\rm GL}}(2,{\mathbb Z}_\ell)$-invariant.
The action of ${{\rm GL}}(2,{\mathbb Z}_\ell)$ fixes vertices of tier 0, and preserves uphill and downhill neighbors, and the tier of each vertex (Lemma \[L:gl2zl-invariant\]). Since these relationships determine the values of $\psi_\ell^0$, the function is ${{\rm GL}}(2,{\mathbb Z}_\ell)$-invariant.
For a prime $\ell$ that splits in $K/{\mathbb Q}$ and does not divide $N$, we now prepare to construct a locally constant homogeneous function $\psi_\ell^0\in F({\mathcal T}_\ell^1)$ that is an eigenfunction of $\Delta_\ell=\Delta_\ell^1$. For the remainder of this section, we will assume that $\ell$ splits in $K$, that $(\ell)=\lambda\lambda'$ and that $\ell\nmid N$, so that $\chi(\lambda)$ and $\chi(\lambda')$ are defined. In this case, the function that we construct will depend not only on the real quadratic field $K/{\mathbb Q}$, but also on the character $\chi$. Since we work in ${\mathcal T}_\ell^1={\mathcal T}_\ell$, the concepts of uphill and downhill neighbors coincide.
We begin by defining some terminology and notation for subsets of ${\mathcal T}_\ell$.
\[base\] We take ${\mathfrak O_\ell}$ as the basepoint of ${\mathcal T}_\ell$ and denote it by $t_0$. A [*descendant*]{} of a vertex $t\in{\mathcal T}_\ell$ is a vertex $t_1\neq t$ such that the path from $t_0$ to $t_1$ passes through $t$. Denote by $C(t)$ the set of all descendants $t'$ of $t$ such that every vertex of the path from $t$ to $t'$ except possibly $t$ is non-idealistic, and let $\overline C(t)=C(t)\cup\{t\}$. We call $C(t)$ the [*open cohort*]{} of $t$, and $\overline C(t)$ the [*closed cohort*]{} of $t$.
A [*simple chain*]{} starting at a vertex $t\in{\mathcal T}_\ell$ is a collection $C$ consisting of $t$ and descendants of $t$ such that for any pair $t',t''\in C$, one of $t',t''$ is a descendant of the other. An [*apartment*]{} in ${\mathcal T}_\ell$ is a union of two infinite simple chains starting at a vertex $t$ and having no other vertices in common.
Let $t$ be an idealistic point in ${\mathcal T}_\ell$.
(1) If $\ell$ is inert, then $t=t_0$.
(2) If $(\ell)=\lambda\lambda'$ splits and $t$ is a distance $k>0$ from $t_0$, then $t=\lambda_\ell^k$ or $t=\lambda_\ell'^k$, and both of these points are a distance $k$ from $t_0$.
(3) If $(\ell)$ splits and $k>0$, then $\lambda_\ell^k$ and $\lambda_\ell'^k$ define distinct points in ${\mathcal T}_\ell$.
(4) No descendant of an non-idealistic point in ${\mathcal T}_\ell$ is idealistic.
(5) The vertices of ${\mathcal T}_\ell$ are partitioned into the closed cohorts $\overline C(t_I)$ as $t_I=I_\ell$ runs over the idealistic points of ${\mathcal T}_\ell$ (where $I$ is an ideal of ${\mathfrak O}$ of $\ell$-power norm.)
(6) In the split case, the set of idealistic points of ${\mathcal T}_\ell$ form an apartment, namely $$\{{\lambda_\ell^k}|k>0\}\cup\{t_0\}\cup \{{\lambda_\ell'^k}|k>0\}.$$
In the discussion following Definition \[D:idealistic\], we proved that the set of idealistic nodes of ${\mathcal T}_\ell$ is $\{t_0\}$ if $\ell$ is inert and $\{{\lambda_\ell^k}|k>0\}\cup\{t_0\}\cup \{{\lambda_\ell'^k}|k>0\}$ if $\ell$ is split. Since $\lambda^k$ has index $\ell$ in $\lambda^{k-1}$, and similarly for the powers of $\lambda'$, (1) and (2) are now clear. As for (3), if $\lambda_\ell^k=\lambda_\ell'^k$, then $\lambda^k=\ell^m\lambda'^k$ for some integer $m$, which is absurd.
If $\ell$ is inert, (4) and (5) are obvious.
Assume then that $\ell$ splits. Then the idealistic point $\lambda_\ell^k$ is at the end of a path containing the nodes $t_0,\lambda_\ell,\ldots,\lambda_\ell^k$. A similar statement holds for $\lambda_\ell'^k$. Since every non-idealistic node is a descendant of $t_0$ and ${\mathcal T}_\ell$ is a tree, no idealistic point can be a descendant of a non-idealistic point. Hence (4) holds.
For any node $u\in{\mathcal T}_\ell$ consider the path from $t_0$ to $u$ (possibly of length 0.) Let $t_I$ be the last idealistic point in this path. Then $u\in\overline C(t_I)$ is in the closed cohort of this idealistic point. If $u$ were in the closed cohort of two distinct idealistic points, there would be a nontrivial loop in ${\mathcal T}_\ell$. Hence, (5) holds.
Finally, (6) is clear, since the set of nonnegative powers of $\lambda$ and of $\lambda'$ each form a simple chain starting at $t_0$.
Let $N$ be a positive integer and $c$ be an ${\mathbb F}$-valued multiplicative function on the group $I_{K}(N)$ of nonzero fractional ideals of $K$ relatively prime to $N$. Fix a prime $\ell$ that does not divide $N$. Assume that $c$ is trivial on the principal fractional ideal $\ell{\mathfrak O}$. Define $\hat c\in F({\mathcal T}_\ell)$ by $$\hat c(t)=\begin{cases}0&\text{if $t$ is non-idealistic,}\cr
c(I)&\text{if $t=I_\ell$, where $I$ is an ideal of $\ell$-power index in ${\mathfrak O}$.}\end{cases}$$
The function $\hat c$ is well defined.
Suppose $I$ and $J$ are both ideals of ${\mathfrak O}$ of $\ell$-power index, and that $I_\ell$ and $J_\ell$ are homothetic in $K_\ell$ by a power of $\ell$. If $\ell$ is inert, then $I$ and $J$ are both powers of $\ell$. They are thus both principal, and we see that $c(I)=c(J)=c({\mathfrak O})$.
If $(\ell)=\lambda\lambda'$ splits, then $I=\ell^q\mu^a$ and $J=\ell^r\nu^b$ for nonnegative integers $a,b,q,r$, and $\mu,\nu\in\{\lambda,\lambda'\}$. The fact that $I_\ell$ and $J_\ell$ are homothetic implies that $\mu=\nu$ and $a=b$, so $I$ and $J$ differ by a factor of $\ell^{q-r}$. Since $c$ is trivial on $\ell{\mathfrak O}$, $c(I)=c(J)$.
Let $t\in{\mathcal T}_\ell$. For any $x$ in the open cohort $C(t)$ of $t$, all of the neighbors of $x$ are in the closed cohort $\overline C(t)$. Hence, the Laplace operator $\Delta_\ell$ defines a linear map from functions on $\overline C(t)$ to functions on $C(t)$.
\[con split\] Assume that $\ell$ is not equal to the characteristic of ${\mathbb F}$. Let $\mu\in {\mathbb F}$, and let $t$ be an idealistic point of ${\mathcal T}_\ell$ with closed cohort $\overline C(t)$. Then there is a unique ${\mathbb F}$-valued function $\theta_{t,\mu}$ on $\overline C(t)$ with the following properties:
(i) \[t-mu1\] $\theta_{t,\mu}(t)=1$,
(ii) \[t-mu2\] $\theta_{t,\mu}(s)=0$ for every $s\in C(t)$ that is distance $1$ from $t$,
(iii) \[t-mu4\] $\theta_{t,\mu}(s)$ depends only on $\ell$, $\mu$, and the distance from $s$ to $t$,
(iv) \[t-mu3\] $\Delta_\ell(\theta_{t,\mu})(s)=\mu\theta_{t,\mu}(s)$ for every $s\in C(t)$.
Define a sequence $a_k\in{\mathbb F}$ for $k\geq 0$ by the recurrence relation $a_0=1$, $a_1=0$, and for $k\geq 2$, $$a_k=\frac{\mu a_{k-1}-a_{k-2}}{\ell}.$$ This clearly defines a unique sequence. For $s$ a distance $k$ from $t$ in $\overline C(t)$, set $\theta_{t,\mu}(s)=a_k$. With this definition, $\theta_{t,\mu}$ satisfies conditions \[t-mu1\], \[t-mu2\], and \[t-mu4\].
Given a point $s\in C(t)$ a distance $k$ from $t$, $s$ has one neighbor a distance $k-1$ from $t$, and $\ell$ neighbors a distance $k+1$ from $t$. Hence $$\begin{aligned}
\Delta_\ell(\theta_{t,\mu})(s)&=a_{k-1}+\ell a_{k+1}\cr
&=a_{k-1}+\ell\left(\frac{\mu a_k-a_{k-1}}\ell\right)\cr
&=\mu a_k\cr
&=\mu\theta_{t,\mu}(s),\end{aligned}$$ so $\theta_{t,\mu}$ satisfies condition \[t-mu3\].
Conversely, if $\theta_{t,\mu}$ is a function on $\overline C(t)$ satisfying condition \[t-mu4\], then for any $s$ a distance $k$ from $t$, we may define $a_k=\theta_{t,\mu}(s)$. If in addition $\theta_{t,\mu}$ satisfies conditions \[t-mu1\], \[t-mu2\], \[t-mu3\], the $a_k$ satisfy the recurrence relation given above. The uniqueness of $\theta_{t,\mu}$ follows from the uniqueness of the sequence $\{a_k\}$.
\[D:split-psi\] Let $\mu\in {\mathbb F}$, and assume $\ell$ does not divide $N$ and does not equal the characteristic of ${\mathbb F}$. We define $\psi_\ell^0\in F({\mathcal T}_\ell)$ by $$\psi_\ell^0(s)=\hat\chi(t)\theta_{t,\mu}(s),$$ where $t\in {\mathcal T}_\ell$ is the unique idealistic vertex with $s\in \overline C(t)$.
\[lemsplit\] Let $\mu\in F$ and assume that $\ell$ does not divide $N$ and does not equal the characteristic of ${\mathbb F}$.
(1) $\psi_\ell^0({\mathfrak O_\ell})=1$.
(2) $\psi_\ell^0$ is locally constant with respect to $T_\ell$ and any $x\in {X_{S,q}}$.
(3) Let $\mu=\chi(\lambda)+\chi(\lambda')$. Then $$\Delta_\ell\psi_\ell^0=\mu\psi_\ell^0.$$
The first assertion is immediate from the definitions.
Let $s$ be any vertex in ${\mathcal T}_\ell$. We wish to show that $\psi_\ell^0$ is constant on all non-idealistic outer downhill neighbors $u$ of $s$. Then, by Lemma \[loc-const\], part (2) will hold.
Let $s\in\bar C(t)$. Then any such $u$ will be in $C(t)$. Since $\hat\chi(t)$ is constant for all points in $C(t)$, we need only show that $\theta_{t,\mu}(u)$ is constant for all such $u$. Letting the distance from $t$ to $s$ be $k-1$, the distance from $t$ to $u$ will be $k$. Hence, the desired constancy follows from Lemma \[con split\]\[t-mu4\].
Now suppose that $s=t=I_\ell$ is idealistic. Then $s$ has exactly two idealistic neighbors, namely $(\lambda I)_\ell$ and $(\lambda' I)_\ell$. The nonidealistic neighbors $u$ of $s$ are all in $C(t)$ and have distance 1 from $t$; hence $\theta_{t,\mu}$ vanishes on them all. Hence $$(\Delta_\ell\psi_\ell^0)(s)=\chi(\lambda I)+\chi(\lambda'I)=(\chi(\lambda)+\chi(\lambda'))\chi(I)=\mu\psi_\ell^0(t).$$
Finally, suppose that $s$ is non-idealistic and belongs to the open cohort $C(t)$. Then $$\begin{aligned}
(\Delta_\ell\psi_\ell^0)(s)&=\sum_u\psi_\ell^0(u)\cr
&=\sum_u\hat\chi(t)\theta_{t,\mu}(u)\cr
&=\hat\chi(t)\sum_u\theta_{t,\mu}(u)\cr
&=\hat\chi(t)(\Delta_\ell\theta_{t,\mu})(s)\cr
&=\mu\hat\chi(t)\theta_{t,\mu}(s)\cr
&=\mu\psi_\ell^0(s),\end{aligned}$$ by Lemma \[con split\]\[t-mu3\], where the sums run over all neighbors $u$ of $s$.
${K_{S,q}}$-invariance {#S:Invariance}
======================
\[aL\] Fix a prime $\ell$ that is unramified in $K$, and let $n=1$ if $\ell$ splits in $K$ and $2$ if $\ell$ is inert. Let $L$ be a ${\mathbb Z}$-lattice in $K$, and let $\alpha\in K^\times$. Let $s^n$ be the vertex in ${\mathcal T}_\ell^n$ corresponding to $L_\ell$, and let $u^n$ be the vertex corresponding to $(\alpha L)_\ell$. Factor the fractional ideal $\alpha{\mathfrak O}=I_1I_2$, where $I_1$ has norm a power of $\ell$ and $I_2$ is prime to $\ell$.
(1) There exists a matrix $g\in {{\rm GL}}(2,{\mathbb Q}_\ell)$ depending only on $\alpha$ (independent of $L$), such that $u^n=gs^n$. If $\ell$ is inert, then $g=\ell^k g'$ with $k\in{\mathbb Z}$ and $g'\in{{\rm GL}}(2,{\mathbb Z}_\ell)$.
(2) The vertex $s^n$ is idealistic if and only if $u^n$ is idealistic. If $s^n$ corresponds to $L_\ell$ with $L$ an ideal, then $u^n$ corresponds $(I_1L)_\ell$.
(3) Suppose $\ell$ is split. Assume that $s^n$ is not idealistic, but lies in the open cohort $C(t)$ of the idealistic point $t^n=M_\ell$, where $M$ is an ideal of $\ell$-power norm. Then $u^n$ lies in the open cohort $C(t_1^n)$, where $t_1^n=(I_1M)_\ell$ and the distance between $s^n$ and $t^n$ is the same as the distance between $u^n$ and $t_1^n$.
\(1) First, suppose that $\ell$ is inert. Via our identification of $K_\ell$ with ${\mathbb Q}_\ell^2$, multiplication by $\alpha$ is a ${\mathbb Q}_\ell$-linear isomorphism from ${\mathbb Q}_\ell^2$ to ${\mathbb Q}_\ell^2$; hence, it is given by a matrix $g\in{{\rm GL}}(2,{\mathbb Q}_\ell)$. We can write $\alpha\in K_\ell$ as $\alpha=\ell^k\eta$ for some $k\in{\mathbb Z}$, and some unit $\eta\in{\mathfrak O_\ell}^\times$; multiplication by $\eta$ is given by a matrix in ${{\rm GL}}(2,{\mathbb Z}_\ell)$.
Now assume that $\ell$ is split. In this case, we identify $K_\ell$ with ${\mathbb Q}_\ell^2$ by mapping $\alpha$ to $(\alpha,\alpha')$. Then multiplication by $\alpha$ is defined by the matrix $$\begin{pmatrix}\alpha&0\cr0&\alpha'\end{pmatrix},$$ which is in ${{\rm GL}}(2,{\mathbb Q}_\ell)$.
\(2) $L$ is a fractional ideal if and only if $\alpha L$ is a fractional ideal. If $L=MP$ with $M$ a fractional ideal of $\ell$-power norm, and $P$ a fractional ideal prime to $\ell$, then $$(aL)_\ell=(I_1M)_\ell=(I_1L)_\ell.$$
\(3) Let $g\in{{\rm GL}}(2,{\mathbb Q}_\ell)$ be the matrix from part (1) corresponding to multiplication by $\alpha$. Multiplication by $g$ is then an isometry of ${\mathcal T}_\ell$ that takes idealistic vertices to idealistic vertices, and non-idealistic vertices to non-idealistic vertices. Let $R$ be a simple path from $t^n$ to $s^n$ whose only idealistic vertex is $t^n$. Then $gR$ is a simple path from $gt^n$ to $u^n$ of the same length as $R$, whose only idealistic vertex is $gt^n$. Moreover, $u^n$ lies in the open cohort $C(gt^n)$ where $gt^n=(I_1M)_\ell$.
\[L:KSQ-invariant\] Let ${\mathbb F}$ be a field of characteristic $0$ or of finite characteristic not equal to two. If ${\mathbb F}$ has characteristic 0, set $p=1$, and otherwise let $p$ be the characteristic of ${\mathbb F}$. Assume that $\chi$ is trivial on principal ideals generated by elements of ${\mathbb Q}^\times\cap K_{S_0}$. Also assume that $\chi$ is trivial on principal ideals generated by elements of ${K_{S,q}}$. Let $\Phi$ be the function from lattices in $K$ to ${\mathbb F}$ defined by $$\Phi(L)=\prod_{w\nmid pdN}\hat\psi_w^0(L_w).$$ Then $\Phi(\alpha L)=\Phi(L)$ for all $\alpha\in{K_{S,q}}$ and all lattices $L$ in $K$.
Moreover, $\Phi(\alpha L)=q(\alpha I)\Phi(L)$ for all $\alpha\in {\mathbb Q}^\times\cap K_{S_0}$.
Let $L=L(c,d)$ be a lattice in $K$ and let $\alpha\in{K_{S,q}}$. Note that $m_L'=m_{\alpha L}'$, since $K^\times$ is commutative. Hence, there is a single integer $m'$, such that for each prime $w\nmid pdN$, we have $$\hat\psi_w^0(L_w)=m'\psi_w^0(L_w)$$ and $$\hat\psi_w^0((\alpha L)_w)=m'\psi_w^0((\alpha L)_w).$$
Assume first that $w$ is inert in $K$. Then we may factor $\alpha{\mathfrak O}$ as $$\alpha{\mathfrak O}=w^jI_2,$$ with $I_2$ a fractional ideal that is relatively prime to $w$. By Lemma \[aL\](1), we have $$(\alpha L)_w=w^jgL_w$$ for some $g\in{{\rm GL}}(2,{\mathbb Z}_w)$. Since $\psi_w^0$ is homogeneous and is ${{\rm GL}}(2,{\mathbb Z}_w)$-invariant on ${\mathcal T}_\ell^2$ (by Lemma \[L:inert-invariant\]), we have $$\hat\psi_w^0((\alpha L)_w)=m'\psi_w^0(w^jgL_w)=m'q^*(w^j)\psi_w^0(L_w)=q^*(w^j)\hat\psi_w^0(L_w).$$
Now assume that $w$ splits in $K$. Let $s\in{\mathcal T}_\ell$ be the vertex corresponding to $L_w$, and let $u$ correspond to $(\alpha L)_w$.
If $s$ is idealistic, so is $u$, and we see that $$\psi_w^0(s)=\hat\chi(s)\theta_{s,\mu}(s)=\hat\chi(s)=\chi(L)=\chi(\alpha L)=\hat\chi(u)=
\hat\chi(u)\theta_{u,\mu}(u)\psi_w^0(u).$$
If $s$ is nonidealistic, then so is $u$, and $u=gs$ for some $g\in{{\rm GL}}(2,{\mathbb Q}_w)$. Suppose $s$ lies in the open cohort $C(t)$ of the idealistic vertex $t$ corresponding to $I_w$, where $I$ is an ideal of $w$-power index in ${\mathfrak O}$. By Lemma \[aL\](3), $u$ is in the open cohort $C(t_1)$ of the idealistic point $t_1$ corresponding to $(I_1I)_w$, where $\alpha{\mathfrak O}=I_1I_2$, with $I_1$ having norm a power of $w$, and $I_2$ having norm relatively prime to $w$. In addition, the distance from $s$ to $t$ is the same as the distance from $u$ to $t_1$. Hence, $$\hat\chi(t_1)=\chi(I_1I)=\chi(I_1)\chi(I)=\chi(I_1)\hat\chi(t)$$ and $$\theta_{t,\mu}(s)=\theta_{t_1,\mu}(u).$$ Therefore, $$\begin{aligned}
\hat\psi_w^0((\alpha L)_w)&=m'\psi_w^0(u)\cr
&=m'\hat\chi(t_1)\theta_{t_1,\mu}(u)\cr
&=m'\chi(I_1I)\theta_{t_1,\mu}(u)\cr
&=m'\chi(I_1)\hat\chi(t)\theta_{t,\mu}(s)\cr
&=\chi(I_1)\hat\psi_w^0(L_w).\end{aligned}$$
In all of this, the fractional ideal $I_1$ depends on $w$; we will call it ${I_\alpha(w)}$. Then ${I_\alpha(w)}$ is a product of powers of primes lying over $w$; if $w$ is inert, it is clear that ${I_\alpha(w)}$ is principal with a generator $\beta_\alpha(w)$ in ${\mathbb Q}^\times\cap K_{S_0}$, so that $\chi({I_\alpha(w)})=1$.
Since $\alpha\in{K_{S,q}}$, $\alpha$ is relatively prime to $pdN$, so that $$\alpha{\mathfrak O}=\prod_{w\nmid pdN}{I_\alpha(w)}=\left(\prod_{w\text{ inert}}{I_\alpha(w)}\right)\left(\prod_{w\text{ split}}{I_\alpha(w)}\right).$$ Setting $\beta=\prod_{w\text{ inert}}\beta_\alpha(w)$, we have $$\beta{\mathfrak O}=\left(\prod_{w\text{ inert}}{I_\alpha(w)}\right).$$ Since $\alpha\in{K_{S,q}}$, $q^*(\alpha)=1$. Because $q^*$ depends only on inert prime factors, and the powers of inert primes dividing $\alpha$ and $\beta$ are equal, we see that $$1=q^*(\alpha)=q^*(\beta).$$ In addition, we have that $\chi(\beta{\mathfrak O})=1$, since $\beta$ is a product of powers of elements of ${\mathbb Q}^\times\cap K_{S_0}$, and we have assumed that $\chi$ is trivial on ideals generated by elements of ${\mathbb Q}^\times\cap K_{S_0}$. Hence, we see that $$\prod_{w\text{ split}}{I_\alpha(w)}$$ is principal, with generator $\alpha/\beta$, so $$\prod_{w\text{ split}}\chi({I_\alpha(w)})=\frac{\chi(\alpha{\mathfrak O})}{\chi(\beta{\mathfrak O})}=\chi(\alpha{\mathfrak O})=1,$$ since we have assumed that $\chi$ is trivial on principal ideals generated by elements of ${K_{S,q}}$.
Hence, we obtain $$\begin{aligned}
\Phi(\alpha L)&=\prod_{w\nmid pdN}\hat\psi_w^0((\alpha L)_w)\cr
&=\left(\prod_{w\text{ inert}}\hat\psi_w^0((\alpha L)_w)\right)\left( \prod_{w\text{ split}}\hat\psi_w^0((\alpha L)_w)\right)\cr
&=\left(\prod_{w\text{ inert}}q^*(\beta_\alpha(w))\hat\psi_w^0((L)_w)\right)\left( \prod_{w\text{ split}}\chi({I_\alpha(w)})\hat\psi_w^0((L)_w)\right)\cr
&=q^*(\beta)\left( \prod_{w\text{ split}}\chi({I_\alpha(w)})\right)\prod_{w\nmid pdN}\hat\psi_w^0((L)_w)
\cr
&=\Phi(L).\qedhere\end{aligned}$$
Finally, if $\alpha\in {\mathbb Q}^\times\cap K_{S_0}$, it is a product of powers of primes not dividing $pdN$. We may thus assume that $\alpha$ is such a prime. The $q$-homogeneity of $\Phi$ then follows by Lemma \[new\] from the homogeneity of the individual $\psi_w^0$ functions (see Theorem \[T:inert-psi\] for inert primes, and note that homogeneity is trivial for split primes).
Galois representations
======================
We now define the Galois representations to which our main theorem below applies.
As before, we let $K$ be a real quadratic field of discriminant $d$, cut out by the Dirichlet character $\theta$. Let ${\mathbb F}$ be a field of characteristic 0 (in which case we set $p=1$) or a field of odd characteristic $p$, let $G_K$ be the absolute Galois group of $K$ (i.e. $\operatorname{Gal}(\bar{\mathbb Q}/K)$), and let $\chi:G_K\to{\mathbb F}^\times$ be a character of $G_K$ with finite image. By class field theory, we can think of $\chi$ as a character on the group of the nonzero fractional ideals of $K$ relatively prime to $N$ for some positive $N\in{\mathbb Z}$. Let $L$ be the fixed field of the kernel of $\chi$. Then $L/K$ is Galois. We fix an $M\geq3$ that divides $pdN$ and define $S_0$ and $S$ as in Definition \[D:S\_0\].
We place the following conditions on the character $\chi$.
(1) $\chi$ is trivial on the principal fractional ideals of $K$ generated by elements of ${K_{S,q}}$.
(2) $\chi$ is trivial on the principal fractional ideals of $K$ generated by elements of ${\mathbb Q}^\times\cap K_{S_0}$.
(3) $[L:K]$ is odd.
(4) $L/{\mathbb Q}$ is Galois.
An example of such a $\chi$ would be any unramified character of $G_K$ of odd order; such a character would be trivial on all principal fractional ideals of $K$, and $L$ would be a subfield of the Hilbert class field of $K$ and hence be Galois over ${\mathbb Q}$.
Let $\rho:G_{\mathbb Q}\to{{\rm GL}}(2,{\mathbb F})$ be the induced representation $$\rho=\operatorname{Ind}_{G_K}^{G_{\mathbb Q}}\chi.$$ Note that this representation will factor through $\operatorname{Gal}(L/{\mathbb Q})$. We have an exact sequence $$1\to\operatorname{Gal}(L/K)\to\operatorname{Gal}(L/{\mathbb Q})\to\operatorname{Gal}(K/{\mathbb Q})\to1;$$ since $[L:K]$ is odd, this sequence splits, so there is an element $\tau$ of order $2$ in $\operatorname{Gal}(L/{\mathbb Q})$ mapping to the nonidentity element of $\operatorname{Gal}(K/{\mathbb Q})$; we can lift it to an element $\tau\in G_{\mathbb Q}$, and we have that $\tau^2$ is the identity modulo $G_L$.
With respect to a suitable basis, it is easy to see that for $g\in G_{\mathbb Q}$, we have the following:
(a) If $g\in G_K$, then $$\rho(g)=\begin{pmatrix}\chi(g)&0\cr0&\chi(g')\end{pmatrix},$$ where $g'=\tau^{-1}g\tau$.
(b) If $g\notin G_K$, then $g=h\tau$ for some $h\in G_K$, and $$\rho(g)=\begin{pmatrix}0&\chi(h')\cr\chi(h\tau^2)&0\end{pmatrix},$$ where $h'=\tau^{-1}h\tau$.
If we now let $g$ be a Frobenius element in $G_{\mathbb Q}$ for some prime $\ell$ of ${\mathbb Q}$ not dividing $pdN$ (so that $\ell$ is unramified in $L/{\mathbb Q}$), then we have the following two cases.
If $\ell$ splits in $K$ and $\ell\nmid N$, then $g\in G_K$. If we write $\ell{\mathfrak O}=\lambda\lambda'$ with $\lambda,\lambda'$ primes in $K$, then we may take $g$ to be a Frobenius in $G_K$ of $\lambda$; a Frobenius of $\lambda'$ will be $g'$. Hence, we have $$\operatorname{Tr}(\rho(g))=\chi(g)+\chi(g')=\chi(\lambda)+\chi(\lambda'),$$ and $$\det(\rho(g))=\chi(g)\chi(g')=\chi(\lambda)\chi(\lambda')=\chi(\lambda\lambda')=\chi(\ell{\mathfrak O})=1$$ by condition (2) on the character $\chi$.
On the other hand, if $\ell$ is inert in $K$ and $\ell\nmid N$, write $g=h\tau$ as above. Then $$\operatorname{Tr}(\rho(g))=0$$ and $\det(\rho(g))=-\chi(h\tau^2)\chi(h')$ with $h'=\tau^{-1}h\tau$. We note that $g^2$ is a Frobenius of $\ell{\mathfrak O}$ in $G_K$. Hence, we have $$\det(\rho(g))=-\chi(h\tau^2)\chi(h')=-\chi(h\tau^2h')=-\chi((h\tau)^2)=-\chi(g^2)=-\chi(\ell{\mathfrak O})=-1,$$ where we have again used condition (2) on $\chi$.
Note that in each case, when $g$ is a Frobenius in $G_{\mathbb Q}$ of $\ell$, we have $\det(\rho(g))=\theta(\ell)$.
Now we check that $\rho$ is even. Let $c\in G_{\mathbb Q}$ be a complex conjugation. Since $c$ has order 2 and $\chi$ has odd order, $\chi(c)=\chi(\tau^{-1}c\tau)=1$. From the formula in (a), since $c\in G_K$, $\rho(c)$ is the identity matrix.
\[T:main\] Let $K$ be a real quadratic field of discriminant $d$, let ${\mathbb F}$ be a field of characteristic 0 or a finite field of odd characteristic. In the first case set $p=1$ and in the second case let $p$ be the characteristic of ${\mathbb F}$. Let $\chi:G_K\to{\mathbb F}^\times$ be a character with finite image. Let $L$ be the fixed field of the kernel of $\chi$ and choose $N\in{\mathbb Z}$ so that $L/K$ is unramified outside primes of $K$ dividing $N$. Let $M\geq 3$, $S_0=S_0(pdN)$, $S=S(M)\cap S_0$, $\theta$ the Dirichlet character cutting out $K$, $q$ the character of $Z$ determined by $q(rI)=\theta(r)$ for all $r\in {\mathbb Z}\cap K_{S_0}$, and ${M_{S,q}}$ the module defined in Definition \[D:MSQ\]. Assume
(1) $\chi$ is trivial on the principal fractional ideals of $K$ generated by elements of ${K_{S,q}}$.
(2) $\chi$ is trivial on the principal fractional ideals of $K$ generated by elements of ${\mathbb Q}^\times\cap K_{S_0}$.
(3) $[L:K]$ is odd.
(4) $L/{\mathbb Q}$ is Galois.
Then $\rho:G_{\mathbb Q}\to{{\rm GL}}(2,{\mathbb F})$ given by $\rho=\operatorname{Ind}_{G_K}^{G_{\mathbb Q}}\chi$ is an even Galois representation, and is attached to a Hecke eigenclass in $H^1({{\rm GL}}(2,{\mathbb Z}),{M_{S,q}}^*)$.
Given $\chi$ satisfying the conditions of the theorem, we define $$\Phi(L)=\prod_{w\nmid pdN}\hat\psi_w^0(L_w)$$ where $\psi_w^0$ is the function constructed in the proof of Theorem \[T:inert-psi\] for $w$ inert in $K$ and prime to $pN$, and the function defined by Definition \[D:split-psi\] for $w$ splitting in $K$ and prime to $pN$.
By Theorem \[L:KSQ-invariant\], $\Phi$ is ${K_{S,q}}$-invariant and $q$-homogeneous. Hence, by Lemma \[Lemma:homothety-to-cohomology\] we may consider it as an element of $H^1({{\rm GL}}(2,{\mathbb Q}),{M_{S,q}}^*)$. By Corollary \[eigenclass\], combined with Lemma \[lemsplit\] and Theorem \[T:inert-psi\] we see that for all $\ell$ unramified in $L/{\mathbb Q}$, $\Phi$ is an eigenvector for $T_\ell$ and $T_{\ell,\ell}$, and that the eigenvalues of $T_\ell$ match the trace of $\rho(\operatorname{Frob}_\ell)$. The $q$-homogeneity of $\Phi$ shows that the eigenvalues of $T_{\ell,\ell}$ match the determinant of $\rho(\operatorname{Frob}_\ell)$ for all $\ell$ unramified in $L/{\mathbb Q}$. Hence, $\Phi$ is attached to $\rho$.
|
---
abstract: |
We show the pointwise version of the Stečkin theorem on approximation by de la Vallée-Poussin means. The result on norm approximation is also derived.
**Key words:** Pointwise** **approximation by de la Vallée-Poussin means
**2000 Mathematics Subject Classification:** 42A24,
author:
- |
**Włodzimierz Łenski**\
University of Zielona Góra\
Faculty of Mathematics, Computer Science and Econometrics\
65-516 Zielona Góra, ul. Szafrana 4a\
P O L A N D\
[email protected]
title: '**Pointwise analog of the Stečkin approximation theorem**'
---
Introduction
============
Let $L^{p}\ (1\leq p<\infty )\ \left[ C\right] $ be the class of all $2\pi $–periodic real–valued functions integrable in the Lebesgue sense with $p$–th power $\left[ \text{continuous}\right] $ over $Q=$ $[-\pi ,\pi ]$ and let $X^{p}=L^{p}$ when $1\leq p<\infty $ or $X^{p}=C$ when $p=\infty $.
Let us define the norms of $f\in X^{p}$ as $$\Vert f\Vert =\Vert f\Vert _{_{X^{p}}}=\Vert f(\cdot )\Vert
_{_{X^{p}}}:=\left\{
\begin{array}{ll}
\left\{ \int_{_{_{Q}}}\mid f(x)\mid ^{p}dx\right\} ^{\frac{1}{p}} & \text{when }1\leq p<\infty \ \\
\sup_{x\in Q}\mid f(x)\mid & \text{when \ }p=\infty
\end{array}\right.$$and$$\begin{aligned}
\Vert f\Vert _{x,\delta } &=&\Vert f\Vert _{X^{p},x,\delta }=\Vert f(\cdot
)\Vert _{X^{p},x,\delta }:=\sup\limits_{0<h\leq \delta }\Vert f(\cdot )\Vert
_{X^{p},x,h}^{\circ } \\
&:&=\left\{
\begin{array}{ll}
\sup\limits_{0<h\leq \delta }\left\{ \frac{1}{2h}\int\limits_{x-h}^{x+h}\mid
f(t)\mid ^{p}dt\right\} ^{\frac{1}{p}} & \text{when }1\leq p<\infty \ \\
\sup\limits_{0<h\leq \delta }\left\{ \sup\limits_{0<\left\vert t\right\vert
\leq h}\mid f(x+t)\mid \right\} & \text{when \ }p=\infty
\end{array}\right. \end{aligned}$$where$$\begin{aligned}
\Vert f\Vert _{x,\delta }^{\circ } &=&\Vert f\Vert _{X^{p},x,\delta }^{\circ
}=\Vert f(\cdot )\Vert _{X^{p},x,\delta }^{\circ } \\
&:&=\left\{
\begin{array}{ll}
\left\{ \frac{1}{2h}\int\limits_{x-\delta }^{x+\delta }\mid f(t)\mid
^{p}dt\right\} ^{\frac{1}{p}} & \text{when }1\leq p<\infty \\
\sup\limits_{0<\left\vert t\right\vert \leq h}\mid f(x+h)\mid & \text{when
\ }p=\infty
\end{array}\right. \text{ \ \ \ }\left( \delta >0\right) .\end{aligned}$$We note additionally that $$\Vert f\Vert _{X,x,0}=\Vert f\Vert _{X,x,0}^{\circ }=\mid f(x)\mid .$$Consider the trigonometric Fourier series of $f$ $$Sf(x)=\frac{a_{o}(f)}{2}+\sum_{k=0}^{\infty }(a_{k}(f)\cos kx+b_{k}(f)\sin
kx)$$with the partial sums $S_{k}f$ .
Let $$\sigma _{n,m}f\left( x\right) :=\frac{1}{m+1}\sum_{k=n-m}^{n}S_{k}f\left(
x\right) \text{ \ \ \ }\left( m\leq n=0,1,2,...\right)$$
As a measure of approximation by the above quantities we use the pointwise characteristics$$\begin{aligned}
w_{x}f(\delta ) &=&w_{x}f(\delta )_{X^{p}}:=\Vert \Delta _{x}f\left( \cdot
\right) )\Vert _{X^{p},x,\delta } \\
&=&\left\{
\begin{array}{ll}
\sup\limits_{0<h\leq \delta }\left\{ \frac{1}{2h}\int_{-h}^{h}\left\vert
\Delta _{x}f\left( t\right) \right\vert ^{p}dt\right\} ^{1/p} & \text{when }1\leq p<\infty \\
\sup\limits_{0<h\leq \delta }\left\{ \sup\limits_{0<\left\vert t\right\vert
\leq h}\mid \Delta _{x}f\left( t\right) \mid \right\} & \text{when \ }p=\infty
\end{array}\right. \end{aligned}$$cf. [@ABT] and $$\begin{aligned}
w_{x}^{\circ }f(\delta ) &=&w_{x}^{\circ }f(\delta )_{X^{p}}:=\Vert \Delta
_{x}f\left( \cdot \right) )\Vert _{X^{p},x,\delta }^{\circ } \\
&=&\left\{
\begin{array}{ll}
\left\{ \frac{1}{2\delta }\int_{-\delta }^{\delta }\left\vert \Delta
_{x}f\left( t\right) \right\vert ^{p}dt\right\} ^{1/p} & \text{when }1\leq
p<\infty \\
\sup\limits_{0<\left\vert t\right\vert \leq \delta }\mid \Delta _{x}f\left(
t\right) \mid & \text{when \ }p=\infty
\end{array}\right. \end{aligned}$$and also$$\Omega _{x}f\left( \frac{\pi }{n+1}\right) =\Omega _{x}f\left( \frac{\pi }{n+1}\right) _{X^{p}}:=\frac{1}{n+1}\sum_{k=0}^{n}w_{x}f(\frac{\pi }{k+1})_{X^{p}}$$and$$\Omega _{x}^{\circ }f\left( \frac{\pi }{n+1}\right) =\Omega _{x}^{\circ
}f\left( \frac{\pi }{n+1}\right) _{X^{p}}=\frac{1}{n+1}\sum_{k=0}^{n}w_{x}^{\circ }f(\frac{\pi }{k+1})_{X^{p}}\text{ ,}$$$$\text{where \ \ }\Delta _{x}f\left( t\right) :=f\left( x+t\right) -f\left(
x\right) \text{ \ ,}$$constructed on the base of definition of $X^{p}-points\ \ (\left[ \text{Lebesgue points}(L^{p}-points)\right] $ or $\left[ \text{points of
continuity}\left( C-points\right) \right] ).$ We also use the modulus of continuity of $f$ in the space $X^{p}$ defined by the formula $$\omega f\left( \delta \right) =\omega f\left( \delta \right)
_{X^{p}}:=\sup_{0<\left\vert h\right\vert \leq \delta }\left\Vert \Delta
_{\cdot }f\left( h\right) \right\Vert _{X^{p}}\text{ \ }$$and its arithmetic mean$$\Omega f\left( \frac{\pi }{n+1}\right) =\Omega f\left( \frac{\pi }{n+1}\right) _{X^{p}}=\frac{1}{n+1}\sum_{k=0}^{n}\omega f(\frac{\pi }{k+1})_{X^{p}}.$$
We can observe that, for $f\in X^{\widetilde{p}}$ and $\widetilde{p}\geq p$, $$\Vert w_{\cdot .}f(\delta )_{X^{p}}\Vert _{C}\leq \omega f\left( \delta
\right) _{C},$$whence$$\Vert \Omega _{\cdot .}f(\delta )_{X^{p}}\Vert _{C}\leq \Omega f\left(
\delta \right) _{C}$$and$$\Vert w_{\cdot .}^{\circ }f(\delta )_{X^{p}}\Vert _{X^{p}}\leq \omega
f\left( \delta \right) _{X^{p}},$$whence$$\Vert \Omega _{\cdot .}^{\circ }f(\delta )_{X^{p}}\Vert _{X^{p}}\leq \Omega
f\left( \delta \right) _{X^{p}}.$$
Let introduce one more measure of poitwise approximation analogical to the best approximation of function $f$ by trigonometric polynomials $T$ of the degree at most $n$ $\left( T\in H_{n}\right) $$$E_{n}\left( f\right) _{X^{p}}:=\inf_{T\in H_{n}}\left\{ \left\Vert f\left(
\cdot \right) -T\left( \cdot \right) \right\Vert _{X^{p}}\right\} ,$$namely$$E_{n}\left( f,x;\delta \right) =E_{n}\left( f,x;\delta \right)
_{X^{p}}:=\inf_{T\in H_{n}}\left\{ \left\Vert f\left( \cdot \right) -T\left(
\cdot \right) \right\Vert _{X^{p},x,\delta }\right\}$$$$=\left\{
\begin{array}{ll}
\inf\limits_{T\in H_{n}}\left\{ \sup\limits_{0<h\leq \delta }\left[ \frac{1}{2h}\int_{-h}^{h}\left\vert f\left( x+t\right) -T\left( x+t\right)
\right\vert ^{p}dt\right] ^{\frac{1}{p}}\right\} & \text{when }1\leq
p<\infty \\
\inf\limits_{T\in H_{n}}\left\{ \sup\limits_{0<\left\vert h\right\vert \leq
\delta }\mid f(x+h)-T\left( x+h\right) )\mid \right\} & \text{when \ }p=\infty
\end{array}\right.$$and$$E_{n}^{\circ }\left( f,x;\delta \right) =E_{n}^{\circ }\left( f,x;\delta
\right) _{X^{p}}:=\inf_{T\in H_{n}}\left\{ \left\Vert f\left( \cdot \right)
-T\left( \cdot \right) \right\Vert _{X^{p},x,\delta }^{\circ }\right\} .$$We will also use its arithmetic mean$$F_{n,m}\left( f,x\right) =F_{n,m}\left( f,x\right) _{X}:=\frac{1}{m+1}\sum_{k=0}^{m}E_{n}\left( f,x;\frac{\pi }{k+1}\right) _{X}$$and$$F_{n,m}^{\circ }\left( f,x\right) =F_{n,m}^{\circ }\left( f,x\right)
_{X^{p}}:=\frac{1}{m+1}\sum_{k=0}^{m}E_{n}^{\circ }\left( f,x;\frac{\pi }{k+1}\right) _{X^{p}}\text{ .}$$
Denote also $$X^{p}\left( w_{x}\right) =\left\{ f\in X^{p}:w_{x}f\left( \delta \right)
\leq w_{x}\left( \delta \right) \right\} ,$$where $\ w_{x}$ is a function of modulus of continuity type on the interval $[0,+\infty ),$ i.e. a nondecreasing continuous function having the following properties: $w_{x}\left( 0\right) =0,$ $w_{x}\left( \delta _{1}+\delta
_{2}\right) \leq w_{x}\left( \delta _{1}\right) +w_{x}\left( \delta
_{2}\right) $ for any $0\leq \delta _{1}\leq \delta _{2}\leq \delta
_{1}+\delta _{2}$.
Using these characteristics we will show the pointwise version of the Stečkin [@S] generalization of the Fejér-Lebesgue theorem. As a corollaries we will obtain the mentioned original result of S. B. Stečkin on norm approximation as well the result of N. Tanović-Miller [TM]{}.
By $K$ we shall designate either an absolute constant or a constant depending on some parameters, not necessarily the same of each occurrence.
Statement of the results
========================
At the begin we formulate the partial solution of the considered problem.
[@WL]If $f\in X^{p}$ then, for any positive integer $m\leq n$ and all real $x,$$$\begin{aligned}
\left\vert \sigma _{n,m}f\left( x\right) -f\left( x\right) \right\vert &\leq
&\pi ^{2}E_{n-m}^{\circ }(f,x,\frac{\pi }{2n-m+1})_{X}+6F_{n-m,m}^{\circ
}\left( f,x\right) _{X} \\
&&+\int_{\frac{\pi }{m+1}}^{\frac{\pi }{2n-m+1}}\frac{E_{n-m}^{\circ
}(f,x,t)_{X}}{t}dt+E_{n-m}^{\circ }\left( f,x;0\right) _{X^{p}}\end{aligned}$$and$$\begin{aligned}
\left\vert \sigma _{n,m}f\left( x\right) -f\left( x\right) \right\vert &\leq
&\left( 6\text{ }+\pi ^{2}\right) F_{n-m,m}\left( f,x\right) _{X^{p}}\left[
1+\ln \frac{n+1}{m+1}\right] \\
&&+E_{n-m}\left( f,x;0\right) _{X^{p}}.\end{aligned}$$
Now, we can present the main result on pointwise approximation.
If $f\in X^{p}$ then, for any positive integer $m\leq n$ and all real $x,$$$\left\vert \sigma _{n,m}f\left( x\right) -f\left( x\right) \right\vert \leq
K\sum_{\nu =0}^{n}\frac{F_{n-m+\nu ,m}\left( f,x\right) _{X^{p}}+F_{n-m+\nu
,\nu }\left( f,x\right) _{X^{p}}}{m+\nu +1}+E_{2n}\left( f,x;0\right)
_{X^{p}}.$$
This immediately yields the following result of Stečkin [@S]
If $f\in C$ then, for any positive integer $n$ and $m\leq n$ $$\left\Vert \sigma _{n,m}f\left( \cdot \right) -f\left( \cdot \right)
\right\Vert _{C}\leq K\sum_{\nu =0}^{n}\frac{E_{n-m+\nu }\left( f\right) _{C}}{m+\nu +1}\text{ .}$$
Theorem also holds if instead of $C$ we consider the spaces $X^{p}$ with $1<p<\infty .$ In the proof we need the Hardy-Littlewood estimate of the maximal function.
At every $X^{p}-point\ $ $x$ of $f$$$\Omega _{x}f\left( \gamma \right) _{X^{p}}=o_{x}\left( 1\right) \text{ \ \
as \ }\gamma \rightarrow 0+$$and thus from Theorem 1 we obtain the corollary which state the result of the Tanović-Miller type[@TM].
If $f\in X^{p}$ then, for any positive integer $m\leq n$ at every $X^{p}-point$ $x$ of $f$,$$\left\vert \sigma _{n,m}f\left( x\right) -f\left( x\right) \right\vert
=o_{x}\left( 1\right) \left[ 1+\ln \frac{n+1}{m+1}\right] \text{ \ as \ }n\rightarrow \infty \text{ .}$$
Auxiliary results
=================
In order to proof our theorems we require some lemmas
If $T_{n}$ is the trigonometric polynomial of the degree at most $n$ of the best approximation of $f\in X^{p}$ with respect to the norm $\left\Vert \cdot \right\Vert _{X^{p}}$ then, it is also the trigonometric polynomial of the degree at most $n$ of the best approximation of $f\in X^{p}$ with respect to the norm $\left\Vert \cdot \right\Vert
_{X^{p},x,\delta }$ for any $\delta \in \left[ 0,\pi \right] .$
From the inequalities$$\begin{aligned}
\left\Vert E_{n}\left( f,\cdot ,\delta \right) _{X^{p}}\right\Vert _{X^{p}}
&\geq &\left\Vert E_{n}^{\circ }\left( f,\cdot ,\delta \right)
_{X^{p}}\right\Vert _{X^{p}} \\
&=&\left\Vert \left\Vert f-T_{n,\delta }\right\Vert _{X^{p},\cdot ,\delta
}^{\circ }\right\Vert _{X^{p}}=\left\Vert f-T_{n,\delta }\right\Vert _{X^{p}}
\\
&\geq &\left\Vert f-T_{n}\right\Vert _{X^{p}}=E_{n}\left( f\right) _{X^{p}}\end{aligned}$$and$$\left\Vert E_{n}^{\circ }\left( f,\cdot ,\delta \right) _{X^{p}}\right\Vert
_{X^{p}}\leq \left\Vert \left\Vert f-T_{n}\right\Vert _{X^{p},\cdot ,\delta
}^{\circ }\right\Vert _{X^{p}}=\left\Vert f-T_{n}\right\Vert
_{X^{p}}=E_{n}\left( f\right) _{X^{p}}\text{ ,}$$where $T_{n,\delta }$ and $T_{n}$ are the trigonometric polynomials of the degree at most $n$ of the best approximation of $f\in X^{p}$ with respect to the norms $\left\Vert \cdot \right\Vert _{X^{p},x,\delta
}^{\circ }$ and $\left\Vert \cdot \right\Vert _{X^{p}}$ respectively$\ ,$ we obtain relation$$\left\Vert f-T_{n,\delta }\right\Vert _{X^{p}}=\left\Vert f-T_{n}\right\Vert
_{X^{p}}=E_{n}\left( f\right) _{X^{p}}\text{ ,}$$whence $T_{n,\delta }=T_{n}$ for any $\delta \in \left[ 0,\pi \right] $ by uniqueness of the trigonometric polynomial of the degree at most $n$ of the best approximation of $f\in X^{p}$ with respect to the norm $\left\Vert \cdot \right\Vert _{X^{p}}$ $\ $(see e.g.[@BN] p. 96)$.$We can also observe that for such $T_{n}$ and any $h\in \left[ 0,\delta \right]
$ $$\left\Vert f-T_{n}\right\Vert _{X^{p},x,h}^{\circ }=E_{n}^{\circ }\left(
f,x,h\right) _{X^{p}}\leq E_{n}\left( f,x,\delta \right) _{X^{p}}\leq
\left\Vert f-T_{n}\right\Vert _{X^{p},x,\delta }\text{ \ .}$$Hence$$E_{n}\left( f,x,\delta \right) _{Xp}=\left\Vert f-T_{n}\right\Vert
_{X^{p},x,\delta }\text{ \ }$$and our proof is complete.
If $n\in
\mathbb{N}
_{0}$ and $\delta >0$ then $\ E_{n}\left( f,x;\delta \right) _{X^{p}}$ is nonincreasing function of $n$ and nondecreasing function of $\delta .$These imply that for $m,n\in
\mathbb{N}
$ the function $\ F_{n,m}\left( f,x\right) _{X^{p}}$ is nonincreasing function of $n$ and $m$ simultaneously.
The first part of our statement follows from the property of the norm $\left\Vert \cdot \right\Vert _{x,\delta }$ and supremum. The second part is a consequence of the calculation$$\begin{aligned}
\frac{F_{n,m+1}\left( f,x\right) _{X^{p}}}{F_{n,m}\left( f,x\right) _{X^{p}}}
&=&\frac{m+1}{m+2}\left( 1+\frac{E_{n}\left( f,x;\frac{\pi }{m+2}\right)
_{X^{p}}}{\sum_{k=0}^{m}E_{n}\left( f,x;\frac{\pi }{k+1}\right) _{X^{p}}}\right) \\
&\leq &\frac{m+1}{m+2}\left( 1+\frac{E_{n}\left( f,x;\frac{\pi }{m+1}\right)
_{X^{p}}}{\sum_{k=0}^{m}E_{n}\left( f,x;\frac{\pi }{m+1}\right) _{X^{p}}}\right) \\
&=&\frac{m+1}{m+2}\left( 1+\frac{1}{m+1}\right) =1.\end{aligned}$$
Let $m,n,q\in
\mathbb{N}
_{0}$ such that $m\leq n$ and $q\geq m+1.$ If $f\in X^{p}$ then$$\left\vert \sigma _{n+q,m}f\left( x\right) -\sigma _{n,m}f\left( x\right)
\right\vert \leq KF_{n-m,m}\left( f,x\right) _{X^{p}}\sum_{\nu =0}^{q-1}\frac{1}{m+\nu +1}.$$
It is clear that$$\begin{aligned}
\sigma _{n,m}f\left( x\right) &=&\frac{1}{m+1}\sum_{k=n-m}^{n}\frac{1}{\pi }\int_{-\pi }^{\pi }f\left( x+t\right) D_{k}\left( t\right) dt \\
&=&\frac{1}{\pi }\int_{-\pi }^{\pi }f\left( x+t\right) V_{n,m}\left(
t\right) dt\end{aligned}$$where$$V_{n,m}\left( t\right) =\frac{1}{m+1}\sum_{k=n-m}^{n}D_{k}\left( t\right)
\text{ \ and \ \ }D_{k}\left( t\right) =\frac{\sin \frac{\left( 2k+1\right) t}{2}}{2\sin \frac{t}{2}}.$$Hence, by orthogonality of the trigonometric system,$$\begin{aligned}
&&\sigma _{n+q,m}f\left( x\right) -\sigma _{n,m}f\left( x\right) \\
&=&\frac{1}{\pi }\int_{-\pi }^{\pi }\left[ f\left( x+t\right) -T_{n-m}\left(
x+t\right) \right] \left( V_{n+q,m}\left( t\right) -V_{n,m}\left( t\right)
\right) dt \\
&=&\frac{1}{\pi \left( m+1\right) }\sum_{k=n-m}^{n}\int_{-\pi }^{\pi }\left[
f\left( x+t\right) -T_{n-m}\left( x+t\right) \right] \left( D_{k+q}\left(
t\right) -D_{k}\left( t\right) \right) dt \\
&=&\frac{1}{\pi \left( m+1\right) }\sum_{k=n-m}^{n}\int_{-\pi }^{\pi }\left[
f\left( x+t\right) -T_{n-m}\left( x+t\right) \right] \\
&&\frac{\sin \frac{\left( 2k+2q+1\right) t}{2}-\sin \frac{\left( 2k+1\right)
t}{2}}{2\sin \frac{t}{2}}dt \\
&=&\frac{1}{\pi \left( m+1\right) }\sum_{k=n-m}^{n}\int_{-\pi }^{\pi }\left[
f\left( x+t\right) -T_{n-m}\left( x+t\right) \right] \frac{\sin \frac{qt}{2}\cos \frac{\left( 2k++q1\right) t}{2}}{\sin \frac{t}{2}}dt\end{aligned}$$with trigonometric polynomial $T_{n-m}$ of the degree at most $n-m$ of the best approximation of $f$ $.$
Using the notations$$\begin{aligned}
I_{1} &=&\left[ -\frac{\pi }{q},\frac{\pi }{q}\right] ,\text{ \ \ \ \ }I_{2}=\left[ -\frac{\pi }{m+1},-\frac{\pi }{q}\right] \cup \left[ \frac{\pi }{q},\frac{\pi }{m+1}\right] \\
I_{3} &=&\left[ -\pi ,-\frac{\pi }{m+1}\right] \cup \left[ \frac{\pi }{m+1},\pi \right]\end{aligned}$$we get$$\begin{aligned}
\sum &=&\frac{1}{\pi \left( m+1\right) }\sum_{k=n-m}^{n}\left(
\int\limits_{I_{1}}+\int\limits_{\text{\ }I_{2}}+\int\limits_{I_{3}}\right) \left[ f\left( x+t\right) -T_{n-m}\left( x+t\right) \right] \\
&&\frac{\sin \frac{qt}{2}\cos \frac{\left( 2k+q+1\right) t}{2}}{\sin \frac{t}{2}}dt \\
&=&\sum\nolimits_{1}+\sum\nolimits_{2}+\sum\nolimits_{3}.\end{aligned}$$and$$\begin{aligned}
\sum\nolimits_{1} &\leq &\frac{1}{\pi \left( m+1\right) }\sum_{k=n-m}^{n}\int\limits_{I_{1}}\left\vert f\left( x+t\right) -T_{n-m}\left( x+t\right)
\right\vert qdt \\
&=&\frac{q}{\pi }\int\limits_{I_{1}}\left\vert f\left( x+t\right)
-T_{n-m}\left( x+t\right) \right\vert dt \\
&\leq &2E_{n-m}\left( f,x;\frac{\pi }{q}\right) _{X^{p}}\end{aligned}$$We next evaluate the sums $\sum_{2}$ and $\sum_{3}$ using the partial integrating and Lemma 1. Thus$$\begin{aligned}
\sum\nolimits_{2} &\leq &\int\limits_{\text{\ }I_{2}}\frac{\left\vert
f\left( x+t\right) -T_{n-m}\left( x+t\right) \right\vert }{t}dt \\
&=&2\left[ \frac{1}{2t}\int_{-t}^{t}\left\vert f\left( x+u\right)
-T_{n-m}\left( x+u\right) \right\vert du\right] _{t=\frac{\pi }{q}}^{t=\frac{\pi }{m+1}} \\
&&+2\int_{\frac{\pi }{q}}^{\frac{\pi }{m+1}}\frac{1}{t}\left[ \frac{1}{2t}\int_{-t}^{t}\left\vert f\left( x+u\right) -T_{n-m}\left( x+u\right)
\right\vert du\right] dt\end{aligned}$$$$\begin{aligned}
&\leq &2E_{n-m}\left( f,x;\frac{\pi }{m+1}\right) _{X^{p}}+2\int_{-t}^{t}\frac{1}{t}E_{n-m}\left( f,x;t\right) _{X^{p}}dt \\
&\leq &4E_{n-m}\left( f,x;\frac{\pi }{m+1}\right) _{X^{p}}\left[ 1+\ln \frac{q}{m+1}\right] \\
&\leq &4E_{n-m}\left( f,x;\frac{\pi }{m+1}\right) _{X^{p}}\left[ 1+\sum_{\nu
=0}^{q-1}\frac{1}{m+\nu +1}\right]\end{aligned}$$and$$\begin{aligned}
\sum\nolimits_{3} &\leq &\frac{1}{m+1}\int\limits_{I_{3}}\frac{\left\vert
f\left( x+t\right) -T_{n-m}\left( x+t\right) \right\vert }{t} \\
&&\left\vert \sum_{k=n-m}^{n}\cos \left( kt+\frac{q+1}{2}t\right)
\right\vert dt\end{aligned}$$$$\begin{aligned}
&\leq &\frac{1}{m+1}\int\limits_{I_{3}}\frac{\left\vert f\left( x+t\right)
-T_{n-m}\left( x+t\right) \right\vert }{t} \\
&&\left\vert \frac{2\sin \frac{\left( n+1\right) t}{2}\cos \frac{\left(
2n-m+q+1\right) t}{2}}{2\sin \frac{t}{2}}\right\vert dt\end{aligned}$$$$\begin{aligned}
&\leq &\frac{\pi }{m+1}\int\limits_{I_{3}}\frac{\left\vert f\left(
x+t\right) -T_{n-m}\left( x+t\right) \right\vert }{t^{2}}dt \\
&=&\frac{\pi }{m+1}\left\{ 2\left[ \frac{1}{2t}\int_{-t}^{t}\left\vert
f\left( x+u\right) -T_{n-m}\left( x+u\right) \right\vert du\right] _{t=\frac{\pi }{m+1}}^{t=\pi }\right. \\
&&+4\int_{\frac{\pi }{m+1}}^{\pi }\frac{1}{t^{2}}\left[ \frac{1}{2t}\int_{-t}^{t}\left\vert f\left( x+u\right) -T_{n-m}\left( x+u\right)
\right\vert du\right] dt\end{aligned}$$$$\begin{aligned}
&\leq &\frac{\pi }{m+1}\left\{ 2E_{n-m}\left( f,x;\pi \right)
_{X^{p}}+4\int_{\frac{\pi }{m+1}}^{\pi }\frac{1}{t^{2}}E_{n-m}\left(
f,x;t\right) _{X^{p}}dt\right\} \\
&=&\frac{2\pi }{m+1}\left\{ E_{n-m}\left( f,x;\pi \right)
_{X^{p}}+2\int_{1}^{m+1}\frac{E_{n-m}\left( f,x;\frac{\pi }{u}\right)
_{X^{p}}}{\pi ^{2}/u^{2}}\frac{\pi du}{u^{2}}\right\} \\
&=&\frac{2\pi }{m+1}\left\{ E_{n-m}\left( f,x;\pi \right) _{X^{p}}+\frac{2}{\pi }\sum_{k=0}^{m-1}\int_{k+1}^{k+2}E_{n-m}\left( f,x;\frac{\pi }{u}\right)
_{X^{p}}du\right\}\end{aligned}$$$$\begin{aligned}
&=&\frac{2\pi }{m+1}\left\{ E_{n-m}\left( f,x;\pi \right) _{X^{p}}+\frac{2}{\pi }\sum_{k=0}^{m-1}E_{n-m}\left( f,x;\frac{\pi }{k+1}\right)
_{X^{p}}\right\} \\
&\leq &\frac{2\pi +\frac{2}{\pi }}{m+1}\sum_{k=0}^{m-1}E_{n-m}\left( f,x;\frac{\pi }{k+1}\right) _{X^{p}}\end{aligned}$$which proves Lemma 2.
Before formulating the next lemmas we define a new difference. Let $m,n\in
\mathbb{N}
_{0}$ and $m\leq n.$ Denote $$\tau _{n,m}f\left( x\right) :=\left( m+1\right) \left\{ \sigma
_{n+m+1,m}f\left( x\right) -\sigma _{n,m}f\left( x\right) \right\} .$$
Let $m,n,\mu \in
\mathbb{N}
_{0}$ such that $2\mu \leq m\leq n$ $.$ If $f\in X^{p}$ then$$\left\vert \tau _{n,m}f\left( x\right) -\tau _{n-\mu ,m-\mu }f\left(
x\right) \right\vert \leq K\mu F_{n-\mu +1,\mu -1}\left( f,x\right)
_{X^{p}}\ln \frac{m}{\mu }.$$
The proof follows by the method of Leindler[@L]. Namely$$\tau _{n,m}f\left( x\right) -\tau _{n-\mu ,m-\mu }f\left( x\right) =\left(
\sum_{k=n+m-2\mu +2}^{n+m+1}-2\sum_{k=n-\mu +1}^{n}\right) \left[
S_{k}f\left( x\right) -f\left( x\right) \right]$$and$$\left\vert \tau _{n,m}f\left( x\right) -\tau _{n-\mu ,m-\mu }f\left(
x\right) \right\vert \leq$$$$\begin{aligned}
&\leq &\left\vert \left( \sum_{k=n+m-2\mu +2}^{n+m-\mu +1}-\sum_{k=n-\mu
+1}^{n}\right) \left[ S_{k}f\left( x\right) -f\left( x\right) \right]
\right\vert \\
&&+\left\vert \left( \sum_{k=n+m-\mu +2}^{n+m+1}-\sum_{k=n-\mu
+1}^{n}\right) \left[ S_{k}f\left( x\right) -f\left( x\right) \right]
\right\vert\end{aligned}$$$$=\mu \left\vert \sigma _{n+m-\mu +1,\mu -1}f\left( x\right) -\sigma _{n,\mu
-1}f\left( x\right) \right\vert$$$$+\mu \left\vert \sigma _{n+m+1,\mu -1}f\left( x\right) -\sigma _{n,\mu
-1}f\left( x\right) \right\vert .$$By Lemma 2 , for $2\mu \leq m,$$$\left\vert \tau _{n,m}f\left( x\right) -\tau _{n-\mu ,m-\mu }f\left(
x\right) \right\vert \leq$$$$\begin{aligned}
&\leq &K\mu F_{n-\mu +1,\mu -1}\left( f,x\right) _{X^{p}}\left[ 1+\ln \frac{\left( n-\mu +1\right) +\mu -1}{\mu }\right] \\
&&+K\mu F_{n-\mu +1,\mu -1}\left( f,x\right) _{X^{p}}\left[ 1+\ln \frac{m+\mu -1}{\mu }\right]\end{aligned}$$$$\leq K\mu F_{n-\mu +1,\mu -1}\left( f,x\right) _{X^{p}}\left[ 1+\ln \frac{m}{\mu }\right]$$and our proof is complete.
Let $m,n\in
\mathbb{N}
_{0}$ and $m\leq n$ $.$ If $f\in X^{p}$ then$$\left\vert \tau _{n,m}f\left( x\right) \right\vert \leq
K\sum_{k=n-m}^{n}F_{k,k-n+m}\left( f,x\right) _{X^{p}}.$$
Our proof runs parallel with the proof of Theorem 1 in [@S].
If $m=0$ then$$\left\vert \tau _{n,0}f\left( x\right) \right\vert =\left\vert \sigma
_{n+1,0}f\left( x\right) -\sigma _{n,0}f\left( x\right) \right\vert \leq
KF_{n,0}\left( f,x\right) _{X^{p}}.$$and if $m=1$ then $$\begin{aligned}
\left\vert \tau _{n,1}f\left( x\right) \right\vert &\leq &2\left\vert \sigma
_{n+1,1}f\left( x\right) -\sigma _{n,1}f\left( x\right) \right\vert \leq
KF_{n-1,1}\left( f,x\right) _{X^{p}} \\
&\leq &K\left[ F_{n-1,1}\left( f,x\right) _{X^{p}}+F_{n-1,1}\left(
f,x\right) _{X^{p}}\right] \\
&\leq &K\left[ F_{n-1,1}\left( f,x\right) _{X^{p}}+F_{n,1}\left( f,x\right)
_{X^{p}}\right]\end{aligned}$$by Lemma 2 and Lemma 3
Next we construct the same decreasing sequence $\left( m_{s}\right) $.of integers that was given by S. B. Stečkin. Let $$m_{0}=m,\text{ \ }m_{s}=m_{s-1}-\left[ \frac{m_{s-1}}{2}\right] \text{ \ \ \
}\left( s=1,2,...\right)$$where $\left[ y\right] $ denotes the integral part of $y$. It is clear that there exists a smallest index $t\geq 1$ such that $m_{t}=1$ and $$m=m_{0}>m_{1}>...>m_{t}=1.$$
By the definition of the numbers $m_{s}$ we have $$m_{s}\geq m_{s-1}/2$$$$m_{s-1}-m_{s}=\left[ \frac{m_{s-1}}{2}\right] \geq \left[ \frac{m_{s-1}}{3}\right] \text{ \ \ \ }\left( s=1,2,...,t\right)$$whence$$m_{t-1}=2,\text{ \ }m_{t-1}-m_{t}=1$$and$$m_{s-1}-m_{s}\leq m_{s}\leq 3\left( m_{s}-m_{s+1}\right) \text{ \ \ \ }\left( s=1,2,...,t-1\right)$$follow.
Under these notations we get the following equality$$\tau _{n,m}f\left( x\right) =\sum_{s=1}^{t}\left( \tau
_{n-m+m_{s-1},m_{s-1}}f\left( x\right) -\tau _{n-m+m_{s},m_{s}}f\left(
x\right) \right) +\tau _{n-m+m_{t},m_{t}}f\left( x\right)$$whence, by $m_{t}=1,$$$\left\vert \tau _{n,m}f\left( x\right) \right\vert \leq
\sum_{s=1}^{t}\left\vert \tau _{n-m+m_{s-1},m_{s-1}}f\left( x\right) -\tau
_{n-m+m_{s},m_{s}}f\left( x\right) \right\vert +\left\vert \tau
_{n-m+m_{t},m_{t}}f\left( x\right) \right\vert$$follows.
It is easy to see that the terms in the sum $\sum_{s=1}^{t}$ , by Lemma 4, with $\mu =m_{s-1}-m_{s}$ and $m=m_{s-1}$ do not exceed $$\begin{aligned}
&&K\left( m_{s-1}-m_{s}\right) F_{n-m+m_{s}+1,m_{s-1}-m_{s}}\left(
f,x\right) _{X^{p}}\ln \frac{m_{s-1}}{m_{s-1}-m_{s}}, \\
&&\text{where }\left( s=1,2,...,t-1\right) .\end{aligned}$$and by Lemma 3 we get $$\left\vert \tau _{n-m+1,1}f\left( x\right) \right\vert \leq 2\left\vert
\sigma _{n-m+2,1}f\left( x\right) -\sigma _{n-m+1,1}f\left( x\right)
\right\vert \leq KF_{n-m,1}\left( f,x\right) _{X^{p}}$$Thus$$\begin{aligned}
\left\vert \tau _{n,m}f\left( x\right) \right\vert &\leq
&K\sum_{s=1}^{t-1}3\left( m_{s}-m_{s+1}\right) F_{n-m+m_{s}+1,m_{s}}\left(
f,x\right) _{X^{p}}\ln 3 \\
&&+KF_{n-m+2,m-2}\left( f,x\right) _{X^{p}}+KF_{n-m,1}\left( f,x\right)
_{X^{p}}\end{aligned}$$whence, by the monotonicity of $F_{\nu ,\mu }\left( f,x\right) _{X^{p}},$$$\begin{aligned}
&&\left\vert \tau _{n,m}f\left( x\right) \right\vert \\
&\leq &K\left( \sum_{s=1}^{t-1}\sum_{\nu =m_{s+1}+1}^{m_{s}}F_{n-m+\nu
+1,\nu }\left( f,x\right) _{X^{p}}+\sum_{\nu =0}^{2}F_{n-m+\nu ,m-\nu
-1}\left( f,x\right) _{X^{p}}\right) \\
&&+KF_{n-m,1}\left( f,x\right) _{X^{p}}\end{aligned}$$$$\begin{aligned}
&\leq &K\sum_{\nu =0}^{m_{1}+1}F_{n-m+\nu ,\nu }\left( f,x\right)
_{X^{p}}+KF_{n-m,1}\left( f,x\right) _{X^{p}} \\
&\leq &K\sum_{\nu =0}^{m}F_{n-m+\nu ,\nu }\left( f,x\right)
_{X^{p}}+KF_{n-m,1}\left( f,x\right) _{X^{p}} \\
&\leq &K\sum_{k=n-m}^{n}F_{k,k-n+m}\left( f,x\right)
_{X^{p}}+KF_{n-m,1}\left( f,x\right) _{X^{p}}\text{ \ .}\end{aligned}$$
Proofs of the results
=====================
Proof of Theorem 2
------------------
The proof follows the lines of the proofs of Theorem 4 in [@S] and Theorem in [@L]. Therefore let $n>0$ and $m\leq n$ be fixed. Let us define an increasing sequence $\left( n_{s}:s=0,1,...,t\right) $ of indices introduced by S. B. Stečki in the following way. Set $n_{0}=n.$ Assuming that the numbers $n_{0},...,n_{s}$ are already defined and $n_{s}<2n,$ we define $n_{s+1}$ as follows: Let $\nu _{s}$ denote the smallest natural number such that $$F_{n_{s}-m+\nu _{s},\nu }\left( f,x\right) _{X^{p}}\leq \frac{1}{2}F_{n_{s}-m,\nu }\left( f,x\right) _{X^{p}}\text{ }\left( \nu
=0,1,...,n\right) \text{.}$$According to the magnitude of $\nu _{s}$ we define$$n_{s+1}=\left\{
\begin{array}{cc}
n_{s}-m+1 & \text{ \ \ for \ }\nu _{s}\leq m\text{\ ,} \\
n_{s}+\nu _{s}\text{ } & \text{for \ \ }m+1\leq \nu _{s}<2n+m-n_{s}\text{ ,}
\\
2n+m & \text{ \ \ for \ }\nu _{s}\geq 2n+m-n_{s}\text{ }\end{array}\right.$$If $n_{s+1}<2n$ we continue the procedure, and if once $n_{s+1}\geq 2n$ then we stop the construction and define $t:=s+1.$
By the above definition of $\left( n_{s}\right) $ we have the following obvious properties:$$t\geq 1,\text{ \ }n=n_{0}<n_{1}<...<n_{t},\text{ \ \ }2n\leq n_{t}\leq 2n+m,$$and $$n_{s+1}-n_{s}\geq m+1\text{ \ \ }\left( s=0,1,...,t-1\right) ,$$and relations$$F_{n_{s+1}-m,\nu }\left( f,x\right) _{X^{p}}\leq \frac{1}{2}F_{n_{s}-m,\nu
}\left( f,x\right) _{X^{p}}\text{ \ \ for \ }s=0,1,...,t-2\text{,}$$and $$\frac{1}{2}F_{n_{s}-m,\nu }\left( f,x\right) _{X^{p}}\leq F_{n_{s+1}-m-1,\nu
}\left( f,x\right) _{X^{p}}\text{ \ \ for \ }s=0,1,...,t-1$$whenever $n_{s+1}-n_{s}>m+1.$
Let us start with$$\begin{aligned}
\left\vert \sigma _{n,m}f\left( x\right) -f\left( x\right) \right\vert
&=&\sum_{s=0}^{t-1}\left[ \left\vert \sigma _{n_{s},m}f\left( x\right)
-f\left( x\right) \right\vert -\left\vert \sigma _{n_{s+1},m}f\left(
x\right) -f\left( x\right) \right\vert \right] \\
&&+\left\vert \sigma _{n_{t},m}f\left( x\right) -f\left( x\right) \right\vert
\\
&\leq &\sum_{s=0}^{t-1}\left\vert \sigma _{n_{s+1},m}f\left( x\right)
-\sigma _{n_{s},m}f\left( x\right) \right\vert +\left\vert \sigma
_{n_{t},m}f\left( x\right) -f\left( x\right) \right\vert \\
&=&\sum_{s=0}^{t-1}\left\vert \frac{1}{m+1}\tau _{n_{s},m}f\left( x\right)
\right\vert +\left\vert \sigma _{n_{t},m}f\left( x\right) -f\left( x\right)
\right\vert .\end{aligned}$$Using Theorem 1 and that $2n\leq n_{t}\leq 2n+m$ we get$$\begin{aligned}
\left\vert \sigma _{n_{t},m}f\left( x\right) -f\left( x\right) \right\vert
&\leq &KF_{n_{t}-m,m}\left( f,x\right) _{X^{p}}\left[ 1+\ln \frac{n_{t}+1}{m+1}\right] +\left\vert f\left( x\right) -T_{n_{t}-m}\left( x\right)
\right\vert \\
&\leq &K\sum_{\nu =0}^{n}\frac{F_{n-m+\nu ,m}\left( f,x\right) _{X^{p}}}{m+\nu +1}+\left\vert f\left( x\right) -T_{n_{t}-m}\left( x\right) \right\vert
\\
&\leq &K\sum_{\nu =0}^{n}\frac{F_{n-m+\nu ,m}\left( f,x\right)
_{X^{p}}+F_{n-m+\nu ,\nu }\left( f,x\right) _{X^{p}}}{m+\nu +1}+\left\vert
f\left( x\right) -T_{n_{t}-m}\left( x\right) \right\vert \\
&\leq &K\sum_{\nu =0}^{n}\frac{F_{n-m+\nu ,m}\left( f,x\right)
_{X^{p}}+F_{n-m+\nu ,\nu }\left( f,x\right) _{X^{p}}}{m+\nu +1}+E_{n_{t}-m}\left( f,x;0\right) _{X^{p}} \\
&\leq &K\sum_{\nu =0}^{n}\frac{F_{n-m+\nu ,m}\left( f,x\right)
_{X^{p}}+F_{n-m+\nu ,\nu }\left( f,x\right) _{X^{p}}}{m+\nu +1}+E_{2n}\left(
f,x;0\right) _{X^{p}}.\end{aligned}$$
The estimate of the sum we derive from the following one$$\left\vert \frac{1}{m+1}\tau _{n_{s},m}f\left( x\right) \right\vert \leq
K\sum_{\nu =0}^{n_{s+1}-n_{s}-1}\frac{F_{n_{s}-m+\nu ,m}\left( f,x\right)
_{X^{p}}+F_{n_{s}-m+\nu ,m}\left( f,x\right) _{X^{p}}}{m+\nu +1}.$$The proof of this inequality we split in two parts. If $\ n_{s+1}-n_{s}=m+1,$ then by Lemma 5, $$\begin{aligned}
\left\vert \frac{1}{m+1}\tau _{n_{s},m}f\left( x\right) \right\vert &\leq &K\frac{1}{m+1}\sum_{k=n_{s}-m}^{n_{s}}F_{k,k-n_{s}+m}\left( f,x\right)
_{X^{p}} \\
&\leq &K\sum_{\nu =0}^{n_{s+1}-n_{s}-1}\frac{F_{n_{s}-m+\nu ,\nu }\left(
f,x\right) _{X^{p}}}{m+\nu +1}.\end{aligned}$$If $n_{s+1}-n_{s}>m+1,$ then, by Lemma 3, $$\left\vert \frac{1}{m+1}\tau _{n_{s},m}f\left( x\right) \right\vert \leq
KF_{n_{s}-m,m}\left( f,x\right) _{X^{p}}\sum_{\nu =0}^{n_{s+1}-n_{s}-1}\frac{1}{m+\nu +1}$$and since $\frac{1}{2}F_{n_{s}-m,m}\left( f,x\right) _{X^{p}}\leq
F_{n_{s+1}-m-1,m}\left( f,x\right) _{X^{p}}$ we have$$\begin{aligned}
\left\vert \frac{1}{m+1}\tau _{n_{s},m}f\left( x\right) \right\vert &\leq
&2KF_{n_{s+1}-m-1,m}\left( f,x\right) _{X^{p}}\sum_{\nu =0}^{n_{s+1}-n_{s}-1}\frac{1}{m+\nu +1} \\
&\leq &2K\sum_{\nu =0}^{n_{s+1}-n_{s}-1}\frac{F_{n_{s}-m+\nu ,m}\left(
f,x\right) _{X^{p}}}{m+\nu +1}\end{aligned}$$
Consequently,$$\begin{aligned}
&&\sum_{s=0}^{t-1}\left\vert \frac{1}{m+1}\tau _{n_{s},m}f\left( x\right)
\right\vert \\
&\leq &2K\sum_{s=0}^{t-1}\sum_{\nu =0}^{n_{s+1}-n_{s}-1}\frac{F_{n_{s}-m+\nu
,m}\left( f,x\right) _{X^{p}}+F_{n_{s}-m+\nu ,\nu }\left( f,x\right) _{X^{p}}}{m+\nu +1}\end{aligned}$$Since $n_{s+1}-n_{s}\leq 2n+m-n-1=n+m-1$ for all $s\leq t-1$ , changing the order of summation we get$$\begin{aligned}
&&\sum_{s=0}^{t-1}\left\vert \frac{1}{m+1}\tau _{n_{s},m}f\left( x\right)
\right\vert \\
&\leq &2K\sum_{\nu =0}^{n+m-1}\frac{1}{m+\nu +1}\sum_{s:n_{s+1}-n_{s}>\nu }\left[ F_{n_{s}-m+\nu ,m}\left( f,x\right) _{X^{p}}+F_{n_{s}-m+\nu ,\nu
}\left( f,x\right) _{X^{p}}\right] .\end{aligned}$$
Using the inequality$$F_{n_{s+1}-m,\nu }\left( f,x\right) _{X^{p}}\leq \frac{1}{2}F_{n_{s}-m,\nu
}\left( f,x\right) _{X^{p}}\text{ \ \ for \ }\left\{
\begin{array}{c}
\nu =0,1,2,...,n_{s+1}-n_{s}-1 \\
s=0,1,2,...,t-2\end{array}\right.$$we obtain$$\begin{aligned}
&&\sum_{s:n_{s+1}-n_{s}>\nu }\left[ F_{n_{s}-m+\nu ,m}\left( f,x\right)
_{X^{p}}+F_{n_{s}-m+\nu ,\nu }\left( f,x\right) _{X^{p}}\right] \\
&=&F_{n_{p}-m+\nu ,m}\left( f,x\right) _{X^{p}}+F_{n_{p}-m+\nu ,\nu }\left(
f,x\right) _{X^{p}} \\
&&+\sum_{s\geq p+1:n_{s+1}-n_{s}>\nu }\left[ F_{n_{s+1}-m+\nu ,m}\left(
f,x\right) _{X^{p}}+F_{n_{s+1}-m+\nu ,\nu }\left( f,x\right) _{X^{p}}\right]
\\
&\leq &F_{n_{p}-m+\nu ,m}\left( f,x\right) _{X^{p}}+F_{n_{p}-m+\nu ,\nu
}\left( f,x\right) _{X^{p}} \\
&&+\sum_{s:s\geq p+1}F_{n_{s}-m,m}\left( f,x\right) _{X^{p}}\left[
F_{n_{s}-m,m}\left( f,x\right) _{X^{p}}+F_{n_{s}-m,\nu }\left( f,x\right)
_{X^{p}}\right] \\
&\leq &F_{n_{p}-m+\nu ,m}\left( f,x\right) _{X^{p}}+F_{n_{p}-m+\nu ,\nu
}\left( f,x\right) _{X^{p}}+2\left[ F_{n_{p+1}-m,m}\left( f,x\right)
_{X^{p}}+F_{n_{p+1}-m,\nu }\left( f,x\right) _{X^{p}}\right] \\
&\leq &3\left[ F_{n_{p}-m+\nu ,m}\left( f,x\right) _{X^{p}}+F_{n_{p}-m+\nu
,\nu }\left( f,x\right) _{X^{p}}\right] \text{ ,}\end{aligned}$$where $p$ denote the smallest index $s$ having the property $n_{s+1}-n_{s}>\nu .$ Hence$$\begin{aligned}
\sum_{s=0}^{t-1}\left\vert \frac{1}{m+1}\tau _{n_{s},m}f\left( x\right)
\right\vert &\leq &K\sum_{\nu =0}^{n+m-1}\frac{F_{n-m+\nu ,m}\left(
f,x\right) _{X^{p}}+F_{n-m+\nu ,\nu }\left( f,x\right) _{X^{p}}}{m+\nu +1} \\
&\leq &K\sum_{\nu =0}^{n}\frac{F_{n-m+\nu ,m}\left( f,x\right)
_{X^{p}}+F_{n-m+\nu ,\nu }\left( f,x\right) _{X^{p}}}{m+\nu +1}.\end{aligned}$$
and our proof follows.
$\blacksquare $
Proof of Theorem 3
------------------
The proof follows by the obvious inequality$$\left\Vert E_{n}\left( f,x;\delta \right) _{C}\right\Vert _{C}\leq
E_{n}\left( f\right) _{C}\text{ .}$$$\blacksquare $
[9]{} S. Aljančič, R. Bojanic and M. Tomić, On the degree of convergence of Fejér-Lebesgue sums, L’Enseignement Mathematique, Geneve, Tome XV (1969) 21-28.
P. L. Butzer, R.J. Nessel, Fourier analysis and approximation, Basel und Stuttgart 1971.
L. Leindler, Sharpening of Stečkin’s theorem to strong approximation, Analysis Math. 16 (1990), 27-38.
W. Lenski, Pointwise best approximation and de la Vallée-Poussin means, submited.
S. B. Stečkin, On the approximation of periodic functions by de la Vallée Poussin sums, Analysis Math. 4 (1978), 61-74.
N. Tanović-Miller, On some generalizations of the Fejér-Lebesgue theorem, Boll. Un. Mat. Ital. B(6) 1 (1982), no. 3, 1217-1233.
|
---
abstract: 'A large star-to-star variation in the abundances of r-process elements, as seen in the \[Eu/Fe\] ratio for Galactic halo stars, is a prominent feature that is distinguishable from other heavy elements. It is, in part, caused by the presence of highly r-process enriched stars, classified as r-II stars (\[Eu/Fe\]$\geq+1$). In parallel, halo stars show that the ratio of a light r-process element (Y) to Eu is tightly correlated with \[Eu/Fe\], giving the lowest \[Y/Eu\] ratio that levels off at r-II stars. On the other hand, recent hydrodynamical simulations of coalescing double neutron stars (cNSNSs) have suggested that r-process sites may be separated into two classes providing different electron-fraction distributions: tidally-driven dynamical ejecta and (dynamical or postmerger) non-tidal ejecta. Here, we show that a widely spanning feature of \[Eu/Fe\] can be reproduced by models that consider the different masses of tidally-driven dynamical ejecta from both cNSNSs and coalescing black hole/neutron star binaries (cBHNSs). In addition, the observed \[Y/Eu\] trend is explained by the combined nucleosynthesis in two kinds of ejecta with varying mass asymmetry in double NS systems. Our scenario suggests that massive tidally-driven dynamical ejecta accompanied by massive non-tidal part from cNSNSs or cBHNSs could alone accommodate r-II abundances, including an actinide boost in some cases. The event rate for cNSNSs estimated from our study agrees with the latest result of $\sim1000$ (90% confidence interval of $110-3840$) Gpc$^{-3}$yr$^{-1}$ by gravitational-wave detection, and a few events per Gpc$^{3}$ per year of cBHNSs associated with r-process production are predicted to emerge.'
author:
- Takuji Tsujimoto
- Nobuya Nishimura
- Koutarou Kyutoku
title: |
r-Process enrichment in the Galactic halo characterized by nucleosynthesis variation\
in the ejecta of coalescing neutron star binaries
---
Introduction
============
Clear answers to the origin and evolution of r-process elements continue to elude us, even after the discovery of gravitational waves from coalescing double neutron star (cNSNS) GW170817 and the subsequent discovery of multi-wavelength electromagnetic counterparts that identified cNSNSs as a promising major source of r-process elements [e.g., @Coulter_17; @Smartt_17; @Pian_17; @Cowperthwaite_17; @Thielemann_17]. The stellar record of r-process abundances leaves open the question of whether cNSNSs are the sole (major) site of the r-process. This is exemplified by two arguments: that the r-process abundance features of very metal-poor stars invoke the contribution from some specific core-collapse supernovae (CCSNe), the so-called magneto-rotational SNe [e.g., @Winteler_12; @Wehmeyer_15; @Nishimura_15], and that disk stars suggest enrichment by another rare CCSNe, collapsars, as the major site of the r-process [@MacFadyen_99; @Fujimoto_08; @Siegel_19]. However, it is not clear whether these specific CCSNe exist if we consider the propagation of strong-magnetic jets in the models for magneto-rotational SNe [@Nishimura_17; @Halevi_18; @Mosta_18] and constraints from stellar abundances of halo stars on collapsars [@Macias_19].
A large scatter seen in the \[Eu/Fe\] ratio, spanning nearly three orders of magnitude among Galactic halo stars, is a feature unique to r-process elements and thus should be connected to r-process sites and their nucleosynthesis [e.g., @Cowan_19]; however, the origin of the scatter has yet to be identified. Among Eu-measured halo stars, special attention has been directed to those stars having highly enhanced Eu abundances, as high as \[Eu/Fe\]$> +1$. These have been classified as r-II stars [@Beers_05] since the discovery of the first r-II star, CS 22892-052 [@Sneden_94]. Recent first detection of r-II stars outside the Galaxy in an ultra-faint dwarf galaxy Reticulumn II [Ret II, @Ji_16] triggered one possible scenario for the origin of Galactic r-II stars; i.e., these stars are accreted from disrupted small satellite galaxies [@Roederer_18; @Brauer_19]. However, as seen in their orbital property that about 70 % of r-II stars reside within the inner regions ($<13$ kpc) of the Galactic halo [@Roederer_18], we have no clear kinematic evidence for their accretion history. Alternatively, observations suggest the possibility that r-II stars differ from other stars in the nucleosynthesis condition of the associated r-process events. These r-II stars are likely to be born in an environment that is neutron-rich enough to produce an actinide (e.g., Th and U) boost for about one third of them [@Mashonkina_14], although the conditions for the actinide production involve uncertainties, including the possibility that actinides could be populated at the not that low electron fraction ($Y_e$) [e.g., @Lippuner_17].
There is another implication for the connection between the variation in \[Eu/Fe\] and the nucleosynthesis of r-process events. The abundance ratios of light r-process elements (Sr, Y, and Zr)[^1] to Eu are correlated with \[Eu/Fe\] in the sense that \[light r-process/Eu\] is larger for smaller \[Eu/Fe\] [@Montes_07; @Tsujimoto_14a]. We need to answer why r-II stars show the lowest \[light r-process/Eu\] ratio among halo stars.
An astrophysical site that is different for light r-process elements than for heavy ones including Eu, has been proposed as the mechanism behind the variation in \[light r-process/Eu\] for individual stars [@Montes_07; @Francois_07]. Possible candidates include regular CCSNe producing Fe together [@Qian_07] through the neutrino-driven wind from the proto-NS [@Arcones_14], electron-capture SNe [@Kitaura_06; @Wanajo_11], and magneto-rotational SNe [@Nishimura_17]. However, our understanding to date should be reexamined in response to recent numerical simulations for r-process nucleosynthesis in cNSNSs. The latest discovery of a signature of synthesized light r-process element (Sr) in the afterglow of GW170817 strengthens this necessity [@Watson_19].
Currently, binary NSs are regarded as being capable of producing various kinds of ejecta during the coalescence. Tidal interaction could allow the NS material to be ejected without experiencing significant shock heating nor neutrino irradiation [e.g., @Freiburghaus_99; @Sekiguchi_16; @Vincent_19], although recent simulations have reported a non-negligible influence of neutrinos on such ejecta [@Radice_18]. These tidal ejecta may keep low $Y_e$ and contribute predominantly to the production of heavy r-process elements such as Eu. Dynamical interaction also produces the shock-heated ejecta, whose $Y_e$ is suggested by some studies to be increased to the extent that Eu is no longer produced efficiently and instead light r-process elements such as Y are produced [e.g., @Wanajo_14]. Following these dynamical ejecta, the merger remnant drives postmerger mass ejection via various possible mechanisms [e.g., @Metzger_14; @Just_15; @Siegel_18; @Fujibayashi_18]. In particular, @Fujibayashi_18 find that the postmerger wind from the torus surrounding a massive NS, which is likely to be formed for typical cNSNS, is primarily characterized by moderate $Y_e$ values and dominantly produces light r-process elements. These updated theoretical inputs of cNSNS may have high potential for altering the interpretation of \[light r-process/Eu\] variation.
A close connection between the star-to-star variation in r-process abundances and cNSNSs is also implied from the wide range of \[Eu/Fe\] ratios. The mass of dynamical ejecta is predicted to vary as widely as two orders of magnitude [$\sim10^{-4}-0.01$ : @Hotokezaka_13 $\sim2\times10^{-4}-0.02$ : Bauswein et al. 2013], where [*the tidally-driven component*]{} increases as the mass asymmetry increases. Further, its mass range widens if we consider the contribution from coalescing black hole/neutron star binaries (cBHNSs); i.e., the mass of dynamical ejecta from cBHNSs can become as massive as $\sim 0.1$ dominated completely by the tidal component [@Kyutoku_15]. As a result, the ejected Eu mass can differ by approximately three orders of magnitude according to the difference in the mass of its production site. Then, it turns out that the degree of variation in an Eu mass broadly coincides with that for the observed \[Eu/Fe\] ratios. Indeed, in the Draco dwarf galaxy, we identified two r-process events with enriched gases at levels that differ by more than one order of magnitude [@Tsujimoto_17a]. These arguments suggest that enrichment driven by both cNSNSs and cBHNSs makes a fundamental feature of Galactic r-process abundance, including the early onset of the r-process at low metallicities [@Wehmeyer_19].
**Nucleosynthesis in tidal and non-tidal ejecta**
=================================================
In this study, we boldly classify the ejecta components into the tidal and non-tidal ejecta, which are responsible for the synthesis of Eu and Y, respectively. Such a classification is possible since the production sites of Eu and Y inside the ejecta can be seen as discrete in terms of the $Y_{\rm e}$ value. To investigate the dependence of nucleosynthesis yields on $Y_{\rm e}$, we perform a set of r-process calculations, using a nuclear reaction network code [@Nishimura_16; @Nishimura_17] and the simplified models of merger ejecta for cNSNSs and cBHNSs. We adopt the adiabatic free-expansion evolution [@Freiburghaus_99; @Farouqi_10; @Wanajo_18], of which the abundance evolution is determined by the initial $Y_{\rm e}$ value, entropy, and expansion velocity.
The results of nucleosynthesis calculations are shown in Figure 1, in which the mass fractions of Eu and Y are plotted as a function of $Y_{\rm e}$. The mass fraction has a band between the minimum and maximum values for the varieties of entropy: 10–35 $k_{\rm B}~{\rm baryon}^{-1}$ and expansion velocity: $0.05$–$0.3$ of the speed of light. These adopted values cover the reasonable range for merger conditions/ejecta suggested by [@Wanajo_18]. We see a clear separation of Eu and Y production depending on the value of $Y_e$. The deviation of the mass fractions of Eu and Y lies within a factor of $\sim 2$ for $Y_{\rm e} < 0.3$. It suggests that the final abundance of Eu, which is a key element of the tidal ejecta, is primarily determined by $Y_{\rm e}$ while the Y abundance shows a bit more complexity of a dependence on other factors. In addition, the uncertainty of the nuclear physics input such as the mass model and fission treatment [@Mendoza_15; @Goriely_15; @Eichler_15; @Cote_18; @Vassh_19] affects calculated abundances of products such as Eu since dynamical ejecta conditions can synthesize nuclei well outside the reach of current experimental data.
![The mass fractions of Eu and Y by the r-process as a function of $Y_{\rm e}$. The nucleosynthesis calculations [@Nishimura_17] are performed with the parametric expansion model of merger ejecta [@Wanajo_18]. The mass fraction has a band between the minimum and maximum values for the varieties of entropy: 10–35 $k_{\rm B}~{\rm baryon}^{-1}$ and expansion velocity: $0.05$–$0.3$ of the speed of light. $Y_{\rm e} \sim 0.22$ is the boundary of the tidal and non-tidal components assumed in this study.](f1){width="\columnwidth"}
We also need to keep in mind that the nucleosynthesis yields in both the dynamical and postmerger ejecta are inevitably dependent on many uncertain physics inputs in hydrodynamical simulations. Examples include neutrino transport scheme [@Caballero_12; @Foucart_16; @Radice_18], neutrino oscillation [@Malkus_16; @Frensel_17; @Tian_17], and neutron star equations of state [@Sekiguchi_15; @Bovard_17; @Radice_18; @Vincent_19]. In particular, the r-process yields associated with the postmerger ejecta are critically important, because they could dominate the mass of ejecta in many cases, given the wide range of accretion disk mass ejection deduced [@Cote_18]. It is possible that the postmerger ejecta also efficiently produce Eu due to low $Y_e$ if the BH is formed promptly formed after merger [@Just_15; @Siegel_18].
Variations in \[E[u]{}/F[e]{}\] and \[Y/E[u]{}\]
================================================
We discuss more comprehensively the origin of variations in both \[Eu/Fe\] and \[light r-process/Eu\] and the correlation between the two. It was first found by @Montes_07 that there exists a downward trend of \[light r-process/Eu\] with increasing \[Eu/Fe\] while \[heavy r-process/Eu\] stays constant with respect to \[Eu/Fe\] (here light r-process elements are Sr, Y and Zr and heavy ones include Ba, La, and Nd). This feature may suggest that the major source of light r-process elements is different from that of Eu [see also @Aoki_05; @Francois_07], and is proposed to be explained by the presence of a weak r-process producing source such as the light element primary process (LEPP) as a consequence of the correlation of light r-process elements with Fe instead of Eu [@Travaglio_04; @Montes_07].
![Observed correlation of \[Y/Eu\] with \[Eu/Fe\] for Galactic halo stars that reflect the r-process Y/Eu abundance ratio. The observed data are selected with \[Fe/H\]$<-2$ and \[Ba/Eu\]$<$0 from a SAGA database [@Suda_08]. For r-II stars, the data are complemented by a JINA database [@Abohalima_18]. In addition, two r-II stars recently discovered [@Sakari_18; @Holmbeck_18] are also added. One extremely Eu-enhanced star (\[Eu/Fe\]=+2.74 and \[\[Y/Eu\]=$-0.84$; Allen et al. 2012) is not shown here. Black dashed line indicates the solar r-process ratio [@Simmerer_04; @Bisterzo_14]. Brief explanations are attached in the upper part (see the text). Here the assumed higher mass ejecta of low $Y_{\rm e}$ tidal ejecta for cBHNS scenarios is motivated by numerical simulation results [@Kyutoku_15]. Red dashed line indicates a mean abundance track.](f2){width="\columnwidth"}
With an update of the observational data, we revisit this matter using Y for light r-process elements, since its abundance determination is generally more reliable than than that of Sr and Zr. To avoid the stars whose Y abundances include s-process contributions, we select halo stars that satisfy \[Fe/H\]$<-2$ and \[Ba/Eu\]$<0$. The \[Y/Eu\] vs. \[Eu/Fe\] diagram thus obtained is shown in Figure 2. The overall correlation between \[Y/Eu\] and \[Eu/Fe\] is summarized as (i) a decreasing \[Y/Eu\] with increasing \[Eu/Fe\] for \[Eu/Fe\]$+1$ and (ii) a broadly constant \[Y/Eu\] for \[Eu/Fe\]$+1$. Thus, r-II stars give the lowest \[Y/Eu\] ($\approx-1.1$).
In this section, we argue that this observed correlation may be naturally explained by assuming that variations in both \[Eu/Fe\] and \[Y/Eu\] for metal-poor halo stars reflect directly those in nucleosynthesis from the merger events. Specifically, the correlation between \[Y/Eu\] and \[Eu/Fe\] is regarded as the result of variations in the properties of cNSNSs or cBHNSs, for which Eu and Y are assumed to be synthesized separately in the tidal and non-tidal ejecta of individual merger events, respectively, as discussed in §2. Here, we assume the solar r-process pattern including the $A\sim 130$ peak (i.e., $Z\geq$52) with no production for $Z<$52 for the matter within the tidal ejecta from both cNSNSs and cBHNSs. On the other hand, for the non-tidal ejecta, we assume the nucleosynthesis pattern following the recent numerical results for the postmerger ejecta which show significantly low production of lanthanide elements and give the mass fraction of $1.64\times10^{-2}$ for Y [@Fujibayashi_18].
We first propose that the degree of Eu enhancement indicated by \[Eu/Fe\] for individual stars ($-0.7$+2) could be basically determined by the different levels of Eu enrichment of gas via associated merger events (cNSNSs or cBHNSs) with widely varying masses ($M_{\rm tid}$) of tidally-driven dynamical ejecta ($10^{-4}$$M_{\rm tid}$$0.1$). Considering the \[Eu/Fe\] distribution of halo stars, we identify two components: a Gaussian distribution in the range of $-0.7$+1.3 with a peak around \[Eu/Fe\]$\approx+0.3$ and an extending Eu-rich tail up to \[Eu/Fe\]$\approx+2$. Then, considering the predicted mass range of dynamical ejecta, cNSNS and cBHNS are likely to be responsible for producing the former and latter distributions, respectively. Approximate mass values of the mass of tidally-driven dynamical ejecta are shown in the upper part of the figure, implying that the average value $\langle M_{\rm tid}\rangle$ for cNSNSs should be a few times $10^{-3}$ . If cBHNS should produce more Eu than cNSNS due to larger tidal mass ejection, the Y/Eu plateau could be connected to such events. Considering very low production of light r-process elements in the tidally dominated dynamical ejecta of cBHNSs owing to a low $Y_e$ (0.1) [@Roberts_17], we argue that Y in r-II stars originates from the postmerger ejecta.
The Y/Eu plateau implies that the production ratio of Y to Eu is quite similar in individual cBHNS events. Since each element owes its origin to non-tidal postmerger ejecta and tidally-driven dynamical ejecta, respectively, the masses of the two type of ejecta are suggested to scale linearly with each other. This linearity may also hold even if the postmerger ejecta of cBHNSs produce Eu due to a smaller $Y_e$ resulting from lower neutrino irradiation [@Fernandez_17], though there are complexities in the influence of the neutrino treatment [e.g., @Caballero_12; @Wu_17]. The required condition of a similar mass ratio of light r-process elements to Eu among the postmerger ejecta of cBHNSs needs to be investigated [see also @Kyutoku_15].
On the other hand, the trend of an increasing \[Y/Eu\] with decreasing \[Eu/Fe\] at the low \[Eu/Fe\] regime is likely to reflect the diversity of nucleosynthesis in cNSNSs with various mass asymmetry in our scheme. The tidal interaction tends to be efficient for asymmetric binaries, and $M_\mathrm{tid}$ becomes large. At the same time, shock interactions are not very efficient for dynamical mass ejection in asymmetric binaries. Thus, the average values of $Y_e$ become lower for more asymmetric binaries [e.g., @Sekiguchi_16]. These combined effects could explain the observed Y/Eu trend. According to this view, cNSNSs with massive ($\gtrsim 0.01 M_\odot$) tidally-driven dynamical ejecta produce a very small amount of Y and share the Y/Eu plateau with cBHNSs. However, it is necessary to comprehensively investigate the dependence of nucleosynthesis in cNSNSs on the mass asymmetry and other binary parameters [@Kiuchi_19], particularly in light of recently discovered highly asymmetric double NSs [@Ferdman_18]. In addition to the above discussion, a contrast in \[Y/Eu\] may, in part, be attributed to the dependence of Y nucleosynthesis in the postmerger ejecta on their different masses. As one possibility, we anticipate that massive postmerger ejecta associated with massive dynamical ejecta produce a smaller amount of Y as a result of low $Y_e$, which is realized by a lower neutrino flux in accordance with a short timescale for collapsing into BHs. We also note that there still exists a downward trend for \[Y/Eu\] with increasing \[Eu/H\], which might support that Y production is connected to an Eu production site instead of being correlated with Fe production as predicted by LEPP [@Travaglio_04; @Montes_07].
In our scheme, r-II stars are interpreted to be born from gas enriched by tidally-driven dynamical ejecta having very low $Y_e$, which result from either highly asymmetric double NS systems or BH-NS binaries. In such highly neutron-rich ejecta, the $Y_e$ distribution inside each ejecta is likely skewed to be so low that the mass fraction with very low $Y_e$ (e.g., $<0.18$) is, in some of them, high enough (e.g., $>30$%) to induce a boost of actinide production [@Holmbeck_19]. This is consistent with the observed fact that among Th-measured halo stars, r-II stars include actinide-boost stars with higher fraction ($\sim$45%) than that ($\sim$25%) of less Eu-enhanced stars (r-I:+0.3$\leq$\[Eu/Fe\]$<$+1) [@Holmbeck_18]. These arguments point to the r-II star’s origin which could demand a specific nucleosynthesis condition rather than the properties such as a mass-scale of protogalaxies (clouds) where these stars originally resided.
Chemical evolution of r-process in the Galactic halo
====================================================
Chemical evolution study to explain an extensive scatter in \[Eu/Fe\] among Galactic halo stars utilizing cNSNSs has been implemented by many authors [e.g., @Argast_04; @Voort_15; @Shen_15; @Hirai_15; @Hirai_17]. The driver of a large scatter in these simulations is local inhomogeneities which can be produced when the limited amounts of interstellar matter are poluted by and mixed with the ejecta of each merger event occurring with a low frequency: inhomogeneous mixing could produce larger \[Eu/Fe\] in strongly polluted areas and smaller values in less polluted ones. On the other hand, our scheme predicts that the variation in \[Eu/Fe\] among halo stars is primarily caused by the different masses of tidally-driven dynamical ejecta from cNSNSs and cBHNSs. To validate this hypothesis, we model the chemical evolution of the Galactic halo and calculate the Eu/Fe evolution.
We consider protogalactic fragments with a baryonic mass of $10^6$, which is the minimum baryonic mass at the epoch of galaxy formation [@Tegmark_97]. This fragment mass is also implied from the threshold of an initial mass of protogalaxies, as implied from Ret II. The observed \[Eu/H\] $\approx-1$ of r-II stars in this galaxy implies that an Eu mass ejected from a cNSNS with $M_{\rm tid}\approx0.01$is mixed with gas having a mass of $\sim 10^6$[see also @Ji_15; @Ji_16]. In each fragment, separate chemical evolutions proceed as building blocks of the halo. Thus, metal-poor stars in the Galaxy are predicted to be an assembly of stars originating from individual fragments, according to the hierarchical galaxy formation scheme. This scenario does not distinguish whether each fragment corresponds to an accreted unit of the Galaxy or one of many pieces of an accreted massive dwarf galaxy, such as Gaia-Enceladus [@Helmi_18]. In fragments, the ejecta are considered to propagate inside the whole volume and thus are well-mixed. This is justified by both a broad uniformity of Eu abundances in Ret II [@Ji_16] and the abundance of $^{244}$Pu in the solar vicinity, which suggests that the ejecta of cNSNSs are mixed with local gas as massive as $10^6$[@Tsujimoto_17b].
![Predicted \[Eu/Fe\]-\[Fe/H\] correlations for the Galactic halo (red lines) compared with the observed data (crosses). For reference, the data of mainly disk stars for \[Fe/H\]$>-1.3$ are shown by small pluses. Predicted feature is the assembly of 100 randomly selected cases for the chemical evolution in protogalactic fragments with a mass of $10^6$. Among them, two representative evolutionary paths are highlighted by green and blue lines (see the text).](f3){width="\columnwidth"}
In each fragment, star formation is assumed to continue for a duration of 300 Myr at a low star formation rate so as to give $\langle$\[Fe/H\]$\rangle \approx -1.6$ as observed for halo stars, which results in a conversion of about 17% of gas to stars. Owing to the short duration considered here, the contribution of Fe from type Ia SNe is not included. For Fe yields from CCSNe, we assume the nucleosynthesis yields tabulated in @Kobayashi_06. We assume that an Eu ejection occurs at a rate of one per $\sim$200 CCSNe (see §4.1), which leads to $\sim$6 events in total. For each event, the ejected mass of Eu is given by randomly assigning the mass of tidally-driven dynamical ejecta from $10^{-4}$to 0.07 , with a Gaussian probability distribution having a peak at $3\times10^{-3}$for $M_{\rm tid}\leq 0.01$. For $M_{\rm tid}\geq 0.01$, we assign a 10% probability in total with uniform mass distribution. Then, the Eu mass ejected from each merger event is calculated from $M_{\rm tid}$ composed of r-process elements with $Z\geq52$ following the solar r-process pattern. For the delay time of the ejection of r-process elements, we adopt a short timescale of $10-30$ Myr to accelerate the early r-process enrichment in the halo [@Tsujimoto_14b].
Since a sequence of r-process events occurs with randomly chosen masses of tidally-driven dynamical ejecta, individual evolutionary paths of Eu/Fe vary among protogalactic fragments. Figure 3 shows a mixture of such paths in 100 fragments. They include two examples that largely differ in Eu production by tidally-driven dynamical ejecta (distinguished by different colors): an initial Eu-boost making r-II stars, followed by a decreasing trend of \[Eu/Fe\] (blue), and a gradual increase in \[Eu/Fe\] via a sequence of mild Eu-enrichment by each event, resulting in no r-II stars (green). This figure excludes the initial generations of stars with no Eu content. It shows broad agreement between our predicted feature and the observation. On the other hand, if we adopt the delay time distribution for the merger objects spanning over gigayears implied from that for short gamma-ray bursts [@Fong_17], there is a poorer agreement with the observation mainly due to a smaller predicted scatter in \[Eu/Fe\] by a reduced number of r-process events within a short duration (300 Myr) of star formation. Note that Eu abundance in disk stars favors a primary r-process production with a short time delay over long-delay merger events [e.g., @Cote_19; @Siegel_19; @Lin_19]. Accordingly, we claim that the large scatter in the abundance ratio that is unique to r-process elements in the Galactic halo is an end result of r-process enrichment driven by merger events under halo formation via mass assembly of small building blocks. In these merger events, widely spanning masses of tidally-driven dynamical ejecta are realized.
Merger Event Rate
=================
Based on our proposed scheme, we can discuss the event rates of cNSNSs and cBHNSs associated with r-process production. On the other hand, Advanced LIGO/Virgo Observing Runs 1 and 2 by the ongoing gravitational-wave search project give the reliable rates of cNSNSs and cBHNSs (The LIGO Scientific Collaboration et al. 2018). Thus, the comparison between the two results will help to verify our argument presented here. In general, the assessment of merger event rates from the analysis of r-process abundance assuming the connection of r-process production with the merger events is highly uncertain unless a mean ejecta mass is narrowed down [@Rosswog_17; @Hotokezaka_18]. Our scheme that assigns each (tidal) ejecta mass to each \[Eu/Fe\] ratio of Galactic halo stars could give a one possible way to resolve this problem through a statistical treatment.
NS-NS
-----
First, we estimate the cNSNS event rate in the local volume (LV) of the Universe, $R_{\rm LV}$. This can be directly compared with the rate deduced from gravitational-wave events in the local Universe. $R_{\rm LV}$ can be converted from the present rate for the Galaxy, $R_{\rm G}$, which is obtained through an analysis of stellar r-process abundances combined with the recent star formation activity in the Galaxy [@Tsujimoto_14b; @Tsujimoto_17b]. Note that $R_{\rm G}$ would be lower than the mean rate for the Galaxy [@Rosswog_17; @Hotokezaka_18] owing to the currently low star formation rate [e.g., @Mor_19], and this tendency of star formation seen in the Galaxy is shared with M31 [@Bernard_15]. In this study, we utilize the most reliably measured stellar abundance: the solar abundance.
The key point is that the relative solar abundance ratio of Eu to Mg ($5.2\times 10^{-7}$) is equivalent to the relative nucleosynthesis yield between the two elements. Here the Eu yield is defined per CCSN basis, that is, as the combination of the Eu mass per cNSNS event and a reciprocal number of CCSNe that is necessary to give birth to one cNSNS event. For the Mg yield from a single CCSN, we adopt a mass of 0.1 . This value is deduced from two arguments: (i) nucleosynthesis calculations for CCSNe give 0.1$-$0.125 as an IMF-weighted average Mg mass for the progenitor’s mass range of 13$-$50 [@Tominaga_07] and (ii) a relative yield with a mean Fe mass of $\sim$0.07 obtained from a light curve analysis of CCSNe [@Hamuy_03] leads to an observed plateau of \[Mg/Fe\]$\approx$+0.4 among halo stars and thus the value of the most metal-poor disk stars [e.g., @Bensby_14]. Accordingly, we obtain the Eu yield of $5.2\times 10^{-8}$.
Then, this together with an average Eu mass per cNSNS event leads to the cNSNS rate. The Eu mass fraction of $6\times10^{-3}$ in the tidally-driven dynamical ejecta, which is obtained from the assumption of the solar r-process pattern for the elements with $Z\geq$52, provides the cNSNS rate per CCSNe of one per 1150$(\langle M_{\rm tid}\rangle$/0.01) CCSNe. This further leads to the Galactic cNSNS rate $R_{\rm G}$, using the present-day Galactic CCSN rate of 2.3 SNe per century [@Li_11]: $$R_{\rm G} \approx 20 \left(\frac{\langle M_{\rm tid}\rangle}{0.01 M_\odot}\right)^{-1} \ \ {\rm Myr}^{-1}.$$
Finally, this rate can be converted to the rate for the local volume of the Universe via the density of a Milky Way-equivalent galaxy of $1.16\times 10^{-2}$ Mpc$^{-3}$ estimated from the blue luminoisty [@Abadie_10]:
$$R_{\rm LV} \approx 230 \left(\frac{\langle M_{\rm tid}\rangle}{0.01 M_\odot}\right)^{-1}\ \ {\rm Gpc}^{-3}{\rm yr}^{-1}.$$
This $R_{\rm LV}$ range is within the rate at the 90% confidence intervals of $110-3840$ Gpc$^{-3}$ yr$^{-1}$, with the most probable value around $\sim 1000$ Gpc$^{-3}$ yr$^{-1}$ by gravitational-wave detection [@LIGO_18]. Therefore, $R_{\rm LV}$ deduced from the solar abundance is in agreement with that from the gravitational-wave event rate if the average mass of tidally-driven dynamical ejecta is assumed to be a few times $10^{-3}$ .
BH-NS producing r-process elements
----------------------------------
The cBHNSs events associated with r-process production are considered rare among all cBHNSs [@Zappa_19] whose event rate is constrained to be $<610$ Gpc$^{-3}$ yr$^{-1}$ [@LIGO_18]. We can predict the approximate rate based on our theoretical scheme that the group of stars enriched by the r-process to the highest level (\[Eu/Fe\]$+1.4$) can be connected to a cBHNS origin, while other stars are associated with a cNSNS origin. The fraction of r-II stars compared to all metal-poor halo stars is estimated to be about $2-4$%, although this estimate likely overrepresents due to incomplete sampling [@Brauer_19]. Then, considering that the fraction of r-II stars with \[Eu/Fe\]$\geq +1.4$ with respect to all r-II stars is about $25$% [10/38: @Roederer_18], the event ratio of cBHNSs producing r-process elements to cNSNSs is implied to be about $0.5-1$% at maximum. Accordingly, we anticipate that the future detection of the gravitational-wave events originating from cBHNSs could be inclined towards no association with electromagnetic counterparts. If we adopt $1000$ Gpc$^{-3}$ yr$^{-1}$ as a fiducial value of the cNSNS event rate, the event rate of cBHNS with the r-process, a few per Gpc$^{3}$ per year, is deduced as the most likely case.
Summary
=======
We show that the large scatter in the \[Eu/Fe\] ratio among Galactic halo stars can be the outcome of varying Eu production, spanning three orders of magnitude in the masses of tidally-driven dynamical ejecta from both cNSNSs and cBHNSs. The origin of r-II stars is predicted to have a close connection to a highly neutron-rich environment, which could be realized in the massive ($\sim0.01-0.02$ ) dynamical ejecta from cNSNSs with highly asymmetric double NS systems or those from cBHNSs. On the other hand, enrichment by cNSNSs with less massive ejecta results in halo stars having an observed variation in the \[light r-process/Eu\] ratio with less enhancement in Eu. We further predict that the typical mass of tidally-driven dynamical ejecta for cNSNSs should be a few times $10^{-3}$ , which is nicely supported by the cNSNS rate deduced from the ongoing gravitational-wave observations.
Our scenario is based on the idea that the correlation between Y/Eu and Eu/Fe emerges from different neutron richness of the ejecta caused by different properties of coalescing binaries. Specifically, the amount of the neutron-rich tidal ejecta and the Eu yield are presumed to increase as the binary becomes asymmetric and tidal deformation of the lighter component is enhanced. Although this trend is consistent with the results of previous studies [e.g., @Sekiguchi_16], whether the observed correlation is reproduced quantitatively should be investigated by future hydrodynamical simulations and nucleosynthetic calculations with systematically varying the binary parameters such as the mass ratio. These inputs would help to promote a thorough understanding of Galactic elemental features for all r-process elements, including light r-process nuclei, lanthanides, and actinides.
The authors thank an anonymous referee for his or her valuable comments. This work was supported by JSPS KAKENHI grant Nos. 16H06342, 17H01131, 18H01258, 18H04593, 18H04595, JP18H05236, and 19K14720, and MEXT Japan (“Priority Issue on Post-K computer: Elucidation of the Fundamental Laws and Evolution of the Universe”). NN thanks S. Fujibayashi for fruitful discussions on the post-merger ejecta.
Abadie, J., Abbott, B. P., Abbott, R., et al. 2010, CQGra, 27, 173001 Abohalima, A., & Frebel, A. 2018, , 238, 36 Allen, D. M., Ryan, S. G., Rossi, S., Beers, T. C., & Tsangarides, S. A. 2012, A&A, 548, A34 Aoki, W., Honda, S., Beers, T. C., et al. 2005, ApJ, 632, 611 Argast, D., Samland, M., Thielemann, F.-K., & Qian, Y.-Z. 2004, A&A, 416, 997 Arcones, A., & Bliss, J. 2014, J. Phys. G: Nucl. Phys., 41, 044005 Bauswein, A., Goriely, S., & Janka, H.-T. 2013, , 773, 78 Beers, T. C., & Christlieb, N. 2005, ARA&A, 43, 531 Bensby, T., Feltzing, S., & Oey, M. S. 2014, A&A, 562, A71 Bernard, E. J., Ferguson, A. M. N., Richardson, J. C., et al. 2015, MNRAS, 446, 2789 Bisterzo, S., Travaglio, C., Gallino, R., Wiescher, M., & Käppeler, F. 2014, , 787, 10 Bovard, L. D., Martin, F., Guercilena, F., Arcones, A., Rezzolla, L., & Korobkin, O. 2017, PhRvD, 96, 124005 Brauer, K., Ji, A. P., Frebel, A., Dooley, G. A., Gómez, F. A., & O’Shea, B. W. 2019, , 871, 247 Caballero, O. L., McLaughlin, G. C., & Surman, R. 2012, , 745, 170 Côté, B., Fryer, C. L., Belczynski, K., et al. 2018, , 855, 99 Côté, B., Eichler, M., Arcones, A., et al. 2019, , 875, 106 Coulter, D. A., Foley, R. J., Kilpatrick, C. D., et a. 2017, Science, 358, 1556 Cowan, J. J., Sneden, C., Lawler, J. E., et al. 2019, arXiv:1901.01410 Cowperthwaite, P. S., Berger, E., Villar, V. A., et al. 2017, , 848, L17 Eichler, M., Arcones, A., Kelic, A., et al. 2015, , 808, 30 Farouqi, K., Kratz, K.-L., Pfeiffer, B., et al. 2010, , 712, 1359 Ferdman, R. D., & PALFA Collaboration 2018, in IAU Symposium, Vol. 337, Pulsar Astrophysics the Next Fifty Years, ed. P. Weltevrede, B. B. P. Perera, L. L. Preston, & S. Sanidas, 146 Fernández, R., Foucart, F., Kasen, D., Lippuner, J., Desai, D., & Roberts, L. F. 2017, CQGra, 34, 154001 Fong, W., Berger, E., & Blanchard, P. K., et al. 2017, , 848, L23 Foucart, F., O’Connor, E., Roberts, L., Kidder, L. E., Pfeiffer, H. P., & Scheel, M. A. 2016, PhRvD, 94, 123016 François, P., Depagne, E., Hill, V., et al. 2007, A&A, 476, 935 Freiburghaus, C., Rosswog, S., & Thielemann, F.-K. 1999, , 525, L121 Frensel, M., Wu, M.-R., Volpe, C., & Perego, A. 2017, PhRvD, 95, 023011 Fujibayashi, S., Kiuchi, K., Nishimura, N., Sekiguchi, Y., & Shibata, M. 2018, , 860, 64 Fujimoto, S., Nishimura, N., & Hashimoto, M. 2008, , 680, 1350 Goriely, S. 2015, EPJA, 51, 22 Halevi, G., & Mösta, P. 2018, MNRAS, 477, 2366 Hamuy, M. 2003, , 582, 905 Helmi, A., Babusiaux, C., Koppelman, H. H., Massari, D., Veljanoski, J., & Brown, A. G. A. 2018, Nature, 563, 85 Hirai, Y., Ishimaru, Y., Saitoh, T. R., Fujii, M. S., Hidaka, J., & Kajino, T. 2015, , 814, 41 Hirai, Y., Ishimaru, Y., Saitoh, T. R., Fujii, M. S., Hidaka, J., & Kajino, T. 2015, MNRAS, 466, 2474 Holmbeck, E. M., Beers, T. C., Roederer, I. U., et al. 2018, , 859, L24 Holmbeck, E. M., Frebel, A., McLaughlin, G. C., Mumpower, M. R., Sprouse, T. M., & Surman, R. 2019, , 881, 5 Hotokezaka, K., Beniamini, P., & Piran, T. 2018, IJMPD, 27, 1842005 Hotokezaka, K., Kiuchi, K., Kyutoku, K., Okawa, H., Sekiguchi, Y., Shibata, M., & Taniguchi, K. 2013, PhRvD, 88, 044026 Ji, A. P., Frebel, A., & Bromm, V. 2015, MNRAS, 454, 659 Ji, A. P., Frebel, A., Chiti, A., & Simon, J. D. 2016, , 531, 610 Just, O., Bauswein, A., Adrevol Pulpillo, R., Goriely, S., & Janka, H.-T. 2015, MNRAS, 448, 541 Kitaura, F. S., Janka, H.-Th, & Hillebrandt, W. 2006, A&A, 450, 345 Kiuchi, K., Kyutoku, K., Shibata, M., & Tanigichi, K. 2019, , 876, L31 Kobayashi, C., Umeda, H., Nomoto, K., Tominaga, N., & Ohkubo, T. 2006, , 653, 1145 Kyutoku, K., Ioka, K., Okawa, H., Shibata, M., & Taniguchi, K. 2015, PhRvD, 92, 044028 Li, W., Chornock, R., Leaman, J., et al. 2011, MNRAS, 412, 1473 Lin, J., Asplund, M., Ting, Y.-S., et al. 2019, arXiv:1911.05221 Lippuner, J., Fernández, R., Roberts, L. F., et al. 2017, MNRAS, 472, 904 Macias, P., & Ramirez-Ruiz, E. 2019, , 877, L24 MacFadyen, A. I., & Woosley, S. E. 1999, , 524, 262 Malkus, A., McLaughlin, G. C., & Surman, R. 2016, PhRvD, 93, 045021 Mashonkina, L., Christlieb, N., & Eriksson, K. 2014, A&A, 569, A43 Mendoza-Temis, J. J., Wu, M.-R., Langanke, K., Martínez-Pinedo, G., Bauswein, A., & Janka, H.-T. 2015, PhRvC, 92, 055805 Metzger, B. D., & Fernández, R. 2014, MNRAS, 441, 3444 Montes, F., Beers, T. C., Cowan, J. J., et al. 2007, ApJ, 671, 1685 Mor, R., Robin, A. C., Figueras, F., Roca-Fábrega, S., & Luri, X. 2019, A&A, 624, L1 Mösta, P., Roberts, L. F., Halevi, G., Ott, C. D., Lippuner, J., Haas, R., & Schnetter, E. 2018, , 864, 171 Pian, E., D’Avanzo, P., Benetti, S., et al. 2017, Nature, 551, 67 Nishimura, N., Takiwaki, T., & Thielemann, F.-K. 2015, , 810, 109 , N., [Podoly[á]{}k]{}, Z., [Fang]{}, D.-L., & [Suzuki]{}, T. 2016, PhLB, 756, 273 Nishimura, N., Sawai, H., Takiwaki, T., Yamada, S., & Thielemann, F.-K. 2017, , 836, L21 Qian, Y.-Z., & Wasserburg, G. J. 2007, Phys. Rep., 444, 237 Radice, D., Perego, A., Hotokezaka, K., Fromm, S. A., Bernuzzi, S., & Roberts, L. F. 2018, , 869, 130 Roberts, L. F., Lippuner, J., Duez, M. D., et al. 2017, MNRAS, 464, 3907 Roederer, I. U., Hattori, K., & Valluri, M. 2018, , 156, 179 Rosswog, S., Feindt, U., Korobkin, O. et al. 2017, CQGra, 34, 104001 Sakari, C. M., Placco, V. M., Hansen, T., et al. 2018, , 854, L20 Sekiguchi, Y., Kiuchi, K., Kyuutoku, K., & Shibata, M. 2015, PhRvD, 91, 064059 Sekiguchi, Y., Kiuchi, K., Kyuutoku, K., Shibata, M., & Taniguchi, K. 2016, PhRvD, 93, 124046 Shen, S., Cooke, R. J., Ramirez-Ruiz, E., Madau, P., Mayer, L., & Guedes, J. 2015, , 807, 115 Siegel, D. M., & Metzger, B. D. , 858, 52 Siegel, D. M., Barnes, J., & Metzger, B. D. 2019, Nature, 569, 241 Simmerer, J., Sneden, C., Cowan, J., Collier, J., Woolf, V. M., & Lawler, J. E. 2004, , 617, 1091 Smartt, S. J., Chen, T.-W., Jerkstrand, A., et al. 2017, Nature, 551, 75 Sneden, C., Preston, G. W., McWilliam, A., & Searle, L. 1994, , 431, L27 Suda, T., Katsuya, Y., Yamada, S., et al. 2008, PASJ, 60, 1159 Tegmark, M., Silk, J., Rees, M. J., Blanchard, A., Abel, T., & Palla, F. 1997, , 474, 1 The LIGO Scientific Collaboration, the Virgo Collaboration, Abbott, B. P., et al. 2018, arXiv:1811.12907 Thielemann, F.-K., Eichler, M., Panov, I. V., & Wehmeyer, B. 2017, ARNPS, 67, 253 Tian, J. Y., Patwardhan, A. V., & Fuller, G. M. 2017, PhRvD, 96, 043001 Tominaga, N., Umeda, H., & Nomoto, K. 2007, , 660, 516 Travaglio, C., Gallino, R., Arnone, E., Cowan, J. J., Jordan, F., & Sneden, C. 2004, , 601, 864 Tsujimoto, T., & Shigeyama, T. 2014a, , 795, L18 Tsujimoto, T., & Shigeyama, T. 2014b, A&A, 565, L5 Tsujimoto, T., Matsuno, T., Aoki, W., Ishigaki, M. N., & Shigeyama, T. 2017a, , 850, L12 Tsujimoto, T., Yokoyama, T., & Bekki, K. 2017b, , 835, L3 van de Voort, F., Quataert, E., Hopkins, P. F., Kereš, D., Faucher-Giguére, C.-A. 2015, MNRAS, 447, 140 Vassh, N., Vogt, R., Surman, R., et al. 2019, JPhG, 46, 065202 Vincent, T., Foucart, F., Duez, M. D., et al. 2019, arXiv:1908.00655 Wanajo, S. 2018, , 868, 65 Wanajo, S., Janka, H.-T., & Müller, B. 2011, , 726, L15 Wanajo, S., Sekiguchi, Y., Nishimura, N., Kiuchi, K., Kyutoku, K., & Shibata, M. 2014, ApJL, 789, L39 Watson, D., Hansen, C. J., Selsing, J., et al. 2019, Nature, 574, 497 Wehmeyer, B., Pignatari, M., & Thielemann, F.-K. 2015, MNRAS, 452, 1970 Wehmeyer, B., Frohlich, C., Côteé, B., Pignatari, M., & Thielemann, F.-K. 2019, MNRAS, 487, 1745 Winteler, C., Käppeli, R., Perego, A., et al. 2012, , 750, L22 Wu, M.-R., Tamborra, I., Just, O., & Janka, H.-T. 2017, PhRvD, 96, 123015 Zappa, F., Bernuzzi, S., Pannarale, F., Mapelli, M., & Giacobbo, N. 2019, PhRv Lett., 123, 041102
[^1]: Although the s-process dominates these neutron-capture elements in the solar abundance pattern, we focus on the chemical enrichment in the Galactic halo where the r-process is the major source for these elements in this study. Therefore, light neutron-capture elements are hereafter referred to as light r-process elements.
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abstract: |
Using the Vakarchuk formulae for the density matrix, we calculate the number $N_{k}$ of atoms with momentum $\hbar k$ for the ground state of a uniform one-dimensional periodic system of interacting bosons. We obtain for impenetrable point bosons $ N_{0} \approx 2\sqrt{N}$ and $N_{k=2\pi j/L} \simeq 0.31N_{0}/\sqrt{|j|}$. That is, there is no condensate or quasicondensate on low levels at large $N$. For almost point bosons with weak coupling ($\beta=\frac{\nu_{0}m}{\pi^{2}\hbar^{2}n} \ll 1$), we obtain $\frac{N_{0}}{N} \approx \left (\frac{2}{N\sqrt{\beta}}\right )^{\sqrt{\beta}/2} $ and $
N_{k=2\pi j/L} \approx \frac{N_{0}\sqrt{\beta}}{4|j|^{1-\sqrt{\beta}/2}}$. In this case, the quasicondensate exists on the level with $k=0$ and on low levels with $k\neq 0$, if $N $ is large and $ \beta$ is small (e.g., for $N \sim 10^{10} $, $ \beta \sim 0.01$). A method of measurement of such fragmented quasicondensate is proposed.
author:
- |
[Maksim Tomchenko]{}\
[Bogolyubov Institute for Theoretical Physics, ]{}\
[14b, Metrolohichna Str., 03680 Kyiv, Ukraine]{}\
[E-mail:[email protected]]{}
title: |
Bose-Einstein condensation in a one-dimensional system\
of interacting bosons
---
**Keywords:** quasicondensate, low dimensions, interacting bosons\
Introduction
============
In the present work, we will study the Bose–Einstein condensation [@einstein1925] for the ground state of a uniform one-dimensional (1D) periodic system of particles with repulsive interaction. In some works, it is asserted that the condensate does not exist in the one-dimensional case. This assertion is true only for infinite systems. But all systems in the Nature are finite. For the finite systems, the macroscopic occupation of the one-particle state is possible, and it corresponds to a condensate [@einstein1925]. The Bose–Einstein condensation in the momentum space depends on the behavior of the one-particle density matrix $F_{1}(R=|\textbf{r}-\textbf{r}^{\prime}|)$ [@penronz], which is the one-particle correlation function. If the function $F_{1}(R)$ approaches a nonzero constant for large $R,$ then the occupation number $N_{0}$ of the lowest one-particle level is of the order of magnitude of the total number of atoms $N,$ and we arrive at the condensate. If $F_{1}(R)$ slowly decreases (by a power law or logarithmically), then the macroscopic occupation of the one-particle state is possible. To distinguish this case from the first one, it is accepted to talk about a quasicondensate [@petrov2004; @pethick2008]. For the fast (e.g., exponential) decrease of $F_{1}(R),$ the macroscopic occupation of the one-particle state is impossible; therefore, there is no condensate or quasicondensate. In the 3D case, the states with condensate and without condensate are possible. In 1D and 2D cases, the quasicondensate is possible additionally; and, as usual, namely the quasicondensate is realized instead of a “true” condensate. We will consider impenetrable point bosons and almost point bosons with weak coupling. In these extreme cases of the strong and weak interactions, the wave functions of the ground state have the same structure.
It was shown in a series of works [@mw1966; @hohenberg1967; @kk1967; @rc1967; @pethick2008] that, at a nonzero temperature and $N, L \rightarrow \infty,$ the condensate on the level with $k=0$ is forbidden for the 1D systems. We will consider the case $T=0$, for which the behavior of the one-particle density matrix $F_{1}(R)$ was determined and it was shown that, in the limit $N, L \rightarrow \infty,$ the condensate is absent [@lenard1964; @popov1980; @schwartz1977; @vaidya1979; @haldane1981; @berkovich1989; @petrov2003; @forrester2003; @mora2003; @petrov2004; @pethick2008]. We will carry out the analysis on the basis of the Vakarchuk formulae for the density matrix [@vak1989; @vak1990]. In a similar approach, the analysis was executed in work [@schwartz1977], but we will use a more accurate formula for the density matrix. We will obtain known results and several new ones.
Regime of infinitely strong coupling
====================================
Consider the system of $N$ impenetrable point bosons located on the periodic interval $[0,L]$. The wave function of the ground state of such system reads [@girardeau1960] $$\Psi_{0}=C \exp{\left (\frac{1}{2}\sum\limits_{j,l=1}^{N\prime} \ln{|\sin{[\pi(x_{j}-x_{l})/L]}|}\right )},
\label{2}$$ where $C=const$, and the prime above the sum means $j\neq l$. Using the collective variables $\rho_{k} = \frac{1}{\sqrt{N}}\sum\limits_{j=1}^{N}e^{-ikx_j}$ and the expansion in the Fourier series $$\frac{1}{2} \ln{|\sin{[\pi(x_{p}-x_{l})/L]}|} = \frac{1}{L}
\sum\limits_{k_{j}}^{(2\pi)}\bar{\lambda}_{j}e^{ik_{j}(x_{p}-x_{l})},
\label{6}$$ $$\begin{aligned}
&&\bar{\lambda}_{j} = \frac{1}{2}\int\limits_{0}^{L}dx
\ln{[\sin{(\pi x/L)}]}e^{-ik_{j}x},
\label{7} \end{aligned}$$ we can write the function $\Psi_{0}$ (\[2\]) in the form [@girardeau1960] $$\Psi_{0}=C^{\prime}e^{\frac{1}{2}\sum\limits_{k\neq 0}^{(2\pi)}a_{2}(k)\rho_{k}\rho_{-k}},
\label{9}$$ where $$a_{2}(k_{j})=2N\lambda_{j}, \quad C^{\prime}=C e^{N^{2}\lambda_{0}-N\sum\limits_{j}\lambda_{j}},
\label{10}$$ $$\begin{aligned}
&&\lambda_{j} =\bar{\lambda}_{j}/L = \frac{1}{2}\int\limits_{0}^{1}dt
\ln{[\sin{(\pi t)}]}\cos{(2\pi jt)}.
\label{8} \end{aligned}$$ Here and below, the symbol $(l\pi)$ above the sum means that $k_{j}$ runs the values $k_{j}=l\pi j/L$, $j=0, \pm 1, \pm 2, \ldots$ Since $\int\limits_{0}^{L}dx \ln{[\sin{(\pi x/L)}]}\sin{(k_{j}x)}=0$, we write in (\[8\]) $\cos{(2\pi jt)}$ instead of $e^{-i2\pi jt}$. It was found [@girardeau1960] that $$\begin{aligned}
\lambda_{0}=-\frac{\ln{2}}{2}, \quad \lambda_{j\neq 0} =-\frac{1}{4|j|}.
\label{lam} \end{aligned}$$ It can be proved by the direct numerical calculation that formulae (\[lam\]) are proper, and series (\[6\]), (\[7\]) restores the function $(1/2) \ln{|\sin{[\pi(x_{p}-x_{l})/L]}|}$ exactly.
I. Vakarchuk [@vak1989] developed a method of calculation of the $s$-particle density matrix, which for the ground state reads $$\begin{aligned}
&&F_{s}(\textbf{r}_{1},\ldots,\textbf{r}_{s}|\textbf{r}_{1}^{\prime},\ldots,\textbf{r}_{s}^{\prime}) =
V^{s} \int d\textbf{r}_{s+1}\ldots d\textbf{r}_{N} \Psi^{*}_{0}(\textbf{r}_{1}^{\prime},\ldots,\textbf{r}_{s}^{\prime},\textbf{r}_{s+1},\ldots , \textbf{r}_{N})
\times \nonumber \\ &\times & \Psi_{0}(\textbf{r}_{1},\ldots,\textbf{r}_{s},\textbf{r}_{s+1},\ldots ,\textbf{r}_{N}).
\label{11} \end{aligned}$$ For $\Psi_{0}$ of the form (\[9\]), the formulae from [@vak1989] yield the following series for the logarithm of the one-particle density matrix [@vak1990]: $$\ln{F_{1}(x,x^{\prime})} = u_{1}(R) + u_{2}(R) + \ldots, \quad R=x-x^{\prime},
\label{13}$$ $$u_{1}(R) =\frac{1}{N}\sum\limits_{k\neq 0}^{(2\pi)}\frac{a_{2}^{2}(k)}{1-2a_{2}(k)}
\left (e^{ikR}-1 \right ),
\label{14}$$ $$u_{2}(R) =\frac{1}{N^{2}}\sum\limits_{k_{1},k_{2}}^{(2\pi)}
\frac{a_{2}(k_{1})a_{2}(k_{2})a_{2}(-k_{1}-k_{2})}{(1-2a_{2}(k_{1}))^{2}(1-2a_{2}(k_{2}))(1-2a_{2}(-k_{1}-k_{2}))}
\left (e^{ik_{1}R}-1 \right ).
\label{15}$$ In sum (\[15\]), $k_{1},k_{2},k_{1}+k_{2}\neq 0$. Two last formulae are true for large $N, L$.
The analysis was performed on the basis of the density matrix also in work [@schwartz1977]. In the approximation of small fluctuations of the density and the current, the following formulae [@schwartz1974] were obtained: $$F_{1}(R)|_{T=0} =e^{\tilde{u}(R)}\left [1-\frac{1}{2N}\sum\limits_{p\neq 0}^{(2\pi)}(S_{p}-1)(1-e^{ipR})\right ],
\label{14s}$$ $$\tilde{u}(R) =\frac{1}{N}\sum\limits_{k\neq 0}^{(2\pi)}\frac{S_{k}^{2}-1}{4S_{k}}(1-\cos{kR}).
\label{15s}$$ In order to compare formulae (\[14s\]) and (\[15s\]) with (\[13\])–(\[15\]), we note the following. For a system of interacting bosons with any *finite* coupling constant (penetrable particles), $ \Psi_{0}$ takes the form [@yuv1980] $$\Psi_{0}=C \exp{\left(\frac{1}{2!}\sum\limits_{k\neq 0}^{(2\pi)}a_{2}(k)\rho_{k}\rho_{-k}+
\frac{1}{3!}\sum\limits_{k_{1},k_{2}\neq 0}^{(2\pi)\prime}a_{3}(k_{1},k_{2})\rho_{k_{1}}\rho_{k_{2}}\rho_{-k_{1}-k_{2}}+\ldots \right ) },
\label{31}$$ where the prime above the sum means $k_{1}+k_{2}\neq 0$. The analysis [@yuv1980] is valid, generally speaking, for nonpoint particles. For point penetrable bosons, we have the Lieb–Liniger solution for $\Psi_{0}$ [@ll1963]. Apparently, the analysis in [@yuv1980] is also proper for the point particles, so that $\Psi_{0}$ given by the Lieb–Liniger solution can be written in the form (\[31\]). But this question was not considered in the literature, to our knowledge (see also [@pointbosonsme2015]). For the point bosons with infinite positive coupling constant (impenetrable bosons), the solution has the form (\[9\]), which follows from (\[31\]) provided $a_{j\geq 3}= 0$. For the penetrable nonpoint bosons, $a_{j\geq 3}\neq 0$. Thus, the penetrable nonpoint bosons and the impenetrable point ones can be described in a unified way, by starting from $\Psi_{0}$ (\[31\]).
For nonpoint bosons in the regime of weak coupling, the relation $2a_{2}(k)\approx 1-1/S_{k}$ holds [@yuv1980], and, for not too small $k,$ the quantity $a_{2}(k)$ is small. We can verify that, in this case, the sum on the right-hand side of (\[14s\]) is small, and it can be raised in the exponent. Then (\[14s\]) and (\[15s\]) are reduced to $F_{1}(R)=e^{u_{1}(R)}$ with $u_{1}$ (\[14\]). That is, the density matrix from [@schwartz1977] at a weak coupling coincides with the first approximation for the density matrix (\[13\])–(\[15\]) [@vak1989]. It is possible to restrict oneself in Eqs. (\[13\])–(\[15\]) to the first approximation ($u_{1}\neq 0, \ u_{2}=0$), if the coupling is weak (see the following section). It follows that the approximation of small fluctuations [@schwartz1977; @schwartz1974] is equivalent to the approximation of weak coupling. Moreover, the Feynman formula $S_{k}=\hbar^{2}k^{2}/2mE(k)$ was used in [@schwartz1977]. For a weak coupling, this formula is close to the exact one for all $k$; for a strong coupling, it is valid only for small $k$. The impenetrable point bosons correspond to the infinitely strong coupling. In this case, $a_{2}(k)$ in Eq. (\[31\]) is set by formulae (\[10\]) and (\[8\]), and $a_{j\geq 3}= 0$. In this case, the density matrices (\[13\])–(\[15\]) and (\[14s\]), (\[15s\]) do not coincide with one another. In work [@vak1989], the perturbation theory is constructed for the logarithm of the density matrix, and it is valid for any coupling (the results given below indicate that, even for a strong coupling, series (\[13\]) is apparently rapidly convergent). Thus, formulae (\[13\])–(\[15\]) [@vak1989; @vak1990] are more accurate than formulae (\[14s\]) and (\[15s\]) [@schwartz1977; @schwartz1974], because the former involve the following correction. In addition, some constants were not determined in [@schwartz1977]. We will find all the constants. In these respects, our analysis is better than the analysis [@schwartz1977].
With regard for (\[10\]) and the equalities $\lambda_{j}=\lambda_{-j}$, $k_{j}=2\pi j/L,$ formulae (\[14\]) and (\[15\]) can be written in the form $$u_{1}(R) =\sum\limits_{j=\pm 1, \pm 2, \ldots}\frac{4N\lambda_{j}^{2}}{1-4N\lambda_{j}}
\left (e^{i2\pi jR/L}-1 \right ),
\label{16}$$ $$u_{2}(R) =\sum\limits_{j_{1},j_{2}=\pm 1, \pm 2, \ldots}^{j_{1}+j_{2}\neq 0}
\frac{8N\lambda_{j_{1}}\lambda_{j_{2}}\lambda_{j_{1}+j_{2}}}{(1-4N\lambda_{j_{1}})^{2}(1-4N\lambda_{j_{2}})(1-4N\lambda_{j_{1}+j_{2}})}
\left (e^{i2\pi j_{1}R/L}-1 \right ).
\label{17}$$ The average number of particles with momentum $\hbar k$ in the 1D case is determined by the well-known formula $$N_{k} =\frac{N}{L}\int\limits_{0}^{L}F_{1}(R)e^{-ikR} dR.
\label{18}$$ Such approach allows one to obtain the reasonable estimates for the condensate in He II [@vak1990; @me2006].
In the literature, the condensate is frequently defined by the formula [@penronz] $$N_{0} =NF_{1}(R\rightarrow \infty),
\label{18b2}$$ which is true in the thermodynamical limit ($N, V \rightarrow \infty, N/V=const)$. We now consider 1D periodic systems of *finite* size $L$. The periodicity yields $F_{1}(0)=F_{1}(L)$. The analysis below indicates that the density matrix $F_{1}(R)$ takes the maximum value ($F_{1}=1$) at the ends of the interval ($R=0, L$) and decreases, while approaching the middle of the interval. The quantity $F_{1}(R)$ is minimal for $R= L/2$. Therefore, for the finite periodic 1D systems, formula (\[18b2\]) should be replaced by $$N_{0} \approx NF_{1}(R\rightarrow L/2).
\label{18c2}$$ Formula (\[18c2\]) underestimates $N_{0}$ as compared with the exact value (\[18\]), which is evident (i) for a strong coupling or (ii) for small $N$ in the case of weak coupling.
Using formulae (\[18\]), (\[13\]), (\[16\]), and (\[17\]), we find now the values of $N_{k}$ for the ground state of $N$ impenetrable point bosons in a cyclic vessel by means of a direct numerical summation. Since sums (\[16\]), (\[17\]) are present in (\[18\]) in the exponent, we need to take rather many terms ($\gsim 10^{6}$ for the summation over each $j$) in order to attain a good accuracy in sums (\[16\]) and (\[17\]). The results for $N=10^2$–$10^4$ are as follows: $$N_{0}= C_{1}\sqrt{N},
\label{19}$$ $$N_{k=2\pi l/L} = C_{2}N_{0}/\sqrt{|l|}, \quad 1\leq |l|\ll N,
\label{19b}$$ where $C_{1}=0.87\pm 0.01, C_{2}=0.33\pm 0.005 $ in the first approximation, and $C_{1}=1.99\pm 0.05, C_{2}=0.31\pm 0.03$ in the second one (we take only $u_{1}$ into account in (\[13\]) in the first approximation and $u_{1}$, $u_{2}$ in the second one).
Estimates (\[19\]) and (\[19b\]) can be obtained analytically for the first approximation. Using formulae (\[16\]) and (\[lam\]), we write the function $u_{1}(R)$ in the form $$u_{1}(R) =\sum\limits_{j= 1, 2, \ldots} \alpha_{j}(\cos{(2\pi jR/L)}-1), \quad \alpha_{j}=\frac{N}{2j(N+j)}.
\label{21}$$ In formula (\[18\]) for $N_{k},$ the function $F_{1}(R)=e^{u_{1}(R)}$ stands under the sign of integral. It follows from formula (\[50\]) below that the value of $|u_{1}(R)|$ is usually large: for example, for $N=3\cdot 10^6,$ we have $|u_{1}(R)|\lsim 8$. Therefore, it is not expedient to expand $e^{u_{1}(R)}$ in a series, since too many terms should be taken into account in order to obtain a proper result. It is better to determine the exponent in $F_{1}(R)=e^{u_{1}(R)}.$ Relation (\[21\]) and the Euler–Maclaurin formula $$\sum\limits_{j=1, 2, \ldots}f(j)\approx \int\limits_{1}^{\infty }f(x)dx+B_{1}(f(\infty)-f(1))+
\frac{B_{2}}{2}(\acute{f}(\infty)-\acute{f}(1))
\label{23}$$ with the Bernoulli numbers $B_{1}=-1/2$ and $B_{2}=1/6$ yield for $1/N\lsim R/L \leq 1/2$: $$F_{1}(R) \approx e^{u_{1}(R)}\approx \left (\frac{f(R)}{nR}\right )^{1/2}, \quad
f(R)\approx 0.089+0.2(R/L)^{2},
\label{50}$$ where $n=N/L.$ The fitting function $f(R)$ was determined by means of the comparison of $u_{1}(R)$ with the results of a numerical summation of (\[21\]). This function allows one to get the numerical values of $u_{1}(R)$ for $R=0.001L$–$0.5L$ and $N=10^3$–$10^6$ with a small error of $\lsim 0.2\% $. For $R> L/2,$ it is necessary to change $R\rightarrow L - R$ on the right-hand side of (\[50\]). Then relation (\[50\]) yields $$N_{k=2\pi l/L} =\frac{N}{L}\int\limits_{0}^{L}F_{1}(R)e^{-ikR} dR \approx
2\sqrt{N}\int\limits_{0}^{1/2}dt\cos{(2\pi l t)}\sqrt{\frac{0.089}{t}+0.2t}.
\label{18b}$$ This gives formula (\[19\]) for $N_{0}$ with constant $C_{1}\approx 0.89$, which is close to the above-given value $C_{1}\approx 0.87$. Integral (\[18b\]) can be easily found numerically, and, for any $l\neq 0,$ the answer is as follows: $$N_{k=2\pi l/L } = (0.295\pm 0.003)\sqrt{N/|l|}\approx 0.331N_{0}/\sqrt{|l|}.
\label{18c}$$ If we eliminate the term $0.2t$ from (\[18b\]), then the law $N_{k=2\pi l/L }\sim 1/\sqrt{|l|}$ is satisfied for small $|l|$ with less accuracy. For $N=400,$ formula (\[18c\]) gives the value, which is overestimated by $ 10\% $ relative to the result of a direct numerical summation in (\[18\]), (\[21\]). But, as $N$ increases, this difference decreases to $5\% $ for $N=2000$ and to $1 \% $ for $N=10^{4}.$
The following results were obtained previously. For $L=N,$ it was shown [@lenard1964] that $$N_{0} < 2\sqrt{e N}, \quad F_{1}(|R|\rightarrow \infty)|_{N\rightarrow \infty} \leq (e/\pi |R|)^{1/2},
\label{501}$$ which agrees with (\[19\]) and (\[50\]). The dependence $F_{1}(|R|\gg n^{-1}) \sim |R|^{-1/2}$ was found in [@schwartz1977]. Formula (\[50\]) with $f(R)= 1$ was deduced in [@petrov2004]. The formulae $F_{1}(|R|\gg n^{-1}) \sim |R|^{-1/2}, N_{k\neq 0}\sim |k|^{-1/2}$ were gotten in [@vaidya1979]. The exact calculation [@forrester2003] gives $$F_{1}(R) = \frac{0.924}{\sqrt{N\sin{(\pi R/L)}}}\approx \left (\frac{0.27+0.45(R/L)^2}{nR}\right )^{1/2},
\label{for1}$$ $$N_{0}\approx 1.543\sqrt{N}, \quad N_{k=2\pi l/L } \approx \frac{0.338N_{0}}{\sqrt{|l|}} \quad (l\neq 0).
\label{for2}$$ In our approach, the direct numerical summation in (\[16\])–(\[18\]) in the second approximation gives $N_{k}$ (\[19\]), (\[19b\]) and the density matrix $$F_{1}(R) = e^{u_{1}(R)+u_{2}(R)}\approx \left (\frac{e^{1.64}f(R)}{nR}\right )^{1/2}\approx
\left (\frac{0.46+(R/L)^2}{nR}\right )^{1/2}.
\label{502}$$ This is in approximate agreement with the results [@schwartz1977; @vaidya1979; @petrov2004; @forrester2003].
It is seen from formulae (\[19\]), (\[19b\]), (\[50\]), and (\[502\]) that, in our approach, the results for $N_{0}$, $N_{k\neq 0 },$ and $F_{1}(R)$ in the second approximation are approximately by a factor $2.3$ larger than in the first approximation. Such significant difference is related to the absence of a small parameter in expansion (\[13\]) and to the fact that this is the expansion of the value in the exponent. However, the results in the second approximation are in better agreement with the exact ones [@forrester2003] (as compared with the results in the first approximation) and differ from the latter by $\leq 30\%$. As for the ratio $N_{k}/N_{0},$ the method gives the result (\[19b\]), which differs from the exact one (\[for2\]) only by $10\%$. That is, the method allows one to get reasonable results, and we expect that, with the account for several following $u_{j}$ in (\[13\]), the results will be close to the exact ones.
It is of interest that, according to our analysis, $u_{1}$ depends weakly on $R$, and $u_{2}(R)$ is a constant $0.82\pm 0.01$ everywhere except for narrow bands $|R|\lsim 1/N$ and $|L-R|\lsim
1/N$. In this case, relation $|u_{2}(R)|\simeq q|u_{1}(R)|$ holds, where $q\approx 0.224\ln{(1925)}/\ln{(3.85N)}$ (e.g., $q\approx
0.22$ for $N=500$). If the same law of decrease of $|u_{j}(R)|$ with increase in $j$ holds for the following $j,$ then even the first approximation (\[19b\]) for $N_{k}/N_{0}$ should be close to the exact value, and the second approximation (\[19\]) for $N_{0}/N$ should differ from the exact value by at most several tens of percents. The comparison of results (\[19\]), (\[19b\]) with the exact solutions (\[for2\]) confirms these properties. This allows us to expect that, though our approach has no small parameter and the correction $u_{2}(R)$ affects considerably the result, the following corrections $u_{j\geq 3}$ will less affect the results.
We note that the condition $$\sum\limits_{k}N_{k}= N
\label{29}$$ holds automatically. This is related to that $LN_{k}/N$ is the Fourier transform of the function $F_{1}(R)$, according to (\[18\]). Therefore, $F_{1}(0)=(1/L)\sum\limits_{k}LN_{k}/N$. In the first and second approximations, $F_{1}(0)=1$, which yields (\[29\]). We note also that the function $F_{1}(R=x-x^{\prime})$ depends on two arguments ($x\in [0,L]$ and $x^{\prime}\in [0,L]$) and is periodic in each argument with period $L$. In this case, the equality $F_{1}(R)= F_{1}(|R|)$ holds. Therefore, $F_{1}(|R|)$ can be expanded in a single Fourier series on the interval $|R|\in[0,L]$. Formula (\[18\]) sets the Fourier transform for such a series. The same is true for expansion (\[6\]), (\[7\]), because $\ln{|\sin{(\alpha)}|}=\ln{|\sin{|\alpha}||}$.
Regime of a weak coupling
=========================
Consider an analogous problem for the ground state of a 1D system with weak coupling (highly penetrable bosons). To simplify the formulae, we consider the interatomic potential $U(x_{i}-x_{j})$ to be an extremely high narrow barrier close to the $\delta$-function with the Fourier transform $\nu(k)=\nu_{0}=const$. For a system of penetrable bosons, we have $\Psi_{0}$ (\[31\]). Under a weak coupling (weak interaction $ \nu_{0}$ or a high concentration, $\beta \ll 1$ in (\[34\])), the correction $a_{3}(k_{1},k_{2})$ in (\[31\]) is small, and the sum with $a_{3}$ can be neglected [@yuv1980]. Therefore, $\Psi_{0}$ takes the form (\[9\]) with $a_{2}(k)$ to be [@bz1955] $$2a_{2}(k)\approx 1-\sqrt{1+\frac{4n\nu_{0}m}{\hbar^{2}k^{2}}},
\label{32}$$ and formulae (\[13\])–(\[15\]) remain valid. With regard for (\[32\]), we get $$u_{1}(R) =\sum\limits_{j= 1, 2, \ldots} \alpha^{p}_{j}(\cos{(2\pi jR/L)}-1),
\label{33}$$ $$\alpha^{p}_{j}=\frac{1+\beta N^{2}/(2j^{2})-\sqrt{1+\beta N^{2}/j^{2}}}{N\sqrt{1+\beta N^{2}/j^{2}}},
\quad \beta=\frac{\nu_{0}m}{\pi^{2}\hbar^{2}n}.
\label{34}$$ Here, $\beta $ is a dimensionless coupling constant. $u_{2}(R)$ (\[15\]) can be represented in the form (\[33\]) too. In this case, $\alpha^{p}_{j}$ is different and much less in modulus (for $\beta \ll 1$). Therefore, the correction $u_{2}(R)$ can be neglected. For sufficiently small $\beta,$ $|u_{1}(R)|\ll 1$ is satisfied (see Eq. (\[40\]) below). Therefore, the exponential function $e^{u_{1}(R)}$ can be expanded in a series. Then we have $$\begin{aligned}
&&\frac{N_{k=2\pi l/L}}{N} = L^{-1}\int\limits_{0}^{L} d R \cos{(2\pi l R/L)}\exp{[\sum\limits_{j= 1}^{\infty } \alpha^{p}_{j}(\cos{(2\pi jR/L)}-1)]}
\approx \label{27a} \\ &\approx & e^{-\sum\limits_{j= 1}^{\infty } \alpha^{p}_{j}}
\int\limits_{0}^{1}dt \cos{(2\pi l t)}\left (1+ \sum\limits_{j= 1}^{\infty } \alpha^{p}_{j}\cos{(2\pi j t)}+\frac{1}{2!}
\sum\limits_{j_{1},j_{2}= 1}^{\infty } \alpha^{p}_{j_{1}}\alpha^{p}_{j_{2}}\cos{(2\pi j_{1}t)}\cos{(2\pi j_{2}t)}
\right ),
\nonumber \end{aligned}$$ where $t=R/L$. This implies $$\begin{aligned}
\frac{N_{0}}{N} \approx e^{-\sum\limits_{j= 1}^{\infty } \alpha^{p}_{j}}
\left (1+ \frac{1}{4}\sum\limits_{j= 1}^{\infty } (\alpha^{p}_{j})^{2}\right )\approx e^{-\sum\limits_{j= 1}^{\infty } \alpha^{p}_{j}},
\label{22} \end{aligned}$$ $$\begin{aligned}
&&\frac{N_{k=2\pi l/L}}{N} \approx e^{-\sum\limits_{j= 1}^{\infty } \alpha^{p}_{j}}
\left (\frac{\alpha^{p}_{l}}{2}+\frac{1}{8}\sum\limits_{j= 1}^{l-1 } \alpha^{p}_{j}\alpha^{p}_{l-j}+
\frac{1}{4}\sum\limits_{j= 1}^{\infty } \alpha^{p}_{j}\alpha^{p}_{l+j}\right )\approx
\frac{\alpha^{p}_{l}}{2}e^{-\sum\limits_{j= 1}^{\infty } \alpha^{p}_{j}},
\label{27} \end{aligned}$$ where $l=\pm 1, \pm 2, \ldots$. It can be verified that, for $|l|\ll N\sqrt{\beta},$ the modulus of each of the sums with $(\alpha^{p})^{2}$ in (\[22\]), (\[27\]) is less that the principal term ($1$ or $\alpha^{p}_{l}/2$) by $\sim \beta^{-1}$ or $\sim \beta^{-1/2}$ times. Therefore, we can neglect these sums, if $\beta$ is small. For $|l| \gsim 0.1N\sqrt{\beta},$ the rejected corrections ($\sim \alpha^{2}, \alpha^{3}, \ldots $) decrease the value of $N_{k}$ significantly. We obtain by formula (\[23\]) that, for $N\sqrt{\beta}\gg 1,$ $$\begin{aligned}
\sum\limits_{j= 1}^{\infty } \alpha^{p}_{j} \approx
\frac{\sqrt{\beta}}{2} \ln{(N\sqrt{\beta}/2)}.
\label{37} \end{aligned}$$ Since the relation $\alpha^{p}_{l}\approx \frac{\sqrt{\beta}}{2|l|}$ holds for $|l|\ll N\sqrt{\beta},$ we finally have $$\begin{aligned}
N_{0} \approx & N\left (\frac{2}{N\sqrt{\beta}}\right )^{\sqrt{\beta}/2},
\quad N_{k=2\pi l/L} \approx \frac{N_{0}\sqrt{\beta}}{4|l|},
\label{38} \end{aligned}$$ where $1\leq |l|\ll N\sqrt{\beta}$.
The values of $N_{k}$ can be found in a different way. For $(\sqrt{\beta}N)^{-1} \lsim R/L \leq 1/2$ and $\sqrt{\beta}N\gg 1$, we have $$u_{1}(R) \approx \frac{\sqrt{\beta}}{2}\ln{\frac{f_{2}(R)}{\sqrt{\beta}nR}}, \quad f_{2}(R)\approx \frac{0.98+2(R/L)^{2}}{3}.
\label{39}$$ The main dependence in (\[39\]) can be found by formula (\[23\]), and the fitting function $f_{2}$ follows from the direct numerical summation of (\[33\]). From whence, we get the density matrix $$F_{1}(R) \approx e^{u_{1}(R)}\approx \left (\frac{f_{2}(R)}{\sqrt{\beta}nR}\right )^{\sqrt{\beta}/2}.
\label{40}$$ For $R> L/2,$ we should replace $R\rightarrow L - R$ on the right-hand sides of (\[39\]) and (\[40\]).
In Fig. 1, we show the calculated density matrix. Two curves are significantly different for $R/L\lsim 0.0001$ and $R/L\gsim 0.9999$. For $0.0001 < R/L < 0.9999,$ the curves practically coincide: that is, formula (\[40\]) with $f_{2}(R)$ (\[39\]) is very close to the exact numerical solution for $F_{1}(R)$. The smallest value of $F_{1}(R)$ is $F_{1}(R=L/2)\approx 0.384$.
![ Density matrix $F_{1}(R)$ for a periodic 1D system of almost point bosons with weak coupling ($N=10^{4}$, $\beta = 0.06$). The direct numerical calculation of $F_{1}(R)$ on the basis of the exact formulae (\[18\]), (\[13\]), (\[33\]), (\[34\]) with $u_{j\geq 2}=0$ ($\circ\circ\circ$) and $F_{1}(R)$ (\[39\]), (\[40\]) with the replacement $R\rightarrow L - R$ for $R> L/2$ ($\star\star\star$). \[fig1\]](fig1.eps){width="85mm"}
Relations (\[18\]) and (\[40\]) yield $$\begin{aligned}
N_{0} \approx & NI_{0}\cdot\left (\frac{2}{N\sqrt{\beta}}\right
)^{\sqrt{\beta}/2}, \quad N_{k=2\pi l/L} \approx
N_{0}\frac{I_{l}}{I_{0}} \quad (l\neq 0),
\label{41} \end{aligned}$$ $$I_{l} =I_{-l}= 2\int\limits_{0}^{1/2}dt\cos{(2\pi l t)}\left (\frac{0.49}{3t}+\frac{t}{3}\right )^{\sqrt{\beta}/2}.
\label{42}$$ For $\beta \lsim 0.01$, we find numerically $$I_{l=0}\equiv I_{0} \approx 1, \quad I_{l\neq 0} \approx \frac{\sqrt{\beta}}{4|l|^{1-\sqrt{\beta}/2}}.
\label{43}$$ Formulae (\[41\]) are close to (\[38\]). For $\beta = 0.1,$ formulae (\[43\]) underestimate the values of $I_{l\neq 0}$ (\[42\]) by $10 \% $ and $I_{0}$ (\[42\]) by $2 \% $. Both formulae (\[41\]), (\[42\]) and formulae (\[38\]) agree with the results of a direct numerical calculation of $N_{k}$ on the basis of (\[18\]), (\[13\]), (\[33\]), and (\[34\]). But formulae (\[41\]) and (\[42\]) are more accurate than (\[38\]) (see below).
It follows from (\[39\]) and (\[40\]) that formula (\[38\]) for $N_{0}$ can be written as $$\begin{aligned}
N_{0} \approx N\cdot 2^{\sqrt{\beta}/2}\cdot F_{1}(L/2).
\label{38b} \end{aligned}$$
![ Fragmented quasicondensate. Values of $100N_{p}/N$ (Eqs. (\[38\])) for $N=10^8$ and $p=0$ (solid line), $p=1$ ($\circ\circ\circ$), $p=2$ ($+++$), $p=3$ ($\bigtriangleup\bigtriangleup\bigtriangleup$), $p=5$ (squares). $N_{0}$ is reduced by $100$ times. \[fig2\]](fig2.eps){width="85mm"}
Formulae (\[38\]), (\[41\]) and (\[43\]) indicate that, for small $\beta$ and large $N,$ the quasicondensate is present not only on the level with $k=0$, but also on low levels with $k\neq 0$ (see Fig. 2; if $\beta$ is very small, then the quasicondensate occupies only the level $k=0$). Such *fragmented quasicondensate* is, in some sense, a corroboration of M. Girardeau ideas [@girardeau1960], but for a weak coupling. It is of interest to mention the work by E. Witkowska et al. [@witkowska2011], where a model of evaporative cooling of a one-dimensional gas in a trap was constructed. It was found that several lower levels are macroscopically filled in the initial nonequilibrium regime and only the lowest level is macroscopically filled in the final equilibrium regime (though, it was not explained in this work how the occupation numbers are calculated; their determination is a complicated task: it is necessary to find the density matrix and then to determine the occupation numbers $\lambda_{j}$ from Eq. (\[46\])). The results [@witkowska2011] imply that the macroscopic occupation of several lower levels is related namely to the absence of equilibrium, i.e. to the disorder. This is not quite clear physically, since a disorder destroys the macroscopic occupation of levels, as usual. Possibly, the effect is related to a comparatively small $N$ ($N\leq 10^{4}$) and will disappear, as $N$ will increase by at least two orders. The equilibrium state [@witkowska2011] corresponds, probably, to small $\beta $: $\beta\leq 0.01$. In this case, formula (\[38\]) yields the macroscopic occupation for the lowest level only. Above, we have found a fragmented quasicondensate for a uniform equilibrium 1D system of interacting spinless bosons, by exactly describing the interaction. Apparently, such solution was not obtained previously. Note that the regimes, in which a generalized condensate appears in the ideal gas, were investigated in [@mullin2012]. Several particular systems with possible fragmented condensates were discussed in [@leggett2006].
According to (\[38\]), the occupation of low levels with $k\neq 0$ is maximal for $N=10^{10}$ and $\beta \approx 0.009$ ($N_{0} \approx 0.388N$, $N_{l} \approx 0.0092N/|l|$). For $N=10^{4},$ the occupation is maximal for $\beta \approx 0.06$ ($N_{0} \approx 0.419N$, $N_{l} \approx 0.0256N/|l|$, see also Fig. 1). Formulae (\[41\]) and (\[43\]) give practically the same values. The numerical calculation on the basis of the exact relations (\[18\]), (\[13\]), (\[33\]), and (\[34\]) for $N=10^{4}$, $\beta \approx 0.06$ gives $N_{0}$ to be by $1 \%$ larger relative to (\[38\]) and (\[41\]), (\[43\]) and $N_{k \neq 0}$ to be by $10 \%$ larger relative to (\[38\]) and by $8 \%$ larger relative to (\[41\]), (\[43\]) (for $|l|\ll N\sqrt{\beta}$). By comparing with the more nearly exact formulae (\[41\]) and (\[42\]), $N_{0}$ is only by $0.1 \%$ larger, and $N_{k \neq 0}$ by $(1\div 2) \%$. As $\beta$ decreases, all these differences decrease as well.
Our results agree with those obtained earlier. From the study of the density matrix [@schwartz1977] and from the study of fluctuations [@haldane1981; @petrov2003; @petrov2004; @pethick2008], the relation $$F_{1}(R \gg 1/n) \approx \left (\frac{l_{c}}{R}\right )^{\frac{cm}{2\pi n \hbar}}
\label{41p}$$ was found. Here, $c$ is the sound velocity, and $l_{c}$ is the healing length ($l_{c}=\hbar/\sqrt{mn\nu_{0}}$ [@petrov2003; @petrov2004]). Since $c=\sqrt{n\nu_{0}/m}$ (under a weak coupling), formula (\[41p\]) coincides with (\[40\]), where $f_{2}(R)= 1/\pi\approx 0.32$. It was shown [@pethick2008] that relation (\[41p\]) yields the formula $N_{k\neq 0}\sim 1/|k|^{1-\sqrt{\beta}/2}$, which agrees with (\[41\]), (\[43\]). If we change $2\pi l/L \rightarrow k$ in formula (\[38\]) for $N_{k\neq 0}$, we obtain a formula [@gn1964; @rc1967] for $N_{k\neq 0}$ in a *three-dimensional* system. We also mention works [@popov1980; @berkovich1989; @mora2003], where formula (\[40\]) with $f_{2}(R)\approx 0.33$ was gotten. Let us write formula (\[40\]) as $F_{1}(R) \approx C(\gamma)(nR)^{-\sqrt{\gamma}/2\pi}$ [@olshanii2009] ($\gamma=\beta \pi^{2}$). Then, in the thermodynamic limit ($L=\infty$) for $\gamma =0.001$ we obtain $C\approx 1.018$, which is in agreement with the values $C\approx 1.016; 1.02$ [@olshanii2009], obtained by two other methods. The additional summand $(2/3)(R/L)^{2}$ in $f_{2}$ (\[39\]) was not obtained earlier. As far as we see, the reason lies in the transition to the thermodynamic limit or in a not quite accurate calculation of sums. We determined numerically a solution for $F_{1}(R)$, by using formulae (2.3.1) and (2.3.2) in [@petrov2003] and formula (15.44) in [@pethick2008]. As a result, for $N=10^{4}$–$10^{6}$ and $\beta =10^{-4}$–$10^{-2},$ we obtain formula (\[40\]) with $$f_{2}(R)\approx \frac{0.5+2(R/L)^{2}}{3}
\label{50n}$$ instead of $f_{2}(R)= 1/\pi$ [@petrov2003; @pethick2008]. Both formulae give the same value for $R=L/2$. But, for other $R,$ formula (\[50n\]) describes the solution better (which is well evident for $\ln F_{1}(R)$). The distinction between $f_{2}(R)$ (\[50n\]) and (\[39\]) is apparently related to the fact that formulae [@petrov2003; @pethick2008] were obtained in the low-energy approximation ($\epsilon (k)<\mu$), whereas our method involves all $k$.
Thus, the term $(2/3)(R/L)^{2}$ in $f_{2}(R)$ is a new result. Its principal meaning consists in that, for the ground state, the decay law of the density matrix turns out to be a not quite power one. In addition, new results are formula (\[38\]) for $N_{0}$ and the constant in formula (\[38\]) for $N_{k\neq 0}$.
It was noticed [@petrov2004; @pethick2008] that, for impenetrable point bosons, the substitution of the value $cm/(2\pi n \hbar)=1/2$ [@girardeau1960] into (\[41p\]) gives the proper formula $F_{1}(R)\sim R^{-1/2}$. On this basis, formula (\[18c\]) (without the constant) was deduced in [@pethick2008]. However, the derivation of formula (\[41p\]) in works [@schwartz1977; @haldane1981; @petrov2003; @petrov2004; @pethick2008] is valid only for a weak coupling. Indeed, the corrections to the logarithm of the density matrix were not taken into account in [@schwartz1977], but these corrections are large for the strong coupling. It follows from the formulae [@petrov2003; @petrov2004] that the fluctuations of a phase are connected with the fluctuations of a concentration ($\delta \hat{n}$), and the smallness of $\delta
\hat{n}$ requires that $\nabla \hat{\varphi }(x)$ be small. Moreover, the smallness of $\langle (\hat{\varphi }(x)-\hat{\varphi
}(0))^{2}\rangle = \sqrt{\beta}\ln{(x/l_{c})}$ [@petrov2003; @pethick2008] for $x \sim l_{c}$ means the smallness of $\beta$. By taking into account only the long-wave fluctuations of a phase, it was obtained for the strong coupling [@petrov2004] $\langle (\hat{\varphi }(x)-\hat{\varphi
}(0))^{2}\rangle \simeq \ln{(x/l_{c})};$ here, the fluctuations are not small. Therefore, such method is approximate.
Thus, formula (\[41p\]) is valid also for a strong coupling. The possible reason consists in that only the two-particle correlations are of importance for the ground state of the system for both strong and weak couplings. Under an intermediate coupling, the higher correlations are significant as well (sums with $a_{3}$, $a_{4},$ etc. in $\Psi_{0}$ (\[31\])), and their consideration can change the function (\[41p\]).
In the two- and three-dimensional cases, we can analogously obtain for low levels under a weak interaction: $$\begin{aligned}
N^{2D}_{\textbf{k}\neq 0} \approx N_{0}\frac{\sqrt{\beta_{2D}}}{4N^{1/2}\sqrt{j_{x}^{2}+j_{y}^{2}}},
\quad \beta_{2D}=\frac{\nu_{0}m}{\pi^{2}\hbar^{2}},
\label{52} \end{aligned}$$ $$\begin{aligned}
N^{3D}_{\textbf{k}\neq 0} \approx N_{0}\frac{\sqrt{\beta_{3D}}}{4N^{2/3}\sqrt{j_{x}^{2}+j_{y}^{2}+j_{z}^{2}}},
\quad \beta_{3D}=\frac{\nu_{0}mn^{1/3}}{\pi^{2}\hbar^{2}},
\label{53} \end{aligned}$$ where $\textbf{k}=2\pi(\frac{j_{x}}{L},\frac{j_{xy}}{L},\frac{j_{z}}{L})$, $\nu_{0}=\int U(\textbf{r})d\textbf{r}$, $n=N/V$, $L_{x}=L_{y}=L_{z}$. Thus, for large $N$ there are no macroscopically filled levels with $\textbf{k}\neq 0$. The plausible reason for this consists in a much larger number of a one-particle levels as compared with the 1D case.
Possible experiment
===================
It is of interest whether it is possible for quasi-1D gases in a trap to enter into the region $\beta \sim 0.01\div 0.1$. In this case, we would be able to reveal experimentally quasicondensates on low levels. We will make some estimates, by using the following parameters of a trap [@druten2008]: $^{87}$Rb atoms ($a_{s}\approx 48\,\mbox{\AA}$ [@pethick2008]), $N=2\cdot 10^7$, $\omega_{\rho}=2\pi\cdot 3280\,Hz$, $\omega_{z}=2\pi\cdot 8.5\,Hz$, $R_{z}\approx 0.54\,\mbox{mm}$, $R_{\rho}\approx 1.4\cdot 10^{-3}\,\mbox{mm}$, and $T \gsim 10^{-7}\,K$. Since $g_{1D}\equiv\nu_{1D}(0)=\frac{2\hbar^{2}a_{s}}{\mu a_{\rho}(a_{\rho}+\zeta(1/2) a_{s})}$ [@olshanii1998] ($\mu=m/2$ is the reduced mass, $a_{\rho}=\sqrt{\hbar/\mu\omega_{\rho}}\approx 2600\,\mbox{\AA}$, $\zeta(1/2)\approx -1.46$), we obtain $$\beta=\frac{\nu_{1D}(0)m}{\pi^{2}\hbar^{2}n} \approx
\frac{4a_{s}}{\pi^{2}na_{\rho}(a_{\rho}+\zeta(1/2)a_{s})}\approx 1.6\cdot 10^{-6}.
\label{45}$$ For the experiment in [@aspect2003], we obtain from (\[45\]) $\beta \simeq 4\cdot 10^{-5}$. For the crude estimate of $N_{k},$ we use formulae (\[38\]) deduced for a uniform system at $T=0$ (for $T > 0,$ the density matrix is multiplied by the factor $\exp{[q_{1}(T)-q_{2}(T)R]}$ [@popov1980; @schwartz1977; @mora2003] (with $q_{1}(T\rightarrow 0)\rightarrow 0$, $q_{2}(T)=\frac{mk_{B}T}{2\hbar^{2}n}$), which is close to 1 for $T\ll T_{f}=\frac{2\hbar^{2}n}{mk_{B}R_{z}}\simeq 4\cdot 10^{-7}\,K$ and has no influence on $N_{k}$). Then, for $\beta$ (\[45\]) and $N=2\cdot 10^7,$ relation (\[38\]) yields $N_{0}\approx 0.994N$. That is, practically all atoms are in the condensate on the low level. In this case, relation (\[39\]) yields $|u_{1}(R)|\lsim 0.007$, $F_{1}(R)\approx const$, i.e., the condensate is close to the true one.
For a nonuniform gas in a trap, the eigenfunctions $f_{j}(x)$ are not plane waves, but are determined from the equation [@leggett2006] $$F_{1}(x_{a},x_{b}) = \sum\limits_{j=0}^{\infty}\lambda_{j}f_{j}(x_{a})f_{j}^{*}(x_{b}).
\label{46}$$ We can establish the connection between $N_{k}$ and $\lambda_{j}$: $$\frac{N_{k}}{N} =\frac{1}{L^2}\int\limits_{-L/2}^{L/2}dx_{a}
\int\limits_{-L/2}^{L/2}dx_{b}F_{1}(x_{a},x_{b})e^{-ik(x_{a}-x_{b})}=\sum\limits_{j=0}^{\infty}\lambda_{j}|\chi_{j}(k)|^{2},
\label{47}$$ $$\chi_{j}(k) =\frac{1}{L}\int\limits_{-L/2}^{L/2}dx f_{j}(x)e^{-ikx},
\label{48}$$ where $k\equiv k_{l}=2\pi l/L$. It is of interest that, for impenetrable point bosons in a trap at $T=0,$ the values of $\lambda_{j}$ [@forrester2003] for $j=0, 1, 2,$ are close to the values of $N_{k_{j}}/N$ for the same uniform system. This is related to the absence of a quasicondensate and to the fact that, for the given $l,$ $|\chi_{l}(k_{l})|$ is maximal among $|\chi_{j}(k_{l})|$. In other words, for the strong coupling, *the values of $\lambda_{j}$ for low levels of a uniform system and a system in a trap are close, and the same is possible for a weak coupling.* The system of bosons with a weak coupling in a trap should contain a quasicondensate. If almost all atoms are on the level $j=0$, then relation (\[47\]) is reduced to $N_{k_{l}}/N\approx \lambda_{0}|\chi_{0}(k_{l})|^{2}$. If a quasicondensate is present on several levels, then all these levels $j$ should be taken into account in (\[47\]). For the systems considered in [@druten2008; @aspect2003], we have $\beta \ll 0.01$. Therefore, the condensate on the level $j=0$ contains, probably, almost all atoms. This means that $F_{1}(x_{a},x_{b}) \approx \lambda_{0}f_{0}(x_{a})f_{0}^{*}(x_{b})$. In this case, by the measured values of $N_{k}$ [@druten2012], it is possible to restore $f_{0}(x)$ by the relations $N_{k} =N\lambda_{0}|\chi_{0}(k)|^{2}$ and $f_{j}(x)=\sum^{(2\pi)}_{k}\chi_{j}(k)e^{ikx}$. Moreover, the normalization conditions (\[29\]), $\int^{L/2}_{-L/2}dxF_{1}(x,x)=L,$ and $\int^{L/2}_{-L/2}dxf^{*}_{l}(x)f_{j}(x)=L\delta_{l,j}$ yield $\lambda_{0}\approx 1$.
It is seen from formula (\[45\]) that the region $\beta \sim 0.01\div 0.1$ can be realized experimentally, by varying $a_{s}$ by means of the Feshbach resonance [@feshbach; @pit2006]. For such $\beta,$ the number of atoms for the states with the smallest $j$ should be macroscopic for each $j$ (analogous result was derived in [@witkowska2011]; we failed to determine $\beta$ by data [@witkowska2011]). In this case, the distribution $N_{k}$ must be essentially different from $N_{k}$ for $\beta \ll 0.01$ (where almost all particles occupy the low level, and $\lambda_{0}\approx 1$). The fragmented quasicondensate can be discovered in the following way: one needs to measure $N_{k}$ [@druten2012; @aspect2003; @bouchoule2012] for $\beta \ll 0.01$ and to restore $f_{0}(x)$ by $N_{k}$ in the above-described way. If this function turns out close to $f_{0}(x)$ for impenetrable bosons in a trap [@forrester2003], then we can use the whole set $f_{j}(x)$ for such bosons and to find the whole set $\lambda_{j}$ for bosons with $\beta \sim 0.1\div 0.01$ from (\[47\]) and the experimental value of $N_{k}$ for $\beta \sim 0.1\div 0.01$. If $f_{0}(x)$ for $\beta \ll 0.01$ would turn out to be considerably different from $f_{0}(x)$ for impenetrable bosons [@forrester2003], then we need to determine the density matrix for a system with $\beta \sim 0.1\div 0.01$ and, by it, to calculate $f_{j}(x)$. Then, from the experimental values of $N_{k},$ we can find $\lambda_{j}$. The observation of a fragmented quasicondensate would be of interest.
Conclusion
==========
Using the formulae for the density matrix [@vak1989; @vak1990], we have determined the average number of atoms with momentum $\hbar k$ on low levels for the ground state of a one-dimensional uniform periodic system of interacting bosons. The solutions agree with previously obtained ones. The new results are as follows: For impenetrable point bosons, the solution in the second approximation is found (earlier, only a solution in the first approximation was obtained). For almost point bosons with weak coupling, we deduced the formula for $N_{0}$ and made the formula for the density matrix $F_{1}(R)$ to be somewhat more accurate. The most interesting result consists in the finding that the uniform system of bosons with weak coupling can possess the quasicondensate on many low levels. Such fragmented 1D-quasicondensate can be investigated by the use of a gas in a trap.
[200]{} A. Einstein, Sitzungsber. Preuss. Akad. Wiss. **1**, 3 (1925). O. Penrose and L. Onsager, Phys. Rev. **104**, 576 (1956). D.S. Petrov, D.M. Gangardt, and G.V. Shlyapnikov, J. Phys. IV Fr. **116**, 5 (2004). C.J. Pethick, H. Smith, *Bose-Einstein Condensation in Dilute Gases* (Cambridge Univ. Press, New York, 2008), Chap. 15. L.D. Mermin and H. Wagner, Phys. Rev. Lett. **17**, 1133 (1966). P.C. Hohenberg, Phys. Rev. **158**, 383 (1967). J.W. Kane and L.P. Kadanoff, Phys. Rev. **155**, 80 (1967). L. Reatto and G.V. Chester, Phys. Rev. **155**, 88 (1967). A. Lenard, J. Math. Phys. **5**, 930 (1964). V.N. Popov, Theor. Math. Phys. **11**, 565 (1972); JETP Letters **31**, 526 (1980). M. Schwartz, Phys. Rev. B **15**, 1399 (1977). H.G. Vaidya and C.A. Tracy, Phys. Rev. Lett. **42**, 3 (1979). F.D.M. Haldane, Phys. Rev. Lett. **47**, 1840 (1981). A. Berkovich, G. Murthy, Phys. Lett. A **142**, 121 (1989). D.S. Petrov, Ph. D. Thesis (FOM Institute for Atomic and Molecular Physics, Amsterdam, 2003). P.J. Forrester, N.E. Frankel, T.M. Garoni, and N.S. Witte, Phys. Rev. A **67**, 043607 (2003). C. Mora, Y. Castin, Phys. Rev. A **67**, 053615 (2003). I.A. Vakarchuk, Theor. Math. Phys. **80**, 983 (1989). I.A. Vakarchuk, Theor. Math. Phys. **82**, 308 (1990). M. Girardeau, J. Math. Phys. (N.Y.) **1**, 516 (1960). M. Schwartz, Phys. Rev. A **10**, 1858 (1974). I.A. Vakarchuk and I.R. Yukhnovskii, Theor. Math. Phys. **42**, 73 (1980). E.H. Lieb and W. Liniger, Phys. Rev. **130**, 1605 (1963). M. Tomchenko, J. Phys. A: Math. Theor. **48**, 365003 (2015). M. Tomchenko, Low Temp. Phys. **32**, 38 (2006). N.N. Bogoliubov and D.N. Zubarev, Sov. Phys. JETP **1**, 83 (1956). E. Witkowska, P. Deuar, M. Gajda and K. Rzazewski, Phys. Rev. Lett. **106**, 135301 (2011). W.J. Mullin and A.R. Sakhel, J. Low Temp. Phys. **166**, 125 (2012). A.G. Leggett, [*Quantum Liquids*]{} (Oxford Univ. Press, New York, 2006), Chap. 2. J. Gavoret and P. Nozières, Ann. Phys. (N.Y.) **28**, 349 (1964). V. Dunjko, M. Olshanii, arXiv:cond-mat/0910.0565. A.H. van Amerongen, J.J.P. van Es, P. Wicke, K.V. Kheruntsyan, and N.J. van Druten, Phys. Rev. Lett. **100**, 090402 (2008). M. Olshanii, Phys. Rev. Lett. **81**, 938 (1998). S. Richard, F. Gerbier, J.H. Thywissen, M. Hugbart, P. Bouyer, and A. Aspect, Phys. Rev. Lett. **91**, 010405 (2003). M.J. Davis, P.B. Blakie, A.H. van Amerongen, N.J. van Druten, and K.V. Kheruntsyan, Phys. Rev. A **85**, 031604R (2012). H. Feshbach, Ann. Phys. (N.Y.) **5**, 357 (1958); *ibid*. **19**, 287 (1962). L.P. Pitaevskii, Phys. Uspekhi **49**, 333 (2006). T. Jacqmin, B. Fang, T. Berrada, T. Roscilde and I. Bouchoule, Phys. Rev. A **86**, 043626 (2012).
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author:
- 'S. Wycech [^1]'
- 'J-P. Dedonder'
- 'B. Loiseau'
title: Search for baryonium and the physics at FAIR
---
Introduction {#s-1}
============
Baryonium understood as a nucleon-antinucleon quasi-bound was searched for at CERN in the days of LEAR. Nothing has been found, but broad states or states close to the threshold were not excluded [@BER82; @ADI86]. One possible reason of the failure is the heavy background due to annihilation processes. Another is the large number of allowed partial vaves. A convincing detection requires selective experiments, and the first such experiment was performed by the BES Collaboration [@BAI03] who studied the decay $$\label{1}
J/\psi \rightarrow \gamma p\overline{p}.$$ The selectivity in this reaction is due to the definite initial state and CP invariance. As shown in figure \[fig-3\] the measured spectrum of $p\overline{p}$ invariant mass displays two prominent maxima named initially $ X(1859)$ and $ X(2170)$. The first one, close to the threshold, is apparently due to strong attraction in the final iso-singlet spin-singlet $ ^{11}S_0$ wave which is one of three partial waves allowed in reaction (\[1\]). Indeed, calculations of final state interactions (FSI) performed with Paris [@LOI05] and Jülich [@HAI06] potential indicate a strong threshold enhancement. The basic difference is that with the Paris potential the threshold peak is due to a bound state and with the Jülich potential it results from a virtual state. To distinguish between these two possibilities, one needs to look directly below the $p\overline{p}$ threshold. Several methods to do this are discussed in section \[sec-2\].
Final state interactions in $J/\psi$ decays {#ss-1}
-------------------------------------------
In this note we report an extension of the FSI calculations of Ref. [@LOI05] which now covers the whole photon spectrum. The basic assumption of this approach (also that in Ref. [@HAI06]) is that the photon is emitted before the baryons are formed. The two related processes are plotted in diagrams \[fig-1\], \[fig-2\] and the FSI is calculated in terms of half-off shell $T$ matrix generated by the Paris potential [@LAC09]. This approach allows to calculate the spectrum but not the absolute decay rate. One free parameter, the radius $R$ (=0.28 fm) of a Gaussian source function is used to describe the creation of a $\gamma p\overline{p}$ state (see figure \[fig-1\]). However, in order to reproduce in a better way both maxima $ X(1859) $ and $X(2170)$, it turns out profitable to assume the radius to be weakly dependent on the photon energy. It was found to change from $0.28$ fm at maximal $k\sim1.2$ GeV to $0.39$ fm at $k=0$.
![The photon is emitted either from $J/\psi$ or during the hadronisation stage of the process and the final baryons are formed in the $S$ wave[]{data-label="fig-1"}](Fig2_jpsi1.eps){width="4cm"}
![Final state interactions described by half-off shell $T$ matrix generated by the Paris potential[]{data-label="fig-2"}](Fig2_jpsi1_FSI.eps){width="5cm"}
![The $p \overline{p}$ invariant mass spectrum obtained under the assumption that photon is emitted before the baryons are formed. The missing strength at large $p\overline{p}$ invariant mass, $M_{p\overline{p}}$, comes from the photon radiated by final hadrons [@DED14] []{data-label="fig-3"}](Fig3_direct_emi_g.eps){width="5cm"}
![The Paris $N \overline{N}$ real potential in the $ ^{11}S_0$ wave. It generates a $50$ MeV broad quasi-bound state at $ \sim 5 $ MeV binding. The well and barrier structure generate the shape resonance visible in the spectrum in figure \[fig-3\] at $ 2170 $ MeV[]{data-label="fig-4"}](Fig1_Vreal.eps){width="5cm"}
Inspection of figure \[fig-3\] shows that both states may be reproduced by the potential $N \overline{N}$ interactions related at large distances via G-parity transformation to the $N N$ interactions. However, the proper description of both peaks involves distant extrapolation of the $T_{N\overline{N}}$ matrices off energy shell, which corresponds to very short ranged interactions. This figure shows also that a sizable portion of the spectrum is missing. The missing part comes from photon emissions by final baryons: $N\overline{N}$ and exchange currents [@DED14]. Similar final state emission model offers a consistent description of the rate and the spectra in cases of $ \pi,\omega,\phi$ mesons with a constant $R=.28$ fm radius. This knowledge of initial formation radius allows to calculate inverse reactions leading to formation of $ J/\psi$ and a meson in nuclear collisions of $\overline{p}$ [@DED14].
Looking below the $N \overline{N}$ threshold {#sec-2}
============================================
Testing the subthreshold amplitudes may be realised in few body systems in particular in light antiprotonic atoms or at extreme nuclear peripheries. In these conditions nucleons are bound and the effective subthreshold energies are composed of binding energies and recoil of the $N\overline{p}$ pair with respect to the rest of the system. For valence nucleons the $ - E_{binding} - E_{recoil} $ may reach down to - 40 MeV. Here, we want to point out one experiment that might be of interest at FAIR/FLAIR. Table \[tab:radio\] shows the ratios of antiproton capture rates on neutrons and protons $ C(n \overline{p})/ C(p \overline{p})$ bound to nuclear peripheries. These reflect the ratios of neutron and proton densities. The second and third columns indicate such ratios extracted from widths of two antiprotonic atomic levels the “lower” and the “upper” one. These widths are determined at nuclear densities $ \sim 10 \% $ and $ \sim 5\%$ of the central density $\rho_0$. The last column is obtained with radiochemical studies of final nuclei with one neutron or one proton removed in the annihilation reaction [@LUB97]. The latter process is localised at densities $\rho \sim 10^{-3} \rho_0$. In standard nuclei shown in the upper part of the table the ratios $ C(n \overline{p})/ C(p \overline{p}) \sim \rho_n /\rho_p $ increase at nuclear peripheries. However, in some nuclei characterised by a small proton binding, indicated in the lower part of the table, the ratio $ C(n \overline{p})/ C(p \overline{p})$ suddenly drops at extreme nuclear peripheries. That effect cannot be explained by the nuclear structure alone and we attribute it to the existence of a narrow bound state in the $N \overline{N}$ system. Such a narrow state is in fact predicted by the Paris potential in the $^{33}P_1$ wave, see table \[tab-1\].
atom lower upper radiochemistry
------------ ---------- ---------- ----------------
$^{96}$Zr 0.95(9) 1.53(29) 2.6(3)
$^{124}$Sn 1.79(10) 2.44(39) 5.0(6)
$^{106}$Cd 1.64(80) 2.10(80) 0.5(1)
$^{112}$Sn 1.90(13) 2.43(49) 0.79(14)
: Ratios of $ N(n \overline{p}) $ and $ N(p \overline{p}) $ capture rates from atomic states. The last column shows experimental numbers from radiochemical experiments. Other columns (see text) give ratios calculated with optical potential and plausible nuclear densities based on experimental results from Ref. [@LUB97][]{data-label="tab:radio"}
Another way to look below threshold is opened by the BES collaboration experiment [@ABL05] detecting radiative $J/\psi$ decays $$\label{2}
\ J/\psi \rightarrow \gamma \pi^+ \pi^- \eta'$$ and observation that the mesons are correlated to another state named $ X(1835)$ with quantum numbers consistent with the $ X(1859)$ state. Within the Paris-potential interpretation, the $ X(1835)$ is due to the interference of the baryonium $^{11}S_0$ state, seen above the threshold, with a background formation amplitude [@DED09].
$^{2T+1\ 2S+1}L_J$ $ E - i \Gamma/2$
-------------------- -------------------
$^{11}S_0$ -4.8-i26
$^{33}P_1$ -4.5-i9.0
: Binding energies in MeV of the close to threshold quasi-bound states in the Paris potential [@LAC09] []{data-label="tab-1"}
Experiments suggested
----------------------
Following the indications from BES experiments the baryonia should be searched in the region of $0 - 60$ MeV below the $N \overline{N}$ threshold. It would be advisable to repeat two old experiments at different energies, possibly with polarized particles :
$\bullet$ Search for narrow signals in the $\gamma$-spectrum from $p \overline{p}$ annihilation was performed at rest [@ADI86], The signals (in the region that we expect them now to exist) were covered by heavy background due to $\pi^0$ decays and $\pi^-p \rightarrow \gamma n $. It would be better to perform this experiment with few hundred MeV antiprotons which would shift the expected signal away from the region of high background.
$\bullet $ The $ \overline{p} d \rightarrow n X $ experiment [@BER82] was performed at $ 1.3 $ GeV/c. This gives rather small chance of $p\overline{p}$ coupling in the statistically insignificant $ ^{11}S_0$ wave. Lower energies would be better.
New instructive experiments that possibly could be performed in the coming years are
$\circ$ Fine structure splitting in light antiprotonic atoms $^1H,^2H,^3H,^3He,^4He$ would allow to trace energy dependence of the selected $ \overline{p}N$ amplitudes in the subthreshold region down to $ \sim - 40 $ MeV.
$\circ$ Studies of mesons emitted from annihilations of $ \overline{p} $ at nuclear peripheries. In particular nuclei with closed shells with one loosely bound valence nucleon could be profitable. In the latter case the baryonium signal would be separated from a complicated background due to other annihilation channels.
This work has been partially supported by a grant from the French-Polish exchange program COPIN/CNRS- IN2P3, collaboration 05-115. SW was also supported by Narodowe Centrum Nauki grant 2011/03/B/ST2/00270
[00]{} R. Bertini *et al.*, *Search for narrow baryonium states near the $N \overline{N}$ threshold*, Nucl. Phys. **B 209**, 269 (1982)
L. Adiels *et al.*, *Search for narrow signals in the $\gamma$-spectrum from $p \overline p$ annihilation at rest*, Phys. Lett. **B 182**, 405 (1986)
J. Z. Bai [ *et al.*]{} (BES Collaboration), *Observation of a near-threshold enhancement in the $p\overline{p}$ mass spectrum from radiative $J/\psi \to \gamma p \overline{p}$ decays*, Phys. Rev. Lett. **91**, 022001 (2003) and M. Ablikim [*et al.*]{} (BES Collaboration), *Analysis of $p \overline p$ threshold structure in $J/\psi$ and $\psi'$ radiative decays*, Phys. Rev. Lett. **108**, 112003 (2012)
B. Loiseau and S. Wycech, *Antiproton-Proton Channels in J$/\psi$ Decays*, Phys. Rev. C **72**, 011001 (2005)
J. Haidenbauer, Ulf-G. Meißner and A. Sibirtsev, *Near threshold $ p \overline{p}$ enhancement in $B$ and $ J/\psi $ decay*, Phys. Rev. D **74**, 017501 (2006)
B. El-Bennich, M. Lacombe, B. Loiseau and S.Wycech, *Paris $N\overline N$ potential constrained by recent antiprotonic-atom data and $\overline{n} p$ total cross sections*, Phys. Rev. C **79**, 054001 (2009)
J-P. Dedonder, B. Loiseau and S. Wycech, *On the photon or meson formation in $J/\psi$ decays into $p \overline p$*, to be published
P. Lubinski [*et al.*]{}, *Composition of the nuclear periphery from antiproton absorption*, Phys. Rev. C [**57**]{}, 2962 (1997)
M. Ablikim [*et al.*]{} (BES Collaboration), *Observation of a resonance X(1835) in $ J/\psi \rightarrow \gamma \pi^{+}\pi^{-}\eta'$*, Phys. Rev. Lett. **95**, 262001 (2005)
J-P. Dedonder, B. Loiseau, B. El-Bennich and S. Wycech, *Structure of X(1835) baryonium*, Phys. Rev. C **80**, 045207(2009)
[^1]:
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---
abstract: 'This work explores dynamics existing in interactions between players. The dynamic system of games is a new attitude to modeling in which an event is modeled using several games. The model allows us to analyze the interplay capabilities and the feasibility objectives of each player after a conflict with other players objectives and capabilities. As an application we model relations between Soviet Union and America after World War II to October 1962, by using the dynamic system of games. The dynamic system of games as an important insight clearly has significant implications for modeling strategic interactions in which player pursue goals for increasing their personal interests. In addition, we introduce a new game in which there is a dilemma which this dilemma occurs in most societies. We investigate depends on the claim that each player in this dilemma is hyper-rational. In concept of hyper-rational, the player thinks about profit or loss of other actors in addition to his personal profit or loss and then will choose an action which is desirable to him. In this dilemma, a weak trust has been created between players, but it is fragile.'
author:
- 'Gholamreza Askari, Madjid Eshaghi Gordji and Ali Zarei'
title: 'The dynamic framework of decision-making'
---
Introduction
============
Game theory provide the theoretical underpinnings of analytical techniques and their application in social research. Indeed more fields of science have benefited from game theoretic models. Dynamic games provide a framework for modeling the behavior of players in situations where there are dynamic strategic interactions. Long provides a survey of models of dynamic games in industrial organization [@Van]. At the *Theory of Games and Economic Behavior*, write: We repeat most emphatically that our theory is thoroughly static. A dynamic theory would unquestionably be more complete and therefore preferable [@Von]. Eshaghi and Askari, recently introduced a new method of modeling in game theory, named the dynamic system of strategic games [@Eshaghi]. In this modeling, new properties of games such as game-maker game, strategy-maker game and the pair of rational actions are presented and with the help of these properties, the dynamics of players behavior are studied. According to this feature, strategic games were divided into two classes, strategy maker games, and games that aren’t strategy maker. Also, Strategy maker games itself are of two groups and games that aren’t strategy maker itself are two groups. Here we present required concepts and terms, some advantages of this modeling and prove two propositions. Exactly similar to everyday life or the process of negotiations, in the dynamic system of games after the choice of strategy or rational action pairs by the players, by entering into the new games the productive game is out of reach. Therefore, the producer games are out of reach of the players and only as a history of the system that can influence the selection of strategies and games in the future.
As an application of the dynamic system of strategic games, we seek modeling of conflict between Soviet Union and America after World War II to October 28, 1962. The Cold War was a state of geopolitical tension after World War II between powers in the Eastern Bloc and powers in the Western Bloc. Prior to 1957, Cuba had for many years been under the complete economic and political control of the United States. In late 1956 a revolution to overthrow the Batista regime was initiated by Fidel Castro, and in 1959 he managed to take control of Cuba. The situation was made more unstable by the unsuccessful Bay of Pigs and a declaration by the Soviet Union of its willingness to aid Cuba in defending itself against the US [@Fraser]. Countries Soviet Union and America also belong to different models of international relation and political systems. For modeling of Cuban missile crisis, we divide the period into 5 parts and in each period we examine static games with perfect information which had occurred. In return for a public US pledge not to invade Cuba and a private assurance that the US controlled missiles in Turkey would eventually be dismantled, the Soviets agreed to withdraw the missiles [@Zagare].
Theory of Game have introduced the variety of games to express the circumstances of the event [@Brams]. The game theorists present the variety of models for interpreting the interaction conditions that players face, which they suggest solutions to resolve these conflicts( see [@Yeung; @Ungureanu]). In the following, we introduced Rostam’s Dilemma. Rostam’s Dilemma is a symmetric $2\times 2$ game that examines difficult conditions of decision making between players. Difference between this dilemma and some of other dilemmas is that in the Rostam’s Dilemma, cooperation is weak dominant over non-cooperation. As an application of this Dilemma, social dilemma in the way of a wife and husband is modeled.
Dynamic system of games {#S.2}
=======================
Maybe an event can’t be modeled completely through one game but there is more chance with several games [@Eshaghi]. The dynamic system of strategic games is a dynamic model of $2\times 2$ games. The advantages of this modeling include:
- Considering the impact of games on each other
- Using several games to model an event
- Dynamic interaction between players
- Performance of players’ rationality within the system
- Selecting an agreed mechanism between players to achieve cooperation
- Finding solutions to conflicts
In dynamics of the decision-making process in traditional game theory by modeling the game in its extensive form, in each node one player was able to make a decision, but in dynamic system of games in each node, two or more players are able to make decisions. The inability to model the dynamical element of game play in static game theory, and the extent to which dynamic system of games naturally incorporates dynamical considerations, reveals an important virtue of dynamic system of games.
Here, we consider strategic $2\times 2$ games with perfect information. If a game produces other games, it is called *game-maker game*. In general, if the games $g_1, g_2, ...,g_n$ generate games $g^{'}_1, g^{'}_2, ...,g^{'}_m$, then $g_i$- and $g^{'}_i$-s are called *producer* and *produced*, respectively. We call the form of displaying game-maker games as *dynamic system of strategic games*.
If a game create one or more strategies is will be called *strategy-maker game*. Each *strategy* has at least two pairs of actions. Each pair of actions includes players’ payoffs. The produced strategies can be dominant strategy, dominated strategy, weakly dominant strategy and weakly dominated strategy. Therefore, each dominant action of a player can be called *dominant strategy* of a player. If a game doesn’t generate any strategy, the game isn’t strategy maker.
In a strategic game with ordinal preferences, player $i$s action $a^{''}_{i}$ strictly dominates her action $a^{'}_{i}$ if $$u_i(a^{''}_{i}, a_{-i})> u_i(a^{'}_{i}, a_{-i}) \;\ for \;\ every \;\ a_{-i}\in A_{-i},$$ where $u_i$ is a payoff function that represents player i’s preferences [@Webb]. If for player $i$ the action $a^{''}_{i}$ is preferred to action $a^{'}_{i}$ per every choice of action of other players, it is called *strictly dominant strategy* and is shown by $S^{j}_{i}$ where ${ _{k}S^{j}_{i}}$ shows $j$-th strategy of $i$-th player from $k$-th game.
In a strategic game with ordinal preferences, player $i$’s action $a^{''}_{i}$ weakly dominates her action $a^{'}_{i}$ if $$u_i(a^{''}_{i}, a_{-i})\geq u_i(a^{'}_{i}, a_{-i}) \;\ for\; every \;\ a_{-i}\in A_{-i}$$ and $$u_i(a^{''}_{i}, a_{-i})> u_i(a^{'}_{i}, a_{-i}) \;\ for \;\ some \;\ a_{-i}\in A_{-i},$$ where $u_i$ is a payoff function that represents player i’s preferences [@Webb]. If for player $i$ the action $a^{''}_{i}$ is preferred over action $a^{'}_{i}$ for each action choice of other players, it is called *weakly dominant strategy* and will be represented by $S^{j}_{i}$.
If a game with $n$ players is strategy maker for $k$ players ($1\leq k \leq n$) it is called *strategy maker game of order* $(n, k)$. If a game with $n$ players isn’t strategy maker it is called *strategy maker game of order* $(n, 0)$. In other word, we can consider a strategy maker game of order $(n, 0)$ as a game which is not strategy maker. In the following, we introduce pair of rational actions. Players’ preferences on pairs of rational actions are based on payoffs that they obtain.
(*Pair of rational actions*) A pair of actions is called *rational* if at least hold true in one of the following conditions:
- would be Nash equilibrium;
- pair of actions, would be Pareto dominant for both players over other pairs of actions;
- for each game that is strategy maker of order $(2, 1)$, pairs of rational actions for one player is responses to dominant strategy or weakly dominant strategy produced for other player.
In a strategy maker game of order $(2, 2)$ where both players have strictly dominant strategy and the game hasn’t Pareto action pairs over Nash equilibrium, the Nash equilibrium of game is the only rational actions pair. In Fig.\[fig:A\], Stag Hunt game $g_1$ is strategy maker of order $(2,0)$. This game has Nash equilibriums $({_{1}C},{_{1}C})$ and $({_{1}D},{_{1}D})$. Also in this game, $({_{1}C},{_{1}C})$ is the dominant Pareto compared to pair of action $({_{1}D},{_{1}D})$. Therefore, set of rational actions pair of players includes $({_{1}C},{_{1}C})$ and $({_{1}D},{_{1}D})$.
(4)
[c|c|c|]{} & &\
& &\
${_{1}C}$ & 4,4 & 1,3\
${_{1}D}$ & 3,1 & 2,2\
;
A tools that can display dynamic system of strategic games is games graph. Within each node of a graph, there is a strategic game in which players can make decisions. Each node of this graph can be generator of the next game through the two following methods and be connected to it:
1. strategies
2. pair of rational actions
Moving to the next node by the made strategy is always preference of one of players but continuing game with rational action pair may be preference of one player or both of them. Graph nodes are considered as *initial node*, *move node* and *final node*. Nodes that players desire to continue the game are called *move node*. *Final node* is a node that players have no desire to continue the games (system). Every dynamic system of games includes players set, strategies set, set of rational actions pairs, system history and node and systemic preferences of players.
A graph $\mathcal{G}$ is given by $(G, E)$, where $G=\{g_1, g_2, ..., g_n\}$ is a finite set of nodes or vertices that each node of this graph is a strategic game and $E=\{g_ig_j, g_sg_r,..., g_kg_l\}$ is a set of pairs of vertices (or 2-subsets of $E$) called branches or edges which indicates which nodes are produced strategies or pair of rational actions. Set of all strategies produced by $k$-th game is represented by ${_{k}\mathcal{S}}={_{k}\mathcal{S}_1}\cup {_{k}\mathcal{S}_2} \cup \emptyset$. Set of all pairs of actions players’ in $k$-th game is shown with ${_{k}\mathcal{A}}={_{k}A_1}\times {_{k}A_2}$. Set of all pairs of rational actions for player $i$ is shown with ${_{k}\mathcal{A}^{'}_{i}}$ that is a subset of ${_{k}\mathcal{A}}$, for all $k\in\{1,2,...,n\}$.
Let $\mathcal{A}={_{1}\mathcal{A}} \cup {_{2}\mathcal{A}} \cup ... \cup {_{n}\mathcal{A}}\cup \emptyset$ and $\mathcal{S}={_{1}\mathcal{S}}\cup {_{2}\mathcal{S}} \cup...\cup {_{n}\mathcal{S}}$ be two set. The set valued functions, rational actions pair $\phi^{'}_{i}: G\to \mathcal{A}$ and strategy maker $\phi_i: G\to \mathcal{S}$ for players $i$’s are defined as follows: $$\phi^{'}_{i}(g_k)={_{k}\mathcal{A}^{'}_{i}}=
\left\{\begin{array}{rl}
\{({_{k}a_i}, {_{k}a_{-i}})_i | ({_{k}a_i}, {_{k}a_{-i}})_i\in {_{k}\mathcal{A}} \ \ \ \ if~g_k~has~pair~of~ \ \ \ \ \ \ \ \ \ \ \\ rational~actions \ \ \ \ \ \ \ \ \ \ \ \ \\
\emptyset \ \ \ \ \ \ \ \ \ \ \ \ \ ~if~g_k~has'nt~pair~of \ \ \ \ \ \ \ \ \ \ \ \\
\ \ \ \ \ \ rational~actions, \ \ \ \ \ \ \ \ \ \ \ \
\end{array}\right.$$
$$\phi_{i}(g_k)={_{k}\mathcal{S}_i}=
\left\{\begin{array}{rl}
\{{_{k}S^{j}_{i}}| {_{k}S^{j}_{i}}\in {_{k}\mathcal{S}}\} \ \ \ \ \ \ if~g_k~is~strategy~maker~for~player~i \\
\emptyset \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ~if~g_k~is'nt~strategy~maker~for~player~i,
\end{array}\right.$$
for all $i\in N$ and $j,k\in I=\{1,2,...,n\}$, where $g_k$ shows $k$-th game, $({_{k}a_i}, {_{k}a_{-i}})_i$ shows rational actions pair of $i$-th player from $k$-th game and ${ _{k}S^{j}_{i}}$ shows $j$-th strategy of $i$-th player from $k$-th game. Every move of system as a member of set $M$ is as follows: $$\begin{aligned}
&M:=\Big\{m^{j}_{k}| \;\ m^{j}_{k}={_{k}S^{j}_{i}}\;\ or \;\ m^{j}_{k}=({_{k}a_i}, {_{k}a_{-i}})_i \ \ or \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ m^{j}_{k}=({_{k}a_i}, {_{k}a_{-i}})_{i,j} \ \ \forall \;\ {_{k}S^{j}_{i}}\in {_{k}\mathcal{S}_i}, \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ({_{k}a_i}, {_{k}a_{-i}})_{i} \in {_{k}\mathcal{A}_i}, \ \ ({_{k}a_i}, {_{k}a_{-i}})_{j} \in {_{k}\mathcal{A}_j} \Big\},\end{aligned}$$ where $m^{j}_{k}$ shows $j$-th move of $k$-th game and $({_{k}a_i}, {_{k}a_{-i}})_{i,j}$ shows the pair of rational action selected by players $i$ and $j$ of $k$-th game. Players’ move function $\varphi_{i}: M \to G^2$ and $\varphi_{i,j}: M \to G^2 \cup \emptyset$ with $ \varphi_{i,j}({_{k}S^{j}_{i}})=\emptyset$ is defined as following: $$\varphi_{i}(m^{j}_{k})=
\left\{\begin{array}{rl}
(g_k, g_p)=g_kg_p & ~~if~~ m^{j}_{k}={_{k}S^{j}_{i}} \\
(g_k, g_q)=g_kg_q & ~~if~~ m^{j}_{k}=({_{k}a_i}, {_{k}a_{-i}})_i,
\end{array}\right.$$ $$\varphi_{i,j}(m^{j}_{k})=
\left\{\begin{array}{rl}
\emptyset \ \ \ \ & ~~if ~~~~ m^{j}_{k}={_{k}S^{j}_{i}}\\
(g_k, g_s)=g_kg_s & ~~if ~~~~ m^{j}_{k}=({_{k}a_i}, {_{k}a_{-i}})_{i,j}.
\end{array}\right.$$
The above function shows by what move two game nodes have been connected to each other by one or both players. Consequently, it can be said that in move $m^{j}_{k}={_{k}S^{j}_{i}}$, nodes $g_k$ and $g_p$ have been connected through the strategy selected by player $i$ to each other. In move $m^{j}_{k}=({_{k}a_i}, {_{k}a_{-i}})_i$ the nodes $g_k$ and $g_q$ have been connected by pair of rational action selected by player $i$ to each other. In move $m^{j}_{k}=({_{k}a_i}, {_{k}a_{-i}})_{i,j}$ the nodes $g_k$ and $g_s$ have been connected through a pair of rational actions selected by players $i$ and $j$ to each other.
Consider that $H$ is a set including all series (finite and infinite) that hold true in the following conditions:
1. $\emptyset$ is member of $H$.
2. Sequence $\big\{m^{j}_{i},\{g_k, m^{j}_{k}\}\big\}_{i,j,k\in I }$ for all $i, j, k\in \{1,2,...,n\}$, is a member of $H$. Each member of $H$ is called a history and is represented by $h$.
3. History $h=\big\{m^{j}_{i},\{g_k, m^{j}_{k}\}\big\}_{i,j,k\in I }$ is called final history if it is infinite or there isn’t $g_{k+1}$ that is a member of h.
The set $H$ is called *system history*. Preferences of each node of a games system that are exactly the same preferences on the pairs of a strategic game actions are called *node preferences* or *tactical preferences*. Preferences on strategies set or set of rational actions pair of a game is called *systemic preferences* or *strategic preferences*.
(*Dynamic system of strategic games*) A dynamic system of strategic games with perfect information including:
- a set of players
- for each player, a set of strategies
- for each player, a set of rational actions pair
- system history
- node preferences (tactical preferences) on set of all actions pairs
- systemic preferences (strategic preferences) on strategies or pairs of rational actions.
This modeling allows players to design games to earn most benefit during the negotiation and based on their bargaining power. In the produced games, players and strategies may change depending on conditions and also players and strategies may increase or decrease. This system will be provided practical freedom in which players can analyze their interaction conditions with others in the future and decide what strategy they choose and at what level has interaction. Also, a player may choose an action that has less outcome for him, in game conditions, but, obtain more benefit in future according to strategic preferences and conversely may he prefer present over future or a player may prefer tactical preferences on strategic preferences. From this modeling, the following propositions can be deduced.
\[P.1\] Each strategy-maker game $G$, has at least one pair of rational actions.
Suppose $G$ is a strategy-maker game. Then game $G$, either is strategy-maker game of order $(2, 1)$ or is strategy-maker game of order $(2, 2)$. If game $G$, is strategy-maker game of order $(2, 1)$, then pairs of rational actions for one player is responses to strictly dominant strategy or weakly dominant strategy produced for other player. If game $G$, is strategy-maker game of order $(2, 2)$, both players either have strictly dominant strategy and game hasn’t Pareto action pairs over Nash equilibrium, then the Nash equilibrium of game is the only rational actions pair or have weakly dominant strategy, then in this case game has more than one rational actions pair.
Contrary to the above proposition, do not be right. That is, there are games that has pair of rational actions, but are not necessarily a strategy-maker.
\[P.2\] Each game $G$ that does not have a pair of rational actions is not strategy-maker.
The proof is by contradiction. First suppose $G$ is a strategy-maker game. Then by proposition \[P.1\], game $G$ has at least one pair of rational actions, which is inconsistent with main assumption of proposition \[P.2\].
Contrary to the above proposition, do not be right. That is, there are games that are not strategy-maker, but has a pair of rational actions. In the following, we propose a different model from coincident attendance Soviet Union and America in Cuban Missile Crisis.
Cuban Missile Crisis
====================
Soviet Union and America relations after World War II sometimes was strained. But the most important and dangerous crisis in Soviet Union and America relations, that put the world in the edge of a destructive war, was on establishment of Soviet Missile bases in Cuba. Soviet Union wanted to establish a massive arsenal in this region of Cuba in which deploy great number of heavy weapons to become the largest military station of this country in Caribbean. But Khrushchev’s trick was ultimately revealed. In October 1962 an American spy plane discovered the site of missile launch in the west of Havana that confirmed establishment of medium-range missiles in Cuba. To confront with danger that threat America’s security, America’s then president also constituted a war council and for one week, a serious and secret discussion was helpful in White House. In this Council, different suggestions including an all-out military attack to Cuba and military occupation of the country or bombarding missile launch platform in Cuba were announced. In continuance, Kenedi, in 22 October 1962 warned American people that Soviet government has established missile bases in Cuba and deploy of these missiles in a distance of one hundred mile from America beaches is a threat for the country’s security. Kennedy at the same time with this warning announced that Soviet government must remove its missile bases from Cuba. Following this, he issued the order of Cuba naval blocked to prevent transfer of Soviet’s new missile equipments to Cuba [@Kennedy]. This issue became a world crisis for one week.
Now, using dynamic system of games, we model relations between Soviet Union and America after World War II to October 28, 1962. To this end, we divide the period into 5 parts and in each period we examine static games with perfect information which had occur. The first period is after World War II to October 13, 1962 that is shown in the form of game $g_1$. Second period from October 14 to 22 includes games $g_2$ and $g_3$. Third period is from October 23 to 25 that includes games $g_4$, $g_5$, $g_6$ and $g_7$. Fourth period is from October 26 to 27 that includes games $g_8$ and $g_9$. And finally, the day October 28 that includes game $g_{10}$. For more information about Cuban Missile Crisis, refer to references (see [@Blaschke; @Crall; @May; @Polletta]).
Relations of Soviet Union and America after World War II and during 1950s were entered in Deadlock game and they hadn’t much desire to cooperation with each other. As a result, both countries had selected non-cooperation. Deadlock game $g_1$ has been shown in Fig. \[fig:Cuba missile crisis\]. Soviets is assumed as row player (player $1$) and America is assumed column player (player $2$). The players’ set of actions include cooperation ${_{1}C}$ and defect ${_{1}D}$. Players’ preferences in this node are the same order preferences of strategic game $g_1$. The game Nash equilibrium is $({_{1}D},{_{1}D})$. In $g_1$, dominant strategy $_{1}S^{1}_{1}$ is defect and dominated strategy $_{1}S^{2}_{1}$ is cooperation for player $1$. Also this game is producer of dominant strategy of defect $_{1}S^{1}_{2}$ and dominated strategy of cooperation $_{1}S^{2}_{2}$ for player $2$. In other words, game $g_1$ is strategy maker of order $(2, 2)$. The only pair of rational actions for players is $({_{1}D},{_{1}D})_{1,2}$. Based on players being rational, they select dominant strategy $_{1}S^{1}_{1}$ and dominant strategy $_{1}S^{1}_{2}$ for the game continuation. According to its dominant strategy, Soviet, was convinced in July 1962 to implement its Atomic Missile Establishment plan in Cuba that in order to confront with America’s growing expansion. This strategy caused Soviet to hostage a part of Havana in Cuba (America’s backyard). But Khrushchev’s trick was ultimately revealed. In October 1962 an American spy plane discovered the missile launch sit in the west of Havana that confirmed establishment of medium-range missiles in Cuba.
Strategy $_{1}S^{1}_{1}$ ends to Hostage game $g_2$. Soviet wanted to surprise Americans. In this game, Soviet has two actions: either cooperates ${_{2}C}$ through removing missile or defect ${_{2}D} $ and maintain missiles. America could reveal the issue ${_{2}C}$ or keep it secret ${_{2}D} $. The game Nash equilibrium is $({_{2}D},{_{2}C})$. This game is producer of dominant strategy of defect $_{2}S^{1}_{1}$ and dominated strategy of cooperation $_{2}S^{2}_{1}$ for player $1$ and also dominant strategy $_{2}S^{1}_{2}$ for player $2$ is reveal the issue and dominated strategy $_{2}S^{2}_{2}$ is keep it secret. The only pair of rational actions for both players is $({_{2}D},{_{2}C})_{1,2}$.
Strategy $_{1}S^{1}_{2}$ ends to Self-Serving game $g_3$. In $g_3$, America has two actions: either it does nothing and makes cooperation ${_{3}C}$ or makes defect ${_{3}D}$ through blockade of Cuba. Also, Soviet also has two actions: either it cooperates ${_{3}C}$ to remove missile through diplomatic canal or makes defect ${_{3}D}$ and maintain missiles. The game Nash equilibrium is $({_{3}C},{_{3}D})$. $g_3$ is strategy maker of order $(2, 1)$. This game is producer of dominant strategy of defect $_{3}S^{1}_{2}$ and dominated strategy of cooperation $_{3}S^{2}_{2}$ for player 2. Pairs of rational actions for players are $({_{3}C},{_{3}D})_{1,2}$ and $({_{3}D},{_{3}D})_{1}$.
Strategy $_{2}S^{1}_{1}$ ends to Chicken $g_4$. In $g_4$, Soviet has two actions: either doesn’t attack to America ${_{4}C}$ or attack to America ${_{4}D}$. Also, America has two actions: either it doesn’t attack to Soviet ${_{4}C}$ or attacks to Soviet for retaliation ${_{4}D}$. The game Nash equilibria are $({_{4}C},{_{4}D})$ and $({_{4}D},{_{4}C})$. $g_4$ is strategy maker of order $(2, 0)$. In other words, this game isn’t strategy maker for players. The players’ pairs of rational actions are $({_{4}C},{_{4}C})_{1,2}$, $({_{4}D},{_{4}C})_{1}$ and $({_{4}C},{_{4}D})_{2}$.
Strategy $_{2}S^{1}_{2}$ ends to Stag Hunt $g_5$. In $g_5$, America has two actions: either through diplomacy seeks solution ${_{5}C}$ or through diplomacy put pressure on Soviet ${_{5}D}$. Also, Soviet has two actions: either it seeks solution ${_{5}C}$ through diplomacy or thinks about crisis intensification ${_{5}D}$. The game Nash equilibria are $({_{5}C},{_{5}C})$ and $({_{5}D},{_{5}D})$. $g_5$ is strategy maker of order $(2,0)$. The players’ pair of rational actions are $({_{5}C},{_{5}C})_{1,2}$ and $({_{5}D},{_{5}D})_{1,2}$.
Pair of rational actions $({_{3}C},{_{3}D})_{1}$ ends to another Stag Hunt $g_6$. In $g_6$, Soviet has two actions: either it doesn’t break blockade ${_{6}C}$ or break the blockade ${_{6}D}$. Also, America has two actions: either it doesn’t conflict ${_{6}C}$ with Soviet ships or conflict with Soviet ships ${_{6}D}$. The game Nash equilibria are $({_{6}C},{_{6}C})$ and $({_{6}D},{_{6}D})$. $g_6$ is strategy maker of order $(2,0)$. The players’ pair of rational actions are $({_{6}C},{_{6}C})_{1,2}$ and $({_{6}D},{_{6}D})_{1,2}$.
Strategy $_{3}S^{1}_{2}$ ends to Chicken $g_7$. In $g_7$, America has two actions: either doesn’t attack to Soviet ${_{7}C}$ or attack to Soviet ${_{7}D}$ . Also, Soviet has two actions: either it doesn’t attack to America ${_{7}C}$ or attacks to America in a retaliatory invasion ${_{7}D}$. The game Nash equilibria are $({_{7}C},{_{7}D})$ and $({_{7}D},{_{7}C})$. $g_7$ is strategy maker of order $(2, 0)$. The players’ pairs of rational actions are $({_{7}C},{_{7}C})_{1,2}$, $({_{7}D},{_{7}C})_{1}$ and $({_{7}C},{_{7}D})_{2}$.
Based rationality of players and strategic preferences, players selecting pairs of rational actions $({_{4}C},{_{4}C})_{1,2}$, $({_{5}C},{_{5}C})^{'}_{1,2}$ and $({_{6}C},{_{6}C})_{1,2}$ end to Coordination game $g_8$. In $g_8$, Soviet has two actions: either it issues the command of ships not to move toward Cuba ${_{8}C}$ or thinks about resolving the crisis through negotiation ${_{8}D}$. Also, America has two actions: either it issues the command that America’s ships not confront with Soviet ${_{8}C}$ or thinks about resolving the crisis through negotiation ${_{8}D}$. The game Nash equilibria are $({_{8}C},{_{8}C})$ and $({_{8}D},{_{8}D})$. $g_8$ is strategy maker of order $(2,0)$. The players’ pairs of rational actions are $({_{8}C},{_{8}C})_{1,2}$ and $({_{8}D},{_{8}D})_{1,2}$.
Based on players’ rationality and strategic preferences, players by selection of rational actions pairs $({_{5}C},{_{5}C})_{1,2}$, $({_{6}C},{_{6}C})^{'}_{1,2}$ and $({_{7}C},{_{7}C})_{1,2}$ ends to Rostam’s Dilemma $g_9$. In $g_9$, Soviet has two actions: either it removes missile from Cuba ${_{9}C}$ or maintains missiles in Cuba ${_{9}D}$. Also, America has two actions: either it removes blockade of Cuba ${_{9}C}$ or maintains blockade of Cuba ${_{9}D}$. The game Nash equilibria are $({_{9}C},{_{9}C})$ and $({_{9}D},{_{9}D})$. $g_9$ is strategy maker of order $(2, 2)$. In $g_9$, the weak dominant strategy $_{9}S^{1}_{1}$ for Soviet is removing missiles and its weakly dominated strategy $_{9}S^{2}_{1}$ is maintaining missiles. Also, for America weakly dominant strategy $_{9}S^{1}_{2}$ is removing blockade and weakly dominated strategy $_{9}S^{2}_{2}$ is maintaining blockade. The players’ pairs of rational actions are $({_{9}C},{_{9}C})_{1,2}$, $({_{9}D},{_{9}D})_{1,2}$, $({_{9}D},{_{9}C})_{1}$ and $({_{9}C},{_{9}D})_{2}$.
Based on players’ rationality and strategic preferences, players by selection of pairs of rational actions $({_{8}C},{_{8}C})_{1,2}$ and $({_{9}C},{_{9}C})_{1,2}$ ends to game Win-Win $g_{10}$. In this game, both players have two actions cooperation ${_{10}C}$ and defection ${_{10}D}$. The game Nash equilibrium is $({_{10}C},{_{10}C})$. In this step, the players have no appetence to continue so this completed the system. Dynamic system of game with strategic games between Soviet and America is represented by graphs in Fig. \[fig:Cuba missile crisis\]. History of system is as follows: $$\begin{aligned}
&H=\Big\{\emptyset, \big\{g_1, {_{1}S^{1}_{1}}, {_{1}S^{1}_{2}} \big\},\big\{{_{1}S^{1}_{1}},\{g_2, {_{2}S^{1}_{1}}, {_{2}S^{1}_{2}}\}\big\}, \big\{{_{1}S^{1}_{2}},\{g_3, {_{3}S^{1}_{2}}, ({_{3}C},{_{3}D})_{1}\}\big\},\\
&\;\;\;\;\;\;\;\;\;\
\big\{{_{2}S^{1}_{1}}, \{g_4, ({_{4}C},{_{4}C})_{1, 2}\}\big\}, \big\{{_{2}S^{1}_{2}},\{g_5, ({_{5}C},{_{5}D})_{1,2}, ({_{5}C},{_{5}D})^{'}_{1,2}\}\big\}, \\
&\;\;\;\;\;\;\;\;\;\
\big\{({_{3}C},{_{3}D})_{1},\{g_6, ({_{6}C},{_{6}D})_{1,2}, ({_{6}C},{_{6}D})^{'}_{1,2}\}\big\}, \big\{{_{3}S^{1}_{2}}, \{g_7, ({_{7}C},{_{7}C})_{1, 2}\}\big\},\\
&\;\;\;\;\;\;\;\;\;\
\big\{({_{4}C},{_{4}D})_{1,2}, ({_{5}C},{_{5}D})^{'}_{1,2}, ({_{6}C},{_{6}D})_{1,2}, \{g_8, ({_{8}C},{_{8}C})_{1, 2}\}\big\}, \\
&\;\;\;\;\;\;\;\;\;\
\big\{({_{5}C},{_{5}D})_{1,2}, ({_{6}C},{_{6}D})^{'}_{1,2}, ({_{7}C},{_{7}D})_{1,2}, \{g_9, ({_{9}C},{_{9}C})_{1, 2}\}\big\}, \\
&\;\;\;\;\;\;\;\;\;\
\big\{({_{8}C},{_{8}C})_{1,2}, ({_{9}C},{_{9}D})_{1,2}, \{g_{10}\}\big\}\Big\}.\end{aligned}$$
![Dynamic system of games between Soviet Union and America[]{data-label="fig:Cuba missile crisis"}](MIS){width=".8\linewidth"}
In Fig. \[fig:Cuba missile crisis\], if Chicken had dominant strategy even for one player, atomic war was probably occurred. This weakness of not having dominant strategy in this game causes the players to consider their strategic preferences and not to use tactical preferences inside the game. In other words, players prefer strategic preferences over tactical preferences. Therefore, it can be concluded that in Cuban missile crisis, the two countries reached to compromise.
Rostam’s Dilemma (Weak Trust Game)
==================================
In section Cuban missile crisis, game $g_9$ in Fig. \[fig:Cuba missile crisis\], is one of the other several dilemmas that we confront in attempt to achieve cooperation. In this game, cooperation can produce the best possible overall result, but there is a non-cooperative Nash equilibrium that wants to draw us toward itself. Difference between this dilemma and some of other dilemmas is that in the Rostam’s Dilemma, cooperation is weak dominant over non-cooperation. Despite Prisoner’s Dilemma that players have no trust in each other, in Rostam’s Dilemma, there is a weak trust between players but yet this weak trust doesn’t ensure that cooperation completely. There has been a view that if players trust in each other, they will obtain better result but this game shows that despite players’ weak trust in each other, there is no definite guarantee to achieve desirable result and this trust is fragile.
Abolghasem Ferdosi Toosi is an Iranian epic poet and composer of Iran national epic Shahnameh that stated conflict between Rostam and Sohrab in the epic form [@Ferdowsi]. Rostam and Sohrab epic is one of the saddest events of Shahnameh. Rostam is one of the Iranian athletes that marry to Samangan king’s daughter, Tahmineh. Some days after marriage, he said goodbye to Tahmineh, and happily came to Iran from there went to Zabolestan. After nine months Tahmineh gave birth to a boy and informed Rostam. He was named Sohrab. One day Sohrab went to his mother and said: who is my father? If someone asks me what I say in answer? Mother said: you are the son of robust athlete Rostam and from Sam and Zal race.
In fact there are several intertwined pieces in Rostam and Sohrab story but the main conflict is between Rostam and Sohrab. After some years Soharb with an army of Tooranian and Samanganian depart for war with Iranians. When Rostam reached to Sohrab on the battle plain said: let go from here to another side and fight. Sohrab agreed and demanded person to person war and said: you are old and not able to fight against me. Rostam said: calm down. Many demons were killed in my hand so wait to see me in fight. I don’t like to fight with you and kill you. Sohrab suddenly asked: who are you and from what race? I think you are Rostam. Rostam said: no I’m not. Both went to battle field and fought to the end of day. Rostam said it is night, tomorrow we wrestle. Rostam said himself: I have a son from Tahmineh who is a youth as old as Sohrab and maybe he is himself.
By sunrise, they went again to battle field. Sohrab said with himself: the more I watch him, the more I think that he is Rostam himself and I have not to fight with him. Sohrab said to Rostam: how was last night? Come on sit down to speak with each other and don’t fight. My heart is drawn to you. I asked to know your name a lot of time but no one told me your name then don’t hide your name. Rostam said: last night, we talked about wrestling you can’t deceive me. Then they wrestled and fought for a while finally Sohrab took Rostam’s belt and threw him on the ground and took out dagger but Rostam said: our tradition is that one who throws a person on the ground doesn’t kill him first time but in the second time kill him. Sohrab agreed because he was both brave and chivalrous. They went again to battle field and grappled, this time Rostam threw Sohrab on the ground then took out his dagger and killed him.
With help of game theory, the game between Rostam and Sohrab is modeled. Rostam has two actions; either he says his name to Sohrab $S$ or not says his name to Sohrab $NS$. Sohrab also has two actions; either he fights against Rostam $F$ or not to fight against Rostam $NF$. We call this game as Rostam’s Dilemma. In Rostam’s Dilemma, Sohrab is considered as row player (player 1) and Rostam is considered as column (player 2)(Fig. \[fig:RS\]G). Because in static game, players choose their action simultaneously.
(4)
[c|c|c|]{} & &\
& &\
NF & 4,4 & 1,3\
F & 3,1 & 1,1\
; ()
[c|c|c|]{} & &\
& &\
C & 4,4 & 1,3\
B & 3,1 & 1,1\
;
The above game is a symmetric game. Row player has weak dominant action $NF$. Column player has weak dominant action $S$. This game has two Nash equilibria $(NF, S)$ and $(F, NS)$. Rostam’s Dilemma is strategy maker of order $(2, 2)$. Row player has weak dominant strategy $NF$ and weak dominated strategy $F$. Column player has weak dominant strategy $S$ and weak dominated strategy $NS$. Also the game has four pairs of rational actions $(NF, S)_{1,2}$, $(F, NS)_{1,2}$, $(NF, NS)_{2}$ and $(F, S)_{1}$.
Recently Eshaghi and Askari introduced a new concept of rational choice called *hyper-rational choice*. In this concept, the actor thinks about profit or loss of other actors in addition to his personal profit or loss and then will choose an action which is desirable to him [@Askari]. This concept explains that, based on the loss of player 2, $F$ is a strictly dominant action for player 1, and based on the loss of player 1, $NS$ is a strictly dominant action for player 2. If interaction between players is based on loss of other player, player 1 prefers: $(F, NS)_{1} \sim^{'} (F, S)_{1} \succeq^{'} (NF, NS)_{1} \succeq^{'} (NF, S)_{1}$, and player 2 prefers: $(F, NS)_{2} \sim^{'} (NF, NS)_{2} \succeq^{'} (F, S)_{2} \succeq^{'} (NF, S)_{2}$. Therefore, for both players two pairs of actions $(NF, S)$, $(F, NS)$ are hyper-rational.
Rostam’s game put a dilemma on the way of players. Yet two players have weak dominant actions and can choose that and gain more payoff. In the other words, each player’s weak dominant action in some kind build a weak trust between players and this weak trust is fragile. But the game is dilemma because players can select weak dominated action and obtain least payoff. In game $g_9$ of Fig. \[fig:Cuba missile crisis\], both players based on system dynamic and weak trust to each other selected weak dominant strategy. In other words, based on collective benefit thinking, both player prefers: $({_{9}C},{_{9}C}) \succeq^{'} ({_{9}D},{_{9}D}) \succeq^{'} (({_{9}D},{_{9}C}) \succeq^{'} ({_{9}C},{_{9}D})$ or $({_{9}C},{_{9}C}) \succeq^{'} ({_{9}D},{_{9}D}) \succeq^{'} (({_{9}C},{_{9}D}) \succeq^{'} ({_{9}D},{_{9}C})$. But in the game between Rostam and Sohrab because a game occur just once, weak trust was broken and both selected weak dominated action. In other words, based on collective loss thinking, both player prefers: $(F, NS) \succeq^{'} (NF, S) \succeq^{'} (F, S) \succeq^{'} (NF, NS)$ or $(F, NS) \succeq^{'} (NF, S) \succeq^{'} (NF, NS) \succeq^{'} (F, S)$. Weak trust between Rostam and Sohrab was broken and turned into distrust to each other and son (Sohrab) was killed by father(Rostam).
Now, as an application of the Rostam’s Dilemma, social dilemma in the way of a wife and husband is modeled. Consider a couple who live together for several years, The couple are disagreeing with each other about a problem. This dispute lead to having trouble between husband and wife in some other problems. If both player condonation $C$ of these disputes and resolve all problems, then each earns reward $4$. If the husband select condonation of disputes, but the wife blame husband and intensify disputes $B$, then the wife gain payoff $3$ and the husband gets $1$ and vice versa. If the wife to blame her husband and the husband to blame her wife, more intensify disputes becomes, then both gain a payoff 1. The wife is considered as row player (player 1) and husband is considered as column (player 2)(Fig. \[fig:RS\]$G^{'}$).
This dilemma occurs in most societies, wife and husband have weakly dominant strategy $C$, but sometimes wife and husband select weakly dominated strategy $B$, which lead to intensify disputes. The game Nash equilibria are $(C,C)$ and $(B,B)$. Eshaghi and Askari introduced a new concept that Called *taxonomy* [@Askari]. Taxonomy of hyper-preference means that if we face an actor with two choices of hyper-preferences, she will necessarily have an opinion on which she likes more. Taxonomy of player’s hyper-preferences depends on environmental condition, the kind of behavior interactive, self-evaluation system and evaluation system of other interacting persons, helps to the wife and husband that based on collective profit prefer $(C, C)$ and based on collective loss prefer $(B, B)$.
Conclusion {#S.5}
==========
The dynamic system of games helps us to analyze an event by dividing it into different courses and dynamically using several games. This will enable the event analyst to evaluate the decisions and strategies that the players have chosen and achieved a reasonable and acceptable result. The system also shows the impact of players’ decisions on each other and the impact of games on each other in any period.
In this study, we used the dynamic system of strategic games to investigate the interaction between Soviet Union and America after World War II until October 28, 1926, that is, the end of Cuban missile crisis. To this end, we divide this time interval into five periods and in each period, we reviewed static games with complete information that has occurred. Each country, based on its forces and capabilities, sought to achieve its goals and objectives in Cuba. Therefore, it can be concluded that in Cuban missile crisis, the two countries reached to compromise. Dynamics system of games is a combination of dynamic and static interactive situations that are moving forward. In many cases, dynamic system of strategic games can provide a mechanism to move towards cooperation between players which helps to find a solution to exit conflicts. Moreover, we introduced Rostam’s Dilemma. Rostam’s Dilemma is a symmetric $2\times 2$ game that examines difficult conditions of decision making between hyper-rational players. In this game, a weak trust has been created between players, but it is fragile. The new attitude in this article toward $2\times 2$ games properties can result in a new and different characterization compared to the topology of $2\times 2$ games. The hyper-rational choice theory suggests that hyper-rational players have considered three classes of hyper-preferences that help determine how to behave in interactive decisions. The goal of a game analysis with two hyper-rationality players is to provide insight into real-world situations that are often more complex than a game with two rational players where the choices of strategy is only based on individual preferences.
Blaschke AM (2016) Running the Cold War: Gender, Race, and Track in Cultural Diplomacy, 1955-1975. [*Diplomatic History*]{} 40(5):826–844.
Brams SJ (1994) [*Theory of Moves*]{} (Cambridge, UK: Cambridge University Press).
Crall DA, Martin TM (2013) Cool Hand Nuke: Lessons From the Quiet Diplomacy of the Cienfuegos Non-Crisis. [*Foreign Policy Analysis*]{} 9(2):189–201.
Eshaghi M, Askari G (2017) Dynamic system of strategic games. [*Int. J. Nonlinear Anal. Appl*]{} DOI: 10.22075/IJNAA.2017.12774.1657.
Eshaghi Gordji, M and Askari G (2018) Theory of Hyper-Rational Choice (October 20, 2017). Available at SSRN: https://ssrn.com/abstract=3099441 or http://dx.doi.org/10.2139/ssrn.3099441.
Ferdowsi A (2006)[*Shahnameh: The Persian Book of Kings*]{}, Translated by Dick Davis, (New York: Viking Penguin).
Fraser NM, and Hipel KW (1982) Dynamic modelling of the cuban missile crisis. [*Conflict Management and Peace Science*]{} 6(2):1–18.
Kennedy RF (1969) [*13 days: the Cuban missile crisis October 1962*]{} (Springer).
May ER, Zelikow P (2002) [*The Kennedy tapes: Inside the White House during the Cuban missile crisis*]{} (WW Norton and Company).
Gibson DR (2012) [*Talk at the brink: Deliberation and decision during the Cuban missile crisis*]{} (Princeton University Press).
Ungureanu S (2017) Dynamic contracting under permanent and transitory private information. [*International Journal of Game Theory*]{} 46(3):667–692.
Van Long N (2011) Dynamic games in the economics of natural resources: a survey. [*Dynamic Games and Applications*]{} 1(1):115–148.
Von Neumann J, Morgenstern O (2007) [*Theory of games and economic behavior*]{} (Princeton university press).
Webb JN (2007) [*Game theory: decisions, interaction and Evolution*]{} (Springer Science and Business Media).
Yeung D, Ovanes P (2017) Infinite Horizon Dynamic Games: A New Approach via Information Updating. [*International Game Theory Review*]{} 19(4):1750020.
Zagare FC (2014) A game-theoretic history of the cuban missile crisis. [*Economies*]{} 2(1):20–44.
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abstract: 'Using 3D numerical hydrodynamical simulations we show that jets launched prior to type Ia supernova (SN Ia) explosion in the core-degenerate (CD) scenario can account for the appearance of two opposite lobes (’Ears’) along the symmetry axis of the SN remnant (SNR). In the double-degenerate (DD) and CD scenarios the merger of the two degenerate compact objects is very likely to lead to the formation of an accretion disk, that might launch two opposite jets. In the CD scenario these jets interact with the envelope ejected during the preceding common envelope phase. If explosion occurs shortly after the merger process, the exploding gas and the jets will collide with the ejected nebula, leading to SNR with axisymmetric components including ’Ears’. We also explore the possibility that the jets are launched by the companion white dwarf prior to its merger with the core. This last process is similar to the one where jets are launched in some pre-planetary nebulae. The SNR ’Ears’ in this case are formed by a spherical SN Ia explosion inside an elliptical planetary nebula-like object. We compare our numerical results with two SNRs - Kepler and G299.2-2.9.'
author:
- Danny Tsebrenko and Noam Soker
title: 'TYPE Ia SUPERNOVAE INSIDE PLANETARY NEBULAE: SHAPING BY JETS'
---
INTRODUCTION {#sec:intro}
============
Type Ia Supernovae (SNe Ia) are thermonuclear detonations of carbon-oxygen white dwarfs (WDs;[ ]{}@Hoyle1960). The exact mechanism leading to the explosion is debated. Four viable scenarios for SNe Ia are currently considered: (a) the single degenerate (SD) scenario (e.g., @Whelan1973 [@Nomoto1982; @Han2004]), in which a WD accretes mass from a non-degenerate stellar companion; (b) the double degenerate (DD) scenario (e.g., @Webbink1984 [@Iben1984]), in which a merger of two WDs takes place. Recent papers discuss violent merger and collision (e.g., @Pakmor2013 [@Kushniretal2013]) as an ignition channel of the DD scenario; (c) the core-degenerate (CD) scenario (e.g., @Livio2003 [@Kashi2011; @Soker2011; @Ilkov2012; @Ilkov2013; @Soker2013]), in which a WD merges with a hot core of a massive asymptotic giant branch (AGB) star; (d) the ’double-detonation’ mechanism (e.g., @Woosley1994 [@Livne1995]), in which a sub-Chandrasekhar mass WD accumulates on its surface a layer of helium-rich material, which can detonate and lead to a second detonation near the center of the CO WD (e.g., @Shenetal2013 for a recent paper).
In the violent merger model [@Pakmor2012] it is possible that in the merger process of the two WDs the helium is ignited first. In this mechanism both the DD and the double detonation operate [@Pakmor2013]. The double detonation might operate in the CD scenario as well, with or without a violent merger. In both the SD and the CD scenarios, a circumstellar shell can be formed by ejecting the companion (to the WD) stellar envelope close to the explosion time. This ejection can be accompanied by the formation of jets launched from an accretion disk. In the SD scenario the disk is formed during the mass accretion process onto the WD, while in the CD scenario the disk is formed either around the core from the destructed WD material, or around the WD from the destructed core. The formation of a disk as a result of merger of two WDs was discussed before, e.g., [@Raskin2013] and [@Ji2013] for recent papers. Another possibility in the CD scenario is that the WD accretes mass from the giant stellar envelope and launches two opposite jets before merging with the giant’s core. Such circumstellar matter (CSM) shells are similar to some planetary nebulae (PNs), as the central WD ionizes the CSM. When the SN is finally ignited, the WD mass in the SD scenario and the merger product mass in the CD scenario is expelled by the explosion at high velocities of up to $V_{\rm SNm}=20,000 \km \s^{-1}$. If the explosion occurs within $\sim 10^5 \yrs$ from the CSM ejection episode, we have a SN Ia exploding inside a PN or a PN-like object. The exploding gas catches up with the dense (relative to the interstellar medium; ISM) CSM and interacts with it.
In this paper we study the interaction of SN Ia ejecta with a PN (or PN-like) shell. Observations show that some of the supernova remnants (SNRs) exhibit non-spherical morphologies, including axi-symmetrical morphologies resembling the structures of some PNe. An example is the Kepler SNR that possesses two opposite protrusions which are termed ‘Ears’ (after the nomenclature used in @Cassam2004). Another example is the G299.2-2.9 SNR. For the general discussion (not limited to Kepler or G299.2-2.9 SNRs), we will refer to these protrusions as ’ears’.
The motivation for this work comes from the expectation that jets will be formed in some SN Ia scenarios, and may suggest an explanation to the observed two ’Ears’ of the Kepler SNR and the G299.2-2.9 SNR. The ’Ears’ are marked on an the X-ray images of these two SNRs shown in Fig. \[fig:observations\].
We perform 3D hydrodynamical simulations of such SN explosions and study the interaction of the expelled material with the CSM. The numerical code and initial setting and conditions are described in Section \[sec:numerical\], while in Section \[sec:results\] we demonstrate that the double ’ears’ and similar morphological features can be explained by jets that are launched during the pre-explosion phase. Our discussion and comparison with the SNRs Kepler and G299.2-2.9 are in Section \[sec:comparison\], and a short summary is given in Section \[sec:summary\].
NUMERICAL SETUP {#sec:numerical}
===============
The simulations are performed using the high-resolution multidimensional hydrodynamics code [@Fryxell2000]. We employ a full 3D uniform grid (all cells have the same size) with Cartesian $(x,y,z)$ geometry. In all runs but one each axis has 512 cells and a total length of $\Delta=3.2 \pc$. In one run each axis still has 512 cells but $\Delta=0.65 \pc$. We use radiative cooling for solar metallicity values, according to [@Sutherland1993], although radiative cooling has a minor role in our simulations. The ejected material is metal-rich (see a recent paper by @Park2013) and the cooling of the shocked ejecta is expected to be more efficient than what the cooling function mentioned above gives. However, this difference in not significant for the duration of our simulations and does not play a significant role in the shaping of the ’Ears’.
There are two scenarios for forming various shapes of the CSM during the stage prior to SN Ia explosion. In the SD scenario the WD grows in mass through accretion from a non-degenerate stellar companion. If the companion is an evolved AGB star it might blow large parts of its envelope, as progenitors of PNe do. The accretion onto the WD takes place via an accretion disk, which enables jets to be formed. The jets can blow two opposite small lobes (’ears’) in the nebula as observed in some PNe. Examples of elliptical PNe with ’ears’ are NGC 6563 [@Schwarz1992], NGC 7139, IC 418, and NGC 3242 (@Balick1987; their images can be seen in the Planetary Nebula Image Catalogue[^1]). We note that in some cases the two ’ears’ are not exactly symmetric to each other. As well, in some PNe there are indications that the ’ears’ are formed by jets.
In the CD scenario [@Soker2013] the common envelope (CE) phase is terminated by a merger of a WD companion with the hot core of a massive AGB star. However, in the cases where the merger takes place when the WD is denser than the core, the core will be disrupted and accreted onto the cooler WD. In such cases the explosion might occur shortly, months to tens of years, after the merger process.
Whereas in the SD scenario the shaping of ’ears’ by jets is only of the CSM shell, in the CD scenario the shaping can occur both in the CSM and/or during the core-WD merger. In the rare cases when the SN explosion occurs within $\simeq 10^5 \yrs$ from the formation of the CSM, jets can be formed prior to the explosion and, together with the high-velocity SN ejecta, collide with the CSM and lead to formation of specific features in the SNR. This type of interaction might form SN Ia remnants that in general possess a large scale spherical symmetry, but have small, notably axisymmteric, departures from sphericity. We examine whether the formation of the ’Ears’ in the SNRs of Kepler and G299.2-2.9 can be explained by such process.
We note that protrusions can also be formed by ’bullets’ ploughing through the shell, as suggested by recent simulations of Tycho’s SNR and SN 1006 [@Warren2013]. [@Warren2013] show that knots of SN ejecta can overtake the forward shock and form low density bubbles around the rim of the SNR. However, these features appear only for one specific simulated case having an adiabatic index of $\gamma = 6/5$, and are absent entirely from other simulations. We aim at two opposite protrusions in an axisymmetrical structure, rather than at randomly distributed protrusions.
Based on the above discussion we simulate the two scenarios for the formation of protrusions. One scenario sets the asymmetry in the PN shell long before the explosion, while the other assumes the launching of two opposite jets very shortly before the explosion. Based on these, we conduct simulations with two types of initial setups of explosion inside a CSM shell. (a) [*CSM-lobes model:*]{} a spherical shell with two hollow small lobes, mimicking the structure of such two opposite small lobes observed in some elliptical PNs. The initial distribution of the SN ejecta is spherically symmetric. (b) [*Pre-explosion jets (PEJ) model:*]{} a completely spherical CSM shell, with two jets added inside the otherwise spherical SN ejecta. This can occur only in the CD scenario during the core-WD merger process [@Soker2013].
The SN ejecta density is modeled by an exponential density profile [@Dwarkadas1998] $$\rho_{\rm{SN}} = A \,{\rm{exp}}{} (-v/v_{\rm{ejecta}})t^{-3},$$ where $v_{\rm{ejecta}}$ is a constant which depends on the mass and kinetic energy of the ejecta, $$v_{\rm{ejecta}} = 2.44 \times 10^8 E_{51}^{1/2} \left(\frac{M_{\rm{SN}}}{M_{\rm{Ch}}}\right)^{-1/2} \cm \s^{-1},$$ $M_{Ch} = 1.4 M_\odot$, $E_{51}$ is the explosion energy in units of $10^{51} \erg$, and $A$ is a parameter given by $$A = 7.67 \times 10^6 \left(\frac{M_{\rm{SN}}}{M_{\rm Ch}}\right)^{5/2} E_{51}^{-3/2} \g \s^{3} \cm^{-3} .$$ The maximum velocity of the SN ejecta is taken to be $v_{\rm SNm}=20,000 \km \s^{-1}$. Each simulation starts roughly $8 \yrs$ after the explosion took place. By this time the fastest ejecta reached $0.16 \pc$ from the center of the explosion. The total energy of the explosion is set to be $E_{\rm SN}=10^{51} \erg$, and the mass ejected in the explosion is $M_{\rm SN}=1.4 M_\odot$. The CSM initial profile is set to be a constant density [@Patnaude2012], $\rho_{\rm CSM}=3.15 \times 10^{-21} \g \cm^{-3}$, shell within radii $0.24-0.27 \pc$, so that the total mass of the CSM shell is $M_{\rm CSM} \simeq 1M_\odot$. The mass is based on the estimated CSM mass in the Kepler SNR [@Borkowski1992; @Borkowski1994; @Kinugasa1999; @Blair2007]. The ambient ISM density is taken to be $\rho_{\rm ISM}= 10^{-24} \g \cm^{-3}$ [@Vink2008].
In the small CSM-lobes model, some of the mass of the CSM envelope is ’pushed outward’ to form the ’Ears’. Each small-lobe has an initial half-spherical shell structure within radii $0.065-0.09 \pc$ from the center of the small-lobe. We assume that the small CSM-lobes are formed by jets with low mass, and hence take the total mass of the CSM in the small-lobes to be the same as the CSM shell segments they replaced. As the width of the small lobes is as that of the CSM, the density in the small lobes is lower than that in the CSM, and amounts to a value of $\rho_{\rm small-lobes}=1.56 \times 10^{-21} \g \cm^{-3}$. The initial setup of the small CSM-lobes is shown in Fig. \[subfigure:lobesscheme\].
In the PEJ model, jets are added to the explosion itself. The reason is that in the CD scenario the jets might occur during the core-WD merger [@Soker2013], and the explosion may occur very shortly after the merger, e.g., as in the violent merger ignition model of [@Pakmor2013]. The jets are expected to be launched at about the escape velocity from the massive and compact WD, $v_{\rm jets}> 10,000 \km \s^{-1}$ , not much different from the SN explosion velocity. To make the model simple, we set the velocity and density radial profiles of the jets to be same as that of the SN ejecta, but take the density in the jets to be 3 times larger that that of the SN ejecta at the same radius. The mass of each jet is $ 0.03 M_{\odot}$ and the half-opening angle of the jets is $\theta = 5^{\rm{o}}$. The initial setup of the PEJ model is shown in Fig. \[subfigure:jetsscheme\].
To have better understanding of the process of interaction of the SN ejecta with the CSM at early times (up to $\simeq 20 \yr$) we perform one additional simulation that focuses on the region within the initial radius of the CSM shell. In this setup each axis still has 512 cells but is only $0.65 \pc$ in length, so that the effective resolution is increased significantly in comparison to the simulation runs described above.
The numerical $x-y$ $(z=0)$ plane is taken to be in the plane containing the central explosion and the two ’Ears’ or jets. Initially this plane is a symmetry plane, but we simulate the entire space, so that we assume no symmetry about the $z=0$ axis at $t>0$.
RESULTS {#sec:results}
=======
As the SN ejecta hits the CSM, part of the kinetic energy of the ejecta is deposited in the CSM shell, and the CSM is accelerated outward. Two shock waves are formed. The first shock runs outward into the CSM, and then passes to the ISM. The second shock is a reverse shock that runs inward into the incoming ejecta. There are two contact discontinuities in the flow, one between the shocked ejecta and the shocked CSM, and the other between the CSM and ISM (initially preshock media, later shocked media). The shocks are clearly seen in all panels of Fig. \[fig:densityPlots\] and \[fig:densityPlots2\], and are marked on panel \[subfigure:densears2\]. By ISM we refer to the medium outside the dense CSM shell. This ISM might contain previous stellar wind, and hence is actually a low-density CSM. The reverse shock is best seen in the temperature and pressure maps in Fig. \[fig:otherplots\], and is marked on panel \[subfigure:jetstemperature\].
In our flow setting the shocked CSM is denser than the shocked ejecta. Therefore, as the shocked ejecta accelerates the shocked CSM, the flow becomes Rayleigh-Taylor (RT) unstable, leading to the formation of ‘RT fingers’ and the penetration of the two media into each other. The fingers are clearly seen in Fig. \[fig:densityPlots\], and are marked on panel \[subfigure:densears2\]. Similar instabilities are present in SN Ia simulations performed by [@Warren2013]. The shocked ISM is of lower density than the shocked CSM. As the shocked ISM slows down the shocked CSM this contact discontinuity also becomes RT unstable. These unstable regions are best presented in panels \[subfigure:jetsrt1\] and \[subfigure:jetsrt2\], as described next. The post shock CSM gas is heated up to temperatures $\rm{of} \simeq 10^9 \K$, and slowly cools down mainly via adiabatic cooling. For our initial conditions the radiative cooling time scale of the shocked CSM is $\gtrsim 1000 \yrs$, whereas the flow time in our simulation is up to $\simeq 170 \yrs$.
The initial departure from spherical symmetry (see Fig. \[subfigure:densears1\] and \[subfigure:densjets1\]) is ’magnified’ in absolute size by the sweeping SN ejecta as the lobes are inflated from within by the ejecta-CSM interaction (see Fig. \[subfigure:densears2\], \[subfigure:densjets2\] and \[fig:densityPlots2\]). In both models, a large-scale deviation from spherical symmetry is formed along the symmetry axis.
\
\
The outflow in the inner part of the SNR at late simulation times is mostly governed by the low-velocity ejecta that was located close to the center of the SN explosion in the beginning of the simulation. Our modeling of this slow ejecta is not accurate due to numerical limitations (pixel size), thus the simulation results of the innermost regions (up to $0.5 \pc$ at $173 \yr$) of the SNR are of limited accuracy. The overall “clumpiness” of the regions dominated by RT fingers resembles the fleecy structure analysed in [@Warren2013] for the interiors of type Ia SNRs.
As the differences between the results of the CSM-lobes (Fig. \[subfigure:densears1\],b) model and the PEJ model (Fig. \[subfigure:densjets1\],d) are small, we choose to focus on the PEJ model for further analysis.
In Fig. \[fig:densityPlots2\] we present the density maps in different planes, and in Fig \[fig:otherplots\] we present more physical quantities of the flow. Both figures emphasize the main features of the flow, in particular the structure and evolution of the instabilities. Fig. \[subfigure:jetstemperature\] and \[subfigure:jetspressure\] show temperature and pressure maps. Fig. \[subfigure:jetsrt1\] and \[subfigure:jetsrt2\] show the ratio of the RT-growth time $\tau_{\rm RT}$, to the present time of the simulation $t_{\rm sim}$. This ratio is calculated as $$\label{eq:rt}
\frac{\tau_{\rm RT}}{t_{\rm sim}} = \frac{1}{t_{\rm sim}} \sqrt{\frac{\lambda \rho}{\vert \vec{\nabla} P \vert}},$$ where $\lambda$ is the typical size of the RT instabilities and $\vec{\nabla} P$ is the pressure gradient in the $x-y$ plane. We scale $\lambda$ as a fraction of the initial CSM shell width, $0.03 \pc$, and take it to be, somewhat arbitrarily, $0.01 \pc$. The exact value is of no significance for our analysis. $\lambda$ equals 1.6 times cell size.
\
In addition to the nominal resolution runs whose results are presented above, we perform a high-resolution simulation that focuses on early times (around the time of the initial interaction of the SN ejecta with the CSM shell), before the CSM shell expanded to higher radii. In this run $\lambda=0.01 \pc$ equals the size of 8 numerical cells. Fig. \[fig:resolutionplots\] shows various quantities at early times of the high-resolution simulation, beside panel \[subfigure:denslow\] that presents the density map of the nominal lower resolution run presented in Fig. \[subfigure:densjets1\], \[subfigure:densjets2\], \[fig:densityPlots2\] and \[fig:otherplots\].
\
While there are differences between the nominal and the high-resolution runs on small scales, in particular the formation of RT instabilities on the high-resolution density map (compare panel \[subfigure:denslow\] with \[subfigure:denshigh\]), the two simulations are similar on large scales. As well, we note that the ’Ears’ that begin to form in the high-resolution run have ’sharper’ structure than the ’Ears’ in the nominal-resolution run.
The early times, when the forward shock front is still inside the CSM, are of particular interest for understanding the interaction between the media. Fig. 7 shows the radial profiles of the density, temperature, pressure and the RT instability growth time, at two early times. In panel \[subfigure:radial1\] we present the profiles at $t=6 \yr$, when the forward shock front is still inside the CSM (not all the CSM has been shocked yet), while in panel \[subfigure:radial2\] at $t=16 \yr$, the shock front is running through the ISM. The profiles are drawn from the center of the grid along the positive $x$ axis.
The formation of the RT instabilities begins at an early time in the simulation, when the SN ejecta first hit the CSM shell. At this time the CSM shell did not move much from its original location at radii of $0.24-0.27 \pc$ (marked in panel \[subfigure:radial1\]). However, the shell’s inner part is hit by the SN ejecta and a shock runs into the CSM and heats the material up to $T_{\rm max} \simeq 10^{10} \K$. As the shock progresses further into the shell, a highly RT-unstable region develops around $0.24-0.26 \pc$ (marked in panel \[subfigure:radial2\]). This region is on the contact discontinuity between the shocked ejecta and the shocked CSM ($0.28 \pc$, where a density jump is seen). Over time, these instabilities develop to the large fingers seen in Fig. \[subfigure:densears2\], \[subfigure:densjets2\] and \[fig:densityPlots2\].
The main results of our simulations presented above are as follows. (1) In both the CSM-lobes model and the PEJ model two ’Ears’ on opposite sides of the SNR are formed. These protrusions are of significant size ($r \simeq 0.5 \pc$ at $170 \yrs$ for the parameters we used), and may appear as ‘Ears’ in observations. In the next section we examine whether the assumed jets can account for the structure of some SNRs. (2) The interaction of the SN ejecta with the CSM shell creates a complex density structure inside the SNR, mainly through evolution of Rayleigh-Taylor instabilities.
COMPARISON WITH SNRs {#sec:comparison}
====================
We attempt to match some of the features in our numerical results with some morphological features in the SNRs Kepler and G299.2-2.9. The X-ray emission from the asymmetrical Kepler’s SNR has been studied in the past, recently by [@Burkey2013]. They interpret the observed distribution of CSM in Kepler’s SNR as resulting from a disk seen edge-on, which extends from the center of the SNR along the line that marks our proposed symmetry axis in Fig. \[fig:observations\]. Our interpretation of the observations is different, in that we propose that the line connecting the two ’Ears’ is actually a symmetry axis and not the equatorial plane. Our conjecture of the direction of the symmetry axis is based on similar interpretations applied to planetary nebulae that possess similar ’double-Ear’ morphologies (see Section \[sec:numerical\]). Following these interpretations we here attribute the ’Ears’ observed in SNRs to jets. In particular, we examine the Kepler and G299.2-2.9 SNRs.
The exact reconstruction of the expected x-ray emission map from our simulations is beyond the scope of this paper. However, as x-ray emission magnitude is proportional to the square of the density we can make a qualitative comparison by constructing an ‘intensity map’ by simply integrating $\rho^2$ along the line of sight (a similar approach was used in @Burkey2013). We take the ’Ears’ to lie in the plane of the sky. This intensity map is shown in Fig. \[fig:integratedDens\].
![ Integrated $\rho^2$ for the spherical CSM with PEJ run. A density map of the same run is presented in panel \[subfigure:densjets2\]. Apparent are the two ’ears’ features, formed by jets close to the time of the SN explosion.[]{data-label="fig:integratedDens"}](KeplerSNDannyFigintegratedDens.eps)
Considering our results presented in Section \[sec:results\] for both the CSM-lobes and the PEJ models at a late simulation time (e.g. the resulting density maps shown in panels \[subfigure:densears2\] and \[subfigure:densjets2\]) and making a qualitative comparison between Fig. \[fig:integratedDens\] and the observed images in Fig. \[fig:observations\], we are led to suggest that both our proposed models - the CSM-lobes model and the PEJ model - might explain the observed ’Ears’ features in Kepler SNR and G299.2-2.9 SNR.
SUMMARY {#sec:summary}
=======
Although many SN Ia remnants exhibit almost spherical large-scale symmetry, there are several remnants of SN Ia (SNRs) that show an axisymmetrical deviation from spherical large-scale structure. The Kepler and G299.2-2.9 SNRs, for example, have two protrusions (’Ears’) on the outer rim of the SNR, positioned exactly opposite to each other (see Fig. \[fig:observations\]). Noting that some planetary nebulae (PNs) have similar axisymmetrical morphologies with two opposite ’ears’, and following interpretations that attribute these features to jets, we propose that the ’Ears’ in these two SNRs can also be attributed to jets.
In Section \[sec:numerical\] we discussed two scenarios for the formation of such protrusions: (a) The CSM-lobes model, in which a SN explosion takes place inside an overall spherical shell with two hollow small lobes, mimicking the structure of such two opposite small lobes observed in some elliptical PNs. This model is plausible if the SN explosion takes place via the SD or the CD scenarios. (b) The pre-explosion jets (PEJ) model, in which the SN explosion is ignited inside a CSM shell (which might be spherical or not), with two jets added inside the SN ejecta. This can occur only in the CD scenario during the core-WD merger process [@Soker2013]. The models are described schematically in Fig. \[fig:schemes\].
The initial conditions of the runs were motivated by the Kepler SNR, although we did not try to reproduce its ’Ears’ one to one. We employed 3D hydrodynamical simulations to follow the interaction of the SN ejecta with the CSM in the two scenarios - CSM-lobes and PEJ.
The main results of our simulations are presented in Section \[sec:results\], and can be summarized as follows. (1) In both the CSM-lobes model and the PEJ model two ’Ears’ on opposite sides of the SNR are formed. These protrusions are of significant size ($r \simeq 0.5 \pc$ at $170 \yrs$ for the parameters we used), and may appear as ‘Ears’ in observations. (2) The interaction of the SN ejecta with the CSM shell creates a complex density structure inside the SNR, mainly through evolution of Rayleigh-Taylor instabilities.
In Section \[sec:comparison\] we compare the results of our simulations with observations, and propose that both our suggested scenarios (CSM-lobes and PEJ) may explain the observed morphologies of the Kepler and G299.2-2.9 SNRs. In general, we propose that some SN Ia may blow jets close to the time of the explosion, offering an explanation to the occasionally observed axisymmetrical morphology for the remnants of such SN.
We thank Alexei Baskin for useful comments. This research was supported by the Asher Fund for Space Research at the Technion, the E. and J. Bishop Research Fund at the Technion, and the USA-Israel Binational Science Foundation. The FLASH code used in this work is developed in part by the US Department of Energy under Grant No. B523820 to the Center for Astrophysical Thermonuclear Flashes at the University of Chicago. The simulations were performed on the TAMNUN HPC cluster at the Technion.
Balick, B. 1987, , 94, 671
Borkowski, K. J., Blondin, J. M., & Sarazin, C. L. 1992, , 400, 222
Borkowski, K. J., Sarazin, C. L., & Blondin, J. M. 1994, , 429, 710
Burkey, M. T., Reynolds, S. P., Borkowski, K. J., & Blondin, J. M. 2013, , 764, 63
Blair, W. P., Ghavamian, P., Long, K. S., et al. 2007, , 662, 998
Cassam-Chena[ï]{}, G., Decourchelle, A., Ballet, J., et al. 2004, , 414, 545
Dwarkadas, V. V., & Chevalier, R. A. 1998, , 497, 807
Fryxell B., Olson K., Ricker P., et al., 2000, , 131, 273
Han, Z., & Podsiadlowski, P. 2004, , 350, 1301 Hoyle, F., & Fowler, W. A. 1960, , 132, 565 Iben, I., Jr., & Tutukov, A. V. 1984, , 54, 335 Ilkov, M., & Soker, N. 2012, , 419, 1695
Ilkov, M., & Soker, N. 2013, , 428, 579
Ji, S., Fisher, R. T., Garcia-Berro, E., et al. 2013, arXiv:1302.5700
Kashi, A., & Soker, N. 2011, , 417, 1466
Kinugasa, K., & Tsunemi, H. 1999, , 51, 239
Kushnir, D., Katz, B., Dong, S., Livne, E., & Fern[á]{}ndez, R. 2013, arXiv:1303.1180
Livio, M., & Riess, A. G. 2003, , 594, L93
Livne, E., & Arnett, D. 1995, , 452, 62
Nomoto, K. 1982, , 253, 798
Park, S., Slane, P. O., Hughes, J. P., et al. 2007, , 665, 1173
Pakmor, R., Kromer, M., Taubenberger, S., et al. 2012, , 747, L10
Pakmor, R., Kromer, M., & Taubenberger, S. 2013, arXiv:1302.2913
Patnaude, D. J., Badenes, C., Park, S., & Laming, J. M. 2012, , 756, 6
Park, S., Badenes, C., Mori, K., et al. 2013, , 767, L10
Raskin, C., & Kasen, D. 2013, arXiv:1304.4957
Reynolds, S. P., Borkowski, K. J., Hwang, U., et al. 2007, , 668, L135
Schwarz, H. E., Corradi, R. L. M., & Melnick, J. 1992, , 96, 23
Shen, K. J., Guillochon, J., & Foley, R. J. 2013, arXiv:1302.2916
Soker, N. 2011, arXiv:1109.4652
Soker, N., Kashi, A., Garcia-Berro, E., Torres, S., & Camacho, J. 2013, , 431, 1541
Sutherland, R. S., & Dopita, M. A. 1993, , 88, 253
Vink, J. 2008, , 689, 231
Warren, D. C., & Blondin, J. M. 2013, , 429, 3099
Webbink, R. F. 1984, , 277, 355
Whelan, J., & Iben, I., Jr. 1973, , 186, 1007
Woosley, S. E., & Weaver, T. A. 1994, , 423, 371
[^1]: http://www.astro.washington.edu/users/balick/PNIC/
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abstract: 'We solve a set of selected exercises on rotational motion requiring a mechanical and thermodynamical analysis. When non-conservative forces or thermal effects are present, a complete study must use the first law of thermodynamics together with the Newton’s second law. The latter is here better expressed in terms of an ‘angular’ impulse-momentum equation (Poinsot-Euler equation), or, equivalently, in terms of a ‘rotational’ pseudo-work-energy equation. Thermodynamical aspects in rotational systems, when e.g. frictional forces are present or when there is a variation of the rotational kinetic energy due to internal sources of energy, are discussed.'
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0.6cm plus 1pt minus 1pt =1500
[**Thermodynamics in rotating systems — analysis of selected examples**]{} 0.4cm [ J. Güémez$^{a,}$[^1], M. Fiolhais$^{b,}$[^2] ]{} 0.1cm [*$^a$ Departamento de Física Aplicada*]{}\
[*Universidad de Cantabria*]{}\
[*E-39005 Santander, Spain*]{}\
0.1cm [*$^b$ Departamento de Física and Centro de Física Computacional*]{}\
[*Universidade de Coimbra*]{}\
[*P-3004-516 Coimbra, Portugal*]{}
Introduction {#sec:intro}
============
Rotations are present everywhere. Their mechanical treatment relies on the Newton’s second law, whose formulation for rotations leads to the so-called Poinsot-Euler equation [@taylor59]. In spite of the analogy that can be established between translations and rotations, some peculiarities associated with the rotational motion makes it sometimes more challenging for students than the translational motion. For instance, the direction of the static friction force, which can be either the direction of the centre-of-mass motion of a rolling object or the opposite direction, is usually presented as an example of such difficulties [@pinto01]. In addition, if there are kinetic frictional forces, or internal sources of energy responsible, e.g. for the production of kinetic rotational energy, another type of problems arise, namely on the use of the energy balance equation (first law of thermodynamics) in addition to the Newton’s second law. Not so surprisingly, and in spite of their presence in everyday life and of their scientific interest, examples of rotational processes with mechanical energy dissipation (e.g. a sliding disc on an incline [@knudsen00]) or with production of mechanical energy (e.g. a fireworks wheel) are scarce in the literature.
In a previous publication [@guemez13] we presented a number of examples of systems in translation, ranging from mechanics to thermodynamics, to illustrate the applicability of the (pseudo)work–energy equation [@sherwood83] (actually, equivalent to Newton’s second law [@penchina78]) versus the first law of thermodynamics [@sousa97]. In this paper, in the same spirit, we present examples of rotational motion whose description also requires both mechanics and thermodynamics.
The classical dynamics of bodies in translation is described using Newton’s second law which can be expressed by [@guemez13; @penchina78] \_[ext]{} t = M \_[cm]{} , [F]{}\_[ext]{} \_[cm]{} = [12]{} M v\_[cm]{}\^2 \[eq-3\] where ${\vec F}_{\rm ext}$ is the resultant of the external forces and ${\vec v}_{\rm cm}$ the center-of-mass velocity of the system of constant mass $M$. The first expression in (\[eq-3\]) is the impulse-momentum equation and, the second one, the centre-of-mass (pseudo-work) equation [@mallin92].
For the rotation of a rigid body around a principal inertia axis containing the centre-of-mass, the Newton’s second law is better expressed by (Poinsot-Euler equation) [@taylor59; @tipler04; @varios] \_[ext]{}t = I , \[eq:in002a\] where $\vec {\Gamma}_{\rm \, ext} = \sum_j \vec \tau_j^{\rm \, ext} $ is the resultant external torque, $\vec \omega$ is the angular velocity around the axis and $I$ the constant moment of inertia with respect to that principal axis. This is an angular impulse-momentum equation whose evident analogy with the first equation (\[eq-3\]) is rather appealing and always worthwhile to point out in a classroom context. However, for a constant $I$, the previous equation is only valid when the internal forces produce a vanishing torque, which is the most common situation. Otherwise equation (\[eq:in002a\]) should include, on the left-hand side, the torque of the internal forces which does not vanish if the two forces of an action-reaction pair are not along the same line but along two parallel lines. After multiplying both sides of equation (\[eq:in002a\]) by $\vec\omega$, and since the rotation is along a principal inertia axis ($\vec {\Gamma}_{\rm ext}$ and $\vec \omega$ are, therefore, co-linear), one obtains the equivalent to the second equation (\[eq-3\]) for the restricted class of rotations considered here, namely \_[ext]{} = [12]{}I \^2 \[eq:in002b\] where $\d \theta = \omega \d t$ is the infinitesimal angular displacement of the body. Similarly to the second expression in equation (\[eq-3\]), one should note that equation (\[eq:in002b\]) is not, in general, an expression of the ‘work-energy theorem’ for rotations [@sherwood84; @walker11c]. Actually, in general, $\Gamma_{\rm ext} \d \theta$ is not always a work [@young12]. The association of $ \Gamma_{\rm ext} \d \theta$ always with real work is as erroneous as the association of the pseudo-work $ {\vec F}_{\rm ext} \cdot \d {\vec r}_{\rm cm}$, in equation (\[eq-3\]), always with real work. Some examples presented in this paper will emphasize this point. It should remain clear that equations (\[eq-3\]) are always valid for translations of a constant mass body, whereas equations (\[eq:in002a\]) and (\[eq:in002b\]) are valid for rotations under the specific circumstances mentioned above.
When thermal effects are present, they cannot be accounted for by any of the previous equations, either in the case of translations or rotations. In such a case one has to consider, additionally, the first law of thermodynamics [@sousa97; @erlichson84; @serway02a]. For the sake of completeness here we very briefly repeat part of the discussion presented in [@guemez13]. The internal energy infinitesimal variation of any system, $\d U$, may receive contributions: i) from the variation of the internal kinetic energy, $\d K_{\rint}$ (it includes rotational kinetic energy and translational kinetic energy with respect to the centre-of-mass); ii) from any internal work, $\d w_{\rint} = -\d \Phi$ ([*i.e.*]{} work possibly done by internal forces) [@mallin92]; iii) from the internal energy variations related to temperature variations, expressed as $ M\, c\, \d T$ ($c$ is the specific heat); iv) from the internal energy variations related to chemical reactions [@atkins10]; v) from any other possible way not explicitly mentioned before.
It is well known that both work and heat contribute to the internal energy variation of a system. For a general process on a macroscopic system, whose analysis needs to combine mechanics and thermodynamics [@jewett08v], besides the centre-of-mass equation, such as the second expression in (\[eq-3\]), or (\[eq:in002b\]), one should also consider the equation [@guemez13] \_[cm]{} + = \_j [ [F]{}\_[[ext]{}, j]{}]{} \_j + \[totale\] which is nothing but an expression of the first law of thermodynamics [@erlichson84; @besson01]: this equation is complementary to the centre-of-mass equations (\[eq-3\]) or (\[eq:in002a\])-(\[eq:in002b\]), but all are valid. (The infinitesimal heat in (\[totale\]) is denoted by $\delta Q $ because it is not an exact differential [@landsberg83].) Each term in the sum on the right-hand side of equation (\[totale\]) is work associated with each [*external*]{} force ${\vec F}_{{\rm ext}, j}$, and $\d {\vec r}_j$ is the infinitesimal displacement of the force ${\vec F}_{{\rm ext}, j}$ itself (not the displacement of the centre-of-mass). Therefore, $\delta {W}_j= {\vec F}_{{\rm ext}, j} \cdot \d {\vec r}_j$ is always real work and not pseudo-work. Note that an external force may have an associated pseudo-work (when there is displacement of the system centre-of-mass) without doing any work, if its application point does not move — a common example is the force acting on the foot of a walking person [@guemez13c].
We stress that, in expressing the first law by equation (\[totale\]), one assumes that any translational kinetic energy with respect to the center-of-mass and the rotational energy should be included in the internal energy of the system, a point that will be illustrated with the examples of next sections. Moreover, the range of applicability of equation (\[totale\]) is larger than the most known expression for the first law, $\d U=\delta W+\delta Q$, since it can also be applied to thermal engines that move by themselves (a steam locomotive, a car, a person, etc.).
If dissipative forces are present, such as kinetic friction or dragging forces, it is not unambiguous what their displacements are [@besson01; @serway]. Therefore, we may have a problem in computing the work from its basic definition [@serway]. However, since dissipative work is equivalent to heat, we may always assume that the energy transfer associated with such forces is accounted by the last term in (\[totale\]): operationally we simply consider that any dissipative work, $\delta W_{\rm D}$, is thermodynamically equivalent to a heat transfer $\delta Q=\delta W_{\rm D}$. We take this point of view and, as we shall see in various examples, this protocol does not affect the mechanical treatment (for which only pseudo-works matter) and even allows for a very clean analysis from both the first and the second laws of thermodynamics. The important point is to consider the energy transfer associated with dissipation and include it on the right-hand side of equation (\[totale\]).
Both the second equation (\[eq-3\]) and equation (\[totale\]) refer to energy balances in processes but they express two quite different fundamental physical laws. The former, the so-called centre-of-mass energy equation, is an alternative way of stating Newton’s second law, whereas the latter expresses the first law of thermodynamics in its most general form. They are both simultaneously general and always valid, therefore each one provides new information with respect to the other. Of course, if the problem is out of the scope of thermodynamics, the two equations are then equivalent or, in other words, they become the same equation. This happens when there is no internal energy variation, no heat transfers and when the displacement of the forces equals the displacement of the centre-of-mass. The sum on the right-hand side of equation (\[totale\]), $ \sum_j { {\vec F}_{{\rm ext}, j}}\cdot \ \d {\vec r}_j$, then simply becomes $\vec F_{\rm ext}\cdot \ \d {\vec r}_{\rm cm}$, where $\vec F_{\rm ext}= \sum_j { {\vec F}_{{\rm ext}, j}}$ is the resultant of the external forces, hence (\[totale\]) becomes equal to the second equation (\[eq-3\]).
Summarizing, within their limits of applicability, all above equations, given in infinitesimal form, are valid and, therefore, they should be compatible. Equation (\[totale\]) may provide the same information as the other energy equations if the problem under consideration is a purely mechanical one. However, if the situation lies in the scope of thermodynamics, that equation, which embodies the first law, provides new information with respect to the center-of-mass equation in (\[eq-3\]) or with respect to equation (\[eq:in002b\]). Of course, for a complete study, one still needs the second law of thermodynamics [@leff12b], which states that only processes compatible with a non-decrease of the entropy of the universe, $\Delta S_\rU \ge 0$, are allowed.
To illustrate the usage of the angular impulse-momentum equation (\[eq:in002a\]) together with (\[eq:in002b\]), or with the more general energy equation (\[totale\]), we discuss, in the next sections, a set of examples combining mechanics and thermodynamics. We selected a series of representative examples, ranging from mechanics to thermodynamics as in [@guemez13], which illustrates the additional information that can be extracted from equation (\[totale\]), with respect to Newton’s equation. It is important to note that in the formulation of Newton’s second law in terms of energy — second equation in (\[eq-3\]) and equation (\[eq:in002b\]) — in general it is the pseudo-work that matters, whereas in the first law of thermodynamics it is the work that matters. This is a clear difference between (\[totale\]) and the preceding equations, which is very natural since they correspond to two distinct physical laws. This subtle difference is probably not always adequately mastered.
This paper, following the structure used in [@guemez13], is organized as follows. In section \[sec:falling\] we study a descending pulley, connected to a rotating axially fixed one, the whole system being a mechanical energy conserving one. In this case, the system is a pure mechanical one and equation (\[totale\]) does not apport additional information with respect to the mechanical equations. In section \[sec:rotatingdisc\] it is analyzed the motion of a disc acted upon by a pair of opposite forces and also subjected to friction forces (with a consequent mechanical energy dissipation). We treat the dissipative work as heat when we apply equation (\[totale\]), as mentioned above. In section \[sec:fire\] we consider a simplified fireworks wheel as an example of a system where there is production of rotational kinetic energy. In Section \[sec:joule\] we discuss, in the mechanical-thermodynamical perspective, the historical wheel paddle Joule’s apparatus to illustrate the ‘mechanical equivalent of heat’ In the final section we present the conclusions.
Falling and rotating pulley {#sec:falling}
===========================
We first consider a translational-rotational motion where there is conservation of mechanical energy. In figure \[fig:rotatingdiscdescendingdisc\] the system and the process are sketched: a pulley of mass $M$ and radius $R$, rotates and falls down attached by a rope to a similar axially fixed rotating pulley ([@knudsen00] p. 307). The rotation of the upper pulley or disc (disc 1) is frictionless and the moment of inertia for the rotation around the axis is assumed as $I={1\over 2}MR^2$.
![\[fig:rotatingdiscdescendingdisc\] Falling rotating pulley attached to an axially fixed rotating pulley, at $t=0$ and at a generic $t$. Indicated are the tension forces on the pulleys transmitted by the rope, $\vec T $ and $\vec T'$, the weights, $\vec G$, and the force, $\vec F$, exerted by the support on the top pulley. ](duasroldanas.eps){width="5.5cm"}
We first consider two separate systems: pulley 1 and pulley 2. The forces are indicated in the figure — the two tensions are constant and equal in magnitude, $T'=T$, which establishes a connection between the two systems; the weight of each disc is $G=Mg$ and, since disc 1 has no translational motion, $F=Mg+T$. For the rotation of disc 1, assuming that the discs are at rest for $t=0$, equations (\[eq:in002a\]) and (\[eq:in002b\]) lead, after a trivial integration, to {
[rl]{} I &= T R t\
[12]{} I \^2 &= T R
. \[8670r\] where $\omega$ is the angular velocity at instant $t$ and $\theta$ is the corresponding angular displacement of the disc. The angular acceleration is constant, $\alpha= TR/I$, and $\omega=\alpha t$. From equations (\[8670r\]) one readily obtains $\omega=2 \theta / t$ and, therefore, $\theta = {1 \over 2} \alpha t^2$. Because $T'=T$, rotation equations for disc 2 are the same as for 1, so all the previous results hold for disc 2 ($\theta$, $\omega$ and $\alpha$ are the same for both pulleys).
Regarding the translation of disc 2, equations (\[eq-3\]) lead to {
[rl]{} M v &= (Mg-T) t\
[12]{} M v\^2 &= (Mg-T) y
. \[8670s\] where $v$ is the velocity of the center of disc 2 at instant $t$ and $y$ is the vertical displacement of that disc. The disc falls down with constant acceleration, $a=(Mg-T)/M$, and $v=a t$. From equations (\[8670s\]) one obtains $v=2y / t$ and, therefore, $y = {1 \over 2} \, a t^2$.
Assuming that the rope does not slide on the pulleys’ rim, the connection between the translation and rotations is expressed by the geometrical relation $y=2R\theta$, which yields $v=2R\omega$ and $a=2R\alpha$. Using these relations in (\[8670r\]) and (\[8670s\]) one obtains T=[15]{}Mg, F=[6 5]{}Mg, a= [45]{} g, v= . \[trgh\]
Next we consider the translation of the system as a whole (pulley 1 + pulley 2). Equations (\[eq-3\]) lead to {
[rl]{} 2 M v\_[cm]{} &= (2Mg-F)t ,\
[12]{} 2 M v\_[cm]{}\^2 &= (2Mg-F)y\_[cm]{} .
. \[8670t\] The acceleration of the center-of-mass, $a_{\rm cm}= v_{\rm cm}/t$, is readily obtained as well as the center-of-mass velocity and the vertical displacement: a\_[cm]{}= [25]{} g, v\_[cm]{}= [v2]{}, y\_[cm]{}= [y2]{} . \[tys\]
Let’s now investigate what does equation (\[totale\]), applied to the whole system, tell us in this process. The integrated form of that equation reads K\_[cm]{} + \_k U\_k = \_j W\_j\^[ext]{} + Q \[totalei\] The first term, $\Delta K_{\rm cm}$, is the variation of center-of-mass kinetic energy, whose calculation is straightforward using (\[tys\]). The rotational energy of the discs is part of the internal energy. We denote the associated energy variation by $\Delta U_{\rm R}$. On the other hand, the translational kinetic energy of the system with respect to its center-of-mass is also part of the internal energy of the system as a whole, and we denote that energy variation by $\Delta U_{\rm K}$. In the absence of dissipative processes or internal sources of energy there is no temperature change, hence the internal energy variation is just due to the variation of the internal kinetic energies, namely $\Delta U=\Delta K_{\rm int}=\Delta U_{\rm R}+\Delta U _{\rm K}$. Regarding the right hand side of (\[totalei\]) the work of the external forces is the work of the gravitational force on disc 2, $W_G$ (the tensions on the rope are internal forces and, moreover, their total work is zero). Altogether, and taking into account the previous results, one has
[rl]{} K\_[cm]{} &= [12]{} 2M v\_[cm]{}\^2 = [14]{} M v\^2\
U\_[K]{} &= [12]{} M (0-v\_[cm]{} )\^2 + [12]{} M (v-v\_[cm]{} )\^2 = [14]{} M v\^2\
U\_[R]{} & = 2 I \^2 = [18]{} M v\^2\
W\_G & = Mgy .
\[8670x\] Inserting in equation (\[totalei\]) one obtains M v\^2 = Mgy + Q . \[aavcf\] Using here the last expression in (\[trgh\]) for $v$, one immediately concludes that $Q=0$, an expected result consistent with the assumption of the non sliding condition. Therefore, one also concludes that the entropy of the universe does not increase, the process is reversible and the mechanical energy remains constant. In this case equation (\[totale\]) and the second equation (\[eq-3\]) are equivalent.
If there were dissipative friction forces, for instance in the rotation of the axially fixed pulley or between the rope and the pulleys, the mechanical treatment would be more complicated (due to the presence of new forces and torques) and so would the thermodynamical treatment. In that case, $Q\not= 0$ corresponding to a heat transfer to the surroundings and a possible temperature increase of the pulleys. This will be the case in the next example where we study a rotating system in which kinetic friction forces are present. Though equation (\[aavcf\]) gets modified if there are dissipative forces, its present form allows for some qualitative analysis if we assume a heat such that $Q<0$ (heat dissipated in the surrounding). In that case, $Mgy>{5 \over 8 } M v^2 $, i.e. for the same distance traveled by centre-of-mass the velocity would be lower than it actually is when no dissipative processes are present.
Rotating disc with friction {#sec:rotatingdisc}
===========================
An axially fixed disc of radius $R$, mass $M$ and moment of inertia $I$ may freely rotate around its axis. It is acted upon by two horizontal forces, $\vec F$ and $\vec F '$, equal in magnitude and opposite, applied on (massless) ropes wrapped around two narrow pulleys of radii $r$ and $r'$, as shown in fig. \[fig:rotatingdiscfriction\]. On the other hand, two vertical friction forces, $\vec f$ and $\vec f '$, act on the rim of the disc at P and P$'$. The cartesian components of the forces are: $\vec F=(-F,0,0)$ and $\vec F '=(F,0,0)$; $\vec f=(0,f,0)$ and $\vec f '=(0,-f,0)$ — hence, there is no translational motion, i.e. $\d K_{\rm cm}=0$.
![\[fig:rotatingdiscfriction\] Rotating disc with friction, at $t=0$ and at a generic $t$. The vectors are the tension forces transmitted by the ropes, $\vec F$ and $\vec F'$, and the friction forces $\vec f$ and $\vec f'$. ](rodacomatrito.eps){width="14cm"}
The system is the disc and the massless ropes. Equations (\[eq:in002a\]) and (\[eq:in002b\]) are readily integrated (all forces are constant) leading to {
[rl]{} I &= t ,\
[12]{} I \^2 &= .
. \[86r\] As in the previous example, $\omega =2 \theta/t$. Since $F$ and $f$ are constants, the angular acceleration is constant, $\alpha=\omega/t$, and given by = For given $F$ and $f$ (which is such that $\alpha$ is non-negative) this acceleration can be determined. On the other hand, the angular velocity and the angular displacement at instant $t$ are $\omega=\alpha t$ and $\theta={1\over 2} \alpha t^2 $, respectively.
Next, let us consider the energy equation (\[totale\]) for the system. There is no center-of-mass displacement and, hence, there is no translational energy with respect to the center-of-mass. Therefore, the internal energy variation solely results from the rotational energy increase, ${1\over 2}I \omega^2 $, and from a possible temperature variation expressed by $Mc\Delta T$. Equation (\[totale\]) allows us to write \[eq:rdf003\] [12]{} I \^2 + M c T = F (r + r’) + Q , where the work of the forces transmitted by the ropes (whose displacements are $\Delta x=r\theta$ and $\Delta x'=r'\theta$) is taking into account on the right-hand side. The friction forces are always applied at the same points and it can be argued that they don’t do any work [@halliday01] accountable by the first term on the right-hand side of (\[totale\]), as mentioned in the Introduction. Instead, we take the point of view that this dissipative work should be taken into account by the second term on the right-hand side of (\[totale\]). Using the second equation (\[86r\]) in (\[eq:rdf003\]) one concludes that $Q= -2fR\theta+ Mc \Delta T$. If we define $W_{\rm D}=2fR\theta>0$, we can still write $W_{\rm D}= Mc \Delta T - Q$ and the interpretation is clear: part of the energy resulting from friction, $W_{\rm D}$, is absorbed by the system which increases its temperature and, therefore, its internal energy; the other part is a heat transfer to the surrounding, which is suppose to be a heat reservoir. If the temperature variation is ignored hence, $Q=-W_{\rm D}=-2fR\theta$ is the heat exchanged with the heat reservoir that surrounds the disc. In that case, the increment of the entropy of the universe is $\Delta S_{\rm U} = -{Q\over T} = {2 f R \theta\over T} > 0$ (if the body’s temperature increase due to friction is considered, one certainly has a different entropy increase but one still has $\Delta S_\rU>0$). We note that the force-displacement product $(2f)(R\theta)$ was not considered as work strictly speaking [@wolfson07] but rather as thermal energy [@besson03]. This is a mechanical energy dissipating process and it is an irreversible one. If the friction forces vanish, $f=0$, there would be no entropy increase and the process would be reversible. In this case the energy equation (\[totale\]) would not provide any additional information in comparison to (\[eq:in002b\]). Finally, we note that if the friction forces were in directions opposite to those represented in figure \[fig:rotatingdiscfriction\], $f<0$, and the entropy of the universe would decrease, which is forbidden by the second law.
Fireworks wheel {#sec:fire}
===============
The next system, illustrating the increase of rotational kinetic energy due to an internal source, belongs to a type of examples that is scarce in the literature. In figure \[fig:fireworks\] we sketch a simple fireworks wheel with two cartridges filled with gunpowder. When the system stars the operation, the rotational energy increase is obviously due to the chemical reactions in the gunpowder.
![\[fig:fireworks\] Fireworks wheel (with two cartridges of gunpowder), at $t=0$ and at a generic $t$. Indicated are the forces on the wheel resulting from the combustion of the gunpowder. ](rodadefogo.eps){width="14cm"}
At a certain time, $t$, the wheel already rotated by an angle $\theta$ due to the binary of forces, $\vec F$ and $\vec F'$, resulting from the combustion of the gunpowder, whose magnitudes are equal and assumed to be constant, from $t=0$ until the exhaustion of the gunpowder. The cartridges containing gunpowder are glued to the wheel, not necessarily equidistant from the center, as shown in figure \[fig:fireworks\]. There is no translational motion, so that $\Delta K_{\rm cm}=0$ in (\[totale\]). Moreover, the rotation around the axis is free (no friction forces are assumed).
The mechanical system is the rotating wheel (including the cartridges and excluding the products of the gunpowder chemical reaction, which anyway are almost massless). For magnitude constant forces it is straightforward to apply equations (\[eq:in002a\]) and (\[eq:in002b\]) to the present example which, after integration, lead to {
[rl]{} I &= F (r+r’) t ,\
[12]{} I \^2 &= F (r+r’) .
. \[86hr\] Once more $\omega=2\theta/t$ and the constant angular acceleration, $\alpha=\omega/t$, is given by = [F(r+r’) I]{} .
The energy equation (\[totale\]) is obviously required to analyze this fireworks wheel. For instance, the role played by the gunpowder has not been explicitly considered up to now but it enters, indirectly, through $F$ and $t$ in equations (\[86hr\]): the maximum $t$ is related to the amount of gunpowder in the cartridges and $F$ is related to its ‘quality’.
Before applying equation (\[totale\]) one should carefully define the thermodynamical system. To derive the previous mechanical equations, the mechanical system was essentially the wheel. The two forces $\vec F$ and $\vec F'$ are “external" since they are, ultimately, produced by the gunpowder which was excluded from the system. However, for the thermodynamical description, it is more interesting to include the gunpowder in the system. As already mentioned, $\Delta K_{\rm cm} = 0$ and the first law of thermodynamics — equation (\[totale\]) — for this process is \[eq:fw003\] U\_[R]{} + U\_T + U\_=W\_+ Q\_ . On the left-hand side we sum the contributions to the internal energy variation due to the increase of rotational kinetic energy, to the temperature variation and to the chemical reactions: {
[rl]{} U\_[R]{} &= [12]{}I \^2\
U\_T &= M c (T - T\_0)\
U\_&= n u\_ ,
. \[ver45a\] where $T_0$ is the initial temperature and $T$ the temperature at time $t$, $c$ is the average specific heat of the system of mass $M $, and $n$ is the number of moles of consumed gunpowder. For simplicity, we firstly neglect the temperature variation in (\[ver45a\]). Let us now analyze the contributions to the right-hand side of (\[totale\]). A fraction of the energy liberated in the chemical reaction, must be spend as work, $W_\xi$, due to gas expansion against the external atmospheric pressure (a work reservoir, at constant pressure $P$), and another fraction is spent as heat, $Q_\xi$, exchanged with the wheel surrounding (a heat reservoir, at constant temperature $T_0$). Denoting by $\Delta v_\xi$ and $\Delta s_\xi$ the gunpowder chemical reaction specific volume and entropy variations, respectively, the two terms on the right-hand side of equation (\[eq:fw003\]) are explicitly given by {
[l]{}W\_= - n P v\_\
Q\_= n T\_0 s\_ .
. \[ver45b\] The molar internal energy, volume and entropy variations associated with the chemical reaction can be determined in some independent experiment. Else, they can be obtained in the framework of a model calculation. For instance, the molar internal energy variation $\Delta u_\xi$ is, in principle, computable from tabulated binding energies, knowing the number of broken and created bonds in the molecules that take part in the gunpowder combustion reaction. One should note that there is, obviously, a contribution to the work in (\[totale\]) from the forces $\vec F$ and $\vec F'$. However, that contribution vanishes with the work of the corresponding reaction forces exerted by the cartridges on the gunpowder. Since the gunpowder is now included in the system, there are two pairs of action-reaction forces, namely $(\vec F, -\vec F)$ and $(\vec F', -\vec F')$, and the total work of each pair vanishes. In conclusion, when the system includes, altogether, the wheel and the gunpowder inside the cartridges, those action-reaction pairs are internal forces whose total work, in the present case, vanishes, since the displacement of the forces in each pair is the same.
Inserting (\[ver45a\]) and (\[ver45b\]) in equation (\[eq:fw003\]) one readily obtains \[eq:fw003b\] [12]{}I \^2 = - n g\_ , where $\Delta g_\xi = \Delta u_\xi + P\Delta v_\xi - T\Delta s_\xi$ is the Gibbs molar free energy variation for the chemical reaction. It is the Gibbs energy variation that naturally appears in the energy balance equation because the process is assumed to take place in contact with a heat and a work reservoirs. In a not so refined calculation it would be simply the internal energy variation and not the free Gibbs energy that would appear in the equations. If the wheel increases its temperature during the process, there is an extra term, $Mc(T-T_0)>0$ on the left-hand side of equation (\[eq:fw003b\]), i.e. ${1\over 2}I \omega^2 < - n \Delta g_\xi\, $ because part of the available energy ought to be used to increase the temperature of the wheel, so that the final kinetic energy would be correspondingly reduced.
But let’s go back to equation (\[eq:fw003b\]), which is the analog, for the rotations, to a similar expression for translations derived in the context of the study of an accelerating car [@guemez13b]. If initially the wheel is at rest, the angular velocity after the consumption of an amount $n$ of gunpowder is = ([2 n |g\_|I]{})\^[1/2]{} . Using the second equation (\[86hr\]) one also concludes that F (r + r’) = - n g\_ .
Finally, the amount of consumed gunpowder can be obtained as a function of time. For a given (constant) $F$ — therefore, a constant angular acceleration, $\alpha$ —, one finds n= [F\^2 (r+r’)\^2 2 I | g\_| ]{} t\^2 . For this idealized process, the entropy of the universe does not increase, $\Delta S_{\rm U} = 0$, if the process is reversible. Actually, the (free Gibbs) energy decrease resulting from the gunpowder reaction is stored in the system as rotational kinetic energy of the wheel. The system’s capacity to produce external work did not decrease. The variation of the Gibbs energy corresponds to the maximum “useful work" that can be obtained. In the idealized situation, the equality holds in equation (\[eq:fw003b\]) and the capacity of the system to produce useful work is totally kept. In the limit, that stored (organized) kinetic energy can be used, in particular to increase by $\Delta G=-n \Delta g_\xi$ the Gibbs function of another chemical reaction elsewhere [@atkins10]. In this sense the process occurs without any entropy increase of the universe.
Joule’s apparatus {#sec:joule}
=================
The historic wheel paddle experiment was firstly performed by Joule to measure the “mechanical equivalent of heat" [@zemansky97]. In a simplified version, it can be described as a wheel paddle that is immersed in a liquid (e.g. water) contained in a vessel with rigid adiabatic walls [@lemons09]. When it is placed in motion, the wheel paddle stirs the liquid. Figure \[fig:pasdejoule\] represents Joule’s apparatus: a spindle attached to the paddles turns around due to the action of a rope, wrapped around its rim. The rope is connected, through a pulley, to a descending block of mass $m$.
![\[fig:pasdejoule\] Joule’s apparatus to measure the mechanical equivalent of heat. On the right side we represent the descending block and the two forces applied to it: the tension, $\vec F$ and the weight $\vec G$; we also represent the force on the pulley transmitted by the rope, $\vec F'$ and the dragging forces exerted by the liquid on the paddles, $\vec F_{\rm D}$ and $\vec F'_{\rm D}$, which are along parallel lines but in opposite directions. ](pasdejoule.eps){width="7cm"}
The tensions $\vec F$ and $\vec F'$ in the (massless) rope are equal in magnitude and this establishes a relation between the mechanical system on the left (spindle attached to paddles) and the mechanical system on the right (block). The block falls down subjected to its weight, $\vec G$, and to the upward tension whose magnitude is variable but such that $F\le mg$. The actual value of $F$ is also determined by the dragging forces ($\vec F_{\rm D}$ and $\vec F'_{\rm D}$) produced by the fluid on the paddles. The torque due to these forces, $\vec \Gamma_{\rm D}$, depends on the paddle’s area, on the viscosity of the fluid and, surely, on the velocity of the rotating paddles. One expects that, after some time, a steady state for which $F=mg$ can eventually be achieved. Though $F$ is a variable force, its actual value is not needed for the forthcoming discussion. The falling object communicates, via the tension in the rope, an external torque $FR$ to the paddle, where $R$ is represented in figure \[fig:pasdejoule\].
Since $F$ is, at least for some time, a variable force, it is better to use the differential form of the equations to describe the process, as given in section 1. Let us first discuss the motion of the block alone. The second equation in (\[eq-3\]) leads, in the present case, to (mg-F) y = [12]{} m v\^2 \[fga23a\] Now let’s consider the other mechanical system: the moving wheel paddle. We assume that the rotation around the axis is frictionless. On the other hand, all vertical forces (weight, normal force) have zero resultant and they do not contribute to the rotation since they are along the rotation axis. For the rotation of the wheel paddle there are two torques to be considered: the one produced by the tension $\vec F'$ and the torque produced by the fluid, in the opposite direction, resulting from the dragging forces represented in figure \[fig:pasdejoule\]. Under the action of the two torques, equation (\[eq:in002b\]) takes the form
( FR - \_[D]{}) = [12]{}I \^2 \[fga23b\] where $I$ is the moment of inertia for the rotation of the wheel paddle around the vertical axis (with the minus sign on the left hand side, $\Gamma_{\rm D}>0$). The connection between the translation of the block, on the right, and the rotation of the wheel paddle, on the left, is established by the equation $\d y = R \d \theta$ (or, equivalently, by $\d v = R \d \omega$).
Regarding equation (\[totale\]) applied to the block, it does not add new information with respect to (\[fga23a\]), but it does when it is applied to the wheel paddle and the liquid. For the thermodynamical discussion of the left part of figure \[fig:pasdejoule\] the system [ is now]{} the wheel paddle [*plus*]{} the liquid inside the vessel. One should note that the forces exerted by the paddles on the fluid and by the fluid on the paddles are equal and opposite (action-reaction pair) and these internal forces produce a vanishing work contribution to the right-hand side of (\[totale\]). (The situation is similar to the one found in section \[sec:fire\] when we included the gunpowder in the system.) Let us denote by $M$ the mass of the liquid in the vessel which is thermally isolated from the exterior. If we denote the specific heat of the fluid by $c$, the variation of the internal energy receives a contribution from the temperature change, $\d U_T= M\,c\, \d T$. The other contribution to the internal energy variation of the system is due to the wheel paddle rotation, $\d U_{\rm R}= {1\over 2} I \d \omega^2$. Since $\delta Q=0$ (vessel with adiabatic walls, so there is no external heat transfer) and the external work reduces to $F R \d \theta=F \d y$, equation (\[totale\]) takes the form (for the system under consideration — paddles and liquid —, $\d K_{\rm cm}=0$) I \^2 +M c T = F y . \[fga23c\] Eliminating $F$ by means of (\[fga23a\]) one concludes that M c T = mg y - [12]{} m v\^2 - [12]{}I \^2 or, eqivalently, $m\, g \, \d y=M\,c \, \d T+\d K$, where the second term here represents the variation of the total kinetic energy of the global system. Is is worthwhile to point out that, before the steady state starts, the potential energy is also transformed into kinetic energy of the block and paddles.
When the steady state is achieved (constant $v$ and constant $\omega$ — in the original Joule experiment $v \approx 0$ and $\omega \approx 0$), the previous equation can be integrated out, yielding, for a constant specific heat, T= T\_0 + [mgMc]{} y \[sdfg\] where $T_0$ is the temperature of the fluid when the steady state starts. This equation expresses the direct transformation of ‘organized’ gravitational potential energy into ‘disorganized’ internal energy with a consequent increase of the entropy of the universe. Of course, we are assuming that the paddles, as well as the vessel walls, do not change their temperature.
The increase of the entropy of the universe is only due to the entropy increase in the fluid. It can be easily calculated using an auxiliary quasi-static process (Clausius algorithm) [@leff12] of heating up the fluid on the vessel: S\_[U]{} = \_[T\_0]{}\^T [McT T]{} = Mc > 0 or, if $T\sim T_0$, and using (\[sdfg\]), S\_[U]{} = [m g y T\_0]{} . The process is clearly irreversible ($\Delta y>0$ for the descending block).
A final remark on equation (\[fga23b\]). For the steady state it can equivalently be expressed, after integration, as W \_[D]{} =mgy [or]{} W \_[D]{}=M c T where $W_{\rm D}=\Gamma_{\rm D}\Delta \theta$ is the torque exerted on the paddles produced by the dragging forces times the angular displacement. This is also equal to the dissipative work of the dragging forces [@anacleto10] that leads to the direct increase of the internal energy of the liquid, through a temperature increase. Therefore, a ‘mechanical equivalent of heat’ ([*i.e.*]{} dissipative work) was used to increase the temperature of a system. The result expressed by (\[sdfg\]) was originally observed experimentally by Joule, and played an important role in the development of thermodynamics.
Conclusions {#sec:conclusions}
===========
We presented and discussed various examples, all involving rotational motion, whose analysis requires the first law of thermodynamics in addition to the Newton’s second law. We started with a situation for which the mechanical treatment is sufficient. Such processes are reversible and do not lead to an entropy increase of the universe. In general, this is not the case when we are in the presence of dissipative forces, also analyzed in this paper. In particular we considered kinetic friction forces whose work is thermodynamically equivalent and considered as heat (therefore that work can set to zero and included as heat). The role of friction forces is subtle and sometimes not adequately addressed in textbooks [@bauman92]. Examples with rotational kinetic energy increase due to internal sources were also considered here, in particular we discussed the motion of a fireworks wheel.
Finally, we discussed the Joule’s wheel paddle apparatus, an historic experiment to measure the mechanical equivalent of heat. Thermodynamics textbooks mention the experiment but usually they do not provide the detailed physics explanation for the process. In particular we stressed the role of the forces of the paddles on the liquid that produce a dissipative positive work.
In this concluding remarks one should also stress that, in the formulation of the first law of thermodynamics, the kinetic rotational energy of a system and the kinetic translational energy with respect to the center-of-mass were always considered as part of the internal energy of the system.
Summarizing, we illustrated that processes undergone by mechanical systems, usually also require a thermodynamical detailed analysis in order to be fully understood, a point of view adopted in the most modern physics textbooks [@chabay11b]. With the variety of examples, all involving rotations, we intended to show the appropriateness of the first law of thermodynamics to complement the pure mechanical description of a system provided by the Newton’s second law.
[99]{} L W Taylor, [*Physics. The Pioneer Science. Volume I. Mechanics, Heat, Sound*]{}, Dover, New York, 1959, pp. 236-8.
[Leonhard Euler (1707-83), through the definition of the moment of inertia, wrote, around 1765, the rotational equation which corresponds to ${\vec F}=m {\vec a}$, namely ${\vec \Gamma}=I{\vec \alpha}$. The analogy of rotation with translation through the identification of the angular variables variables ($\theta$, $\omega$, $\alpha$, etc.) with the translational ones ($r$, $v$, $a$, etc.), was established by Louis Poinsot (1777-1859), around 1838.]{}
A. Pinto, M. Fiolhais, [*Rolling cylinder on a horizontal plane*]{}, Phys. Edu. [**36**]{} 250-254 (2001);
P. S. Carvalho, A. S. Sousa, [*Rotation in secondary school: teaching the effects of frictional force*]{}, Phys. Edu. [**40**]{} 257-265 (2005) J M Knudsen, P G Hjorth, [*Elements of Newtonian Mechanics. Including Nonlinear Dynamics*]{}, Springer Heidelberg 2000, pp. 290-291. J. Güémez, M. Fiolhais, [*From mechanics to thermodynamics: analysis of selected examples*]{}, Eur. J. Phys. [**34**]{} 345-357 (2013) B A Sherwood, [*Pseudowork and real work*]{}, Am. J. Phys. [**51**]{}, 597-602 (1983) C M Penchina, [*Pseudowork-energy principle*]{}, Am. J. Phys. [**46**]{}, 295-296 (1978) C A de Sousa, E P Pina, [*Aspects of mechanics and thermodynamics in introductory physics: an illustration in the context of friction and rolling*]{}, Eur. J. Phys. [**18**]{}, 334-337 (1997) A J Mallinckrodt, H. S. Leff, [*All about work,*]{} Am. J. Phys. [**60**]{} 356-365 (1992) P A Tipler, G Mosca, [*Physics for Scientists and Engineers*]{} 5th Ed. W H Freeman and Co. New York 2004, Sec. 9-4 M Alonso, E J Finn [*University Physics Vol. I, Mechanics*]{} Addison Wesley Publishing Company (1967); D Kleppner, R Kolenkow [*An introduction to mechanics*]{}, Mc Graw-Hill Publishing Company (1973) B A Sherwood and W H Bernard, [*Work and heat transfer in the presence of sliding friction*]{}, Am. J. Phys. [**52**]{}, 1001-1007 (1984). p. 1006 J Walker, [*Halliday, and Resnick Principles of Physics. International Student Version* ]{}, 9th Ed. John Wiley and Sons, 2001, pp. 262. H D Young and R A Freedman, [*Sears and Zemansky’s University Physics. With Modern Physics*]{}, 13th Ed. Addison-Wesley, San Francisco 2012. Sec. 10.4, p. 321. H Erlichson, [*Internal energy in the First Law of thermodynamics*]{}, Am. J. Phys. [**52**]{} 623-625 (1984) R A Serway, J W Jewett, Jr., [*Principles of Physics. A Calculus-Based Text*]{}, 3rd Ed. Harcourt College Publishers, p. 194. P Atkins, [*The Laws of Thermodynamics. A very Short Introduction*]{}, Oxford U. Press, Oxford UK 2010, pp. 65-72 J W Jewett, Jr. [*Energy and the Confused Student V: The Energy/momentum approach to problems involving rotating and deformable systems*]{}, Phys. Teach. [**46**]{}, 269-274 (2008) U Besson, [*Work and energy in the presence of friction: the need for a mesoscopic analysis*]{}, Eur. J. Phys. [**22**]{} 613-622 (2001) P T Landsberg, [*The Born Centenary: Remarks about classical thermodynamics*]{}, Am. J. Phys. [**51**]{}, 842-845 (1983)
[ In the ‘Born formulation of the first law of thermodynamics’ (with $\d K_{\rm cm}=0$) internal energy variations are related with an adiabatic work, $\d U=\delta W_{\rm ad.}$ (this term might include dissipative work, $\delta W_{\rm D}$, that can always be measured), and then heat, $\delta Q$, exchanged during a process is operationally defined as the difference $\delta Q = \d U - \delta W$, $\delta W$ being the actual work done during the process.]{}
J. Güémez, M. Fiolhais, [*The physics of a walking robot*]{}, Phys. Educ. [**48**]{} 455-458 (2013) J W Jewett Jr., R A Serway [*Physics for Scientists and Engineers with Modern Physics*]{} 8th Ed. Brooks/Cole Cengage Learning (2010), p. 209 H S Leff, [*Removing the mystery of entropy and thermodynamics - Part II*]{}, Phys. Teach. [**50**]{} 87-90 (2012) D Halliday, R Resnick, J Walker, [*Fundamentals of Physics*]{}, 6th Ed. John Wiley and Sons, New York 2001, p. 153 R Wolfson, [*Essential University Physics. Vol. 1*]{} 2nd ed. Addison-Wesley San Francisco 2007. p. 166 U Besson, [*The distinction between heat and work: an approach based on a classical mechanical model*]{}, Eur. J. Phys. [**24**]{} 245-252 (2003) J. Güémez, M. Fiolhais, [*Forces on wheels and fuel consumption in cars*]{}, Eur. J. Phys. [**34**]{} 1005–1013 (2013) M W Zemansky, R H Dittman, [*Heat and Thermodynamics: An Intermediate Textbook,*]{} 7th Ed., McGraw-Hill International Editions (1997), p. 73, p. 79. D S Lemons, [*Mere Thermodynamics*]{}, The Johns Hopkins University Press, Baltimore 2009. p. 12. J Anacleto, [*Work reservoirs in thermodynamics*]{}, Eur. J. Phys. [**31**]{} 617-624 (2010) H S Leff, [*Removing the mistery of entropy and thermodynamics - Part I*]{}, Phys. Teach. [**50**]{} 28-31 (2012). p. 30 R P. Bauman, [*Physics that textbook writers usually get wrong. I. Work*]{}, Phys. Teach. [**30**]{}, 264-269 (1992) R W Chabay and B A Sherwood, [*Matter and Interactions*]{}, 3th ed. Wiley and Sons, Hoboken 2011, Sec. 11.6
[^1]: [email protected]
[^2]: [email protected]
|
---
abstract: 'In this paper, we study the behaviour of the output of pure entangled states after being transformed by a product of conjugate random unitary channels. This study is motivated by the counterexamples by Hastings [@hastings] and Hayden-Winter [@hayden-winter] to the additivity problems. In particular, we study in depth the difference of behaviour between random unitary channels and generic random channels. In the case where the number of unitary operators is fixed, we compute the limiting eigenvalues of the output states. In the case where the number of unitary operators grows linearly with the dimension of the input space, we show that the eigenvalue distribution converges to a limiting shape that we characterize with free probability tools. In order to perform the required computations, we need a systematic way of dealing with moment problems for random matrices whose blocks are i.i.d. Haar distributed unitary operators. This is achieved by extending the graphical Weingarten calculus introduced in [@cn1].'
address:
- ' Département de Mathématique et Statistique, Université d’Ottawa, 585 King Edward, Ottawa, ON, K1N6N5 Canada and CNRS, Institut Camille Jordan Université Lyon 1, 43 Bd du 11 Novembre 1918, 69622 Villeurbanne, France'
- 'Zentrum Mathematik, M5, Technische Universität München, Boltzmannstrasse 3, 85748 Garching, Germany'
- 'CNRS, Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, UPS, 31062 Toulouse, France and Zentrum Mathematik, M5, Technische Universität München, Boltzmannstrasse 3, 85748 Garching, Germany'
author:
- Benoît Collins
- Motohisa Fukuda
- Ion Nechita
title: Low entropy output states for products of random unitary channels
---
Introduction
============
In Quantum Information Theory, random unitary channels are completely positive, trace preserving and unit preserving maps that can be written as $$\Phi(\rho) = \sum_{i=1}^kw_i U_i \rho U_i^* ,$$ where $U_i$ are unitary operators acting on $\mathbb C^n$ and $w_i$ are positive weights that sum up to one.
This class of quantum channels is very important, not only because the action of such a channel can be understood as randomly applying one of the unitary transformations $U_i$ with respective probabilities $w_i$, but also because it has highly non-classical properties. For example, random unitary channels have been used to disprove the additivity of minimum output entropy [@hastings].
In this paper, we are interested in a natural setting in which we take a convex combination of $k$ *random* unitary evolutions; in other words, we choose the unitary operators $U_i$ at random and independently from the unitary group, following the Haar distribution. The behaviour of this kind of quantum channels has been extensively studied in the literature [@aubrun; @hlsw; @hastings].
We are principally interested in the study of the moments of typical outputs for products of conjugated random unitary channels. One of the main results of our paper is that the typical outputs are deterministic when one takes a product of such a channel and its complex conjugate and applies it to entangled input states, in the spirit of [@cn1; @cn3; @cfn1]. More precisely, our main results can be stated informally as follows (for precise statements, see Theorems \[thm:bell-phenomenon\] and \[thm:k-sim-n\]):
Consider the output state $Z_n = [\Phi \otimes \bar \Phi](\psi_n \psi_n^*)$, given as the image of a “well behaved” pure state $\psi_n$ under the product of conjugate random unitary channels.
If $k$, the number of unitary operators $U_i$, is fixed and $n \to \infty$, then the set of the $k^2$ non-zero eigenvalues of $Z_n$ is $$\{w_iw_j \, : \, i,j = 1, \ldots, k, i \neq j\} \cup \{s_i \, : \, i=1, \ldots, k\},$$ where the numbers $s_i$ depend on $w$ and some parameter $m$ quantifying the overlap between the input state $\psi_n$ and the Bell state ${\varphi}_n$.
If both dimensions $k$ and $n$ grow to infinity, but at a constant ratio $k/n \to c$, then the largest eigenvalue of $Z_n$ behaves as $\mathrm{Const.}/k$ and the rest of the spectrum, when properly rescaled, converges towards a compound free Poisson distribution that can be characterized in terms of the behaviour of the ratio $c$ and the weights $w_i$.
We strengthen previous results on the eigenvalues of the output states $Z_n$, obtained by linear algebra techniques, by computing exactly the asymptotic spectrum of the output states. Such precise results, along with results for single channels [@cfn2], may improve existing violations of the additivity of the minimum output entropy or similar quantities. For the purpose of studying these random channels, we extend the graphical calculus introduced in [@cn1] to the case of multiple independent, identically distributed ( i.i.d. ) random Haar unitary matrices, see Theorem \[thm:V-graphical-calculus\].
Our paper is organised as follows. In Section \[sec:random-unitary-channels\] we recall facts about quantum channels and give the definition of random unitary channels. In Section \[sec:graphical-calculus\], we extend Weingarten calculus to include the case of several independent Haar unitary matrices. Section \[sec:measures-isometry\] provides a comparison of Haar random isometries and block random isometries. Even though it is intuitively clear that the two are quite different from each other, we give explicit instances of the differences via a few explicit moment computations. These moment computations serve also as a warm-up for Sections \[sec:fixed-k\] and \[sec:k-sim-n\], where the behaviour of output of entangled pure states is considered under different scalings; these sections contain the main results of the paper. In Section \[sec:fixed-k\] we treat the case of a fixed number of unitary operators, while in Section \[sec:k-sim-n\] we consider the scaling $k/n \to c$. We analyze thoroughly the entropies of the limiting objects, finding the parameter values which yield the output states with the least entropy. Finally, we make a few concluding remarks in Section \[sec:conclusions\].
Random unitary channels {#sec:random-unitary-channels}
=======================
Our model
---------
Random unitary channels are natural models among quantum channels and have played a key role in the research around violations of different quantities related to classical capacities of quantum channels.
Random unitary channels are unit-preserving quantum channels that can be written as \[eq:def-Phi\] () = \_[i=1]{}\^k w\_i U\_i U\_i\^\* where $U_i \in \mathcal{U} (n)$ are independent random unitary operators with *uniform* (or *Haar*) distribution and $w_i$ are fixed positive weights that sum up to one. Hastings used a similar model to prove additivity violation of minimal output entropy [@hastings]. In our model, a unitary operator $U_i$ is applied to an input with some fixed probability $w_i$, while in Hastings’ model those probabilities were chosen randomly.
Stinespring picture and complementary channel
---------------------------------------------
The above random channel $\Phi:M_n(\mathbb C) \to M_n(\mathbb C)$ is described in the Stinespring picture as () = \_[k]{} where \[isometry\] V = \[(,…,) \_n\] V =
U\_1\
\
U\_k
: \^n \^k \^n is an isometry, $$V = \sum_{i=1}^k e_i \otimes U_i$$ is a stack of unitary operators and $\operatorname{Tr}_{k}$ is the partial trace operator $\operatorname{Tr}\otimes\mathrm{id}$ acting on $M_k(\mathbb C) \otimes M_n(\mathbb C)$. In matrix representation, V V\^\* =
w\_1 U\_1 U\_1\^\* & …& U\_1 U\_k\^\*\
& &\
U\_k U\_1\^\* & …& w\_k U\_k U\_k\^\*
As one sees, tracing out the first space (the environment space ${\mathbb{C}}^k$) gives the sum of diagonal blocks, which is the quantum channel $\Phi$. On the other hand, if we switch the roles of output and environment spaces, i.e., if we trace out the second space (the output space ${\mathbb{C}}^n$), we get the complementary channel $\Phi^C: M_n(\mathbb C) \to M_k(\mathbb C)$ [@Holevo05; @KMNR]: ( \^C () )\_[i,j]{}= A quantum channel and its complementary channel share the same output eigenvalues, up to zeroes, for any pure input via Schmidt decomposition, see [@Holevo05; @KMNR] for details.
Recall that the Shannon entropy functional, defined for probability vectors $p=(p_i)$ $$H(p) = -\sum_i p_i \log p_i,$$ extends to unit trace, positive matrices (or quantum states) $\rho$ via functional calculus $$H(\rho) = -\operatorname{Tr}(\rho \log \rho).$$
For a quantum channel $\Phi$, one can define its *minimal output entropy* $$H_\text{min}(\Phi) = \min_{\rho \text{ state}} H(\Phi(\rho)).$$
By using convexity properties, one can show that the minimum above is reached on pure states (i.e. rank one projectors) so that the value of the minimum output entropy does not change when by replacing $\Phi$ by $\Phi^C$. Our interest in complementary channels is motivated by the fact that, often, the size of the environment is smaller than the output size, so output states are easier to study for $\Phi^C$.
In the seminal paper of Hastings [@hastings], violation of additivity was found when the dimensions of input and output spaces are much larger than that of the environment space of the channel. In this paper, we shall study two asymptotic regimes, $k$ fixed and $n \to \infty$ and then $k,n \to \infty$, with a linear growth $k/n \to c$.
Weingarten calculus for several independent unitary matrices {#sec:graphical-calculus}
============================================================
The method of graphical calculus was introduced in [@cn1] and later used in [@cn3; @cn-entropy; @cnz; @cfn1] to study random matrix models in which Haar distributed unitary operators played a major role. In particular, the limiting eigenvalue distribution of $[\Phi \otimes \bar \Phi] (\psi_n\psi_n^*)$, where $\Phi$ is random quantum chanel and $ \psi_n $ is a generalised Bell state, was calculated in [@cfn1]. The aim of this paper is to perform similar computations for random unitary channels instead of random channels.
In the previous work in which random quantum channels were analysed using the graphical calculus method [@cn1; @cn3; @cn-entropy; @cfn1], the isometry defining the channel in the Stinespring picture was a truncation of a Haar-distributed random unitary matrix. Hence, only one random unitary operator was needed to perform the computations. In this paper, we need to develop a new technique of graphical calculus for $k$ i.i.d. Haar-distributed unitary matrices.
We start with the usual, 1-matrix, graphical Weingarten calculus and then generalise it to cover the case of block-isometries built up from independent Haar-distributed unitary matrices.
Weingarten formula for a single matrix
--------------------------------------
Let us start by recalling the definition of a combinatorial object of interest, the unitary Weingarten function.
The unitary Weingarten function $\operatorname{Wg}(n,\sigma)$ is a function of a dimension parameter $n$ and of a permutation $\sigma$ in the symmetric group ${\mathcal{S}}_p$. It is the inverse of the function $\sigma \mapsto n^{\# \sigma}$ under the convolution for the symmetric group ($\# \sigma$ denotes the number of cycles of the permutation $\sigma$).
Notice that the function $\sigma \mapsto n^{\# \sigma}$ is invertible when $n$ is large, as it behaves like $n^p\delta_e$ as $n\to\infty$. Actually, if $n<p$ the function is not invertible any more, but we can keep this definition by taking the pseudo inverse and the theorems below will still hold true (we refer to [@collins-sniady] for historical references and further details). We shall use the shorthand notation $\operatorname{Wg}(\sigma) = \operatorname{Wg}(n, \sigma)$ when the dimension parameter $n$ is clear from context.
The function $\operatorname{Wg}$ is used to compute integrals with respect to the Haar measure on the unitary group (we shall denote by ${\mathcal{U}}(n)$ the unitary group acting on an $n$-dimensional Hilbert space). The first theorem is as follows:
\[thm:Wg\] Let $n$ be a positive integer and $i=(i_1,\ldots ,i_p)$, $i'=(i'_1,\ldots ,i'_p)$, $j=(j_1,\ldots ,j_p)$, $j'=(j'_1,\ldots ,j'_p)$ be $p$-tuples of positive integers from $[n]= \{1, 2, \ldots, n\}$. Then $$\begin{gathered}
\label{eq:Wg}
\int_{{\mathcal{U}}(n)}U_{i_1j_1} \cdots U_{i_pj_p}
\overline{U_{i'_1j'_1}} \cdots
\overline{U_{i'_pj'_p}}\ dU=\\
\sum_{\sigma, \tau\in {\mathcal{S}}_{p}}\delta_{i_1i'_{\sigma (1)}}\ldots
\delta_{i_p i'_{\sigma (p)}}\delta_{j_1j'_{\tau (1)}}\ldots
\delta_{j_p j'_{\tau (p)}} \operatorname{Wg}(n,\tau\sigma^{-1}).\end{gathered}$$
If $p\neq p'$ then $$\label{eq:Wg_diff} \int_{{\mathcal{U}}(n)}U_{i_{1}j_{1}} \cdots
U_{i_{p}j_{p}} \overline{U_{i'_{1}j'_{1}}} \cdots
\overline{U_{i'_{p'}j'_{p'}}}\ dU= 0.$$
Since we shall perform integration over *large* unitary groups, we are interested in the values of the Weingarten function in the limit $n \to {\infty}$. The following result encloses all the data we need for our computations about the asymptotics of the $\operatorname{Wg}$ function; see [@collins-imrn] for a proof.
\[thm:mob\] For a permutation $\sigma \in {\mathcal{S}}_p$, we write $c \in \sigma$ to denote the fact that $c$ is a cycle of $\sigma$. Then $$\label{Wg formula 1}
\operatorname{Wg}(n,\sigma)=\prod_{c\in \sigma}\operatorname{Wg}(n,c)(1+O(n^{-2}))$$ and $$\operatorname{Wg}(n,(1,\ldots ,d) ) = (-1)^{d-1}c_{d-1}\prod_{-d+1{\leqslant}j {\leqslant}d-1}(n-j)^{-1}$$ where $c_i=\frac{(2i)!}{(i+1)! \, i!}$ is the $i$-th Catalan number.
As a shorthand for the quantities in Theorem \[thm:mob\], we introduce the function $\operatorname{Mob}$ on the symmetric group. $\operatorname{Mob}$ is invariant under conjugation and multiplicative over the cycles; further, it satisfies for any permutation $\sigma \in {\mathcal{S}}_p$: $$\label{weingartenapprox}
\operatorname{Wg}(n,\sigma) = n^{-(p + |\sigma|)} (\operatorname{Mob}(\sigma) + O(n^{-2}))$$ where $|\sigma |=p-\# \sigma $ is the *length* of $\sigma$, i.e. the minimal number of transpositions that multiply to $\sigma$; we shall also use the notation $|\cdot |$ for the cardinality of sets. We refer to [@collins-sniady] for details about the function $\operatorname{Mob}$. We shall make use of the following explicit formulas.
\[lem:Mob\] The Möbius function is multiplicative with respect to the cycle structure of permutations $$\operatorname{Mob}(\sigma) = \prod_{b \in \sigma} (-1)^{|b|-1} \operatorname{Cat}_{|b|-1},$$ where $\operatorname{Cat}_n$ is the $n$-th Catalan number. In particular, if $\sigma$ is a product of disjoint transpositions, then \[mob formula\] () =(-1)\^[||]{}.
We finish this section by a well known lemma which we will use several times in the proofs of the main theorems of this paper. This result is contained in [@nica-speicher].
\[lem:S\_p\] The function $d(\sigma,\tau) = |\sigma^{-1} \tau|$ is an integer valued distance on ${\mathcal{S}}_p$. Besides, it has the following properties:
- the diameter of ${\mathcal{S}}_p$ is $p-1$;
- $d(\cdot, \cdot)$ is left and right translation invariant;
- for three permutations $\sigma_1,\sigma_2, \tau \in {\mathcal{S}}_p$, the quantity $d(\tau,\sigma_1)+d(\tau,\sigma_2)$ has the same parity as $d(\sigma_1,\sigma_2)$;
- the set of geodesic points between the identity permutation $\operatorname{id}$ and some permutation $\sigma \in {\mathcal{S}}_p$ is in bijection with the set of non-crossing partitions smaller than $\pi$, where the partition $\pi$ encodes the cycle structure of $\sigma$. Moreover, the preceding bijection preserves the lattice structure.
Weingarten formulas for several independent unitary matrices
------------------------------------------------------------
In this paper we want to treat random matrix models in which several i.i.d. Haar random unitary matrices appear. We shall use the independence property and the Weingarten formula together and unify these properties into an unique statement, as in Theorems \[thm:colored-Weingarten\] and \[thm:colored-Weingarten-2\]. Before we can do this, we introduce the following notation (we put $[n]=\{1,2,\ldots,n\}$):
Consider two integer functions $l:[p] \to [k]$ and $l':[p'] \to [k]$. Whenever $p=p'$, we denote by $\mathcal S_p^{l \to l'}$ the set of permutations of $p$ objects which map the level sets of $l$ to the level sets of $l'$: $$\mathcal S_p^{l \to l'} = \{\sigma \in \mathcal S_p \, | \, \sigma(l^{-1}(s)) = \sigma(l'^{-1}(s))\}.$$ Note that this set is empty iff $$| \{t:l_t = s\} | \not = |\{t:l'_t = s\}|$$ for some $s \in [k]$. If $p \neq p'$, we put $\mathcal S_p^{l \to l'} = \emptyset$.
Note that permutations $\sigma \in \mathcal S_p^{l \to l'}$ admit a decomposition along the level sets of $l,l'$ $$\label{eq:decomp-perm-level-sets}
\sigma = \sigma_1 \times \cdots \times \sigma_k,$$ where $$\sigma_s: l^{-1}(s) \to l'^{-1}(s).$$
The main result of this section is the following generalization of the Weingarten formula for moments in several independent Haar unitary matrices.
\[thm:colored-Weingarten\] Let $n$ and $k$ be positive integers and $i=(i_1,\ldots ,i_p)$, $i'=(i'_1,\ldots ,i'_{p'})$, $j=(j_1,\ldots ,j_p)$, $j'=(j'_1,\ldots ,j'_{p'})$ be tuples of positive integers from $[n]$ and $l=(l_1,\ldots ,l_p)$, $l'=(l'_1,\ldots ,l'_{p'})$ be tuples of positive integers from $[k]$. Then \[eq:colored-Weingarten\] &&\_[(n)\^k]{} U\_[i\_1,j\_1]{}\^[(l\_1)]{} U\_[i\_p,j\_p]{}\^[(l\_p)]{} dU\^[(1)]{} dU\^[(k)]{}\
&=& \_[,S\_p\^[l l’]{}]{} \_[i\_1 i’\_[(1)]{}]{}\_[i\_[p]{} i’\_[(p)]{}]{} \_[j\_1 j’\_[(1)]{}]{} \_[j\_[p]{} j’\_[(p)]{}]{} \^[l l’]{}(n,, ), where the modified Weingarten function $\operatorname{Wg}^{l \to l'}$ is defined via the product formula $$\operatorname{Wg}^{l \to l'}(n,\alpha, \beta) = \prod_{s=1}^k \operatorname{Wg}(n,\alpha_s^{-1}\beta_s).$$
We start by factoring the integral using the independence of the random unitary matrices $U^{(s)}$: \[Colored Weingarten proof\] &&\_[(n)\^k]{} U\_[i\_1,j\_1]{}\^[(l\_1)]{} U\_[i\_p,j\_p]{}\^[(l\_p)]{} dU\^[(1)]{} dU\^[(k)]{}\
&=& \_[U(n)]{} \[\] dU\^[(1)]{} \_[U(n)]{} \[\] dU\^[(k)]{} The above product vanishes whenever $|l^{-1}(s)| \neq |l'^{-1}(s)|$ for some $s \in [k]$. The value of each factor is computed using the usual Weingarten formula in Theorem \[thm:Wg\] and the result follows.
One can reformulate the above result in a more practical way, eliminating the restriction on the sum indices $\alpha, \beta$, as follows.
\[thm:colored-Weingarten-2\] Let $n$ and $k$ be positive integers and $i=(i_1,\ldots ,i_p)$, $i'=(i'_1,\ldots ,i'_{p'})$, $j=(j_1,\ldots ,j_p)$, $j'=(j'_1,\ldots ,j'_{p'})$ be tuples of positive integers from $[n]$ and $l=(l_1,\ldots ,l_p)$, $l'=(l'_1,\ldots ,l'_{p'})$ be tuples of positive integers from $[k]$. Then $$\begin{aligned}
\label{eq:colored-Weingarten-2}
&\qquad\int_{\mathcal{U}(n)^k}
U_{i_1,j_1}^{(l_1)} \cdots U_{i_p,j_p}^{(l_p)} \overline{ U_{i'_1,j'_1}^{(l'_1)}} \cdots \overline{ U_{i'_p,j'_p}^{(l'_p)}} dU_1 \cdots dU_k \\
\notag
&= \sum_{\alpha, \beta \in S_p}
\underbrace{
\delta_{i_1 i'_{\alpha(1)}}\cdots\delta_{i_{p} i'_{\alpha(p)}}
\delta_{j_1 j'_{\beta(1)}} \cdots \delta_{j_{p} j'_{\beta(p)}}
}_{(\star)}
\underbrace{
\delta_{l_1 l'_{\alpha(1)}}\cdots\delta_{l_{p} l'_{\alpha(p)}}
\delta_{l_1 l'_{\beta(1)}} \cdots \delta_{l_{p} l'_{\beta(p)}}
}_{(\star\star)}
\tilde \operatorname{Wg}(n, \alpha^{-1} \beta).\end{aligned}$$ Here, $$\tilde \operatorname{Wg}(n, \alpha^{-1} \beta) = \prod_{s=1}^k \operatorname{Wg}(n,\alpha_s^{-1} \beta_s) \quad \text{ when } \alpha, \beta \in \mathcal S_p^{l \to l'},$$ and it can have any other value when one of $\alpha$ or $\beta$ is not in $\mathcal S_p^{l \to l'}$.
This is a direct consequence of Theorem \[thm:colored-Weingarten\]. Indeed, because of the factors $(\star\star)$ in the above formula, only $\alpha, \beta \in \mathcal S_p^{l \to l'}$ survive, and the value of the Weingarten weight is the same as in Theorem \[thm:colored-Weingarten\].
For the concrete applications that follow, it is useful to have a simple equivalent for the quantity $\tilde \operatorname{Wg}(n, \alpha^{-1} \beta)$.
\[prop:equivalent-modified-Wg\] Assuming that $p$ is fixed, for every $l,l'$ and $\alpha, \beta \in \mathcal S_p^{l \to l'}$, we have $$\label{eq:Weingarten-function}
\operatorname{Wg}^{l \to l'}(n,\alpha, \beta) = \tilde \operatorname{Wg}(n, \alpha^{-1} \beta) = \operatorname{Wg}(n, \alpha^{-1} \beta)(1+O(n^{-2}))$$
We shall use the fact that one can approximate the usual Weingarten function by a polynomial times the Möbius function, which are multiplicative. Start from the left hand side of the above equality, and use the fact that both $\alpha$ and $\beta$ decompose as products of smaller permutations acting on the level sets of $l$ and $l'$, see : $$\begin{aligned}
\operatorname{Wg}^{l \to l'}(n,\alpha, \beta) &= \prod_{s=1}^k \operatorname{Wg}(n,\alpha_s, \beta_s)\\
&= \prod_{s=1}^k n^{-|l^{-1}(s)|-|\alpha_s^{-1}\beta_s|}\operatorname{Mob}(\alpha_s^{-1}\beta_s)(1+O(n^{-2}))\\
&= n^{-p-|\alpha^{-1}\beta|}\operatorname{Mob}(\alpha^{-1}\beta)(1+ O(n^{-2}))\\
&= \operatorname{Wg}(n, \alpha^{-1} \beta)(1+O(n^{-2})).\end{aligned}$$
Note that the product over $k$ above has at most $p$ non trivial factors, because at most $p$ of the permutations $\alpha_s, \beta_s$ are non trivial.
Graphical Weingarten calculus for several independent unitary matrices
----------------------------------------------------------------------
In this section we extend the graphical formalism introduced in [@cn1] to encompass integrals over several independent unitary Haar-distributed matrices. We first recall briefly the single-matrix case. For more details on this method, we refer the reader to the paper [@cn1] and to other work which make use of this technique [@cn3; @cn-entropy; @cnz; @cfn1].
In the graphical calculus, matrices (or, more generally, tensors) are represented by boxes. To each box, one attaches symbols of different shapes, corresponding to vector spaces. The symbols can be empty (white) or filled (black), corresponding to spaces or their duals.
Wires connect these symbols, each wire corresponding to a tensor contraction $V \times V^* \to \mathbb C$. A diagram is a collection of such boxes and wires and corresponds to an element in a tensor product space.
The main advantage of such a representation is that it provides an efficient way of computing expectation values of such tensors when some (or all) of the boxes are random unitary matrices. In [@cn1], the authors proposed an efficient way to apply the Weingarten formula to a diagram, which has a nice graphical interpretation.
The delta functions in each summand in the RHS of (\[eq:Wg\]) describes how one should connect boxes. Each pair of permutations $(\alpha,\beta)$ in (\[eq:Wg\]) connects the labels of $U$ and $\overline U$ boxes and then, after erasing those boxes and keeping the wires, one obtains a new diagram. The permutation $\alpha$ is used to connect the white (or empty) labels of the boxes in the following way: the white decorations of the $i$-th $U$ box are connected to the corresponding white decorations of the $\alpha(i)$-th $\overline U$ box. Permutation $\beta$ is used in a similar manner to connect the black (or filled) decorations.
This process for a fixed permutation pair $(\alpha, \beta)$ is called a *removal* and the whole process which sums all the new graphs over the all permutations is called the graph expansion. Importantly, the graphical calculus works linearly and separated components are related by tensor products, as is assumed implicitly above. The above procedure is summarized in the following important result.
\[thm:graphical-calculus\] If $\mathcal D$ is a diagram containing boxes $U, \overline U$ corresponding to a Haar-distributed random unitary matrix $U \in \mathcal U(n)$, the expectation value of $\mathcal D$ with respect to $U$ can be decomposed as a sum of weighted diagrams ${\mathcal{D}}_{\alpha, \beta}$ obtained by connecting the white labels of the $U$ boxed along the permutation $\alpha$ and the black labels along $\beta$. The weights of the diagrams are given by the Weingarten functions. $${\mathbb{E}}_U({\mathcal{D}})=\sum_{\alpha, \beta} {\mathcal{D}}_{\alpha, \beta} \operatorname{Wg}(n, \alpha\beta^{-1}).$$
We now extend the graphical calculus by adding a new box corresponding to a column-block matrix made of independent unitary matrices and generalizing the removal procedure in order to allow to take expectations of diagrams containing such boxes.
Consider i.i.d. Haar unitary matrices $U^{(1)},U^{(2)}, \ldots, U^{(k)} \in \mathcal U(n)$ and stack them up into a block-column $V \in M_{kn \times n}(\mathbb C)$ $$V = \sum_{i=1}^k e_i \otimes U^{(i)},$$ where $\{e_i\}$ is an orthonormal basis of $\mathbb C^k$ used to index the unitary blocks. Note that, up to a constant, $V$ is an isometry from $\mathbb C^n \to \mathbb C^k \otimes \mathbb C^n$, i.e. $V^*V=k\mathrm{I}_n$. Graphically, one can represent $V$ as in Figure \[fig:V-box\]. With this correspondence, one can read the Weingarten formulas and in terms of the matrix $V$ by using the identification $U^{(l)}_{ij} = V_{(l,i),j}$. Moreover, the diagram of the (true) isometry $\tilde V$ used to defined random unitary channels can be easily obtained from $V$ and the diagonal matrix $$\label{eq:W}
W = \mathrm{diag}(w_1, \ldots, w_k),$$ see Figure \[fig:tilde-V-box\].
Consider now a diagram $\mathcal D$ containing a box $V$ corresponding to a stack of i.i.d. Haar unitary operators. We are interested in computing the expectation value of $\mathcal D$ with respect to $V$. In the spirit of Theorem \[thm:graphical-calculus\], we are going to express this expectation value as a sum over diagrams obtained by a new removal procedure, weighted by Weingarten coefficients. One of the main differences between the Weingarten formula for a single Haar unitary matrix and the Weingarten formula for several such Haar random matrices is the fact that the indices $l$ and $l'$ in the latter equation are connected by *both* permutations $\alpha$ and $\beta$. Hence, we need to redefine the removal procedure when computing expectation values with respect to $V$ boxes. Since the $l,l'$ indices are connected using both permutations, we shall add a *duplication* symbol inside the box of $V$ of $V^*$, next to the label corresponding to $\mathbb C^k$, see Figure \[fig:V-random\]. During the removal procedure, labels associated to the duplication symbols should be connected with both permutations $\alpha$ and $\beta$, as in Figure \[fig:expectation-block-isometry\].
![Removal procedure for boxes corresponding to stacks of i.i.d. Haar unitary operators. Labels with duplication symbols must be connected using both permutations.[]{data-label="fig:expectation-block-isometry"}](expectation-block-isometry)
After the removal procedure, the duplication symbols remaining must be interpreted in terms of the duplication operator $T:\mathbb C^k \to \mathbb C^k \otimes \mathbb C^k$ and its adjoint $T^*$ (see Figure \[fig:duplication\]): $$\begin{aligned}
T &= \sum_{i=1}^k e_i^* \cdot (e_i \otimes e_i), \\
T^* &= \sum_{i=1}^k (e_i^* \otimes e_i^*) \cdot e_i. \end{aligned}$$
![Duplication operators $T$ and $T^*$.[]{data-label="fig:duplication"}](duplication.eps)
\[thm:V-graphical-calculus\] Let $\mathcal D$ be a diagram containing boxes $V, \bar V$ corresponding to a random matrices having a block-Haar distribution. The expectation value of $\mathcal D$ with respect to $V$ can be decomposed as a sum of weighted diagrams ${\mathcal{D}}_{\alpha, \beta}$ obtained by connecting the white labels of the $V$ boxed along the permutation $\alpha$, the black labels along $\beta$, and the labels with a duplication symbol along both $\alpha$ and $\beta$. The weights of the diagrams are given by modified Weingarten functions. $${\mathbb{E}}_V({\mathcal{D}})=\sum_{\alpha, \beta} {\mathcal{D}}_{\alpha, \beta} \,\tilde \operatorname{Wg}(n, \alpha,\beta).$$
Several applications of this result shall be discussed in Sections \[sec:fixed-k\] and \[sec:k-sim-n\].
Haar random isometries versus block random isometries {#sec:measures-isometry}
=====================================================
In this section we perform a very basic analysis of the two probability measures we considered earlier on the set of isometries from $\mathbb C^n$ to $\mathbb C^k \otimes \mathbb C^n$. Let us first recall their definitions. The easiest way to introduce these measures is via an image measure construction, starting from the Haar measure on unitary groups. Recall that both these objects are probability measures on the set of isometries $$\mathrm{Isom}(\mathbb C^n,\mathbb C^{kn}) = \{V \in M_{kn \times n}(\mathbb C) \, | \, V^*V = \mathrm{I}_n\}.$$
The Haar measure $\mu_\text{Haar}$ is the easiest to define: it is the image measure of the Haar measure on the unitary group $\mathcal U(kn)$ via the truncation operation which erases the last $(k-1)n$ columns of a $kn \times kn$ matrix. In other words, if $P \in M_{kn \times n}(\mathbb C)$ is the truncation operator $P_{ij} = \delta_{ij}$ and $U \in \mathcal U(kn)$ is a Haar-distributed unitary matrix, then $UP \sim \mu_\text{Haar}$.
An isometry $\tilde V \in \mathrm{Isom}(\mathbb C^n,\mathbb C^{kn})$ has distribution $\mu_{ w}$ if it is obtained by stacking $k$ independent Haar unitary matrices one on top of the other, with weights $\sqrt{w_i}$: $$\label{eq:V-def}
\tilde V = \sum_{i=1}^k \sqrt{w_i} e_i \otimes U^{(i)}.$$ The probability distribution $\mu_{ w}$ can also be seen as the image measure of the product of $k$ Haar measures on $\mathcal U(n)$ via the weighted stacking procedure described above.
We gather next some basic properties of these measures, whose proofs are left to the reader.
The measures $\mu_\text{Haar}$ and $\mu_\text{block}$ have the following invariance properties:
1. If $V \sim \mu_\text{Haar}$ and $U \in \mathcal U(kn)$, $U' \in \mathcal U(n)$ are fixed unitary matrices, then $UVU' \sim \mu_\text{Haar}$.
2. If $V \sim \mu_{ w}$ and $U' \in \mathcal U(n)$ is a fixed unitary matrix, then $VU' \sim \mu_{ w}$.
3. If $V \sim \mu_{ w}$ and $U \in \mathcal U(kn)$ is such that $$U = \sum_{j=1}^k e_{\sigma(j)}e_j^* \otimes U^{(j)},$$ where $\sigma \in \mathcal S_k$ is a permutation that leaves the vector $w$ invariant and $U^{(j)} \in \mathcal U(n)$ are unitary matrices, then $UV \sim \mu_{ w}$.
One can easily discriminate statistically between the two measures by computing moments or covariances for different matrix entries. In what follows, for the sake of simplicity, we shall consider the equiprobability vector $w_*=(1/k, \ldots, 1/k)$. Let us start by computing the moments of a single matrix element. It is well known (see [@hiai-petz]) that, for a Haar unitary matrix $U$ of size $n$, one has $$\mathbb E |U_{11}|^{2p} = \binom{n+p-1}{n-1}^{-1}.$$ It follows that, for the two ensembles we consider, we have $$\begin{aligned}
\mathbb E_\text{Haar} |V_{11}|^{2p} &= \binom{kn+p-1}{kn-1}^{-1},\\
\mathbb E_{w_*} |V_{11}|^{2p} &= k^{-p} \binom{n+p-1}{n-1}^{-1}.\end{aligned}$$
Although the above expressions agree at $p=1$, they are different at $p=2$, showing a statistical difference between the two ensembles.
More striking examples come from covariance computations: matrix elements $V_{11}$ and $V_{1, n+1}$ are independent under $\mu_{w_*}$, while this is obviously not true for $\mu_\text{Haar}$, see [@hiai-petz Proposition 4.2.3].
Product of conjugate channels with bounded output dimension {#sec:fixed-k}
===========================================================
We start by representing random unitary channels (and the corresponding complementary channel) in the graphical formalism we introduced. In terms of the random block-Haar map $V$, the channel has the following form: $$\Phi(X) = \sum_{i=1}^k w_i U_i X U_i^* = \mathrm{Tr}_k ((\sqrt W \otimes \mathrm{I}_n) VXV^*(\sqrt{W} \otimes \mathrm{I}_n)),$$ where $X \in M_n({\mathbb{C}})$ is the input matrix and $\tilde V = (\sqrt{W} \otimes \mathrm{I}_n) V$ is the isometry in the Steinespring picture. As before, we define the weighting matrix $W = \mathrm{diag}(w_1, w_2,\ldots w_k)$ and $V$ is obtained by stacking the unitary matrices $U_i$ one on top of the other, as in equation . Graphically, the diagram corresponding to the channel $\Phi$ is presented in Figure \[fig:Phi\], whereas the complementary channel is depicted in Figure \[fig:Phi-c\].
Next, we want to describe the limiting output eigenvalues of a fixed input state going through a random unitary channel. We are interested in the following $k^2 \times k^2$ random matrix: Z\_n =\[\^C \^C\] (\_n\_n\^\* ) Here, $\psi_n \in \mathbb C^n \otimes \mathbb C^n$ is a fixed input vector for each $n$; notice that we are considering only rank-one inputs, since these states are known to yield minimal entropy outputs.
To represent the input $\psi_n$ in the graphical calculus, we add $A_n$ and $A_n^*$ boxes on the wires of the Bell input, as in Figure \[AAstar\].
![Generalized Bell states are used as inputs.[]{data-label="AAstar"}](AAstar.eps)
Algebraically, we consider a sequence of inputs \[input-formula\] \_n = \_[i,j =1]{}\^[n]{} a\_[ij]{} e\_i e\_j, a\_[ij]{} . In the matrix version, this reads A\_n = \_[i,j =1]{}\^[n]{} a\_[ij]{} e\_i e\_j\^\*. We require the normalization relation $\|\psi_n\|=1$, which is equivalent to \[A\_n A\_n\^\*\] =1.
The input vectors $\psi_n$ generalize Bell states $${\varphi}_n = \frac{1}{\sqrt{n}} \sum_{i=1}^n e_i \otimes e_i,$$ which correspond to the trivial choice $A_n = \mathrm{I}_n / \sqrt{n}$.
Calculation of the limiting eigenvalues
---------------------------------------
In order to define *well-behaved* inputs, as in [@cfn1], we introduce two assumptions on the asymptotic behavior of the sequence of input states $A_n$.\
[**Assumption 1:**]{} \[eq:ass1\] = m + O() for some $m\in{\mathbb{C}}$. Note that a similar relation holds for $A_n^*$, with $\bar m$ replacing $m$. Note that one has $$m = \lim_{n \to \infty} \langle \psi_n, {\varphi}_n \rangle,$$ so that one can say that $m$ (or rather $|m|^2$) measures the overlap between the input state $\psi_n$ and the Bell state ${\varphi}_n$.\
[**Assumption 2:**]{} $$\label{eq:ass2}
\|A_n\|_\infty = O\left(\frac{1}{\sqrt{n}}\right)$$
Recall that the empirical eigenvalue distribution of a self-adjoint matrix $Z \in M_{k^2}(\mathbb C)$ is the probability measure $$k^{-2} \sum_{i=1}^{k^2} \delta_{\lambda_i},$$ where $\lambda_1, \ldots, \lambda_{k^2}$ are the eigenvalues of $Z$.
Before we state our result, we introduce one more notation, essential to what follows.
\[def:S\] Let $S:\mathbb R^2 \times \mathbb R^k \to \mathbb R^k$ be the function defined by $$S(x,y; w) = \mathrm{spec}^\downarrow(H_\Sigma(x,y;w)),$$ where $\mathrm{spec}^\downarrow$ denotes the ordered spectrum of a self-adjoint matrix and $$\label{eq:H-Sigma}
\forall i,j \in [k], \qquad H_\Sigma(i,j)=
\begin{cases}
(x+y)w_i^2 \qquad &\text{ if } i=j\\
y w_i w_j \qquad &\text{ if } i \neq j.
\end{cases}$$
\[thm:bell-phenomenon\] Under the assumptions and above, the empirical eigenvalue distribution of the matrix $Z_n$ converges *almost surely*, as $n \rightarrow \infty$, to the probability measure: \[limitdistUbar\] , where $s = S(1-|m|^2, |m|^2 ; w)$.
Before we prove this theorem, let us state some of its corollaries and analyze the limit entropy of the matrix $Z_n$ (which is the entropy of the probability vector appearing in the conclusion of the theorem) as a function of the parameters $|m|^2$ and $w_i$.
We analyze first the “extremal” cases for the weight vector $w$.
In the case where the weighting vector is uniform, $W=\mathrm{I}/k$, the $S$ function can be evaluated to give s\_1 = + ; s\_i = (2 i k). This implies that the output state has asymptotically the following eigenvalues:
- $\frac{|m|^2}{k} + \frac{1-|m|^2}{k^2}$, with multiplicity one;
- $\frac{1-|m|^2}{k^2}$ with multiplicity $k-1$;
- $\frac{1}{k^2}$, with multiplicity $k^2-k$.
The entropy of the probability vector $$\left( \frac{|m|^2}{k} + \frac{1-|m|^2}{k^2}, \underbrace{\frac{1-|m|^2}{k^2}, \ldots, \frac{1-|m|^2}{k^2}}_{k-1 \text{ times}} , \underbrace{\frac{1}{k^2}, \ldots, \frac{1}{k^2}}_{k^2-k \text{ times}} \right)$$ is a decreasing function of $|m|^2$, the asymptotic overlap between the input vector $\psi_n$ and the Bell state ${\varphi}_n$.
In the case where the weighting vector is trivial, $w=(1,0,\ldots, 0)$ the channel $\Phi$ is a unitary conjugation and the output matrix $Z_n$ is a pure state of null entropy.
We now turn to the “extreme" values of the parameter $m$, $|m|=1$ (the input state is, up to a phase, a Bell state) and $m=0$ (the input state is orthogonal to the Bell state).
In the case where the input state is equal, up to a phase, to the Bell state, i.e. $|m|^2=1$, the matrix $H_\Sigma$ is, up to a constant, a rank one projector and thus $$s_1 = \sum_{i=1}^k w_i^2, \qquad s_2 = \cdots = s_k = 0.$$
In the case where the input state is orthogonal to the Bell state, i.e. $m=0$, the matrix $H_\Sigma$ is diagonal and thus $$s_i=w_i^2, \qquad \forall i \in [k].$$ The limiting eigenvalue vector of the output state $Z_n$ is $w \otimes w$ and its entropy is thus $$\lim_{n \to \infty} H(Z_n) = H(w \otimes w) = 2H(w).$$
The proof uses the moment method and consists of two steps. First, we compute the asymptotic moments of the output density matrix $Z_n$ and then, by a Borel-Cantelli argument, we deduce the almost sure convergence of the spectral distribution and of the eigenvalues.
[**Step 1:**]{} We calculate the limit moments of $Z_n$, using the graphical calculus, see Figure \[fig:Zn\]. Here, $\includegraphics{circle_w.eps}$ and $\includegraphics{circle_b.eps}$ correspond to the $n$-dimensional space, and $\includegraphics{square_w.eps}$ to the $k$-dimensional output space.
![The diagram for the output state $Z_n$.[]{data-label="fig:Zn"}](Zn)
In order to compute the $p$-th moment of the matrix $Z_n$, we use the graphical calculus on a diagram obtained by connecting $p$ copies of the diagram in Figure \[fig:Zn\] in a tracial manner. For fixed $p \in {\mathbb{N}}$ the Weingarten sum in Theorem \[thm:V-graphical-calculus\] are indexed by pairs of permutations $(\alpha, \beta) \in {\mathcal{S}}_{2p}^2$. We label the $V$ and ${\overline}V$ boxes in the following manner: $1^T, 2^T, \ldots, p^T$ for the $V$ boxes of the first channel (T as “top”) and $1^B, 2^B, \ldots, p^B$ for the $V$ boxes of the second channel (B as “bottom”). We shall also order the labels as $\{1^T, 2^T, \ldots, p^T, 1^B, 2^B, \ldots, p^B\} {\simeq}\{1, \ldots, 2p\}$. A removal $r=(\alpha, \beta) \in {\mathcal{S}}_{2p}^2$ of the $V$ and ${\overline}V$ boxes connects the decorations in the following way:
1. the round white decoration  of the $i$-th $V$-block is paired with the round white decoration of the $\alpha(i)$-th ${\overline}V$ block by a wire;
2. the round black decoration  of the $i$-th $V$-block is paired with the round black decoration of the $\beta(i)$-th ${\overline}V$ block by a wire;
3. the square white decoration  of the $i$-th $V$-block is paired with both the square white decorations of the $\alpha(i)$-th and $\beta(i)$-th ${\overline}V$ blocks by wires. This double pairing is suggested by the duplication symbol associated to the square label.
We also introduce two fixed permutations $\gamma, \delta \in {\mathcal{S}}_{2p}$ which represent wires appearing in the diagram before the graph expansion. The permutation $\gamma$ represents the initial wiring of the $\includegraphics{square_w.eps}$ decorations (corresponding to the trace operation) and $\delta$ accounts for the wires between the $\includegraphics{circle_b.eps}$ decorations connecting boxes $A$ or $A^*$. More precisely, for all $i$, $$\label{eq:def-gamma-delta}
\gamma(i^T) = (i-1)^T, \quad \gamma(i^B) = (i+1)^B,\quad
and \quad
\delta(i^T) = i^B, \quad \delta(i^B) = i^T.$$ After the removal procedure, for each pair of permutations $(\alpha,\beta)$, we obtain a diagram $\mathcal D_{\alpha,\beta}$ consisting of:
1. $\includegraphics{circle_w.eps}$-*loops*; $n^{\#(\alpha)}$
2. $\includegraphics{square_w.eps}$-*nets*; $f_W(\alpha,\beta)$
3. $\includegraphics{circle_b.eps}$-*necklaces*; $f_A(\beta)$
First, one can easily see that the number of $\includegraphics{circle_w.eps}$-loops is exactly $n^{\# \alpha}$. Next, for $f_W(\alpha,\beta)$, since square labels are connected by wires with the box $T$’s and $T^*$’s, the graph they yield is not a collection of loops, but can be more general, where the boxes $W$’s are “caught in nets” which are made of $T$’s and $T^*$’s. The general formula for $f_W$ can be found in Lemma \[W-in-nets\]. Finally, the contribution of $\includegraphics{circle_b.eps}$-necklaces depends on the moments of the matrices $A_n$ and is encoded in a function $f_A(\beta)$ (see [@cfn1] for a more detailed treatment of a similar situation): f\_A() &=& \_[c (\^[-1]{})]{} Here, $|c|$ is the number of elements in $c$ and $s_{c,1} \ldots ,s_{c,|c|}$ are defined such that s\_[c,i]{}=
1 &\
\* &
Note that the above function $f(\beta)$ is well-defined in spite of the ambiguity of $s_{c,i}$, because of the circular property of the trace.
Therefore, the Weingarten formula in Theorem \[thm:V-graphical-calculus\] reads \[moments\] \[Z\_n\^p\] = \_[,S\_[2p]{}]{} n\^[\# ]{} f\_W(,) f\_A() (n,,). Using the moment growth assumptions , for the matrices $A_n$, we get that, for all cycle $c$ of $\beta$, we have &&|| A\^[s\_[c,1]{}]{} \_ A\^[s\_[c,|c| -1]{}]{}\_ A\^[s\_[c,|c|]{}]{} \_1\
&&( )\^[c-1]{} = n\^[1-|c|/2]{} , where the notation $f(x) \lesssim g(x)$ means that there exists some constant $C>0$ such that $f(x) {\leqslant}Cg(x)$ for $x$ large enough. The above inequality is the only place where Assumption 2 (see (\[eq:ass2\])) is used. Hence it yields the following asymptotic bound for the factor $f(\beta)$; \[roughbound\_f\] |f()| n\^[\#(\^[-1]{}) -p ]{}.
Using the equivalent for the (modified) Weingarten function in Proposition \[prop:equivalent-modified-Wg\], we get (note that the factors depending on the fixed parameter $k$ are hidden in the $\lesssim$ notation) \[Z\_n\^p\] \_[,S\_[2p]{}]{} n\^[\#]{} n\^[\#(\^[-1]{}) -p]{} n\^[-2p - |\^[-1]{}|]{} . \[bound\_f\] The power of $n$ in the RHS of (\[bound\_f\]) is bounded by using Lemma \[lem:S\_p\] as follows. 2p- || +p-|\^[-1]{}| -2p - |\^[-1]{}| = p- (|| + |\^[-1]{}| + |\^[-1]{}|) 0 Here, equality holds if and only if $\operatorname{id}\rightarrow \alpha \rightarrow \beta \rightarrow \delta$ is a geodesic: \[eq:geodesic-a-b\] = \_[iA]{} \_i, = \_[iB]{} \_i where $\tau_i = (i^T,i^B)$ and $A\subseteq B\subseteq \{1, \ldots,p\}$; we refer to [@cn1] for a proof of this fact. Importantly, for such geodesic permutations $\beta$, the following asymptotic behavior follows form the first assumption on the growth of the trace of $A_n$: \[behaviour\_f\] f\_A () = (n|m|\^2)\^[||]{} + O (n\^[-2]{}). Note that $|\beta|=|B|$. This implies that the power of $n$ in (\[moments\]) in fact becomes $0$ for all the $\alpha,\beta$ which satisfy the geodesic condition $\operatorname{id}\rightarrow \alpha \rightarrow \beta \rightarrow \delta$: \#+ |B| -2p - |\^[-1]{}| = 2p - |A| + |B| -2p - |B A| =0 Here, $|B| = |B \setminus A|+ |A|$.
Hence, by using (\[weingartenapprox\]), (\[mob formula\]) and (\[behaviour\_f\]), we have the following approximation on ${\mathbb{E}}\operatorname{Tr}[Z_n^p] $ (note that the estimate on the error order is not necessary here but will be so in Step 2): \[momentsapprox\] = (1 + O (n\^[-2]{})) \_ f\_W(,)|m|\^[2|B|]{} (-1)\^[|B A|]{} For the above error term, note that Lemma \[lem:S\_p\] implies that permutations $(\alpha,\beta)\in S_{2p} \times S_{2p}$ off the geodesic make the power of $n$ less by two or more; only even powers are allowed.
Using Lemma \[lem:counting-loops\], we can further process the moment expression &&\_[n]{} \[Z\_n\^p\] = (\[W\^p\])\^2\
&+&\_[()]{} +\_[()]{}. The multinomial identities: $$\begin{aligned}
\sum_{\emptyset \subseteq A \subseteq \{1, \ldots, p\}} \!\!\!\! x^{|A|} &= (1+x)^p \\
\sum_{\emptyset \subseteq A \subseteq B \subseteq \{1, \ldots, p\}} \!\!\!\!\!\!\!\! x^{|A|}y^{|B \setminus A|} &= (1+x+y)^p.\end{aligned}$$ give further calculations () = \[W\^[2p]{}\] . Set $C=B\setminus A$ such that $\emptyset \subseteq C \subseteq [p]\setminus A$ and use Lemma \[technical-identity\] with $x=1-|m|^2$ and $y=|m|^2$ to get () &=& \_[ A = ]{} \_[i=1]{}\^[|A|]{} \_[C A]{} |m|\^[2(|C|+|A|)]{} (-1)\^[|C|]{}\
&=& \_[A = ]{} \_[i=1]{}\^[|A|]{} |m|\^[2|A|]{} (1-|m|\^2 )\^[p-|A|]{}\
&=&\_[i=1]{}\^k s\_i\^p - (1-|m|\^2)\^p, where, by Lemma \[technical-identity\] and Definition \[def:S\], the vector $s \in \mathbb R^k$ is given by $s = S(1-|m|^2, |m|^2 ; w)$. Therefore, as a whole, \[the limit\] \_[n]{} \[Z\_n\^p\] &=& (\[W\^p\])\^2 + () + ()\
&=& (\[W\^p\])\^2 - \[W\^[2p]{}\] +\_[i=1]{}\^k s\_i\^p\
&=& \_\^k (w\_iw\_j)\^p + \_[i=1]{}\^k s\_i\^p, expression in which one can recognize the limiting eigenvalues announced in the theorem.
[**Step 2:**]{} We now move on to prove the almost sure convergence. Since this part of proof is very similar to that of Theorem 6.3 in [@cn1] or Theorem 3.1 in [@cfn1], we only sketch here the main ingredients. Using Borel-Cantelli Lemma, it is enough to prove that the covariance series converges: \[series\] \_[n=1]{}\^= \_[n=1]{}\^- ()\^2 < which will imply that for all $p{\geqslant}1$ (\[the limit\]) .
Also, note that by Carleman’s condition, equation (\[the limit\]) uniquely determines the measure as in (\[limitdistUbar\]).
First, (\[momentsapprox\]) implies that $$\begin{aligned}
\label{moment^2}
\left(\operatorname*{\mathbb{E}}\operatorname{Tr}\left[ Z_n^p \right] \right)^2
=
\left(k^{-2p} \sum_{\operatorname{id}\rightarrow \alpha \rightarrow \beta \rightarrow \delta}
f_W(\alpha,\beta)|m|^{2|\beta|}
(-1)^{|\alpha^{-1}\beta|}\right)^2
+ O(n^{-2}) \end{aligned}$$
On the other hand, we use Theorem \[thm:V-graphical-calculus\] to calculate $\operatorname*{\mathbb{E}}[(\operatorname{Tr}\left[ Z_n^p] \right)^2 ]$. In the diagram we have two identical copies of $\operatorname{Tr}Z_n^p$, which amounts to a total of $4p$ pairs of $V$ and $\overline V$ boxes. As a result, removals $(\bar\alpha,\bar\beta)$ are defined for $\bar\alpha,\bar\beta \in S_{4p}$. However, importantly those two copies are initially separated. Namely, initial wires $\bar \gamma, \bar \delta \in S_{2p}\oplus S_{2p} = S_{4p}$ are written as direct sums: |= \_1 \_2 |= \_1 \_2, where the indices 1 and 2 refer to the first or the second group of $2p$ boxes appearing in the diagram. Then, as before, we calculate the power of $n$, which is 2p- (||| + ||\^[-1]{}|| + ||\^[-1]{}||) 0. Equality holds if and only if $\operatorname{id}\rightarrow \bar\alpha \rightarrow \bar\beta \rightarrow \bar\delta$ is a geodesic. Moreover, this geodesic condition implies that $\bar\alpha$ and $\bar\beta$ can be written as |= \_1 \_2 |= \_1 \_2 Here, pairs $(\alpha_1, \beta_1)$ and $(\alpha_2, \beta_2)$ are defined as in (\[eq:geodesic-a-b\]).
Therefore, in the diagram all removals which matter as $n\rightarrow \infty$ keep those two copies separated. Also, these removals have the following properties: $$\begin{aligned}
& f_W(\bar\alpha, \bar\beta )
=f_W(\alpha_1,\beta_1)
\times f_W(\alpha_2,\beta_2), \\
&|\bar\beta| = |\beta_1|+|\beta_2|, \qquad
|\bar\alpha^{-1} \bar\beta |= |\alpha_1^{-1} \beta_1 | + |\alpha_2^{-1} \beta_2 | \end{aligned}$$ The first statement says that $f_W$ can be calculated for the each copy independently. Then, for the same reasons as before, we get an approximation with the error of order $1/n^2$: $$\begin{aligned}
\label{2ndmoment}
&\operatorname*{\mathbb{E}}\left(\operatorname{Tr}\left[ Z_n^p\right] \right)^2 = O(n^{-2}) \\
&+ k^{-4p}
\sum_{\substack{\operatorname{id}\rightarrow \alpha_1 \rightarrow \beta_1 \rightarrow \delta \\
\operatorname{id}\rightarrow \alpha_2 \rightarrow \beta_2 \rightarrow \delta}}
\left[
f_W(\alpha_1,\beta_1) f_W(\alpha_2,\beta_2)
|m|^{2|\beta_1|+2|\beta_2|}
(-1)^{|\alpha_1^{-1} \beta_1 | + |\alpha_2^{-1} \beta_2 | } \right]. \notag\end{aligned}$$
Finally, we see from and that - ()\^2 = O(n\^[-2]{}) which proves (\[series\]), and finalizes the proof of the theorem.
The following rather technical lemmas are needed in the proof of the result above when dealing with nets containing $W$ boxes.
\[W-in-nets\] For fixed permutations $\alpha, \beta \in \mathcal S_{2p}$ f\_W(,) = \_[b (\^[-1]{}\^[-1]{})]{} W\^[|b|]{} Here, we understand the notation $\gamma^{-1}\alpha \vee \gamma^{-1}\beta$ in terms of partitions: both permutations $\gamma^{-1}\alpha$ and $\gamma^{-1}\beta$ naturally induce partitions on $\{1,\ldots,2p\}$, and $\vee$ is the join operation on the poset of (possibly crossing) partitions. Also, $b$ and $|b|$ stand for a block of a partition and its cardinality, respectively.
First, we observe that because of the cyclic structure of $Z_n$ we can put two $\sqrt{W}$-boxes into one $W$-box and associate it to the neighboring $U$-box. Next, we claim that since $W$ is a diagonal matrix, we can slide $W$ along wires within the net. Indeed, we can show it algebraically: \_i (e\_i\^\* W) e\_i e\_i = \_i w\_i e\_i\^\* e\_i e\_i =\_i e\_i\^\* (W e\_i) e\_i = \_i e\_i\^\* e\_i (W e\_i). Finally, by using the above fact, we collect the $W$ matrices together. However, the rest which is composed of $T$’s and $T^*$’s can be contracted to a point. Therefore, each connected component $b$ in $\gamma^{-1}\alpha \vee \gamma^{-1}\beta$ gives the factor $\operatorname{Tr}W^{|b|}$.
The above general formula must be studied in details in order to complete the proof of Theorem \[thm:bell-phenomenon\]. We need precise values of $f_W(\alpha,\beta))$ when $\alpha,\beta$ lie on the geodesic $\operatorname{id}\rightarrow \alpha \rightarrow \beta \rightarrow \delta$.
\[lem:counting-loops\] Suppose permutations $\alpha, \beta$ lie on the geodesic $\operatorname{id}\rightarrow \alpha \rightarrow \beta \rightarrow \delta$. As in , they admit decompositions as products of disjoint transpositions indexed by subsets $A \subseteq B$. Then:
- When $A = \emptyset$ we have f\_W(,) =
\^2 & B=\
( W\^[2p]{}) & B=.
- When $A = \{a_1 < \cdots < a_{|A|} \} \not = \emptyset$ we have f\_W(,) = \_[i=1]{}\^[|A|]{} . Here, we understand $a_{|A|+1} -a_{|A|}$ to be equal to $p+a_1 - a_{|A|}$.
First, we consider the case $A = \emptyset$. If $B = \emptyset$, the net associated to $W$ is composed of two cycles containing each $p$ $W$ boxes (the top and the bottom cycles). However, if $B$ is nonempty, the top and bottom cycles become connected and one obtains a large cycle of length $2p$. In conclusion, these cases yield respectively $f_W(\operatorname{id},\operatorname{id})=[\operatorname{Tr}(W^p)]^2$ and $f_W(\operatorname{id},\beta)=\operatorname{Tr}(W^{2p})$, by using Lemma \[W-in-nets\].
Next, let us assume that $A = \{a_1, \ldots, a_{|A|} \} \not = \emptyset$. The structure of $\alpha$ and $\beta$ as in with $A \subseteq B$ implies that the connected components of the net are determined by indices $i \in [p]$ such that $i \in A$ and $i \in B$, see Figure \[fig:net-W\] for a proof. Since the first condition implies the second and each such connected components carries $2(a_{i+1} -a_i)$ boxes $W$, one obtains the announced formula.
\
\
\
The following result is needed to simplify the formulas in the lemma above. It can be, however, interesting on its own, from a combinatorial perspective.
\[technical-identity\] For a real diagonal matrix $W = \mathrm{diag}(w_1,\ldots,w_k)$ and real numbers $x,y \in \mathbb R$, we have \_[= A ]{} x\^[p-|A|]{}y\^[|A|]{} \_[i=1]{}\^[|A|]{} = \_[i=1]{}\^k s\_i\^p - x\^p, where $A=\{a_1 < \cdots < a_{|A|}\}$, $(a_{|A|+1}-a_{|A|})$ should be understood as $(p+a_1 - a_{|A|})$ and $s=S(x,y;w)$ is as in Definition \[def:S\].
To the diagonal operator $W = \sum_{i=1}^k w_i e_i e_i^*$, we associate the vector $$\mathbb R^k \otimes \mathbb R^k \ni \tilde W = \sum_{i=1}^k w_i e_i \otimes e_i$$ and $P_{\tilde W} \in M_{k^2}(\mathbb C)$, the orthogonal projection on $\tilde W$. The idea of the proof is to consider the self-adjoint operator $$H = x W \otimes W + y P_{\tilde W}$$ and to expand $\mathrm{Tr} (H^p)$. Since $xW \otimes W$ and $yP_{\tilde W}$ do not commute in general, one has to consider general words in these two matrices. Such words can be indexed by the positions $A=\{a_1 < \cdots < a_{|A|}\}$ where $P_{\tilde W}$ appears in the word; let us call $\mathcal W_A$ the word corresponding to a subset $A \subseteq [p]$. Oviously, one has $\mathrm{Tr}(\mathcal W_\emptyset) = x^p[\mathrm{Tr}(W^p)]^2$, and thus $$\mathrm{Tr}(H^p) = x^p[\mathrm{Tr}(W^p)]^2 + \sum_{\emptyset \neq A \subseteq [p]} \mathrm{Tr}(\mathcal W_A).$$ Using the graphcal notation, it follows from Figure \[fig:trace-H-p\], that the general term $\mathrm{Tr}(\mathcal W_A)$ factorizes along the intervals defined by the set $A$ and one has, for all $A \neq \emptyset$, $$\label{eq:trace-W-A}
\mathrm{Tr}(\mathcal W_A) = x^{p-|A|}y^{|A|} \prod_{i=1}^{|A|} \operatorname{Tr}\left[W^{2(a_{i+1}-a_i)}\right].$$
\
\
More precisely, each conected component in the diagram for $\mathcal W_A$ corresponds to an interval $a_{i+1}-a_i$ and, for such a trace (see Figure \[fig:trace-H-p-interval\]), one has a contribution of $x^{a_{i+1}-a_i-1}y\mathrm{Tr}\left[W^{2(a_{i+1}-a_i)}\right]$. Multiplying all these contribution gives .
It follows that $$\label{eq:sum-A-intermediate}
\sum_{\emptyset \not= A \subseteq [p]} x^{p-|A|}y^{|A|} \prod_{i=1}^{|A|} \operatorname{Tr}\left[W^{2(a_{i+1}-a_i)}\right]
= \operatorname{Tr}\left( H^p \right) - x^p\left[\operatorname{Tr}(W^p)\right]^2.$$ One can further simplify this equation by using the explicit form of the operator $H$. Indeed, either by using the graphical notation or by simple algebra, the action of $H$ on some basis vectors can be easily computed as follows. For $i \neq j$, it is obvious that $P_{\tilde W}e_i \otimes e_j = 0$ and thus $H e_i \otimes e_j = xw_iw_j e_i \otimes e_j$, proving that $e_i \otimes e_j$ are eigenvectors of $H$ for the eigenvalues $x w_i w_j$.
Let $\Sigma = \oplus_{i=1}^k \mathbb R e_i \otimes e_i$ be the subspace orthogonal to the space spaned by $e_i \otimes e_j$ with $i \neq j$. The restriction of $H$ to $\Sigma$ is exactly the operator $H_\Sigma$ defined in , Definition \[def:S\]. Thus, the eigenvalue vector of $H_\Sigma$ is $s = S(x,y;w)$. We now have computed all the $k^2$ eigenvalues of $H$ and we have $$\label{eq:trace-H-p}
\mathrm{Tr}(H^p) = \sum_{\substack{i,j=1\\ i \neq j }} x^p (w_iw_j)^p + \sum_{i=1}^p s_i^p.$$ The conlcusion of the lemma follows now easily from , , and the following equality $$\left[\mathrm{Tr}(W^p)\right]^2= \sum_{\substack{i,j=1\\ i \neq j }} (w_iw_j)^p +\sum_{i=1}^k w_i^{2p}.$$
Product of conjugate channels with unbounded output dimension {#sec:k-sim-n}
=============================================================
In this section, we consider the case where the output dimension grows with the input dimension of the channel, in a linear manner: $$k/n \rightarrow c,$$ where $c$ is a positive constant that we consider as a parameter of the model. Since both $n$ and $k$ grow to infinity, there is no incentive to consider complementary channels, so we focus on the output of the original channels Z\_n = \[\] ([\_n]{}[\_n\^\*]{} ), where $\psi_n$ is the generalised Bell state introduced in the previous section. We shall make the same assumptions , on the growth of the matrices $A_n$ appearing in the definition of $\psi_n$. Moreover, since the number of unitary matrices in $\Phi$ grows with $n$, we introduce the following assumptions on the growth of the weight matrices $W_n$.\
[**Assumption 3:**]{} $$\label{eq:ass3}
\forall p {\geqslant}1, \qquad \lim_{n \to \infty} \frac{1}{k} \operatorname{Tr}\left[(kW_n)^p \right] = t_p = \int x^p d\nu(x).$$ where $t_p$ are the moments of some given compactly supported measure $\nu$. The probability measure $\nu$, or, equivalently, the moment sequence $(t_p)_{p {\geqslant}1}$, are parameters of the model and they are fixed. The trace-preserving condition for the channel $\Phi$, $\operatorname{Tr}(W_n) = 1$, implies that $t_1 = 1$.
We first compute the moments of the $n^2 \times n^2$ output matrix $Z_n$.
\[thm:moments-ZnC\] Under the assumptions , , and , the output matrix $Z_n$ has the following asymptotic moments: \[(cnZ\_n)\^2\] &=& t\_2\^2 + c\^2 + t\_2\^2|m|\^4 + O(n\^[-1]{}) ;\
\[(cnZ\_n)\^p\] &=& t\_2\^p|m|\^[2p]{} + O(n\^[-1]{}), p 3.
We start by applying the graphical expansion procedure described in Theorem \[thm:V-graphical-calculus\] to the diagram for $\operatorname*{\mathbb{E}}\operatorname{Tr}(Z_n^p)$, obtained by connecting $p$ copies of the diagram for $Z_n$, displayed in Figure \[fig:ZnC\] in a tracial way. We obtain a formula which is very close to , the only differences coming from the fact that we are not using complementary channels. The notation is the same as the one in the proof of Theorem \[thm:bell-phenomenon\].
![The diagram for the output state $Z_n$.[]{data-label="fig:ZnC"}](ZnC)
\[eq:moments-Wn\] \[(cnZ\_n)\^p\]= (cn)\^[p]{} \_[,S\_[2p]{}]{} n\^[\# (\^[-1]{} ) ]{} f\_W(,) f\_A() (n,,).
Let us first upper bound the factor $f_W(\alpha,\beta)$. The boxes $W_n$ appearing in a diagram $\mathcal D_{\alpha,\beta}$ are connected in a net whose connected components are given by the blocks of the partition $\alpha \vee \beta$. In the spirit of Lemma \[lem:counting-loops\], each such connected component $b$ contributes a factor of $\mathrm{Tr}(W_n^{|b|})$, which, by , is equivalent to $k^{1-|b|}t_{|b|}$. Hence, $$\label{eq:equiv-f-W}
f_W(\alpha,\beta) \sim k^{\#(\alpha \vee \beta) - 2p} \prod_{b \in \alpha \vee \beta} t_{|b|}.$$ Replacing $k \sim cn$ and using the bound for $f_A(\beta)$, we can express everything in terms of $c$ and $n$. The power of $n$ appearing in the general term of the moment formula can then be bounded by $$\begin{aligned}
& p+ \# (\gamma^{-1} \alpha) + \#(\beta^{-1}\delta) -p + \#(\alpha \vee \beta) -2p -2p- |\alpha^{-1}\beta| \\
&\qquad {\leqslant}2p - (|\gamma^{-1}\alpha|+|\alpha^{-1}\beta|+|\beta^{-1}\delta|+|\beta| ) \\
& \qquad {\leqslant}2p - (|\gamma^{-1}\beta|+|\beta^{-1}\delta|+|\beta| ) {\leqslant}0.
\label{S_1}\end{aligned}$$ Here, the first inequality holds because \#() \# \[ab-relation\] the second one is true by the triangle inequality (Lemma \[lem:S\_p\]) |\^[-1]{}|+|\^[-1]{}| |\^[-1]{}|. \[bg-line\] Finally, the last inequality follows from the proof of Theorem 6.8 in [@cn3], which we recall as a lemma.
\[b-value\] For any permutation $\beta \in \mathcal S_{2p}$, one has |\^[-1]{}|+|\^[-1]{}|+|| 2p equality holding if and only if =
, , & p =2;\
& p 3.
Next, we analyze when the power of $n$ becomes $0$. When $\beta = \delta$, the condition (\[ab-relation\]) enforces $\alpha$ to be on the geodesic: $\operatorname{id}\rightarrow \alpha \rightarrow \beta = \delta$. However, with the equality condition for (\[bg-line\]), we conclude that $\alpha = \delta$. It is easy to see that $\beta = \operatorname{id}$ implies $\alpha = \operatorname{id}$ via (\[ab-relation\]) and that $\beta = \gamma$ results in $\alpha = \gamma$ because of the equality condition for (\[bg-line\]). One can easily check that for these values of $\beta$, the bound is saturated from the calculation below.
Since we identified the dominating terms in the moment equation , it is now easy to compute the limits; we just plug the following equivalents into . $$\begin{aligned}
\alpha =\beta&= \delta &f_W(\delta, \delta) &\sim (cn)^{-p} t_2^p
&f_A(\delta) &\sim n^p |m|^{2p} \\
\alpha =\beta&= \operatorname{id}&f_W(\mathrm{id}, \mathrm{id}) &= 1
&f_A(\mathrm{id}) &= 1\\
\alpha =\beta&= \gamma & f_W(\gamma, \gamma) &\sim (cn)^{2-2p}t_p^2
&f_A(\gamma) &= 1.\end{aligned}$$
When $p {\geqslant}3$, we get \[(cnZ\_n)\^p\]= t\_2\^p|m|\^[2p]{} + O(n\^[-1]{}), while when $p=2$, we obtain \[(cnZ\_n)\^2\] = t\_2\^2 + c\^2 + t\_2\^2|m|\^4 + O(n\^[-1]{}).
The fact that two different behaviours appear in the limits above depending on the value of $p$, can be explained by the presence of eigenvalues on different scales. We first apply the Hayden-Winter trick [@hayden-winter; @hastings] to our weighted random unitary channel setting. Note that the following proposition applies to any random unitary channel and any input state, at fixed dimension $n$.
\[prop:hayden-winter-trick\] Consider the output of a rank one input state through a product of conjugated random unitary channels $$Z_n = [\Phi \otimes \bar \Phi] (\psi_n \psi_n^*),$$ where the vector $\psi_n \in \mathbb C^n \otimes \mathbb C^n$ is defined as in and the channel $\Phi$ has weights as in .Then, one has the following lower bound for the largest eigenvalue of $Z_n$: $$\lambda_1(Z_n) {\geqslant}\frac{|\mathrm{Tr} A_n|^2}{n} \sum_{i=1}^k w_i^2.$$
After expanding the sums, one has $$Z_n = \sum_{i,j=1}^k w_i w_j (U_i \otimes \bar U_j) \psi_n \psi_n^* (U_i \otimes \bar U_j)^*.$$ If $\varphi_n$ is the vector corresponding to the Bell state $$\label{eq:Bell-state}
{\varphi}_n = \frac{1}{\sqrt n} \sum_{i=1}^n e_i \otimes e_i,$$ one has $$\langle \varphi_n, Z_n \varphi_n \rangle {\geqslant}\sum_{i=1}^k w_i^2 \langle \varphi_n, (U_i \otimes \bar U_i) \psi_n \psi_n^* (U_i \otimes \bar U_i) \varphi_n \rangle.$$ Using the fact that, for all unitary transformations $U$, one has $(U \otimes \bar U)\varphi_n = \varphi_n$, we get $$\lambda_1(Z_n) {\geqslant}\langle \varphi_n, Z_n \varphi_n \rangle {\geqslant}|\langle \varphi_n, \psi_n \rangle|^2 \sum_{i=1}^k w_i^2 = \frac{|\mathrm{Tr} A_n|^2}{n} \sum_{i=1}^k w_i^2.$$
This Theorem \[prop:hayden-winter-trick\] gives the following lemma as a corollary:
\[prop:hayden-winter-trick2\] Take input states $A_n$ satisfying assumption and weights $W_n$ satisfying the scaling . Then, for all realizations of the random matrix $V$, the largest eigenvalue of the matrix $Z_n$ can lower bounded, asymptotically, as follows: $$\liminf_{n\rightarrow \infty}\lambda_1(cnZ_n) {\geqslant}t_2 |m|^2.$$
First, Theorem \[prop:hayden-winter-trick\] implies that cn \_1(Z\_n) cn W\^2 Then, we use and with $p=2$.
Note that this behaviour is consisted with the moments computed in Theorem \[thm:moments-ZnC\], for $p {\geqslant}3$. In order to investigate the smaller eigenvalues, we analyse the matrix $Q_nZ_nQ_n$, where $Q_n=I_n- \tilde E_n$. Here, $\tilde E = {\varphi}_n {\varphi}_n^*$ is the projection on Bell state . Precise statements about the spectrum will be made later, using Cauchy’s interlacing theorem. Before we state our theorem, let us recall the definition of a compound free Poisson distribution.
Compound free Poisson distributions were introduced in [@sp2] by Speicher and the theory was further developed in [@hiai-petz] and [@nica-speicher Prop. 12.15]. Traditionally, they are defined via a limit theorem that mimics the classical Poisson limit theorem.
\[def:compound-Poisson\] Let $\lambda$ be a positive real number and $\mu$ a compactly supported probability measure. The limit in distribution, as $N \to \infty$ of the probability measure $$\left[ \left( 1-\frac{\lambda}{N} \right) \delta_0 + \frac{\lambda}{N}\mu \right]^{\boxplus N}$$ is called a *compound free Poisson distribution* of rate $\lambda$ and jump distribution $\mu$ , which is denoted by $\pi_{\lambda,\mu}$; it has free cumulants given by $$\kappa_p(\pi_{\lambda,\mu}) = \lambda m_p(\mu),$$ where $m_p(\mu)$ denotes the $p$-th moment of the probability distribution $\mu$.
Note that the usual free Poisson (or Marchenko-Pastur) distributions $\pi_c$ are special cases of the above definition, obtained by letting $\lambda = c$ and $\mu = \delta_1$. We also introduce the notation for the distribution of the square of a random variable: if $X$ has distribution $\mu$, then $\mu^{\times 2}$ is the distribution of the random variable $X^2$. It follows that the moments of $\mu^{\times 2}$ are $$m_p(\mu^{\times 2}) = m_p(\mu)^2.$$
Before going to the the main result of this section, we make a remark on free cumulants. The free cumulants and moments of a random variable satisfy the following relation, called the moment-cumulant formula: \[cumulant-formula\] m\_p = \_[NC(p)]{} \_[b ]{} \_[|b|]{} Here, $NC(p)$ is the non-crossing partition and $b \in \sigma$ is a block of the partition $\sigma$. For more details, please see, for example, [@nica-speicher Proposition 1.4].
\[thm:compound-Poisson-limit\] The empirical eigenvalue distribution of the matrix $(cn)^2Q_nZ_nQ_n$ converges in moments to a compound free Poisson distribution with rate $c^2$ and jump distribution $\nu^{\times 2}$.
This proof is similar with the one of Theorem 6.10 in [@cn3] and uses the method of moments. ((cn)\^2Q\_nZ\_nQ\_n)\^p &=& c\^[2p]{} n\^[2p-2]{} \_p (I\_n-E\_n )Z\_n\
&=& c\^[2p]{} n\^[2p-2]{} \_[g F\_p]{} (-1)\^[|g\^[-1]{}(E\_n)|]{} n\^[-|g\^[-1]{}(E\_n)|]{} \_[()]{} Here, $E_n = n \tilde E_n$ and $g \in \mathcal F_p = \{h: \{1,2,\ldots,p\} \rightarrow \{I_n, E_n\}\}$. To calculate $(\diamondsuit)$, we need to set the natural correspondence of $g$ in $S_{2p}$, which is denoted by $\hat g$: When $i \in g^{-1}(I)$, g ((i+1)\^T) = i\^T g (i\^B) = (i+1)\^B and when $i\in g^{-1}(E)$, g ((i+1)\^T) = (i+1)\^B g (i\^B) = i\^T With this notation we have () = \_[,S\_[2p]{}]{} f\_W(,) n\^[\# (g\^[-1]{} )]{} f\_A() (\^[-1]{}) Putting everything together and interchanging the two sums, we obtain $$\begin{aligned}
\frac{1}{n^2} \operatorname*{\mathbb{E}}& \operatorname{Tr}((cn)^2Q_nZ_nQ_n)^p = \\
&= c^{2p} n^{2p-2} \sum_{g \in \mathcal F_p} (-1)^{|g^{-1}(E_n)|} n^{-|g^{-1}(E_n)|}
\sum_{\alpha,\beta \in S_{2p}}
f_W(\alpha,\beta) n^{\# (\hat g^{-1} \alpha)} f_A(\beta) \tilde \operatorname{Wg}(\alpha^{-1}\beta) \\
&= c^{2p} n^{2p-2}
\sum_{\alpha,\beta \in S_{2p}}
f_W(\alpha,\beta)f_A(\beta) \tilde \operatorname{Wg}(\alpha^{-1}\beta)
\underbrace{\sum_{g \in \mathcal F_p} (-1)^{|g^{-1}(E_n)|}n^{-|g^{-1}(E_n)| +\# ( \hat g^{-1} \alpha)}}_{(\clubsuit)}\end{aligned}$$
Importantly, it was shown in [@cn3] that $(\clubsuit)$ vanishes unless $\alpha$ belongs to the following set: S\_[2p]{} = {S\_[2p]{} : } Moreover, it follows also from [@cn3] that for such $\alpha \in \tilde S_{2p}$ one has \[bound-ad\] || p.
With the estimates and for $f_W$ and $f_A$ respectively, we have $$\begin{aligned}
\frac{1}{n^2} &\operatorname*{\mathbb{E}}\operatorname{Tr}((cn)^2Q_nZ_nQ_n)^p \lesssim c^{2p} n^{-p-2} \times \\
&\sum_{\substack{\alpha \in \tilde S_{2p} \\ \beta \in S_{2p} \\g \in \mathcal F_p } }
(cn)^{\# (\alpha \vee \beta)-2p} \; t_{\alpha \vee \beta} \;
n^{\# (\beta^{-1}\delta) -|\alpha^{-1}\beta| -|g^{-1}(E_n)| +\# (\hat g^{-1} \alpha)} \;
(-1)^{|g^{-1}(E_n)|} \;
\operatorname{Mob}(\alpha^{-1}\beta),\end{aligned}$$ where $t_{\alpha \vee \beta}$ is defined multiplicatively over the cycles of the partition $\alpha \vee \beta$: $$t_{\alpha \vee \beta} = \prod_{b \in \alpha \vee \beta} t_{|b|}.$$ The power of $n$ in the expression above is bounded by the triangle inequality and (\[bound-ad\]): && p-2 + \#() -|\^[-1]{}| -|\^[-1]{}| -|g\^[-1]{}(E\_n)| - |g\^[-1]{} |\
&& 3p -2 - (|| + |\^[-1]{}| + |\^[-1]{}| + |g\^[-1]{}(E\_n)| + |g\^[-1]{} | )\
&& 3p -2 - (|| + |\^[-1]{}| + |g\^[-1]{}(E\_n)| + |g\^[-1]{} | )\
&& 2p -2 - (|| + |g\^[-1]{}(E\_n)| + |g\^[-1]{} | )\
&& 2p -2 - ( g\^[-1]{}(E\_n) + |g\^[-1]{}| ) In the above each ${\leqslant}$ is $=$ respectively if and only if \# () = \# , , |\^[-1]{}| =p g
We also claim that 2p -2 - ( g\^[-1]{}(E\_n) + |g\^[-1]{}| ) 0 and ${\leqslant}$ becomes $=$ if and only if $g=I$, called the condition (v). This is from the following fact: \# g =
2 & g I\
|g\^[-1]{}(E\_n)| & g I
Hence we conclude that the power of $n$ becomes $0$ if and only if the conditions (i) to (v) are satisfied.
The condition (v) implies that $\hat g = \gamma$, which in turn via (iv) forces $\alpha$ to have the following structure: \[structure-a\] = \^T \^B Here, $\operatorname{id}\rightarrow \alpha^T \rightarrow \gamma^T$ and $\operatorname{id}\rightarrow \alpha^B \rightarrow \gamma^B$ where $\gamma^T = (p^T, (p-1)^T, \ldots, 1 )$ and $\gamma^B = (1^B,2^B, \ldots, p^B)$. This result also enforces a similar structure to $\beta$ via (i): \[structure-b\] = \^T \^B Moreover, the special structure of $\delta$ shows that $\# ((\alpha^T \oplus \alpha^B)\delta) = \#(\alpha^T \alpha^B)$. Here, for the RHS, we identify $i^T = i^B$. Then, (iii) imposes $\# (\alpha^T\alpha^B)=p$, i.e., $\alpha^T\alpha^B = \operatorname{id}$.
The condition (iii) with $\alpha \in \tilde S_{2p}$ implies that $\alpha^{-1} \delta$ is paring. However, note that $\alpha$ moves each point within T-group and B-group, and on the other hand, $\delta$ moves each point between these groups. So, $\alpha^{-1} \delta$ has the following structure: \^[-1]{} = \_[i ]{} \_i Here, $\Lambda$ is an index set and $\tau_i = (a_i^T, b_i^B)$. Now, we reinterpret (ii) as \^[-1]{} \^[-1]{} which implies, by using (\[structure-a\]) and (\[structure-b\]), that $\alpha = \beta$.
Hence we now have three conditions: g I, = , = \^T \^B and check that these conditions really gives the leading power. The condition (v) implies that $|g^{-1}(E_n)| =0$ and the conditions (vi) and (vii) lead to f\_A() = f\_B() = 1 Note that the conditions (v), (vi) and (vii) are in fact neccessary and surficient for the power of $n$ to be $0$ because $f(\beta)$ achieves the bound under these conditions.
Therefore, $$\begin{aligned}
\lim_{n \to \infty} \frac{1}{n^2} &\operatorname*{\mathbb{E}}\operatorname{Tr}((cn)^2Q_nZ_nQ_n)^p\\
&=\sum_{\alpha^T \text{ in (vii)}}
c^{2\#(\alpha^T)}t_{\alpha^T \oplus \alpha^B}\operatorname{Mob}(\operatorname{id})
= \sum_{\sigma \in NC(p)} c^{2\#\sigma}m^2_\sigma(\nu) \\
&=\sum_{\sigma \in NC(p)} \prod_{b \in \sigma} c^2 m_p (\nu^{\times 2}) =m_p(\pi_{c^2,\nu^{\times 2}}).\end{aligned}$$ Here, we used the fact that $\alpha\in S_p$ such that $\operatorname{id}\rightarrow \alpha \rightarrow (1,2,\ldots,p)$ corresponds to a non-crossing partion of $[p]$. The last equality above follows from and the moment-cumulant formula for the compound Poisson distribution of rate $c^2$ and jump distribution $\nu^{\times 2}$, finalising the proof.
\[thm:k-sim-n\] The ordered eigenvalues $\lambda_1 {\geqslant}\lambda_2 {\geqslant}\ldots {\geqslant}\lambda_{n^2}$ of the output matrix $Z_n$ have the following asymptotic behaviour, as $n \to \infty$:
1. In probability, $cn \lambda_1 \rightarrow t_2|m|^2$.
2. The empirical eigenvalue distribution $\frac{1}{n^2-1} \sum_{i=2}^{n^2} \delta_{(cn)^2\lambda_i}$ converges weakly to the compound free Poisson distribution $\pi_{c^2,\nu^{\times 2}}$ of rate $c^2$ and jump distribution $\nu^{\times 2}$.
Here, $\nu$ is a probability measure defined in .
In the case where the weighting vector is uniform, $W_n=\mathrm{I}/k$, the limiting measure is the Dirac mass $\nu = \delta_1$. The largest eigenvalue of $Z_n$ behaves as $|m|^2/(cn)$ and the asymptotic shape of the lower spectrum is given by a free Poisson distribution of parameter $c^2$, $\pi_{c^2}$. Then, almost surely, one obtains the following asymptotic behavior for the entropy of the output matrix $Z_n$ (see [@cn3 Proposition 6.12] for a proof): $$H(Z_n) =
\begin{cases}
2\log n - \frac{1}{2c^2}+o(1) \qquad &\text{ if } c{\geqslant}1,\\
2\log(cn) - \frac{c^2}{2}+o(1) \qquad &\text{ if } 0<c<1.
\end{cases}$$
First, let $\tilde \lambda_1 {\geqslant}\ldots {\geqslant}\tilde \lambda_{n^2-1}$ be the ordered eigenvalues of $Q_nZ_nQ_n$. By Cauchy’s interlacing theorem (for example, see Corollary III.1.5 of [@bhatia]) one has \_1 \_1 \_2 …\_[n\^2-1]{} \_[n\^2-1]{} \_[n\^2]{}, which, together with the conclusion of Theorem \[thm:compound-Poisson-limit\], proves the second statement.
Next, we prove the first part. Since $ c |\operatorname{Tr}A_n|^2 \operatorname{Tr}W^2 \rightarrow t_2 |m|^2 $ as $n\rightarrow \infty$, we define \_n =| c |A\_n|\^2 W\^2 - t\_2 |m|\^2 | Then, Proposition \[prop:hayden-winter-trick\] implies that 1 . After taking expectations and using Theorem \[thm:moments-ZnC\], we obtain 1 , Then, there exists a sequence of positive numbers $\{{\varepsilon}_n^\prime\}$ such that ${\varepsilon}_n^\prime \rightarrow 0$ as $n \rightarrow \infty$, and \[cn \_1\] t\^2 |m|\^2 + \_n\^Markov’s inequality implies that for any $\delta >0$, ¶(cn\_1 - t\_2 |m|\^2 ) && ¶(cn\_1 - t\_2 |m|\^2 + \_n )\
&& = The other bound is obvious from Lemma \[prop:hayden-winter-trick2\].
Conclusions and final remarks {#sec:conclusions}
=============================
In this final section, we would like to compare the results obtained in the current paper with similar results for non unit-preserving random quantum channels studied in [@cn1; @cn3; @cfn1].
Before going into details, let us first note that the weights appearing in the definition of random unitary channels represent more parameters that can be chosen to one’s convenience. This explains why the limiting objects in the current paper are more general than the ones in [@cn1; @cn3; @cfn1].
Let us first analyze the case where $k$ is fixed. We shall compare the results in Theorem \[thm:bell-phenomenon\] to the ones in [@cfn1 Theorem 3.1]. To do this, we must first make sure that the channels we compare have the same input and output spaces. We must thus enforce the condition $t=1/k$ in [@cfn1 Theorem 3.1]. We find that the $k^2$ limiting eigenvalues are *different* in the two cases: $$\begin{aligned}
\lambda^{RC}(Z_n) &\to \left( \frac{|m|^2}{k} + \frac{1}{k^2}-\frac{|m|^2}{k^3}, \underbrace{\frac{1}{k^2}-\frac{|m|^2}{k^3}, \ldots, \frac{1}{k^2}-\frac{|m|^2}{k^3}}_{k^2-1 \text{ times}} \right),\\
\lambda^{RUC}(Z_n) &\to \left( \frac{|m|^2}{k} + \frac{1-|m|^2}{k^2}, \underbrace{\frac{1-|m|^2}{k^2}, \ldots, \frac{1-|m|^2}{k^2}}_{k-1 \text{ times}} , \underbrace{\frac{1}{k^2}, \ldots, \frac{1}{k^2}}_{k^2-k \text{ times}} \right).\end{aligned}$$
Finally, let us compare output spectra in the regime $k/n \to c$. We shall make the same assumption $t=1/k$ for the results of [@cn3 Theorem 6.11] and also we shall consider usual Bell states, which imposes $|m|^2=1$ for Theorem \[thm:k-sim-n\]. In the case where the coefficients $w$ are “flat”, i.e. $\nu=1$ in Theorem \[thm:k-sim-n\], the results are *identical* for random channels and random unitary channels: the largest eigenvalue behaves like $1/(cn)$ and the lower spectrum has a limiting shape $\pi_{c^2}$ on the scale $1/n^2$.
In order to understand whether different weights for the unitary operators in are more interesting for the purpose of finding counterexamples to additivity relations, one needs to understand how the minimal output entropy behaves for a single copy of such channels - this is the subject of future work [@cfn2].
Acknowledgments {#acknowledgments .unnumbered}
===============
The three authors would like to thank the organizers of the workshop “Probabilistic Methods in Quantum Mechanics” in Lyon, where the project was initiated, and the ANR project HAM-MARK. B. C. ’s research was supported by an NSERC Discovery grant and an ERA at the University of Ottawa. I. N. acknowledges financial support from a PEPS grant from the CNRS, the AO1-SdM project ECIU and the ANR project OSvsQPI 2011 BS01 008 01.
[99]{}
Aubrun, G. [*On almost randomizing channels with a short Kraus decomposition.*]{} Comm. in Math. Phys. 288, 1103-1116 (2009).
Banica, T. and Nechita, I. [*Block-modified Wishart matrices and free Poisson laws.*]{} arXiv:1201.4792.
Belinschi, S., Collins, B. and Nechita, I. [*Laws of large numbers for eigenvectors and eigenvalues associated to random subspaces in a tensor product.*]{} To appear in Invent. Math.
Bengtsson, I. and Życzkowski, K. (2006) [*Geometry of quantum states. An introduction to quantum entanglement.*]{} Cambridge University Press, Cambridge. xii+466 pp.
Bhatia, R. (1997) [*Matrix Analysis*]{}. Graduate Texts in Mathematics, 169. Springer-Verlag, New York.
Bo[ż]{}ejko, M., Krystek, A. and Wojakowski, L. (2006) [*Remarks on the [$r$]{} and [$\Delta$]{} convolutions.*]{} Math. Z. 253, no. 1, 177–196.
Brandao, F., Horodecki, M. S. L. [*On Hastings’s counterexamples to the minimum output entropy additivity conjecture.*]{} Open Systems & Information Dynamics, 2010, 17:01, 31–52.
Braunstein, S. L. [*Geometry of quantum inference.*]{} Phys. Lett. A 219, no. 3-4, 169–174.
Bryc, W. [*Asymptotic normality for traces of polynomials in independent complex Wishart matrices.*]{} Probab. Theory Related Fields 140, no. 3-4, 383–405.
Coecke, B. [*Kindergarten quantum mechanics — lecture notes*]{} Quantum theory: reconsideration of foundations—3, 81–98, AIP Conf. Proc., 810, Amer. Inst. Phys., Melville, NY.
Collins, B. [*Moments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability* ]{} Int. Math. Res. Not., (17):953-982.
Collins, B., Fukuda M., and Nechita, I. [*Towards a state minimizing the output entropy of a tensor product of random quantum channels.*]{} J. Math. Phys. 53, 032203 (2012)
Collins, B., Fukuda M., and Nechita, I. [*Characterizing output states of random unitary channels.*]{} In preparation.
Collins, B. and Nechita, I. [*Random quantum channels I: Graphical calculus and the Bell state phenomenon.*]{} Comm. Math. Phys. 297 (2010), no. 2, 345-370.
Collins, B. and Nechita, I. [*Random quantum channels II: Entanglement of random subspaces, Rényi entropy estimates and additivity problems.*]{} Advances in Mathematics 226 (2011), 1181�1201.
Collins, B. and Nechita, I. [*Gaussianization and eigenvalue statistics for Random quantum channels (III).*]{} Ann. Appl. Probab. Volume 21, Number 3 (2011), 1136–1179.
Collins, B. and Nechita, I. [*Eigenvalue and Entropy Statistics for Products of Conjugate Random Quantum Channels.*]{} Entropy, 12(6), 1612-1631.
Collins, B., Nechita, I.; Życzkowski, K. [*Random graph states, maximal flow and Fuss-Catalan distributions.*]{} J. Phys. A: Math. Theor. 43, 275303.
Collins, B. and Śniady, P. [*Integration with respect to the Haar measure on unitary, orthogonal and symplectic group.*]{} Comm. Math. Phys. 264, no. 3, 773–795.
Fukuda, M. and King, C. [*Entanglement of random subspaces via the Hastings bound.*]{} J. Math. Phys. 51, 042201 (2010).
Fukuda, M., King, C. and Moser, D. [*Comments on Hastings’ Additivity Counterexamples.*]{} Commun. Math. Phys., vol. 296, no. 1, 111 (2010).
Graczyk, P., Letac, G. and Massam, H. [*The complex Wishart distribution and the symmetric group.*]{} Ann. Statist. 31, no. 1, 287–309.
Guionnet, A. [*Large Random Matrices: Lectures on Macroscopic Asymptotics.*]{} Lecture Notes in Mathematics, Ecole d’Eté Probabilit. Saint-Flour, Vol. 1957, Springer.
Hanlon, P. J., Stanley, R. P., Stembridge, J. R. [*Some combinatorial aspects of the spectra of normally distributed random matrices.*]{} Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991), 151–174, Contemp. Math., 138, Amer. Math. Soc., Providence, RI.
Hastings, M.B. [*Superadditivity of communication capacity using entangled inputs*]{} Nature Physics 5, 255.
Hayden, P., Leung, D., Shor, P.W., and Winter, A. [*Randomizing quantum states: constructions and applications.*]{} Comm. Math. Phys. 250 (2004), 371Ð391.
Hayden, P. and Winter, A. [*Counterexamples to the maximal p-norm multiplicativity conjecture for all $p>1$*]{}. Comm. Math. Phys. 284, no. 1, 263–280.
Hiai, F. and Petz, D. [*The Semicircle Law, Free Random Variables and Entropy*]{}. AMS Press, 2006.
Holevo, A. S. “On complementary channels and the additivity problem”, Probab. Theory and Appl., [**51**]{}, 133-143, (2005).
Jones, V.F.R. [*Planar Algebras*]{} arXiv:math/9909027v1
C. King, K. Matsumoto, M. Nathanson, M. B. Ruskai, “Properties of Conjugate Channels with Applications to Additivity and Multiplicativity”, Markov Processes and Related Fields, volume [**13**]{}, no. 2, 391 – 423 (2007).
Mingo, J. A. and Nica, A. [*Annular noncrossing permutations and partitions, and second-order asymptotics for random matrices.*]{} Int. Math. Res. Not., no. 28, 1413–1460.
Nechita, I. [*Asymptotics of random density matrices.*]{} Ann. Henri Poincaré 8, no. 8, 1521–1538.
Nica, A and Speicher, R. [*Lectures on the combinatorics of free probability*]{} volume 335 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge.
Page, D. [*Average entropy of a subsystem.*]{} Phys. Rev. Lett., [**71**]{}, no. 9, pp. 1291–1294.
H.-J. Sommers and K. [Ż]{}yczkowski [*Statistical properties of random density matrices.*]{} J. Phys. A, [**37**]{}, no. 35, pp. 8457–8466.
R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, [*Mem. Amer. Math. Soc.*]{} [**132**]{} (1998).
Stanley, R. [*Enumerative combinatorics, vol. I.*]{} Cambridge University Press.
Życzkowski, K., Sommers, H.-J. [*Induced measures in the space of mixed quantum states.*]{} J. Phys. A 34, no. 35, 7111–7125.
Zvonkin, A. [*Matrix Integrals and Map Enumeration: An Accessible Introduction.*]{} Mathl. Comput. Modelling 26, No. 8-10, 281–304.
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abstract: 'Intelligent terminals support a large number of multimedia, such as picture, audio, video, and so on. The coexistence of various multimedia makes it necessary to provide service for different requests. In this work, we consider interference-aware coordinated multi-point (CoMP) to mitigate inter-cell interference and improve total throughput in the fifth-generation (5G) mobile networks. To select the scheduled edge users, cluster the cooperative base stations (BSs), and determine the transmitting power, a novel dynamic bargaining approach is proposed. Based on affinity propagation, we first select the users to be scheduled and the cooperative BSs serving them respectively. Then, based on the Nash bargaining solution (NBS), we develop a power control scheme considering the transmission delay, which guarantees a generalized proportional fairness among users. Simulation results demonstrate the superiority of the user-centric scheduling and power control methods in 5G CoMP systems.'
author:
- 'Liying Li, Chungang Yang, Mbazingwa E. Mkiramweni, and Lei Pang [^1] [^2]'
title: 'Intelligent Scheduling and Power Control for Multimedia Transmission in 5G CoMP Systems: A Dynamic Bargaining Game'
---
5G, CoMP, cooperative game, delay, NBS, power control, transmission scheduling
Introduction
============
With the development of the fifth-generation (5G) Generation mobile networks, the distance between two base stations (BSs) decreases significantly, which leads to the severe interference among edge users. Currently, the deployment of BSs in 5G is as follows. Both the standalone (SA) and non-standalone (NSA) deployment are considered in the development of 5G. The NSA deployment has three phases. First, the eNB and gNB are all connected to the evolved packet core (EPC), and the control signaling of BSs is reported to EPC through the eNB. Then, the eNB and gNB are connected to EPC, and the control signaling of BSs is reported to the next-generation core network (NGCN) through the eNB; Finally, the eNB and gNB are all connected to NGCN, and the control signaling of BSs is reported to NGCN through the gNB. The NSA deployment has option3/3A, option7/7A, and option4/4A, while the SA deployment has option2 and option5. Option2 is expected to be deployed in the end stage of 5G, and its standard version was frozen in September 2018. Each deployment approach corresponds to the development phase of 5G.
With the development of 5G, BSs are deployed more densely. The ultra-dense network has led to more severe inter-cell interference, which results to serious edge users performance degradation. There are some enlightenment in the 3rd generation partnership project Rel-15. To improve the throughput of edge users, coordinated multi-point (CoMP), a promising technology, becomes more popular [@S1]. This technology can utilize the interference to increase the total throughput by cooperating with adjacent BSs. From the current research results, CoMP technology can by divided into coordinate schedule / beamforming and joint processing. Joint transmission (JT) is one key technology mode of JP, which enables to transmit data to one or multiple mobile stations (MSs) by multiple cooperative base stations (CBSs), and it is our research focus.
It is crucial to design an effective transmission scheduling and power control approach for increasing the throughput of edge users[@S2; @S3]. In this paper, we optimize the transmission scheduling by studying how to cluster the BSs and how to select the CBSs serving for users. The power control is to determine the transmitting power of each BS on the overall physical resource blocks (PRBs).
Among the existing research, several traditional static transmission scheduling schemes are studied in [@S4; @S5]. To select the edge users, an algorithm exploiting the signal processing within each cell in the interference-limited region to increase the total throughput is proposed in [@S4]. To enhance the performance of MSs, authors in [@S5] design a low-complexity algorithm combining the specific knowledge and channel state information (CSI) at the receiver for downlink CoMP. However, in real-world systems, the static clustering cannot adapt to the real channel status. Hence, some research are aimed at designing the dynamic clustering [@S6; @S7]. Authors in [@S6] consider the target of sum-rate maximization in the scenario of uplink transmission. To guarantee the proportional fairness of the throughput among all users, BSs select the MSs in a round-robin way. Authors in [@S7] propose a distributed dynamic transmission scheduling algorithm exploiting reference signal receiving power (RSRP). In addition, they also propose a power control scheme based on the water filling algorithm. Authors in [@S8] analyze the dynamic power allocation for downlink CoMP-NOMA (Non-orthogonal multiple access) in multi-cell networks. BSs usually allocate a great amount of resources to the edge users in NOMA, thus ensuring the fairness among users. But this algorithm is difficult to implement due to the challenge of successive interference cancellation. Meanwhile, the user who is allocated fewer resources needs to decode the better signal, which causes the larger delay. Distributed dynamic transmission scheduling in CoMP is discussed in [@S7], however, it is carried out in the framework of the non-cooperative game for power allocation. Authors in [@S9] propose a Nash bargaining solution (NBS) fairness scheme based on cooperative game theory in orthogonal frequency division multiple access (OFDMA). A Hungarian method is proposed to allocate the optimal bargaining pairs among users.
Most existing research is based on LTE network. With the increasing bandwidth and more intelligent services, mobile applications providing videos attract more users[@S10]. The network becomes denser and the quality of service(QoS) requirements in 5G are getting higher and higher. [@S11] includes a general framework for the evolution of limited-feedback CoMP systems from 4G to 5G. If the applied information is out of data before reaching the helping eNB, the performance of users will decrease. There are a lot of research on the academic field about delay. To decrease the startup and switching delays, authors in [@S12] propose a solution for low delay dynamic adaptive streaming over HTTP streaming exploiting the low delay web real-time communication protocol data channel as a transport vehicle for carrying dynamic adaptive streaming over HTTP video sessions. To avoid the applied information outdating and improve QoS, the authors in [@S13] study the backhaul delay problem. Authors in [@S13] propose a solution for the Gaussian broadcast channels that utilizes the latest channel information available. Small cell BSs will be deployed very densely and play a major role in minimizing delay in 5G [@S14]. In this work, we mainly focus on how to decrease the transmission delay when users need to transmit video, therefore the backhaul, queuing, and other delays will be ignored.
Based on the above observations and our previous work[@S15], to increase the throughput of edge users, and consider the transmission delay combining with the throughput, a new framework for intelligent scheduling and power control of multimedia transmission is proposed in 5G CoMP system. We select the edge users according to the RSRPs and determine the cluster for each edge user in each PRB based on affinity propagation(AP) clustering algorithm [@S16]. A novel power allocation game is developed based on NBS fairness. Then, we prove that there is a unique Nash equilibrium in the cooperative game. The simulation results show the effectiveness of the proposed user-centric algorithm. The contributions of this paper are summarized as follows:
- **AP clustering algorithm**: We model the dynamic framework based on the AP cluster algorithm in CoMP to cluster BSs for the edge users. With this method, you do not need to manually set the center of cluster, the size of cluster, and other members of cluster. More importantly, the algorithm can automatically adjust the cluster results according to the channel conditions.
- **Power allocation with NBS**: We consider fairness among edge users. Therefore, a power allocation scheme using NBS fairness is proposed in CoMP. Especially, the utility function takes the NBS fairness into consideration. At the same time, we define the types of edge users according to RSRPs.
- **The existence of maximum for utility function**: When the power received from the interference links is greater than $(1 + \sqrt 2 )$ times the system noise power, then the utility function exists maximum value. Specifically, if the ${\rm{M}}{{\rm{S}}_1}$ and ${\rm{M}}{{\rm{S}}_2}$ belong to the same type, the power of the cluster serving ${\rm{M}}{{\rm{S}}_1}$ and ${\rm{M}}{{\rm{S}}_2}$ takes the upper bound. If the ${\rm{M}}{{\rm{S}}_1}$ belongs to high-type and the ${\rm{M}}{{\rm{S}}_2}$ belongs to low-type, the power of the cluster serving ${\rm{M}}{{\rm{S}}_1}$ takes the upper bound. Power of the cluster serving the ${\rm{M}}{{\rm{S}}_2}$ takes $\frac{{{\sigma ^2}}}{{n{g_3}}}$. If the ${\rm{M}}{{\rm{S}}_1}$ is low-type and the ${\rm{M}}{{\rm{S}}_2}$ is high-type, the power of the cluster which serves the ${\rm{M}}{{\rm{S}}_1}$ takes $\frac{{{\sigma ^2}}}{{n{g_4}}}$. Power of the cluster, which serves the ${\rm{M}}{{\rm{S}}_2}$ takes the upper bound.
- **Combining throughput and delay**: Edge users not only need to increase throughput, but also require shorter transmission delays. There is almost no function to consider throughput and transmission delay simultaneously. In this paper, to emphasize the importance of delay in 5G, we optimize the transmission delay through the utility function. Then the simulation results prove that the proposed algorithm can reduce the transmission delay and increase throughput sharply.
The rest of this paper is structured as follows. In Section , the system model combined with NBS fairness and the problem description are given. In Section , we present the algorithm of transmission scheduling based on AP. In Section , the formulations of power control using NBS in the framework are studied. In Section , numerical results are demonstrated. Conclusions are given in Section .
System Model and Problem Formulation
====================================
System Model
------------
To increase the throughput of edge users, we focus on studying the performance of edge users. In this paper, we consider the homogeneous networks CoMP with downlink transmissions and JT in the JP scenario. Fig. 1 shows the flow of JT: First, the primary BS shares data with the helping BSs. The primary BS is the cluster center, and the helping BSs are the other members in this cluster. Second, the BSs transmit the CSI reference signals to the user. Third, the user transmits the CSI feedback to primary BS. Then, the primary BS transmits the clusters relationship to the helping BSs. Final, the BSs in the cluster transmit data to the user served by the BSs.
In the case of JT, CSI is shared among all BSs simultaneously. $\textbf{\emph{N}} = \{ 1,...,N\} $ represents the set of BSs. We consider all the MSs and the BSs work in a single antenna model, and a CBS only serves a single user.
In this work, we consider that two clusters cooperate with each other, and the BSs in the same cluster have the same transmitting power in the ultra-dense network [@S17; @S18]. As shown in Fig. 2a, ${\rm{M}}{{\rm{S}}_1}$ is associated with ${\rm{B}}{{\rm{S}}_2}$, and the ${\rm{M}}{{\rm{S}}_2}$ is in the coverage of ${\rm{B}}{{\rm{S}}_3}$. The solid lines denote the interference links, and the dotted lines denote the links of the edge users receiving signal power from the associated BSs. For example, ${\rm{B}}{{\rm{S}}_{\rm{1}}} \to {\rm{M}}{{\rm{S}}_1}$ denotes the interference wireless link from ${\rm{B}}{{\rm{S}}_1}$ to ${\rm{M}}{{\rm{S}}_1}$. The dotted lines denote the links of the edge users that receive signal power from their associated BSs. For instance, ${\rm{B}}{{\rm{S}}_{\rm{2}}} \to {\rm{M}}{{\rm{S}}_1}$ denotes the wireless link from ${\rm{B}}{{\rm{S}}_2}$ to ${\rm{M}}{{\rm{S}}_1}$. In this scenario, ${\rm{M}}{{\rm{S}}_1}$ receives interference from ${\rm{B}}{{\rm{S}}_1}$, ${\rm{B}}{{\rm{S}}_3}$, and ${\rm{B}}{{\rm{S}}_4}$, and ${\rm{M}}{{\rm{S}}_2}$ receives the interference from ${\rm{B}}{{\rm{S}}_1}$, ${\rm{B}}{{\rm{S}}_2}$, and ${\rm{B}}{{\rm{S}}_4}$. In Fig. 2b, the BSs are clustered into several CBSs. The ${\rm{M}}{{\rm{S}}_1}$ is in the coverage of ${\rm{B}}{{\rm{S}}_2}$ and the ${\rm{M}}{{\rm{S}}_2}$ is in the coverage of ${\rm{B}}{{\rm{S}}_3}$. ${\rm{M}}{{\rm{S}}_1}$ receives interference from ${\rm{B}}{{\rm{S}}_3}$ and ${\rm{B}}{{\rm{S}}_4}$, ${\rm{M}}{{\rm{S}}_2}$ receives interference from ${\rm{B}}{{\rm{S}}_1}$ and ${\rm{B}}{{\rm{S}}_2}$. We define the gain of link from BS $j$ to the user $m$ as $G_{j,m}$ and the power of BS $j$ as $P_{j}$. Meanwhile, the gain from BS $j$ to the user $u$ is defined as ${G_{ju}}$. Let the signal to interference plus noise ratio (SINR) of ${\rm{M}}{{\rm{S}}_1}$ in the scenario of Fig. 2b be given as follows:
$$\label{Eq:eq5}
{\rm{SINR}_{1a}} = \frac{{{P_2}{G_{21}}}}{{{P_1}{G_{11}} + {P_3}{G_{31}}{\rm{ + }}{P_4}{G_{41}}}},$$
and the SINR of ${\rm{M}}{{\rm{S}}_1}$ in the scenario of Fig. 2b is shown as follows:
$$\label{Eq:eq5}
{\rm{SINR}_{1b}} = \frac{{{P_1}{G_{11}} + {P_2}{G_{21}}}}{{{P_3}{G_{31}}{\rm{ + }}{P_4}{G_{41}}}},$$
By comparing (1) and (2), we find that $$\label{Eq:eq5}
\begin{aligned}
{\rm{SINR}_{1a}} =& \frac{{{P_2}{G_{21}}}}{{{P_1}{G_{11}} + {P_3}{G_{31}} + {P_4}{G_{41}}}} \\
<& \frac{{{P_1}{G_{11}} + {P_2}{G_{21}}}}{{{P_1}{G_{11}} + {P_3}{G_{31}} + {P_4}{G_{41}}}} \\
<& \frac{{{P_1}{G_{11}} + {P_2}{G_{21}}}}{{{P_3}{G_{31}} + {P_4}{G_{41}}}} = {\rm{SINR}_{1b}},
\end{aligned}$$ it can be seen that ${\rm{SINR}_{1a}}<{\rm{SINR}_{1b}}$. Hence, the interference of the users with CoMP technology is lower than the users without CoMP technology.
Problem Formulation
-------------------
CoMP requires the processing devices of multiple sites to work closely together. This is precisely the advantage of centralized radio access network (C-RAN). C-RAN is an architecture based on centralized processing, cooperative radio, and real-time cloud infrastructure. In addition, the architecture based on centralized unit (CU) and distributed unit (DU) has also been recognized by the industry. Therefore, the deployment with CU / DU separation is widely used in 5G deployment due to its unique advantages. The advantages are mainly reflected in the different deployment locations of CU to allocate different resources for traffic diversity. Besides, as networks become denser, the signaling interacting cannot balance for so many points, and when there is a decision conflict, the authority of peer points is not enough. But the balance and authority problem will be solved through pooling DU.
As for dense networks, to satisfy users’ high bandwidth requirements, it is necessary that there are not only macro base station (MBS) but also small base station (SBS), as depicted in Fig. 3. Users can be served by MBS or SBS separately or simultaneously. Because SBSs can get more comprehensive coverage, so they are commonly used.
\(A) no CoMP:
For BSs serve MSs under non-CoMP scenario, one MS only served by one BS. In general, center users are more suitable for this service mode as shown in Fig.3(A);
\(B) CoMP with NSA:
For CoMP under NSA deployment, MBSs and SBSs serve MSs simultaneously as shown in Fig.3(B). A large amount of information needs to interact in ultra-dense networks. Therefore, the integration of CU is profitable;
\(C) CoMP with SA:
For CoMP under SA deployment, with the development of 5G, the ultimate goal is to develop into SA deployment as shown in Fig.3(C).
We define $\textbf{\emph{R}} = \{ 1,...,R\} $ as the set of PRBs, and $\textbf{\emph{M}}=\{1,...,M\}$ as the set of users. Let the BSs in the k${th}$ CBS be a set ${\rm CBS}_{b,k}$ for $ k_b \ge k \ge 1$, where ${k_b}$ is the total number of CBSs in the PRB $b$. We set ${P_{bj}}$ as transmission power allocation of BSs in ${\rm{CB}}{{\rm{S}}_{b,k}}$. Let the system noise power be ${\sigma ^2}$. In this scenario, ${g_{b,j,k}}$ is defined as the power gain of the link from BS $j$ to the relevant user in PRB $b$. SINR of user scheduled in PRB $b$ and served by ${\rm{CB}}{{\rm{S}}_{b,k}}$ as ${\gamma _{b,k}}$ is presented in (4).
$$\label{Eq:eq5}
{\gamma _{b,k}} = \frac{{\sum\limits_{j \in {\rm{CB}}{{\rm{S}}_{b,k}}} {{P_{b,j}}{g_{b,j,k}}} }}{{\sum\limits_{j^\prime \notin {\rm{CB}}{{\rm{S}}_{b,k}}} {{P_{b,j}}{g_{b,j^\prime k}}} + {\sigma ^2}}}.$$
We define the rate of the user served by ${\rm{CB}}{{\rm{S}}_{b,k}}$ in the PRB $b$ as ${{\rm{R}}_{b,k}}$, which is $$\label{Eq:eq5}
{{\rm{R}}_{b,k}} = \frac{B}{R}{\log _2}(1 + {\gamma _{b,k}}),$$ where the system bandwidth defined as $B$, and $R$ represents the number of PRBs.
Initial Analysis
----------------
We focus on increasing the throughput of edge users. It can be described precisely as follows: $$\label{Eq:eq5}
\max \sum\limits_{b \in \textbf{\emph{R}}} {\sum\limits_{1 \le k \le {k_b}} {{\rm{R}}_{b,k}} },$$
$$\label{Eq:eq5}
{\rm{s}}{\rm{.t}}{\rm{. }}\sum\limits_{b \in \textbf{\emph{R}}} {{P_{b,j}}} \le {p_{\max }},\forall j \in \textbf{\emph{N}},$$
where ${P_{b,j}}$ represents the power of BS $j$ on the PRB $b$, and $p_{max}$ is the maximal power constraint at each BS $j$.
Since the formulated problem is non-concave, it is hard to solve. To solve this problem, it can be divided into two sequential sub-problems, including how to cluster and determine the transmitting power of the BSs transmitting to users in the clusters. The former subproblem will be solved the heuristic algorithms in Section , and the latter will be discussed in Section .
NBS for the CoMP system
------------------------
We focus on the particular strategy of proportional fairness [@S19], which is closely linked to the game-theoretic concept of NBS fairness. In this section, we will introduce how to use the NBS fairness in CoMP [@S9]. We formulate the power allocation problem using NBS as follows:
$$\label{Eq:eq2}
\begin{array}{l}
\max \prod\limits_{i = 1}^L {({U_i} - U_i^{\min })} \\
\begin{aligned}
s.t.\quad&{\rm{C1:}}\quad {p_m} \ge 0,m = 1,2, \cdots ,L\\
&{\rm{ C2: }}\quad {p_m}{g_n} \ge {p_0},m = 1,2, \cdots ,L , n = 1,2, \cdots,N \\
&{\rm{ C3: }}\quad {p_m} \le {p_{\max }},m = 1,2, \cdots ,L\\
&{\rm{ C4: }}\quad {U_i} \ge U_i^{\min },\\
\end{aligned}
\end{array}$$
where $L$ is the number of players and the ${U_i}$ is the profit of $i$th player. Let $U_i^{\min }$ represent the minimal payoff that the $i$th player.
Constraint C1 is to ensure the transmission power is non-negative. Constraint C2 shows RSRPs that each MS receives is no less than a threshold ${p_0}$. Otherwise, the MS cannot decode the signal. Constraint C3 shows the total maximal transmitting power constraint ${p_{max}}$ at each BS. Constraint C4 guarantees the utility function is always greater than $U_i^{\min }$.
Our work is to solve the optimization problem (8), that is how to increase the throughput of edge users and decrease the transmission delay of edge users overall BSs around PRBs, and how to determine the transmitting power of each BS in the clusters.
Transmission Scheduling
=======================
In this section, we propose a user-centric dynamic transmission scheduling scheme.
First, input a set of BSs, a set of PRBs, a set of edge users, RSRP, and ${\rho _0}$. Second, the nodes in the AP cluster represent BSs, including MBSs or SBSs. We define the real-valued similarity of $S(m,n)$ as the BS $n$ and the BS $m$ transmit a signal to the edge user in the cell $n$. Third, MS calculates the responsibility based on $R(m,n)=S(m,n)-\mathop {\max }\limits_{n' \ne n} \{ A(m,n') + S(m,n')\} $. We define availabilities as $A(m,n)=\min \{ 0,R(n,n) + \sum\limits_{m' \notin \{ m,n\} } {\max (0,R(m',n))} \} $ when ${m \ne n }$. Otherwise, we define $A(m,n) = \sum\limits_{m' \ne n} {\max \left( {0,R(m',n)} \right)} $. To prevent large variations of parameters, we introduce the damping factor ${\lambda}$. In this paper, damping factor is used in $R = \lambda {R^{old}} + (1 - \lambda ){R^{new}}$ and $A = \lambda {A^{old}} + (1 - \lambda ){A^{new}}$ [@S20]. Then, stop iterating when the result of the clusters is constant. At any point find the value of maximal $(A(m,n)+R(m,n))$, where the point $n$ is the cluster center of the point $m$, and the point $m$ is the cluster member of the cluster center of $n$. Final, if RSRP of the MS received from BSs in the cluster is lower than the threshold, the BS is removed from the cluster which served the MS. Then, the center of the cluster is the edge user, and the others in the cluster are scheduled BSs which serve the edge user.
Problem Formulation Based on NBS
================================
Two-User Multi-BS Case Power Control under NBS
-----------------------------------------------
For this part, we consider that there are two clusters in the CoMP system, and the transmission power of each BS is equal in the same cluster. That is, the power of ${\rm{B}}{{\rm{S}}_1}$ is the same as the power of ${\rm{B}}{{\rm{S}}_2}$ in Fig. 5. We define the transmitting power of the BSs in cluster 1 as ${p_1}$, and the transmitting power of BSs in cluster 2 as ${p_2}$. To simplify the system model, let the channel gain is approximate consistent among the interference link, such as define the gain of ${\rm{U}}{{\rm{E}}_1}$ received the ${\rm{B}}{{\rm{S}}_2}$ covered in the cluster as ${g_{\rm{1}}}$ in Fig. 5. We define the $i$th user minimal payoff as $T _{i \min }$. Under the model of NBS, we consider that the utility function of the users in cluster 1 is ${T _1}$, the utility function of the users in cluster 2 is ${T _2}$.
**Input**\
RSRPs, ${\rho _0}$, scheduled MSs, n(number of BSs in the CBSs served MSs), the types of the scheduled MSs\
**Output**\
Transmission power\
**1. Initialization**\
Set $N \to \textbf{\emph{B}}$ and $M \to \textbf{\emph{A}}$\
**2. while** existence the maximal value\
**do**\
**if** the ${\rm{M}}{{\rm{S}}_1}$ and ${\rm{M}}{{\rm{S}}_1}$ are the same types.\
Power of the cluster, which serves the ${\rm{M}}{{\rm{S}}_1}$, takes the upper bound.\
Power of the cluster, which serves the ${\rm{M}}{{\rm{S}}_2}$, takes the upper bound.\
**end**\
**if** the ${\rm{M}}{{\rm{S}}_1}$ is high-type and the ${\rm{M}}{{\rm{S}}_1}$ is low-type.\
Power of the cluster serving the ${\rm{M}}{{\rm{S}}_1}$ takes the upper bound.\
Power of the cluster serving the ${\rm{M}}{{\rm{S}}_2}$ takes $\frac{{{\sigma ^2}}}{{n{g_3}}}$.\
**end**\
**if** the ${\rm{M}}{{\rm{S}}_1}$ is low-type and the ${\rm{M}}{{\rm{S}}_1}$ is high-type.\
Power of the cluster serving the ${\rm{M}}{{\rm{S}}_1}$, takes $\frac{{{\sigma ^2}}}{{n{g_4}}}$.\
Power of the cluster serving the ${\rm{M}}{{\rm{S}}_2}$ reaches the upper bound.\
**end**\
The utility function in the model of NBS is shown as follows: $$\label{Eq:eq2}
\begin{array}{l}
U = {({T _1} - {T _{1 \min}})^{{w_1}}}{({T _2} - {T _{2 \min }})^{{w_2}}}\\
\begin{aligned}
s.t.\quad&{\rm{C1:}}\quad {\rm{ }}{p_m} \ge 0,m = 1,2\\
&{\rm{ C2: }}\quad {\rm{ }}{p_m}{g_{n}} \ge {p_0},n = 1,2,3,4,m = 1,2\\
&{\rm{ C3: }}\quad {\rm{ }}{p_m} \le {p_{\max },m = 1,2}\\
&{\rm{ C4: }}\quad {T_i} \ge T _{i \min}
\end{aligned}
\end{array},$$ where the $w{}_i$ is the weight of user $i$. The RSRP is defined as $\rho $. We divide these users into the high-type and low-type based on ${\rho _0}$, which depends on the threshold value of RSRP [@S21]. If the user’s RSRP is higher than ${\rho _0}$, we define the user as a high-type user, otherwise we define the user as a low-type user.
Considering that the weight of high-type user is ${w_H} = 2$, and the weight of low-type user is ${w_L} = 1$, i.e., $$\label{Eq:eq6}
{w_H} - {w_L} = 1.$$
In ultra-dense networks, to guarantee the QoS of users, transmission delay plays a major role in 5G. To take the throughput and the transmission delay into consideration simultaneously, we define the utility function as follows: $$\label{Eq:eq4}
\begin{aligned}
T = {\rm{SINR}} \times {{\rm{e}}^{\frac{1}{{Delay}}}}
\end{aligned},$$ where the delay of user can be estimated through dividing the size of files by the user’s transmission rate. The utility function of user 1 and user 2 can be expressed in detail as follows:
$$\label{Eq:eq4}
\begin{aligned}
{T_1} &= \frac{{n{p_1}{g_1}}}{{n{p_2}{g_3} + {\sigma ^2}}} \times {e^{\frac{{\frac{B}{R}\log (1 + \frac{{n{p_1}{g_1}}}{{n{p_2}{g_3} + {\sigma ^2}}})}}{M}}}\\
&= \frac{{n{p_1}{g_1}}}{{n{p_2}{g_3} + {\sigma ^2}}} \times {(1 + \frac{{n{p_1}{g_1}}}{{n{p_2}{g_3} + {\sigma ^2}}})^{\frac{B}{{RM\ln 2}}}}
\end{aligned},$$
$$\label{Eq:eq4}
\begin{aligned}
{T_2} &= \frac{{n{p_2}{g_2}}}{{n{p_1}{g_4} + {\sigma ^2}}} \times {e^{\frac{{\frac{B}{R}\log (1 + \frac{{n{p_2}{g_2}}}{{n{p_1}{g_4} + {\sigma ^2}}})}}{M}}}\\
&= \frac{{n{p_2}{g_2}}}{{n{p_1}{g_4} + {\sigma ^2}}} \times {(1 + \frac{{n{p_2}{g_2}}}{{n{p_1}{g_4} + {\sigma ^2}}})^{\frac{B}{{RM\ln 2}}}}
\end{aligned},$$
In addition, to ensure proportion fairness and simplify the model, let ${T _{i \min}=0},\forall i \in \left\{ {1,2, \cdots ,n} \right\}$. Then the utility function can be expressed as follows:
$$\label{Eq:eq7}
\begin{aligned}
{U_1} &= {(\frac{{n{p_1}{g_1}}}{{n{p_2}{g_3} + {\sigma ^2}}} \times {(1 + \frac{{n{p_1}{g_1}}}{{n{p_2}{g_3} + {\sigma ^2}}})^{\frac{B}{{RM\ln 2}}}})^{{w_1}}}\\
&\times {(\frac{{n{p_2}{g_2}}}{{n{p_1}{g_4} + {\sigma ^2}}} \times {(1 + \frac{{n{p_2}{g_2}}}{{n{p_1}{g_4} + {\sigma ^2}}})^{\frac{B}{{RM\ln 2}}}})^{{w_2}}}
\end{aligned}.$$
In the first quadrant, the result of multiplying multiple increasing functions is also an increasing function. Otherwise, as for $f = x{(1 + x)^a}{\rm{ }}x > 0,{\rm{ }}a > 0$, we can learn that $\frac{{\partial f}}{{\partial x}} = {(1 + x)^a} + ax{(1 + x)^{a - 1}} > 0$, so the utility function grows with x, $f(x)$ is a monotonically increasing function. To simplify the system model, we change formula ${U_1}$ into ${F_1}$, which means that the results obtained in our former work [@S15] can also be applied to the model considering transmission delay. $$\label{Eq:eq7}
{F_1} = {(\frac{{n{p_1}{g_1}}}{{n{p_2}{g_3} + {\sigma ^2}}})^{{w_1}}}{(\frac{{n{p_2}{g_2}}}{{n{p_1}{g_4} + {\sigma ^2}}})^{{w_2}}}.$$
*Theorem 1*. If there are $n$ BSs, when (28) and (31) are satisfied, that is $(1 + \sqrt 2 )$ times the interference power of the user received from BSs is less than the system noise power, then (15) has the maximum value.
$Proof$: Define ${F_2}={\rm{log}}({F_1})$, which is $$\label{Eq:eq8}
{F_2} = \log ({F_1}) = {w_1}\log (\frac{{n{p_1}{g_1}}}{{n{p_2}{g_3} + {\sigma ^2}}}) + {w_2}\log (\frac{{n{p_2}{g_2}}}{{n{p_1}{g_4} + {\sigma ^2}}}).$$
To maximize (15), we should prove (16) has the maximum value firstly.
$$\label{Eq:eq8}
\frac{{\partial {F_2}}}{{\partial {p_1}}} = \frac{{{w_1}}}{{{p_1}}} - \frac{{n{g_4}{w_2}}}{{n{p_1}{g_4} + {\sigma ^2}}},$$
$$\label{Eq:eq8}
\frac{{\partial {F_2}}}{{\partial {p_2}}} = \frac{{{w_2}}}{{{p_2}}} - \frac{{n{g_3}{w_1}}}{{n{p_2}{g_3} + {\sigma ^2}}},$$
$$\label{Eq:eq8}
A = \frac{{{\partial ^2}{F_2}}}{{\partial p_1^2}} = - \frac{{{w_1}}}{{p_1^2}} + \frac{{{n^2}{g_4}^2{w_2}}}{{{{(n{p_1}{g_4} + {\sigma ^2})}^2}}},$$
$$\label{Eq:eq8}
B = 0,$$
$$\label{Eq:eq8}
C = \frac{{{\partial ^2}{F_2}}}{{\partial p_2^2}} = \frac{{{w_1}{n^2}{g_3}^2}}{{{{(n{p_2}{g_3} + {\sigma ^2})}^2}}} - \frac{{{w_2}}}{{p_2^2}}.$$
Above all, we should satisfy: $$\label{Eq:eq8}
AC - {B^2} > 0,$$ $$\label{Eq:eq8}
A < 0.$$ By solving (22)-(23), we can see that, $$\label{Eq:eq8}
ng_4^2({w_2} - {w_1})p_1^2 - 2n{w_1}{p_1}{g_4}{\sigma ^2} - {w_1}{\sigma ^4} < 0,$$ $$\begin{gathered}
\label{Eq:eq8}
[{n^2}g_4^2({w_2} - {w_1})p_1^2 - 2n{w_1}{g_4}{\sigma ^2}{p_1} - {\sigma ^4}{w_1}]\\
*[{n^2}g_3^2({w_1} - {w_2})p_2^2 - 2n{w_2}{g_3}{\sigma ^2}{p_2} - {\sigma ^4}{w_2}] > 0.\end{gathered}$$
When all users in the cluster 1 and cluster 2 are high-type (or low-type), ${w_1} = {w_2} = {w_H}({w_L})$, (24) can be simplified as: $$\label{Eq:eq16}
- 2n{g_4}{\sigma ^2}{w_1}{p_1} - {\sigma ^4}{w_1} < 0.$$
By simplifying (25), we can see that: $$\label{Eq:eq17}
( - 2n{w_1}{g_4}{\sigma ^2}{p_1} - {\sigma ^4}{w_1})( - 2n{w_2}{g_3}{\sigma ^2}{p_2} - {\sigma ^4}{w_2}) > 0.$$
It is obvious that (26) and (27) are true. In conclusion, when the users are all high-type (or low-type), (26) and (27) always satisfy (22) and (23).
When $w{}_1 = {w_2}$, (17) and (18) are always greater than zero. Transmission power allocation takes the upper bound, the utility function reaches the maximum value.
When the user in the cluster 1 is high-type, and the user in the cluster 2 is low-type, i.e., ${w_1} = {w_H}=2$, ${w_2} = {w_L}=1$. Under these circumstances, we should satisfy: $$\label{Eq:eq18}
n{g_3}{p_2} < (1 + \sqrt 2 ){\sigma ^2}.$$
To maximize the utility function, that is $$\label{Eq:eq19}
{p_1} = \frac{{{w_1}{\sigma ^2}}}{{({w_2} - {w_1})n{g_4}}},$$ $$\label{Eq:eq19}
{p_2} = \frac{{{w_2}{\sigma ^2}}}{{n({w_1} - {w_2}){g_3}}}.$$
Since the value of ${{w_2} - {w_1}}$ is negative, (29) is negative. When ${p_2} = \frac{{{\sigma ^2}}}{{n{g_3}}}$ and ${p_1}$ takes the upper bound, the utility function reaches the maximum value.
When the user in the cluster 1 is low-type, and the user in the cluster 2 is high-type, ${w_1} = {w_L}=1$, ${w_2} = {w_H}=2$. In this case, we should satisfy: $$\label{Eq:eq20}
n{g_4}{p_1} < (1 + \sqrt 2 ){\sigma ^2}.$$
Since the value of ${{w_1} - {w_2}}$ is negative, (30) is negative. When ${p_1} = \frac{{{\sigma ^2}}}{{n{g_4}}}$ and ${p_2}$ takes upper bound, the utility function gets maximum value.
Multi-User Multi-BS Case Power Control under NBS
------------------------------------------------
In this subsection, we consider that there are $M$ MSs and $N$ BSs in a cluster. Under the NBS model, the utility function is\
$$\label{Eq:eq20}
\begin{array}{l}
U = \prod\limits_{i = 1}^M {{{({T_i} - {T_{i\min }})}^{{w_i}}}} \\
s.t.{P_{b,j}} > 0(b = 1,2,...,R;j = 1,2,...,N)\\
\ \ \ \ {\rm{ }}{P_{b,j}} \le {P_{\max }}\\
\ \ \ \ {\rm{ }}{P_{b,j}}{g_{b,j,k}} > {p_0}(b = 1,2,...,R;j = 1,2,...,N;\\
\ \ \ \ {\rm{ }}k = 1,2,...)\\
\ \ \ \ {\rm{ }}{T_i} > {T_{i\min }}
\end{array}$$
where the utility function of user $i$ can be expressed as follows: $$\label{Eq:eq21}
\begin{aligned}%\begin{array}{l}
{T_i} &= \frac{{\sum\limits_{j \in CB{S_{b,k}}} {{P_{b,j}}{g_{b,j,k}}} }}{{\sum\limits_{j' \notin CB{S_{b,k}}} {{P_{b,j'}}{g_{b,j',k}}} + {\sigma ^2}}}\\
&\times {{\rm{e}}^{{\raise0.7ex\hbox{${\frac{B}{R}{\rm{log}}(1 + \frac{{\sum\limits_{j \in CB{S_{b,k}}} {{P_{b,j}}{g_{b,j,k}}} }}{{\sum\limits_{j' \notin CB{S_{b,k}}} {{P_{b,j'}}{g_{b,j',k}}} + {\sigma ^2}}})}$} \!\mathord{\left/
{\vphantom {{\frac{B}{R}{\rm{log}}(1 + \frac{{\sum\limits_{j \in CB{S_{b,k}}} {{P_{b,j}}{g_{b,j,k}}} }}{{\sum\limits_{j' \notin CB{S_{b,k}}} {{P_{b,j'}}{g_{b,j',k}}} + {\sigma ^2}}})} M}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{$M$}}}}\\
&= \frac{{\sum\limits_{j \in CB{S_{b,k}}} {{P_{b,j}}{g_{b,j,k}}} }}{{\sum\limits_{j' \notin CB{S_{b,k}}} {{P_{b,j'}}{g_{b,j',k}}} + {\sigma ^2}}}\\
&\times {(1 + \frac{{\sum\limits_{j \in CB{S_{b,k}}} {{P_{b,j}}{g_{b,j,k}}} }}{{\sum\limits_{j' \notin CB{S_{b,k}}} {{P_{b,j'}}{g_{b,j',k}}} + {\sigma ^2}}})^{\frac{B}{{RM\ln 2}}}}
\end{aligned}.$$
Under this scenario, we define the utility function of NBS as follows: $$\label{Eq:eq22}
\begin{aligned}
U = \prod\limits_{i = 1}^M \begin{array}{l}
(\frac{{\sum\limits_{j \in CB{S_{b,k}}} {{P_{b,j}}{g_{b,j,k}}} }}{{\sum\limits_{j' \notin CB{S_{b,k}}} {{P_{b,j'}}{g_{b,j',k}}} + {\sigma ^2}}}\\
\times {(1 + \frac{{\sum\limits_{j \in CB{S_{b,k}}} {{P_{b,j}}{g_{b,j,k}}} }}{{\sum\limits_{j' \notin CB{S_{b,k}}} {{P_{b,j'}}{g_{b,j',k}}} + {\sigma ^2}}})^{\frac{B}{{RM\ln 2}}}}{)^{{w_i}}}
\end{array}
\end{aligned}.$$
Similarly, the result will be an increasing function in the first quadrant. To simplify the system model, we consider that ${T _{\min 1}}=0$. Therefore, ${U}$ can be changed into ${F_1}$. $$\label{Eq:eq80}
{F_1} = {\prod\limits_{i = 1}^M {(\frac{{\sum\limits_{j \in CB{S_{b,k}}} {{P_{b,j}}{g_{b,j,k}}} }}{{\sum\limits_{j' \notin CB{S_{b,k}}} {{P_{b,j'}}{g_{b,j',k}}} + {\sigma ^2}}})} ^{{w_i}}}.$$
Define ${F_2}={\rm{log}}({F_1})$, which is $$\label{Eq:eq80}
{F_2} = \log ({F_1}) = \sum\limits_{i = 1}^M {{w_i}\log (\frac{{\sum\limits_{j \in CB{S_{b,k}}} {{P_{b,j}}{g_{b,j,k}}} }}{{\sum\limits_{j^\prime \notin CB{S_{b,k}}} {{P_{b,j^\prime }}{g_{b,j^\prime ,k}}} + {\sigma ^2}}})}.$$
To maximize (34), we need to prove (36) has the maximum value firstly.
$$\label{Eq:eq23}
\frac{{\partial {F_2}}}{{\partial {P_{b,j}}}} = \sum\limits_{i = 1}^M {\frac{{{w_i}}}{{{P_{b,j}}}}} > 0,$$
we have $$\label{Eq:eq24}
\frac{{{\partial ^2}{F_2}}}{{\partial P_{b,j}^2}} = - \sum\limits_{i = 1}^M {\frac{{{w_i}}}{{{P_{b,j}}^2}}} < 0.$$
From (37) and (38), we can see that the utility function is a monotonically increasing function, although the growth rate of this utility function drops. Hence, the transmitting power should be the maximum power of the BSs under the multi-BSs multi-users scenario.
Numerical Results
=================
Simulation Setup
----------------
Simulations were done using MATLAB 2010. By comparing the performance of the proposed scheme, the throughput and transmission delay are both improved significantly. To manifest the effectiveness of the algorithm proposed in the ultra-dense networks, we compare the throughput of edge users at different distances. The simulation parameters are detailedly described as follows: The CoMP system consists of 19 cells, a frequency reuse factor of 1 [@S22], and the cell radius ranges from 0.05 km to 0.5 km. There are 100 users randomly distributed in each cell. In this paper, we set the system bandwidth as $B=3$MHz and the pathloss as $148.1 + 37.6{\log _{10}}d$, where $d$ is the distance from BS to UE. We calculate the throughput and transmission delay assuming the size of video is 100MB.
\[Tab1\]
Parameter Value
------------------------------ -----------------------------
Number of cells 19
System bandwidth 3MHz
BS maximum power 43dBm
Number of PRBs 15
Number of users per cell 100
Cell radius 0.05Km-0.5Km
Path loss model $148.1 + 37.6{\log _{10}}d$
Frequency reuse factor 1
The size of the video file 100MB
Link Down link
Antenna gain 5dB
Noise power spectral density -174dBm/Hz
: Simulation parameters
Simulation Results
------------------
The edge users are served by several neighboring BSs separately, as shown in Fig. 6. To indicate the effect of the algorithm exploiting AP cluster, we consider that there is just one BS per cell which is at the center of the cell. The green point shown in Fig. 6 represents the BS in each cell. The points that the lines cross are the center of clusters, which represent the edge users of the cells are served. Different kinds of colours stand for different clusters.
Due to the advantages of SBS, 5G introduces SBS into the ultra-dense networks. Fig. 7 shows the scenario where the green point at the center of the cell represents MBS and the neighbouring blue points around the MBS represent SBSs. The distance between the MBS and the SBS in the same cell is 50m and the cell radius is 200m. The centers of the clusters represent edge users. The lines connected to the center express the service relationship. As can be seen in Fig.6 and Fig.7, the sizes of clusters obtained by the AP clustering algorithm are different.
The interference becomes more severe due to the denser networks. To compare the average rate of edge users under these algorithms, we simulate the scenario of 19 cells, and the cell radius is 50m. It can be observed from Fig.8 that the throughput changes according to the size of clusters. The line which expresses performance of the algorithm with CoMP tends to be gentle when the cluster size becomes 3. The average throughput of the edge users using AP cluster algorithm in CoMP is higher than the highest throughput of the edge users using algorithm without AP cluster in CoMP.
From the numerical results presented in Fig. 9, we can conclude that the algorithm exploiting AP cluster and CoMP can increase the throughput of edge users drastically. The simulation scenario is the same as that of Fig. 7. The performance gain of the algorithm exploiting AP cluster for 5G NSA is better than the performance gain in the scenario for 5G SA. For example, when the CBS size is 4 in common CoMP, the average throughput of edge user is almost $4*10^5$ bps which uses common transmission schedule scheme. The average throughput of edge user is $5.5*10^5$ bps which uses the scheme we proposed, while it is only $1.4*10^5$ bps of the edge user without CoMP. It can be concluded that the AP cluster algorithm can adapt to the complex scenario dynamically.
Fig. 10 shows the impact of damping factor. But the lines do not have the same trend. To reduce the impact of random errors, we simulate via cycling experiment. Hence, we consider that the damping cannot impact the throughput of edge users.
To compare the fairness among these algorithms. Let the average users’ transmission rates be ${\bar R_1},...,{\bar R_K}$. Define the Jain’s fairness index as $\frac{{{{(\sum\nolimits_{k = 1}^K {{{\bar R}_k}} )}^2}}}{{K\sum\nolimits_{k = 1}^K {\bar R_k^2} }}$ [@S23]. NBS can improve the fairness of users, which is achieved by power control. However, the simulation results shown in Fig. 11 indicate that the NBS fairness is not obviously higher than the users using other algorithms, which can be attributed to the similarity of transmission power of BSs between the common CoMP algorithm and the algorithm under multi-user multi-BS conditions.
Note that the transmission delay can be obtained via dividing the size of a video file by the transmission rate of the edge users. To compare the delay of these algorithms, we set the size of a video file as 100MB and simulate the transmission scenario. It is evident from Fig. 12 that the algorithm we proposed has less transmission delay than others because it can observably improve the throughput of edge users.
The transmission delay in 5G NSA deployment is shown in Fig. 13, we can see that our algorithm has the lowest transmission delay among these algorithms. Because the algorithm we proposed can increase the transmission rate of edge users.
With the development of the network, the distance between BSs becomes smaller, the interference among edge users gets stronger, and the analysis becomes more complex. We simulate the performance of the average throughput of edge users in 5G NSA deployment. As shown in Fig. 14, we can find that the performance of the algorithm with AP cluster is 1.2 times more than the algorithm with common CoMP regardless of the change of cell radius. In addition, Fig. 14 also proves the algorithm is effective in the ultra-dense networks.
Conclusion
==========
In this paper, we proposed a user-centric dynamic framework including downlink transmission scheduling and power control for improving the performance of edge users in 5G CoMP systems. Specifically, a distributed dynamic transmission algorithm was firstly proposed. Based on this, a transmission scheduling scheme took AP cluster algorithm into consideration. And we utilized NBS to deduce the power control scheme to increase the fairness of users. We built the utility function of the system model based on users’ throughput and transmission delay and proved the existence of the unique Nash equilibrium solution of this function. Finally, the simulation results showed the proposed approach can dramatically improve the performance of edge users.
[99]{} P. Marsch, M. Grieger, and G. Fettweis, “Field Trial Results on Different Uplink Coordinated Multi-Point (CoMP) Concepts in Cellular Systems,” in *Proc. IEEE GLOBECOM 2010*, Miami, 2010, pp. 1-6.
A. Lozano, R. W. Heath, and J. G. Andrews,“Fundamental Limits of Cooperation,” *IEEE Trans. Inf. Theory*, vol. 59, no. 9, pp. 5213-5226, Sep. 2013.
P. Marsch, M. Grieger, and G. Fettweis,“Large Scale Field Trial Results on Different Uplink Coordinated Multi-Point (CoMP) Concepts in an Urban Environment,” in *Proc. WCNC 2011*, Cancun, 2011, pp. 1858-1863.
U. Jang, H. Son, J. Park, and S. Lee,“CoMP-CSB for ICI Nulling with User Selection” *IEEE Trans. Wireless Commun.*, vol. 10, no. 9, pp. 2982-2993, Sep. 2011.
Y. Yang, B. Bai, W. Chen, and L. Hanzo,“A Low-Complexity Cross-Layer Algorithm for Coordinated Downlink Scheduling and Robust Beamforming Under a Limited Feedback Constraint,” *IEEE Trans. Vehic. Tech.*, vol. 63, no. 1, pp. 107-118, Jan. 2014.
A. Papadogiannis, D. Gesbert, and E. Hardouin,“A Dynamic Clustering Approach in Wireless Networks with Multi-Cell Cooperative Processing,” in *Proc. ICC 2008*, Beijing, May. 2008, pp. 4033-4037.
S. Fu, B. Wu, H. Wen, P. H. Ho, and G. Feng, “Transmission Scheduling and Game Theoretical Power Allocation for Interference Coordination in CoMP,” *IEEE Trans. Wireless Commun.*, vol. 13, no. 1, pp. 112-123, Jan. 2014.
M. S. Ali, E. Hossain, A. Al-Dweik and D. I. Kim, “Downlink Power Allocation for CoMP-NOMA in Multi-Cell Networks,” *IEEE Trans. Commun.*, vol. 66, no. 9, pp. 3982-3998, Sep. 2018.
Z. Han, Z. Ji, and K. J. R. Liu, “Fair Multiuser Channel Allocation for OFDMA Networks Using Nash bargaining Solutions and Coalitions,” *IEEE Trans. Commun.*, vol. 53, no. 8, pp. 1366-1376, Aug. 2005.
B. Chen, W. Ji, F. Jiang and S. Rho, “QoE-Enabled Big Video Streaming for Large-Scale Heterogeneous Clients and Networks in Smart Cities,” *IEEE Access*, vol. 4, pp. 97-107, Dec. 2015.
Q. Cui, H. Wang, P. Hu, X. Tao, P. Zhang, J. Hamalainen, L. Xia, “Evolution of Limited-Feedback CoMP Systems from 4G to 5G: CoMP Features and Limited-Feedback Approaches,” *IEEE Veh. Technol. Mag.*, vol. 9, no. 3, pp. 94-103, Sept. 2014.
S. Zhao, Z. Li and D. Medhi, “Low delay MPEG DASH streaming over the WebRTC data channel,” in *Proc.2016 IEEE ICMEW*, Seattle, Sep. 2016, pp. 1-6.
K. Kwak, H. W. Je, S. Park and S. Choi, “CoMP Joint Transmission for Gaussian Broadcast Channels in Delay-Limited Networks,” *IEEE Trans. Veh. Technol.*, vol. 66, no. 3, pp. 2053-2058, Mar. 2017.
S. Chen, T. Zhao, H. Chen and W. Meng, “Downlink Coordinated Multi-Point Transmission in Ultra-Dense Networks with Mobile Edge Computing,” *IEEE Network*, pp.1-8, Sep. 2018.
L. Li, C. Yang, J. Xiao, X. Shao, and T. Younas, “Joint Scheduling and Power Control in CoMP: A Dynamic Bargaining Approach”, in *Proc. 2018 IEEE APCC*, Ningbo, Nov. 2018
K. Nabar and G. Kadambi, “Optimising gateway selection using node lifetime and inter-node interference in cluster-based MANETs,” in *Proc. WOCN 2017*, Mumbai, 2017, pp. 1-5.
D. L¨®pez-P¨¦rez, M. Ding, H. Claussen and A. H. Jafari, “Towards 1 Gbps/UE in Cellular Systems: Understanding Ultra-Dense Small Cell Deployments” *IEEE Commun. Surveys and Tutorials*, vol. 17, no. 4, pp. 2078-2101, Fourthquarter 2015.
B. Soret, K. I. Pedersen, N. T. K. J rgensen, and V. Fern¨¢ndez-L¨®pez, “Interference coordination for dense wireless networks,” *IEEE Commun. Mag.*, vol. 53, no. 1, pp. 102-109, Jan. 2015.
Y. Zhang, S. R. Kang, and D. Loguinov, “Delay-Independent Stability and Performance of Distributed Congestion Control,” *IEEE/ACM Trans. Netw.*, vol. 15, no. 4, pp. 838-851, Jul. 2007.
K. Nabar and G. Kadambi, “Affinity Propagation-driven Multiple Weighted Clustering in MANETs,” in *Proc. AICTC 2016*, Bikaner, Aug. 2016.
T. Zhang, H. Wang, X. Chu, and J. He, “A Signaling-Based Incentive Mechanism for Device-to-Device Content Sharing in Cellular Networks,” *IEEE Commun. Lett.*, vol. 21, no. 6, pp. 1377-1380, Jun. 2017.
S. Fu, H. Zhou, J. Qiao, L. Liang, Y. Jia, and B. Wu, “Distributed Transmission Scheduling and Power Allocation in CoMP,” *IEEE Syst. J.*, pp.1-12. Sep. 2017.
L. Lei, D. Yuan, C. K. Ho, and S. Sun, “Power and Channel Allocation for Non-Orthogonal Multiple Access in 5G Systems: Tractability and Computation,” *IEEE Trans. Wireless Commun.*, vol. 15, no. 12, pp. 8580-8594, Dec. 2016.
[^1]: L. Li, C. Yang. M. E. Mkiramweni, and L. Pang are with the State Key Laboratory on Integrated Services Networks, Xidian University, Xi’an, 710071 China (emails: {[email protected]; [email protected]; [email protected]; [email protected]}).
[^2]: This work is supported in part by the National Science Foundation of China (61871454); by the Fundamental Research Funds for the Central Universities(2018); by the CETC Key Laboratory of Data Link Technology (CLDL-20182308); by the ISN02080001 and ISN90106180001; by the 111 Project under Grant B08038; and by the National Science Foundation of China under Grant 61671062 and Grant 91638202.
|
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abstract: 'The time delay between flux variations in different wavelength bands can be used to probe the inner regions of active galactic nuclei (AGN). Here, we present the first measurements of the time delay between optical and near-infrared (NIR) flux variations in H0507+164, a nearby Seyfert 1.5 galaxy at $z$ = 0.018. The observations in the optical $V$-band and NIR $J$, $H$ and $K_s$ bands carried over 35 epochs during the period October 2016 to April 2017 were used to estimate the inner radius of the dusty torus. From a careful reduction and analysis of the data using cross-correlation techniques, we found delayed responses of the $J$, $H$ and $K_s$ light curves to the $V$-band light curve. In the rest frame of the source, the lags between optical and NIR bands are found to be $27.1^{+13.5}_{-12.0} \, \mathrm{days}$ ($V$ vs. $J$), $30.4^{+13.9}_{-12.0} \, \mathrm{days}$ ($V$ vs. $H$) and $34.6^{+12.1}_{-9.6} \, \mathrm{days}$ ($V$ vs. $K_s$). The lags between the optical and different NIR bands are thus consistent with each other. The measured lags indicate that the inner edge of dust torus is located at a distance of 0.029 pc from the central UV/optical AGN continuum. This is larger than the radius of the broad line region of this object determined from spectroscopic monitoring observations thereby supporting the unification model of AGN. The location of H0507+164 in the $\tau$ - $M_V$ plane indicates that our results are in excellent agreement with the now known lag-luminosity scaling relationship for dust in AGN.'
author:
- |
Amit Kumar Mandal$^{1,2}$[^1], Suvendu Rakshit$^{1, 3}$[^2], Kshama S. Kurian$^{1}$, C. S. Stalin$^{1}$[^3], Blesson Mathew$^{2}$, Sebastian Hoenig$^{4}$, Poshak Gandhi$^{4}$, Ram Sagar$^{1}$ and M. B. Pandge$^{5}$[^4]\
\
$^{1}$Indian Institute of Astrophysics, Block II, Koramangala, Bangalore 560 034, India\
$^{2}$Department of Physics, Christ University, Hosur Road, Bangalore 560 029, India\
$^{3}$Astronomy Program, Department of Physics and Astronomy, Seoul National University, Seoul 151-742, Republic of Korea\
$^{4}$Department of Physics & Astronomy, University of Southampton, Southampton SO17 1BJ, UK\
$^{5}$Dayanand Science College, Barshi Road, Latur, Maharashtra 413512, India
bibliography:
- 'ref.bib'
date: 'Accepted 2018 January 19. Received 2018 January 11; in original form 2017 September 5'
title: Determination of the size of the dust torus in H0507+164 through optical and infrared monitoring
---
\[firstpage\]
galaxies: active $-$ galaxies: Seyfert $-$ (galaxies:) quasars: individual (H0507+164)
Introduction
============
Active Galactic Nuclei (AGN) are among the most luminous objects in the Universe that produce very high luminosity in a very concentrated volume, powered by the accretion of matter onto super-massive black hole (SMBH) located at the centers of galaxies. According to the unified model of AGN, a dusty torus that surrounds the central SMBH, the accretion disk and the broad line region (BLR) play a key role in the identification of Seyfert galaxies, a category of AGN, into Seyfert 1 and Seyfert 2 galaxies . Evidence for their presence is seen in the broad band spectral energy distribution (SED) of AGN, such as the presence of the big blue bump in the optical-UV wavelength region, which is a signature of the accretion disk [@1987ApJ...321..305C; @1982ApJ...254...22M; @1978Natur.272..706S] and a bump in the IR region [@1989ApJ...347...29S; @1987ApJ...320..537B; @1993ApJ...404...94K] which is a signature of the presence of the dusty torus.
The torus, which obscures the central engine, is the dominant source of IR radiation in most of the AGN. It is not easy to spatially resolve the components of the AGN such as the BLR and torus by any current imaging techniques. The IR radiation of AGN coming from the torus that extends in size from sub-pc to pc size scales could in principle be resolvable using IR interferometric observations. However, such interferometric observations as of today are in a minority and limited by the current diameters of optical/IR telescopes. Alternatively, the extent and nature of the torus can be studied via the technique of reverberation mapping [@1982ApJ...255..419B]. This technique relies on the flux variability of AGN, which has been known since their discovery. The continuum emission from AGN is believed to arise from the accretion disk and it is expected to vary over a range of time scales from hours to days. The variations in the continuum flux are observed at a later time in the fluxes of their broad emission lines. Similarly, there is also delayed response of the IR continuum emission (from the torus) to the changes in the optical continuum from the central AGN, which indicates that the NIR continuum and UV/optical continuum emission are causally connected. This delay between the optical and IR variation when measured can give the radius of the inner most hot dust torus $r_{\mathrm{dust}}$ = $\tau \times c$, where c is the speed of light and $\tau$ is the time delay between optical and IR variations. This method of measuring the delayed response of the IR emission relative to the optical emission and consequently determine the size of the torus is referred to as dust reverberation mapping (DRM).
Studies of DRM in few Seyfert galaxies [@2004ApJ...600L..35M; @2006ApJ...639...46S; @2014ApJ...788..159K] have led to the establishment of a correlation between the inner radius of the dust torus and the UV luminosity as $r_{\mathrm{dust}}$ $\varpropto$ $L^{0.5}$ . Also, established a relation between the sublimation radius ($R_{\mathrm{sub}}$) which is defined as the radius at which dust particle sublimates and the dust grain size ($a$) as $$\begin{aligned}
R_{\mathrm{sub}} = 1.3 \left(\frac{L_{UV}}{10^{46} \mathrm{erg s^{-1}}}\right)^{0.5} \left(\frac{T_{\mathrm{sub}}}{1500 K}\right)^{-2.8} \left(\frac{a}{0.05\mu \mathrm{m}}\right)^{-0.5} \mathrm{pc}
\label{eq:1}\end{aligned}$$ Considering sublimation temperature ($T_{\mathrm{sub}}$) to be the dust temperature $T_{\mathrm{dust}}$= 1700 K which was evaluated from the NIR colors of the variable flux component for Seyfert 1 galaxies [@2006ApJ...652L..13T; @2014ApJ...788..159K] and grain size $a = 0.1 \mu$m, the sublimation radius from equation \[eq:1\] can be written as log $R_{\mathrm{sub}}$/pc = $-0.80 + 0.5 \log(L_V/10^{44}$ erg s$^{-1}$), where $L_{UV} = 6 \,L_V$ . This is in excellent agreement with the relation of log $r_{\mathrm{dust}}$/pc = $-0.88 + 0.5 \log(L_V/10^{44}$ erg s$^{-1}$) obtained by a linear regression fit to the reverberation mapping data of a few Seyfert galaxies by [@2014ApJ...788..159K]. Thus, DRM observations using the $K$-band in the infrared region can define the inner radius of the dust torus for an AGN.
The emission line lag and dust reverberation lag [e.g., @1999AstL...25..483O; @2001ASPC..224..149O; @2014ApJ...784L...4H; @2014ApJ...784L..11Y; @2017MNRAS.464.1693H] have been proposed to serve as standard candles. However, the later shows a stronger correlation between lag and luminosity than former and needs only photometric monitoring making it preferable for standard candle although the emission line lags can be measured up to $z\sim 4$ [see @2011ApJ...740L..49W]. Unfortunately, only handful number of objects has dust lag measurement , which motivated @2017MNRAS.464.1693H to carry out a large dust reverberation mapping (DRM) program, “VEILS” (VISTA Extragalactic Infrared Legacy Survey) to use dust lag as standard candle for cosmology and constrain cosmological parameters. The “VEILS” survey will target about 1350 Seyfert 1 galaxies in the redshift range $0.1<z<1.2$, however, missing the objects in the local universe which are crucial in determining the normalization parameter of the AGN distance moduli [see @2017MNRAS.464.1693H]. Thus, dust reverberation mapping of local AGN will not only allow to estimate the extent of the torus, dust morphology and verify orientation dependent unification model but also help to estimate the normalization constant of the Hubble function.
To complement the VEILS program, in the nearby Universe, we are carrying out monitoring observations of a carefully selected sample of AGN with redshift less than 0.1 as part of our long term project called REMAP (REverberation Mapping of AGN Program). Observations towards this project has been carried out using the 2 m Himalayan Chandra Telescope at Hanle, India. Such a systematic study in conjunction with VEILS will serve as an excellent database to probe the complex interplay between optical continuum and dust emission in AGN. Here, we present the results of DRM monitoring of one local ($z=0.018$) Seyfert 1.5 galaxy H0507+164. The structure of this paper is as follows. In section \[sec:observation\] we briefly describe the observation and data reduction processes. The analysis is presented in section \[sec:Analysis\]. The results are discussed in section \[sec:discussion\] followed by the summary in section \[sec:summary\]. For the cosmological parameters we assumed $H_0=73$ km $\mathrm{s^{-1}Mpc^{-1}}$, $\Omega_m=0.27$ and $\Omega_{\lambda}=0.73$ [@2014ApJ...788..159K].
Observation and data reduction {#sec:observation}
==============================
The Sample
----------
The sample of sources for our DRM project were drawn from [@2015PASP..127...67B] that have BLR lag from spectroscopic reverberation observations. From this list, we have selected only those sources that are accessible for observations using the 2m Himalayan Chandra Telescope (HCT) located at Hanle and operated by the Indian Institute of Astrophysics[^5]. In this paper, we present the result of the first source monitored for DRM using the HCT, namely H0507+164, a local Seyfert 1.5 galaxy at $z$ = 0.018 with RA = 05:10:45.5 and DEC = +16:29:56. It is bright with an optical $g$ band magnitude of 10 mag from the SIMBAD database[^6]. It has a black hole mass of $9.62^{+0.33}_{-3.73}\times 10^6 M_{\odot}$ deduced from spectroscopic reverberation observations [@2011MNRAS.416..225S]. It is detected in the NRAO/VLA Sky Survey (NVSS[^7]) with a flux density of $6\pm0.5$ mJy at 1.4 GHz. However, it is radio quiet with a radio loudness parameter, $R$ (ratio of flux density in the radio-band at 1.4 GHz to the optical $V$-band flux density) of $2.26\pm0.26$.
Observations
------------
The observations were carried out for a total of 35 epochs during the period October 2016 to April 2017 using HCT The telescope is a Ritchey-Chretien system with an f/9 beam. The optical observations were carried out using the Himalayan Faint Object Spectrograph and Camera (HFOSC) mounted at the Cassegrain focus and equipped with a 2K $\times$ 4K SiTe CCD system. This CCD has a readout noise and gain of 4.8 electrons/ADU and 1.22 electrons respectively. The observations were carried out in binned mode using only the central 2K $\times$ 2K region. Each pixel of the CCD in this mode corresponds to 0.3 $\times$ 0.3 arcsec$^2$, and for the imaging observations reported here has a field of view of 10 $\times$ 10 arcmin$^2$. The typical exposure time in $V$-band is about 50 seconds. The IR observations in $J$, $H$ and $K_s$ bands were done subsequently to the $V$-band observations at each epoch were carried out using the TIFR Near Infrared Spectrometer (TIRSPEC) mounted on one of the side ports of HCT [@2014JAI.....350006N]. The detector used in TIRSPEC is a 1024 $\times$ 1024 HgCdTe array with a pixel size of 18 $\mu$m covering a field of view of 5 $\times$ 5 arcmin$^2$. It has a readout noise and gain of 25 electrons and 6 electrons/ADU respectively. The IR observations were performed in dithered mode consisted of five exposures each of 20 sec at three dither positions for each of three IR filters namely $J$, $H$ and $K_s$. Apart from the science frames, sky regions were also observed in the same dithering pattern as the object to generate master sky frame.
Data reduction
--------------
The optical data reduction was done using IRAF (Image Reduction and Analysis Facility[^8]) and MIDAS (Munich Data Analysis System[^9]). For image reduction, we followed the standard procedure, such as bias $\&$ dark subtraction and flat-fielding. After removing the cosmic rays, the optical frames were aligned and then combined using [imalign]{} and [imcombine]{} tasks in IRAF for each day. Point spread function (PSF) photometry was carried out on the combined images using the [daophot]{} and [allstar]{} packages in MIDAS to find the instrumental magnitudes of the object and the comparison stars present in the CCD frames. The observed $V$-band image is shown in Figure \[Fig:FOV\].
Reduction of the NIR images was performed using TIRSPEC NIR Data Reduction Pipeline [@2014JAI.....350006N] for each band. The pipeline subtracts the dark obtained for the same exposure time as the science exposures and then uses twilight flats for flat fielding and produce final combined images. PSF photometry on the combined images was carried out by using MIDAS to get the instrumental magnitudes.
The object appears as a point source in our optical and IR data, thereby, making it difficult to model the host galaxy from the observed data and remove its contribution to the measured brightness. Therefore, subtraction of the host galaxy contribution to the measured brightness of the source was not carried out. Also, we have not corrected the observed flux values for galactic extinction. These effects will be small and a constant fraction of the measured flux values, thus having negligible effects on the cross-correlation analysis.
![Observed $V$-band image of H0507+164. The integration time is 50 seconds and each side is 10 arcmin. The target source is shown with a circle.[]{data-label="Fig:FOV"}](image.jpeg)
JD$_V$ $F_V$ $\sigma_V$ JD$_J$ $F_J$ $\sigma_J$ JD$_H$ $F_H$ $\sigma_H$ JD$_{K_s}$ $F_{K_s}$ $\sigma_{K_s}$
-------------- ------- ------------ -------------- ------- ------------ -------------- -------- ------------ -------------- ----------- ----------------
2457687.4272 3.295 0.260 2457687.4429 4.994 0.302 2457687.4374 7.628 0.517 2457687.4316 12.582 0.490
2457693.2663 3.129 0.247 2457693.2765 5.519 0.312 2457693.2819 8.521 0.457 2457693.2875 12.835 0.399
2457697.2763 3.115 0.246 2457697.2802 6.027 0.313 2457697.2858 9.339 0.432 2457697.2950 13.585 0.387
2457699.4455 2.784 0.221 2457699.4277 5.161 0.346 2457699.4219 7.520 0.399 2457699.4164 12.783 1.174
2457700.2714 2.999 0.238 2457700.2536 6.287 0.521 2457700.2473 9.262 0.429 2457700.2416 13.473 0.386
2457704.2750 3.019 0.238 2457704.2595 6.008 0.305 2457704.2541 9.459 0.428 2457704.2486 13.589 0.403
2457710.2058 3.331 0.262 2457710.2274 6.596 0.317 2457710.2219 9.989 0.455 2457710.2165 14.439 0.422
2457711.3872 3.064 0.241 2457711.3110 6.768 0.357 2457711.3051 9.962 0.447 2457711.2986 13.995 0.621
2457712.3179 3.041 0.240 2457712.3339 6.710 0.453 2457712.3279 8.612 0.415 2457712.3219 12.946 0.416
2457717.4375 3.129 0.247 2457717.4540 5.734 0.324 2457717.4488 9.297 0.441 2457717.4435 14.027 0.536
2457719.2792 3.365 0.265 2457719.3544 7.811 0.384 2457719.3489 10.171 0.543 2457719.3433 15.470 0.668
2457721.2253 2.668 0.212 2457721.2420 5.685 0.289 2457721.2364 9.295 0.526 2457721.2308 13.682 0.440
2457727.1733 2.690 0.213 2457727.1922 5.747 0.394 2457727.1868 9.270 0.518 2457727.1812 14.099 0.472
2457728.4147 2.596 0.206 2457728.3990 6.519 0.523 2457728.3928 8.346 0.411 2457728.3874 12.835 0.422
2457732.2453 2.451 0.194 2457732.2619 5.852 0.318 2457732.2559 9.073 0.401 2457732.2499 14.018 0.423
2457739.4652 2.560 0.205 2457739.4483 5.804 0.276 2457739.4421 9.983 0.415 2457739.4363 13.936 0.382
2457745.2972 1.974 0.158 2457745.2543 5.352 0.374 2457745.2485 8.704 0.364 2457745.2420 13.121 0.365
2457750.3023 1.985 0.158 2457750.0752 5.175 0.327 2457750.0811 8.368 0.371 2457750.0890 12.969 0.394
2457752.3102 2.004 0.162 2457752.3535 6.695 0.262 2457752.3476 10.556 0.411 2457752.3415 14.664 0.403
2457768.0595 2.651 0.213 2457768.0408 5.642 0.307 2457768.0331 8.345 0.385 2457768.0276 12.681 0.376
2457773.3053 3.086 0.243 2457773.2892 6.372 0.302 2457773.2830 9.025 0.413 2457773.2769 12.876 0.389
2457784.1059 2.163 0.173 2457784.1302 3.666 0.316 2457784.1220 7.208 0.372 2457784.1136 10.684 0.355
2457788.2195 2.171 0.173 2457788.2422 4.591 0.392 2457788.2362 7.938 0.422 2457788.2302 14.780 0.528
2457794.2058 2.065 0.164 2457794.2221 5.169 0.386 2457794.2160 6.960 0.363 2457794.2104 10.702 0.350
2457801.1541 1.982 0.158 2457801.1683 4.076 0.226 2457801.1626 6.604 0.338 2457801.1568 11.091 0.356
2457807.0827 1.868 0.149 2457807.0662 4.963 0.350 2457807.0598 6.646 0.314 2457807.0538 10.771 0.339
2457808.2595 2.497 0.197 2457808.2429 5.374 0.244 2457808.2372 8.461 0.369 2457808.2315 11.685 0.324
2457812.2031 2.052 0.166 2457812.1793 4.063 0.238 2457812.1733 6.533 0.356 2457812.1673 10.096 0.439
2457817.1344 2.169 0.173 — — — 2457817.1214 4.961 0.317 2457817.0465 10.255 0.356
2457829.0926 2.326 0.186 2457829.0757 4.414 0.261 2457829.0698 6.161 0.344 2457829.0639 9.997 0.359
2457832.1700 2.586 0.206 2457832.1545 4.882 0.251 2457832.1483 8.705 0.425 2457832.1470 10.143 0.306
2457838.1930 2.972 0.235 2457838.1716 4.650 0.287 2457838.1656 7.371 0.374 2457838.1601 10.626 0.333
2457847.1147 3.061 0.243 2457847.1010 4.989 0.422 2457847.0918 7.763 0.389 2457847.0861 10.814 0.411
2457851.1541 3.289 0.265 2457851.1399 5.507 0.306 2457851.1344 8.052 0.399 2457851.1289 11.519 0.436
2457858.1201 3.078 0.244 2457858.1044 6.078 0.356 2457858.0989 8.170 0.414 2457858.0933 11.727 0.373
\[table2\]
Subtraction of the accretion disk component from the NIR bands
--------------------------------------------------------------
The observed NIR fluxes are dominated by emission from the torus due to re-processing of the central UV/optical emission by the hot dust. However, it is contaminated by variable flux from the accretion disk component [@2006ApJ...652L..13T; @2008Natur.454..492K; @2011MNRAS.415.1290L]. This would make the time lag calculated by cross correlation analysis between the optical and IR light curves shorter than the actual lag of the dust-torus emission [@2014ApJ...788..159K]. Thus, to get the true time lag, between optical and NIR flux variations, the contribution of accretion disk to the measured NIR fluxes needs to be removed. We estimated the contribution of the accretion disk in the $J$, $H$, $Ks$-bands in each of the epochs of observations by considering a power-law spectrum of the accretion disk following @2014ApJ...788..159K and given by $$f_{\mathrm{NIR,disk}}(t) = f_{V}(t)\left(\frac{\nu_{\mathrm{NIR}} }{\nu_V}\right)^{\alpha_{\nu}}
\label{eq:nir_eq}$$ where, $f_{V}(t)$ is the $V$-band flux at time ‘$t$’, $\nu_V$ and $\nu_{\mathrm{NIR}}$ are effective frequencies of $V$-band and NIR bands respectively and $\alpha_{\nu}$ is the power-law index. At any given epoch, the observations in the optical and each of the NIR bands differ by less than 300 seconds, and therefore, for the purpose of removing the contribution of accretion disk to each of the NIR bands, they were considered as simultaneous observations. To calculate $f_{\mathrm{NIR,disk}}$($t$), we assumed power-law index $\alpha_{\nu}$ to be equal to 1/3 following @2014ApJ...788..159K. AGN do show spectral variability with a bluer-when-brighter behavior which demands adoption of a time dependent $\alpha_{\nu}$ to correct for the contribution of accretion disk emission to NIR. Our single optical $V$-band observations hinder determination of $\alpha_{\nu}$ for each epoch of our observations. Use of single $\alpha_{\nu}$ might have some effect on the corrected $J$-band fluxes, however, it will have negligible effect on the corrected $K_s$-band flux values.
Prior to the calculation and subsequent subtraction of the accretion disk component from the observed NIR fluxes, the observed instrumental magnitudes were converted to apparent magnitudes via differential photometry of few stars in the field whose apparent magnitudes were taken from the SIMBAD database. The apparent magnitudes were then corrected for Galactic extinction taken from the NASA/IPAC Extragalactic database (NED[^10]). These magnitudes were then converted into fluxes and the contribution of accretion disk evaluated using equation \[eq:nir\_eq\] were subtracted. The final fluxes thus obtained were used to generate optical and NIR light curves for further analysis. The errors in the final flux values in different filters were obtained through propagation of errors during the various steps of flux determination.
Analysis {#sec:Analysis}
========
Light curves
------------
The light curves of H0507+164 in optical $V$-band and NIR $J$, $H$ and $K_s$ bands are shown in Figure \[fig:lc\]. As the source is radio-quiet, not much contamination to the observed flux variations is expected from the jet of the source. The variability nature of the source was characterized using the normalized excess variance [@2002ApJ...568..610E; @2003MNRAS.345.1271V; @2017MNRAS.466.3309R] defined as $$F_{\mathrm{var}} = \frac{\sqrt{S^2-\delta^2}}{\langle f \rangle} \\$$
Here $\langle f \rangle$, $S^2$ , $\delta$ are the mean flux, variance and mean error respectively for N observations and are given as $$\begin{gathered}
\langle f \rangle = \frac{1}{N}\sum_{i=1}^{N}f_i \\
S^2 = \frac{1}{N-1} \sum_{i=1}^{N}(f_i - \langle f \rangle)^2 \\
\delta^2 = \frac{1}{N} \sum_{i=1}^{N}\delta_i^2, \\\end{gathered}$$ where $f_i$ and $\delta_i$ are the flux and error for the $i^{th}$ measurement respectively. The uncertainties in the $F_{\rm{var}}$ are evaluated following @2002ApJ...568..610E and given as $$\begin{gathered}
\mathrm{err}(F_{\mathrm{var}}) = \sqrt{ \bigg(\sqrt{\frac{1}{2N}}\frac{\delta^2}{\langle f \rangle^2F_{\mathrm{var}}}\bigg)^2 + \bigg(\sqrt{\frac{\delta^2}{N}}\frac{1}{\langle f \rangle}\bigg)^2}\end{gathered}$$ In addition to calculating $F_{\mathrm{var}}$ we also calculated the ratio $R_{\mathrm{max}}$ between the maximum and minimum flux in the light curves. The results of variability analysis are given in Table \[table:var\]. It is noticed that $F_{\mathrm{var}}$ in the optical $V$-band and NIR J and H-bands are similar, while the $F_{\mathrm{var}}$ in $K_s$ band is lower than that in the other bands. An anti-corelation between variability amplitude and wavelength has been found recently based on analysis of NIR variability of a large sample of AGN by [@2017ApJ...849..110S]. Also, compared to the optical bands, amplitude of NIR variation is expected to be smaller [@2002ApJS..141...45E]. Though our data do not reveal a clear anti-correlation of variability amplitude with wavelength as found recently [@2017ApJ...849..110S], the $F_{\mathrm{var}}$ in the NIR bands are lower than of the optical V-band as expected [@2002ApJS..141...45E]. Therefore, the low amplitude of variations seen in the $K_s$ band do not point to any inherent bias in the $K_s$ band data.
are in units of 10$^{-26}$ erg s$^{-1}$ cm$^{-2}$ Hz$^{-1}$.
-------- -------------------------- ---------------------- ---------- -------------------- --------------------
Filter $\lambda_{\mathrm{eff}}$ $ \langle f \rangle$ $\sigma$ $F_{\mathrm{var}}$ $R_{\mathrm{max}}$
$V$ 5510 2.66 0.48 0.159$\pm0.005$ 1.801
$J$ 12200 5.56 0.89 0.149$\pm0.003$ 2.131
$H$ 16300 8.39 1.36 0.154$\pm0.002$ 2.360
$K_s$ 21900 12.56 1.56 0.119$\pm0.001$ 1.548
-------- -------------------------- ---------------------- ---------- -------------------- --------------------
: Variability statistics in $VJHK_s$ bands in observer’s frame. Here, $\lambda_{\mathrm{eff}}$ is the effective wavelength in Angstroms. The average values ($\langle f \rangle$) and the standard deviation $\sigma$ of the $VJHK_s$ light curves
\[table:var\]
Cross-Correlation Analysis
--------------------------
Our observations indicate that the source H0507+164 show flux variations in the optical and NIR bands enabling us to test for the presence/absence of time delay between flux variations in different bands. To calculate the time lag between the variations in the optical $V$-band emission from the accretion disk and the IR emission from the torus, we employed two well-known methods, namely the interpolated cross-correlation function [ICCF; @1986ApJ...305..175G; @1987ApJS...65....1G] and the discrete correlation function [DCF; @1988ApJ...333..646E]. In this work, cross-correlation analysis was performed between the $V$-band light curve and each of the NIR ($J$, $H$ and $K_s$) light curves. Figure \[fig:ccf\] shows the result of the cross-correlation analysis between $V$ and $J$ (upper panel), $V$ and $H$ (middle panel), $V$ and $K_s$ (lower panel). In each panel, the solid line shows the cross-correlation function (CCF) obtained by ICCF method, and the data points with error bars are those obtained by DCF method. Both ICCF and DCF are found to be in excellent agreement. The autocorrelation function (ACF) is also plotted for $V$-band (dashed-dot) and corresponding NIR band (dashed) in different panels. The ACFs show zero lag as expected, but the CCFs show an overall shift along the positive lag indicating that NIR continuum lags behind the optical $V$-band emission. The amount of lag can be estimated either by the lag corresponding to the peak of the CCF ($\tau_{\mathrm{peak}}$) or by the lag corresponding to the centroid of CCFs ($\tau_{\mathrm{cent}}$). The later has been found to be a better representative of the lag, particularly in cases when the light curves are either noisy and/or have less number of points [@1998PASP..110..660P]. The centroids of the CCF was calculated as $$\tau_{\mathrm{cent}}=\frac{\sum_{i} \tau_{i} \mathrm{CCF}_{i}}{\sum_{i} \mathrm{CCF}_{i}}.$$ To quantify the lag, we have used here the centroid of the CCF, which is evaluated by considering all the points that are within 60 per cent of the maximum of the CCF. This cut off was selected so as to have sufficient cross-correlation coefficients for the centroid determination.
The uncertainties in the derived lag were evaluated using a model-independent Monte Carlo simulation based on the flux randomization (FR) and random subset selection (RSS) described in @1998PASP..110..660P with the additional improvement as suggested by @1999ApJ...526..579W and summarized by @2004ApJ...613..682P. In each Monte Carlo iteration, we first randomly took $N$ independent points from a parent light curve of $N$ data points regardless of whether any point has previously been selected. The new light curve after RSS method contains $M$ data points. To take into account the uncertainty in the measured flux values, we then randomly modified the fluxes of the $M$ data points by adding the uncertainties of the measured flux multiplied with a random Gaussian value. For each Monte Carlo iteration, we computed the CCF of the modified light curve and calculated the $\tau_{\mathrm{cent}}$ using the points within 60 per cent of the CCF peak. This process was repeated for 20,000 iterations retaining only those CCF having peak value $>0.5$ so that the cross correlation result is significant. We built the cross correlation centroid distribution (CCCD), which is shown by the histogram plot in each panel of Figure \[fig:ccf\]. The median of the CCCD is taken as a representation of the lag. Since the distribution has a non-Gaussian shape, we calculated uncertainties within 68 % confidence interval around the median value. The result of the Monte Carlo simulation using different CCF analysis method is given in Table \[Table:corr1\]. The centroid lag obtained using DCF method was calculated for different bin sizes and the results are found to be consistent within error bars. The lags obtained by ICCF method are also given in Table \[Table:corr1\] which agrees well with the DCF values within error bars. Though the time delays obtained from both DCF and ICCF agree with each other, for all further analysis we consider the lag obtained using the DCF with a bin size of $\Delta\tau=5$ days as the typical sampling of our light curves is about 5 days. Based on this we find the rest frame (corrected for the redshift) time lags of $27.1^{+13.5}_{-12.0}$ days between $V$ and $J$ bands, $30.4^{+13.9}_{-12.0}$ days between $V$ and $H$ bands and $34.6^{+12.1}_{ -9.6}$ days between $V$ and $K_s$ bands. Though there is an indication of wavelength dependent lag which increases with wavelength, because of the larger error bars, we conclude that within errors, the derived lags are consistent with each other. Considering only $V$ and $K_s$ light curves, we obtained a lag of $34.6^{+12.1}_{ -9.6}$ days where $Ks$ band variation lagging the $V$-band variations. This lag is larger than the lag of $3.01^{+0.42}_{-1.84}$ days obtained between the optical and emission line flux variations from optical spectroscopic monitoring observations [@2011MNRAS.416..225S] and in agreement with what is expected from the unification model of AGN. Using this time delay we infer that the inner edge of the dusty torus is at a distance of $\approx0.029^{+0.010}_{-0.008}$ pc from the central UV/optical AGN continuum source.
\[Table:corr1\]
Discussion {#sec:discussion}
==========
Infrared lag and central luminosity correlation
-----------------------------------------------
Using the rest frame time lag, the inner radius of the dusty torus in H0507+164 was found to be 0.029 pc. Similar measurements of the radius of the dust torus in about two dozen AGN are available in the literature based on DRM observations [see @2014ApJ...788..159K and the references therein]. These sources are shown in Figure \[fig:lag-lum\] as empty circles in the lag - luminosity place. Also, shown in the same Figure is the location of the source H0507+164 indicated by a filled circle. The best-fitted regression line $\log\Delta\tau = -2.11 -0.2 M_V$ of @2014ApJ...788..159K which was based on a systematic and homogeneous analysis of DRM data of 17 Seyfert 1 galaxies is shown by a dashed line. Our lag measurement of H0507+164 is in excellent agreement with the lag expected for its luminosity and closely follows the dust lag-optical luminosity correlation of $\Delta\tau_{\mathrm{dust}}\varpropto L^{0.5}$.
H0507+164 which lies close to the regression line (dashed line), obtained by @2014ApJ...788..159K. The lag times were corrected for the time dilation effect using the object redshift. \[fig:lag-lum\]
Structure of the BLR and the Dust Torus
---------------------------------------
@2011MNRAS.416..225S carried out spectroscopic reverberation observations of BLR of H0507+164 and estimated a lag between the optical/UV continuum and the H$\beta$ line emission of $3.01^{+0.42}_{-1.84}$ days in the rest frame of the object. Our DRM observations for the same object in different NIR wavelengths together with the result obtained by @2011MNRAS.416..225S provide important information about the structure of the BLR and the dust torus of H0507+164. The BLR lag is found to be smaller than the dust-lag (lag between $V$-band and $K_s$-band) which is as expected from the unified scheme of AGN. Though there are indications of a wavelength dependent lag in our multi-wavelength data reported here, they are not statistically significant due to the large errors in their lags. From an analysis of a sample of 17 Seyfert galaxies, @2014ApJ...788..159K found that dust reverberation radius of their sample is 4-5 times larger than their BLR radius and typically a factor 2 smaller than their interferometric radius. Similarly, the BLR radius observed by reverberation mapping is smaller than the same observed by NIR interferometry [see @2012SPIE.8445E..0WP]. Such difference is expected as the reverberation mapping is a response weighted radius originating from the compact region, while the interferometric radius is flux weighted sensitive to the flux coming from the outer region . However, there are exceptions. For the source Mrk 335 [@2014ApJ...782...45D] NGC 4151 [@2006ApJ...651..775B] and NGC 4593 [@2013ApJ...769..128B], the dust radius [@2014ApJ...788..159K] is about 10 times larger than the BLR radius. Such varied differences between the mean dust radius and BLR radius known in AGN will hint for variable dust emission . To establish this, we need more precise measurements of dust radius for a large sample of AGN. Our ongoing multi-band DRM project on a larger sample will help to understand dust geometry.
Conclusion {#sec:summary}
==========
We have presented the first measurements of the time delays between optical and NIR bands. This is based on optical $V$ and NIR $J$, $H$ and $K_s$ bands observations spanning a time period of about 170 days. Significant variations have been observed in all bands. The rest frame lags between $V$-band and NIR bands are found to be $27.1^{+13.5}_{-12.0} \, \mathrm{days}$ ($V$ vs. $J$), $30.4^{+13.9}_{-12.0}
\, \mathrm{days}$ ($V$ vs. $H$) and $34.6^{+12.1}_{-9.6} \, \mathrm{days}$ ($V$ vs. $K_s$). From the present analysis it is difficult to probe the existence of different lag between $V$ and $J$, $V$ and $H$ and $V$ and $K_s$, solely due to the large errors in the determined lags because of the limited amount of data analysed here. The presence of wavelength dependent lags if any can be established only with the accumulation of good quality monitoring data over a long period of time. It is likely that a combined analysis of the data would reduce the error in the lag between optical and NIR . Given the limited number of observations in the present work, we have not attempted a combined analysis.
Using the rest frame lag between $V$ and $Ks$-band, we found that the inner radius of the dust torus is at a distance of 0.029 pc, from the central UV/optical continuum source. As expected from the unified model of AGN the dust inner radius is larger than the BLR radius known for this source from spectroscopic reverberation monitoring observations. Our estimate of $R_{\mathrm{dust}}$ and the $V$-band absolute magnitude is in good agreement with the dust lag-optical central luminosity relationship found by [@2014ApJ...788..159K] from a large sample of AGN.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank the referee for his/her valuable comments that helped us to improve our manuscript. S.R. acknowledges the support by the Basic Science Research Program through the National Research Foundation of Korea government (2016R1A2B3011457). This research used the SIMBAD data base, which is operated at CDS, Strasbourg, France, the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA and data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the NASA and the National Science Foundation. We thank to the supporting staff at the Indian Astronomical Observatory (IAO), Hanle, and CREST, Hoskote. AKM and RS thank the National Academy of Sciences, India for providing the required fund for this project. SH acknowledges support from the European Research Council Horizon 2020 grant DUST-IN-THE-WIND (677117). P.G. thanks STFC for support (grant reference ST/J003697/2). MBP gratefully acknowledges the support from Department of Science and Technology (DST), New Delhi under the INSPIRE faculty Scheme (sanctioned No: DST/INSPIRE/04/2015/000108).
\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: E-mail: [email protected]
[^3]: E-mail: [email protected]
[^4]: DST INSPIRE Faculty
[^5]: <http://www.iiap.res.in/centres/iao>
[^6]: <http://simbad.u-strasbg.fr/simbad/>
[^7]: <http://www.cv.nrao.edu/nvss/NVSSlist.shtml>
[^8]: IRAF is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
[^9]: MIDAS is the trade-mark of the European Southern Observatory
[^10]: <https://ned.ipac.caltech.edu/>
|
---
abstract: 'Parity-nonconserving two-body currents due to vector meson exchange are considered with the aim to determine the related contributions to the anapole moment. A particular attention is given to the requirement of current conservation which is essential for a reliable estimate of this quantity. An application is made for the deuteron case.'
author:
- 'C.-P. Liu$^{1}$'
- 'C. H. Hyun$^{2}$'
- 'B. Desplanques$^{3}$'
bibliography:
- 'AM.bib'
- 'MEC.bib'
- 'QEPV.bib'
title: Deuteron Anapole Moment with Heavy Mesons
---
Introduction \[sec:intro\]
==========================
First introduced by Zel’dovich [@Zeld57], the anapole moment (AM) of a quantum system is a quantity that involves both the electromagnetic interaction and parity nonconservation (PNC). Ignored for a long time, it is not before studies by Flambaum and Khriplovich [@FlKr80] that the concept acquires practical interest. In their work, these authors especially emphasize that the AM would grow with the size of the nucleus, making heavy nuclei natural candidates for an observation. Implying the hyperfine structure, a first measurement was performed a few years ago in the $^{133}$Cs nucleus [@Colorado97].
The deuteron AM has also received some attention recently [@SaSp98; @SaSp01; @Sa01; @KhKo00; @HyDe03]. Though its interest is largely academic (an experiment is not feasible in a near future), it offers the advantage of a laboratory where methods and ingredients relative to an estimate can be studied in details. These studies have been concerned with the pion-exchange component of the PNC nucleon-nucleon (NN) force. Calculations were based on assuming an effective-field-theory description [@SaSp98; @SaSp01], zero-range NN strong forces [@KhKo00], or more realistic NN strong forces [@HyDe03]. The use of an alternative field-theory description was also proposed [@Sa01]. The next step concerns the extension of these results to include the component of the PNC force due to vector-meson exchange that could contribute as much as, if not more than the pion exchange.
Determining the AM supposes to calculate the effective current that couples to the photon. As it involves parity nonconservation, the current has necessarily an axial character, making the requirement of current conservation non-trivial. Individual contributions are proportional to the weak coupling, $G_F$, while fulfilling the above property implies that the effective current contains the factor $G_F\,q^2$, which vanishes in the limit of a zero momentum transfer. Getting this result demands particular care with contributions ensuring gauge invariance. When dealing with vector mesons, this task becomes essential. In particular, it has to be done consistently with the PNC interaction model that is employed in calculations. Some contributions were given in Ref. [@HLRM02]. In the present work, we intend to complete this work with the double aim to satisfy gauge invariance and consistency with the PNC interaction model; DDH potential, given by Desplanques, Donoghue, and Holstein, is our case [@DDH80]. Some estimates of the vector-meson-exchange contributions to the anapole moment will be presented in the deuteron case.
The plan of the paper is as follows. In Section \[sec:MECs\], we first present the various ingredients pertinent to the interaction: parity-conserving (PC), parity-nonconserving and electromagnetic (EM) ones. We subsequently provide the expressions for the PNC two-body currents at the lowest $1/\mN$ order and show how they allow one to fulfill current conservation. Section \[sec:d AM\] is devoted to applications involving the deuteron. This includes the deuteron description, especially the determination of the PNC components; the expression of the anapole matrix elements from both the one- and two-body currents; and a numerical estimate in terms of the PNC meson-nucleon coupling constants. A discussion of the results is given in Section \[sec:dis\]. This is completed by Appendix \[sec:MECs in x\] that contains expressions of the two-body currents in configuration space.
PNC $NN$ Interaction, Currents, and Current Conservation \[sec:MECs\]
=====================================================================
The anapole moment is a special electromagnetic property of a system in which parity conservation is violated; therefore, our first step in calculation is formulating the EM current operators. Throughout this work, we assume for a nuclear system the validity of non-relativistic (NR) limit and keep only terms of leading order.
The one-body term $(\rho^{(1)},\bm j^{(1)})$, that is, contributions from each individual nucleon (for deuteron, there are two), include the charge density:$$\rho^{(1)}=e\sum_{i=1}^{2}\frac{1+\tau_{i}^{z}}{2}(2\pi)^{3}\delta^{(3)}
(\bm k+\bm p'{}_{i}-\bm p{}_{i})\,,$$ and the 3-current densities from spin and motion of nucleons:
$$\begin{aligned}
\bm j_{spin} & = & e\sum_{i=1}^{2}\frac{\mu_{\ssst{N}}^{i}}{2\mN}i\bm\sigma_{i}\times
(\bm p'_{i}-\bm p_{i})(2\pi)^{3}\delta^{(3)}(\bm k+\bm p'{}_{i}-\bm p{}_{i})\,,\\
\bm j_{conv} & = & e\sum_{i=1}^{2}\frac{1+\tau_{i}^{z}}{4\mN}(\bm p'_{i}+\bm p_{i})(2\pi)^{3}\delta^{(3)}(\bm k+\bm p'{}_{i}-\bm p{}_{i})\,,\end{aligned}$$
in momentum space, where $\mu_{\ssst{N}}^{i}$ is defined as $$\mu_{\ssst{N}}^{i}\equiv\frac{1}{2}\left(\mu_{\ssst{S}}+\tau_{i}^{z}
\mu_{\ssst{V}}\right)$$ with $\mu_{\ssst{S}}=0.88$ and $\mu_{\ssst{V}}=4.71$; and $\bm k$, $\bm p'_{i}$, and $\bm p{}_{i}$ denote the 3-momentum of outgoing photon, outgoing $i$th nucleon, and incoming $i$th nucleon, respectively.
As the canonical picture of nucleon-nucleon ($NN$) interaction is realized through the exchange of mesons, the nuclear EM currents should also contain these exchange effects, which are two-body in character. Since we are still lacking of a fundamental understanding about the $NN$ interaction, the exact exchange currents (ECs) are also unknown. However, in order to reduce theoretical uncertainties in the studies of nuclear EM processes, it is important, given a chosen model for the $NN$ interaction, to construct these two-body currents which are constrained by both current conservation and phenomenology.
For the PC $NN$ interaction, many high-quality model potentials exist, and our choice for calculation in this work is the Argonne $v{}_{18}$ potential (A$v{}_{18}$) [@AV18]. Although it gives good fits to the scattering data and deuteron properties, it is not straightforward to construct the corresponding ECs because the connection with the meson exchange picture is not clear for some parts in this potential. One traditional way to construct the ECs is implementing the NR minimal coupling (MC) to the potential, i.e.,$$\bm p\rightarrow\bm p-\frac{e}{2}(1+\tau^{z})\bm A\,;\quad H\rightarrow H+\frac{e}{2}(1+\tau^{z})A^{0}\,,$$ then identifying the EM currents from the interaction Hamiltonian density, $ej_{\mu}A^{\mu}$. However, this procedure only constrains the longitudinal components, while giving no information about the transverse components which are conserved by themselves. Some uncertainty about these last terms comes from potentials involving quadratic velocity-dependent components as discussed in Ref. [@Risk89]. Moreover, the derivation of exchange currents for a model like A$v_{18}$, employed in Ref. [@Scetal03] for instance, requires further elaboration. This model contains a Gaussian type component while the above prescription is usually applied to Yukawa potentials. Therefore, we leave this as an open question for future work and follow the treatment of Ref. [@HyDe03] to examine: (1) to what degree current conservation is broken by the omission of PC ECs, and (2) how much the inclusion of PC ECs due to the one pion exchange, which gives the long-range part in A$v{}_{18}$, could restore the conservation. By this exercise, one can get some qualitative handle on this problem. The detail is be to discussed in Section \[sec:dis\].
For the PNC $NN$ interaction, our choice is the potential suggested by Desplanques, Donoghue, and Holstein (DDH) [@DDH80]. Since this potential, based on the one-boson exchange scheme involving $\pi$, $\rho$, and $\omega$ mesons, has a close tie with the exchange picture, a more field-theoretical formalism, the so-called $S$-matrix approach [@ChRh70; @Chem79; @Town87; @HLRM02], is used to construct all the corresponding ECs. As we will show later, some transverse components arise naturally in this derivation. For clarity, we divide the following discussion into three parts: first, the model Lagrangian, consistent with the DDH scheme, is constructed; second, the PNC ECs are derived; and finally, we show how these exchange currents allow one to fulfill current conservation, given the DDH potential.
Model Lagrangian
----------------
The total Lagrangian density we consider is expressed as
$$\mathcal{L}=\mathcal{L}_{0}+\mathcal{L}_{PC}+\mathcal{L}_{PNC}+\mathcal{L}_{EM}
+\delta\mathcal{L}\,,$$
where $\delta\mathcal{L}$ contains all the terms not relevant for this discussion. The free Lagrangian densities for nucleon, $N$; pion, rho, and omega mesons, $\bm\pi$, $\bm\rho^{\mu}$, and $\omega^{\mu}$, are$$\begin{aligned}
\mathcal{L}_{0} & = & \bar{N}'(i\partial\!\!\!/-\mN)N+\frac{1}{2}(\partial_{\mu}\bm\pi)\cdot
(\partial^{\mu}\bm\pi)-\frac{1}{2}m_{\pi}^{2}\bm\pi^{2}-\frac{1}{4}\bm F_{\mu\nu}^{(\rho)}\cdot\bm F^{(\rho)\mu\nu}+\frac{1}{2}m_{\rho}^{2}\bm\rho_{\mu}\cdot\bm\rho^{\mu}-
\frac{1}{2\xi}(\partial_{\mu}\bm\rho^{\mu})\cdot(\partial_{\nu}
\bm\rho^{\nu})\nonumber \\
& & -\frac{1}{4}F_{\mu\nu}^{(\omega)}F^{(\omega)\mu\nu}+\frac{1}{2}
m_{\omega}^{2}\omega_{\mu}\omega^{\mu}-\frac{1}{2\xi}(\partial_{\mu}\omega^{\mu})
(\partial_{\nu}\omega^{\nu})\,,\end{aligned}$$ where $\bm F_{\mu\nu}^{(\rho)}$ and $F_{\mu\nu}^{(\omega)}$ are the field tensors of the vector mesons, and we keep the $R_{\xi}$ gauge-fixing terms of vector mesons explicit for the moment. The PC and PNC meson-nucleon interaction Lagrangian densities are
$$\begin{aligned}
\mathcal{L}_{PC} & = & i\gpiNN\bar{N}'\gamma_{5}\bm\tau\cdot\bm\pi N-\grhoNN\bar{N}'(\gamma_{\mu}-i\frac{\chi_{\ssst{V}}}{2\mN}
\sigma_{\mu\nu}q^{\nu})\bm\tau\cdot\bm\rho^{\mu}N\nonumber \\
& & -\gomegaNN\bar{N}'(\gamma_{\mu}-i\frac{\chi_{\ssst{S}}}{2\mN}\sigma_{\mu\nu}
q^{\nu})\omega^{\mu}N\,,\\
\mathcal{L}_{PNC} & = & -\frac{h_{\pi}^{1}}{\sqrt{2}}\bar{N}'(\bm\tau\times\bm\pi)^{z}N+
\bar{N}'[h_{\rho}^{0}\bm\tau\cdot\bm\rho^{\mu}+h_{\rho}^{1}\rho^{z\mu}+
\frac{h_{\rho}^{2}}{2\sqrt{6}}(3\tau^{z}\rho^{z\mu}-\bm\tau\cdot\bm\rho^{\mu})]
\gamma_{\mu}\gamma_{5}N\nonumber \\
& & +\bar{N}'(h_{\omega}^{0}\omega^{\mu}+h_{\omega}^{1}\tau^{z}\omega^{\mu})
\gamma_{\mu}\gamma_{5}N\,,\end{aligned}$$
where $q^{\mu}$ is the 4-momentum carried by the outgoing boson; the strong couplings, $g_{\ssst{XNN}}$, as well as the weak couplings, $h_{\ssst{X}}^{(i)}$, are of DDH’s definition (except their PNC pion coupling, $f_{\pi}$, is renamed as $h_{\pi}^{1}$ here); and the anomalous strong isoscalar and isovector magnetic moments of nucleon, $\chi_{\ssst{S}}$ and $\chi_{\ssst{V}}$, are assumed to be the same as the EM values, -0.12 and 3.70, by vector meson dominance.
Now we apply the covariant MC:$$p_{\mu}\rightarrow p_{\mu}-\frac{e}{2}(1+\tau^{z})A_{\mu}\,,$$ to the above Lagrangian densities to obtain the EM interactions. For our purpose, only terms of first-order in $e$ are included in $\mathcal{L}_{EM}$. From $\mathcal{L}_{0}$, we get
$$\begin{aligned}
\mathcal{L}_{EM}^{(\ssst{NN}\gamma)} & = & -e\bar{N'}[(F_{1}^{\ssst{(S)}}(Q^{2})\frac{1}{2}+F_{1}^{\ssst{(V)}}(Q^{2})
\frac{\tau^{z}}{2})\gamma_{\mu}-i\frac{1}{2\mN}(F_{2}^{\ssst{(S)}}(Q^{2})
\frac{1}{2}+F_{2}^{\ssst{V}}(Q^{2})\frac{\tau^{z}}{2})\sigma_{\mu\nu}q^{\nu}]N
A^{\mu}\,,\\
\mathcal{L}_{EM}^{(\pi\pi\gamma)} & = & -e(\bm\pi\times\partial_{\mu}\bm\pi)^{z}A^{\mu}\,,\\
\mathcal{L}_{EM}^{(\rho\rho\gamma)} & = & -e(\bm\rho^{\nu}\times\bm F_{\nu\mu}^{(\rho)})^{z}A^{\mu}-\frac{1}{\xi}e(\bm\rho_{\mu}\times\partial_{\nu}
\bm\rho^{\nu})^{z}A^{\mu}\,.\end{aligned}$$
Note that in order to account for the nucleon structure, nucleon EM form factors, $F_{1,2}^{\ssst{(S,V)}}$ (superscript “$S$” for isoscalar and “$V$” for isovector; subscript “$1$” for Dirac and “$2$” for Pauli) have to be added. At $Q^{2}=-q^{2}=0$, $F_{1}^{\ssst{(S)}}(0)=F_{1}^{\ssst{(V)}}(0)=1$, $F_{2}^{\ssst{(S)}}(0)=-0.12$, and $F_{2}^{\ssst{(V)}}(0)=3.70$. In principle one should also take into account the meson structures; however, they are still poorly constrained so we simply assume them elementary.
Due to the momentum-dependent coupling of $\rho$ meson to the nucleon anomalous magnetic moment, a Kroll-Ruderman type contact interaction [@KrRu54], reading as$$\mathcal{L}_{EM}^{(\ssst{NN}\rho\gamma)}=-e\frac{\grhoNN\chi_{\ssst{V}}}{2\mN}
\bar{N'}\sigma_{\mu\nu}(\bm\tau\times\bm\rho^{\nu})^{z}NA^{\mu}\,,$$ also arises. This will lead to a seagull current which is important for current conservation, but was ignored in Ref. [@HLRM02].
It is worthwhile to point out that the EM interactions obtained above depend on the model Lagrangians we start out. For example, compared with the result of QHD II [@SeWa86], a gauge theory approach which first models pion from a global $SU(2)$ symmetry group and then adds $\rho$ meson and photon by a local $SU(2)\otimes U(1)$ symmetry group (both steps involve Higgs mechanism to generate meson masses), one observes a larger $\rho\rho\gamma$ interaction in QHD II by the amount of $$\Delta\mathcal{L}_{EM}^{(\rho\rho\gamma)}=
\frac{1}{2}e(\bm\rho_{\mu}
\times\bm\rho_{\nu})^{z}F^{(\gamma)\mu\nu}\,,$$ which modifies the $\rho\rho\gamma$ vertex in an interesting way as we are going to explain.
By fixing the gauge parameter $\xi=1$, i.e., ’t Hooft-Feynman gauge, we obtained the same $\mathcal{L}_{EM}^{(\rho\rho\gamma)}$ as in Ref. [@Hyetal80], and this gives a vertex factor (see Fig. \[fig:vertex\]):$$\epsilon_{3ij}[(q_{1}-q_{2})^{\mu}g^{\alpha\beta}+k^{\alpha}g^{\beta\mu}-
k^{\beta}g^{\mu\alpha}]\,,\label{eq:rho MC}$$ with $k+q_{1}+q_{2}=0$. The latter two terms are combined to give amplitudes conserved by themselves when dotted by $k_{\mu}$, which means they are purely transverse (actually, they correspond to magnetic dipole couplings which explain the transversality). By adding $\Delta\mathcal{L}_{EM}^{(\rho\rho\gamma)}$, it simply doubles the self-conserved terms, so that we have$$\epsilon_{3ij}[(q_{1}-q_{2})^{\mu}g^{\alpha\beta}+2k^{\alpha}g^{\beta\mu}-
2k^{\beta}g^{\mu\alpha}]\,,\label{eq:rho chiral}$$ for the vertex. As $\rho$ meson is a spin-one particle, it has charge ($c$), magnetic dipole ($\mu$), and charge quadrupole ($Q$) couplings to the EM field. When assuming it is an elementary particle, $c=e$, $\mu=e/m_{\rho}$, and $Q=-e/m_{\rho}^{2}$, the vertex factor appears to be Eq. (\[eq:rho chiral\]) [@Aretal80; @BrHi92], so the MC result under-predicts the $\rho$ meson magnetic moment by two. This factor of two difference in self-conserved terms between MC and chiral Lagrangian approaches has been pointed out in Refs. [@AdTr84; @Town87]. It is also obtained from a quark model calculation of the charged $\rho$ meson magnetic moment (in the limit where the $\rho$ mass is taken as twice the constituent quark one). Although the factor two originates from different models, in order to have a closer contact with phenomenology, we make this particular modification to the $\rho\rho\gamma$ vertex.
The modification mentioned above is just an example of model-dependences in constructing ECs. Since these purely transverse terms, often called non-Born (NB) terms, could not be constrained by current conservation, it is not easy to set up criteria *a priori* to judge which should be included or not, unless compared with experiments [^1]. For our calculation, we choose to include $\rho\pi\gamma$ and $\omega\pi\gamma$ interactions $$\begin{aligned}
\mathcal{L}_{EM}^{(\rho\pi\gamma)} & = & e\frac{g_{\rho\pi\gamma}}{2\, m_\rho}\epsilon_{\alpha\beta\gamma\delta}F^{(\gamma)
\alpha\beta}(\bm\rho^{\gamma}\cdot\partial^{\delta}\bm\pi)\,,\\
\mathcal{L}_{EM}^{(\omega\pi\gamma)} & = & e\frac{g_{\omega\pi\gamma}}{2\, m_\omega}\epsilon_{\alpha\beta\gamma\delta}F^{(\gamma)
\alpha\beta}(\omega^{\gamma}\partial^{\delta}\pi^{z})\,,\end{aligned}$$ where the total anti-symmetric tensor is defined as $\epsilon_{0123}=-1$ [@Adle66; @Chem79], because these coupling constants could be determined from the decay data (except for signs); and we ignore all the nucleon isobaric excitations for they are not the main theoretical emphasis of this work. The present work could be easily extended to them if necesssary.
To sum up, the total EM Lagrangian density we consider is$$\mathcal{L}_{EM}=L_{EM}^{(\ssst{NN}\gamma)}+L_{EM}^{(\pi\pi\gamma)}+\left\{ L_{EM}^{(\rho\rho\gamma)}+\frac{1}{2}e(\bm\rho_{\mu}\times\bm\rho_{\nu})^{z}
F^{(\gamma)\mu\nu}\right\} +\mathcal{L}_{EM}^{(\rho\pi\gamma)}+\mathcal{L}_{EM}^{(\omega\pi\gamma)}
\,.$$
PNC Meson Exchange Currents
---------------------------
Diagrammatically, the ECs could be classified, according to Fig. \[fig:MECs\], as: (a) norm-recoil, (b) pair, (c) mesonic, (d) seagull, (e) isobaric, and (f) NB mesonic types. The division of norm-recoil and pair terms simply comes from the separation of positive- and negative-energy components in the covariant nucleon propagator. Sometimes confusion arises when it comes to pair and seagull diagram. If the PC $\pi NN$ coupling is formulated as pseudo-vector, the seagull term is $O(1/\mN)$, while the pair term is higher-order in $1/\mN$. On the other hand, if the pseudo-scalar coupling is adopted as we do here, there is no seagull term, however, at the leading order, the pair term looks exactly as the seagull term in the pseudo-vector scheme. Therefore, as far as the NR approximation is valid, these two formalisms are equivalent [@Math89; @Risk89]. For the case of $\rho$ meson, both pair and seagull diagrams have $O(1/\mN)$ contributions. As for the NB contributions from (e) and (f), we only consider the latter as explained above.
Before one applies Feynman rules to evaluate these diagrams and extract the corresponding ECs, the gauge parameter has to be fixed. Though the physical results should be gauge-independent, a proper choice may greatly simplify the calculation. Here, we adopt the ’t Hooft-Feynman gauge, $\xi=1$, for the following reasons. First, the propagator is simpler$$\begin{aligned}
\langle A_{\mu}A_{\nu}\rangle & = & \left(g_{\mu\nu}-(1-\xi)\frac{q_{\mu}q_{\nu}}{q^{2}-\xi m^{2}+i\epsilon}\right)\frac{-i}{q^{2}-m^{2}+i\epsilon}\\
& = & \frac{-ig_{\mu\nu}}{q^{2}-m^{2}+i\epsilon}\quad(\textrm{for }\xi=1).\end{aligned}$$ Second, the PNC potential, constructed from NR reduction of the one-boson exchange diagrams, corresponds to the form given by DDH. The last and most important of all, in combination with the previous statement, the contribution from the norm-recoil diagram represents how the one-body EM matrix element is modified by the presence of this NR potential [@Town87]. Therefore, it should not be double-counted if one has already taken care of this by using the perturbed wave function, the route we will follow.
In momentum space, to the order of $1/\mN$, we have the following results for the pair and $\rho$-seagull (KR) 3-currents
$$\begin{aligned}
\bm j_{pair}^{\pi} & = & \frac{-e\gpiNN h_{\pi}^{1}}{2\sqrt{2}\mN}(\bm\tau_{1}\cdot\bm\tau_{2}-\tau_{1}^{z}
\tau_{2}^{z})[\bm\sigma_{1}]\frac{(2\pi)^{3}\delta^{(3)}(\cdots)}{\bm q_{2}^{2}+m_{\pi}^{2}}+(1\leftrightarrow2)\,,\label{eq:PNC MEC first}\\
\bm j_{pair+\ssst{KR}}^{\rho} & = & \frac{-e\grhoNN}{2\mN}\left\{ \left(h_{\rho}^{0}(\bm\tau_{1}\cdot\bm\tau_{2}+\tau_{2}^{z})+
\frac{h_{\rho}^{2}}{2\sqrt{6}}(3\tau_{1}^{z}\tau_{2}^{z}-
\bm\tau_{1}\cdot\bm\tau_{2}+2\tau_{2}^{z})\right)[\bm\sigma{}_{1}-
\bm\sigma_{2}]\right.\nonumber \\
& & \left.+h_{\rho}^{1}(1+\tau_{1}^{z})[\tau_{2}^{z}\bm\sigma_{1}-
\tau_{1}^{z}\bm\sigma_{2}]+(1+\chi_{\ssst{V}})\left(h_{\rho}^{0}-
\frac{h_{\rho}^{2}}{2\sqrt{6}}\right)(\bm\tau_{1}\times\bm\tau_{2})^{z}
[\bm\sigma_{1}\times\bm\sigma_{2}]\right\} \nonumber \\
& & \times\frac{(2\pi)^{3}\delta^{(3)}(\cdots)}{\bm q_{2}^{2}+m_{\rho}^{2}}+(1\leftrightarrow2)\,,\\
\bm j_{pair}^{\omega} & = & \frac{-e\gomegaNN}{2M_{N}}(1+\tau_{1}^{z})\big(h_{\omega}^{0}[\bm\sigma_{1}-
\bm\sigma_{2}]+h_{\omega}^{1}[\tau_{1}^{z}\bm\sigma_{1}-\tau_{2}^{z}
\bm\sigma_{2}]\big)\frac{(2\pi)^{3}\delta^{(3)}(\cdots)}{\bm q_{2}^{2}+m_{\omega}^{2}}+(1\leftrightarrow2)\,,\end{aligned}$$
where the $\rho$-seagull 3-current corresponds to the term involving $\chi_{\ssst{V}}$; $\bm q_{1,2}=\bm p'_{1,2}-\bm p_{1,2}$; and the $\delta$ function imposes the total 3-momentum conservation, $\bm k+\bm q_{1}+\bm q_{2}=0$. All the pair charges are of higher order in $1/\mN$ compared with the nucleon charge, which is $O(1)$, so they are neglected.
For the mesonic 3-currents, we obtain$$\begin{aligned}
\bm j_{mesonic}^{\pi} & = & \frac{-e\gpiNN h_{\pi}^{1}}{2\sqrt{2}\mN}(\bm\tau_{1}\cdot\bm\tau_{2}-\tau_{1}^{z}\tau_{2}^{z})
[\bm q_{2}-\bm q_{1}]\bm\sigma_{1}\cdot\bm q_{1}\frac{(2\pi)^{3}\delta^{(3)}(\cdots)}{(\bm q_{1}^{2}+m_{\pi}^{2})(\bm q_{2}^{2}+m_{\pi}^{2})}+(1\leftrightarrow2)\,,\\
\bm j_{mesonic}^{\rho} & = & \frac{-e\grhoNN}{2\mN}\left(h_{\rho}^{0}-\frac{h_{\rho}^{2}}{2\sqrt{6}}\right)
i(\bm\tau_{1}\times\bm\tau_{2})^{z}\Big\{[\bm q_{1}-\bm q_{2}]\bm\sigma_{2}\cdot\big((\bm p'_{2}+\bm p_{2})-(\bm p'_{1}+\bm p_{1})-i(1+\chi_{\ssst{V}})\bm\sigma_{1}\times\bm q_{1}\big)\nonumber \\
& & +2[(\bm p'_{1}+\bm p_{1})+i(1+\chi_{\ssst{V}})\bm\sigma_{1}\times\bm q_{1}]\bm\sigma_{2}\cdot\bm k+2[\bm\sigma_{2}]\Big(2\mN k_{0}-\bm k\cdot\big((\bm p'_{1}+\bm p_{1})+i(1+\chi_{\ssst{V}})\bm\sigma_{1}\times\bm q_{1}\big)\Big)\Big\}\nonumber \\
& & \times\frac{(2\pi)^{3}\delta^{(3)}(\cdots)}{(\bm q_{1}^{2}+m_{\rho}^{2})(\bm q_{2}^{2}+m_{\rho}^{2})}+(1\leftrightarrow2)\,,\label{eq:rho mesonic}\\
\bm j_{mesonic}^{\rho\pi} & = & \frac{e\grhoNN g_{\rho\pi\gamma}h_{\pi}^{1}}{\sqrt{2}m_{\rho}}(\bm\tau_{1}\times\bm\tau_{2})^{z}
[\bm q_{1}\times\bm q_{2}]\frac{(2\pi)^{3}\delta^{(3)}(\cdots)}{(\bm q_{1}^{2}+m_{\rho}^{2})(\bm q_{2}^{2}+m_{\pi}^{2})}+(1\leftrightarrow2)\,,\\
\bm j_{mesonic}^{\omega\pi} & \approx & 0\,,\end{aligned}$$ where $k_{0}=E_{1}+E_{2}-E_{1}'-E_{2}'=E_{i}-E_{f}$. Specially note that there is a contribution to the charge density at the same order as the nucleon charge:$$\begin{aligned}
\rho_{mesonic}^{\rho} & = & -2e\grhoNN\left(h_{\rho}^{0}-\frac{h_{\rho}^{2}}{2\sqrt{6}}\right)i(\bm\tau_{1}
\times\bm\tau_{2})^{z}\bm\sigma_{2}\cdot\bm k\frac{(2\pi)^{3}\delta^{(3)}(\cdots)}{(\bm q_{1}^{2}+m_{\rho}^{2})(\bm q_{2}^{2}+m_{\rho}^{2})}+(1\leftrightarrow2)\,.\label{eq:PNC MEC last}\end{aligned}$$
PNC $NN$ Interaction and Current Conservation
---------------------------------------------
By considering the one-boson exchange diagrams where one of the meson-nucleon couplings is PC and the other PNC, the DDH potential in momentum space could be expressed as
$$\begin{aligned}
V_{\ssst{PNC}}^{\pi} & = & \frac{\gpiNN h_{\pi}^{1}}{2\sqrt{2}\mN}i\left(\vtau_{1}\times\vtau_{2}\right)^{z}
\left(\vsig_{1}+\vsig_{2}\right)\cdot\bm u_{\pi}\,,\label{eq:vpnc-pi-p}\\
V_{\ssst{PNC}}^{\rho} & = & \frac{-\grhoNN}{\mN}\left[\left(h_{\rho}^{0}\vtau_{1}\cdot\vtau_{2}+
\frac{h_{\rho}^{1}}{2}(\tau_{1}^{z}+\tau_{2}^{z})+\frac{h_{\rho}^{2}}
{2\sqrt{6}}(3\tau_{1}^{z}\tau_{2}^{z}-\vtau_{1}\cdot\vtau_{2})\right)
\right.\nonumber \\
& & \left.\times\big(i(1+\chi_{\ssst{V}})\,(\vsig_{1}\times\vsig_{2})\cdot\bm u_{\rho}+(\vsig_{1}-\vsig_{2})\cdot\bm v_{\rho}\big)-\frac{h_{\rho}^{1}}{2}(\tau_{1}^{z}-\tau_{2}^{z})(\vsig_{1}+
\vsig_{2})\cdot\bm v_{\rho}\right]\,,\label{eq:vpnc-rho-p}\\
V_{\ssst{PNC}}^{\omega} & = & \frac{-\gomegaNN}{\mN}\left[\left(h_{\omega}^{0}+\frac{h_{\omega}^{1}}{2}
(\tau_{1}^{z}+\tau_{2}^{z})\right)\big(i(1+\chi_{\ssst{S}})(\vsig_{1}
\times\vsig_{2})\cdot\bm u_{\omega}+(\vsig_{1}-\vsig_{2})\cdot\bm v_{\omega}\big)\right.\nonumber \\
& & \left.+\frac{h_{\omega}^{1}}{2}\,(\tau_{1}^{z}-\tau_{2}^{z})(\vsig_{1}
+\vsig_{2})\cdot\bm v_{\omega}\right]\,,\label{eq:vpnc-omega-p}\end{aligned}$$
where$$\bm u_{\ssst{X}}=\frac{\bm q}{\bm q^{2}+m_{\ssst{X}}^{2}}\,;\quad\bm v_{\ssst{X}}=\frac{(\bm p'_{1}+\bm p_{1})-(\bm p'_{2}+\bm p_{2})}{2\, ( \bm q^{2}+m_{\ssst{X}}^{2} ) }\,,$$ with $\bm q$ denotes the meson 3-momentum.
To prove the conservation of these PNC currents at the operator level, we showed explicitly the following matrix element identities (with bra $\langle\bm p'_{1},\bm p'_{2}|$ and ket $|\bm p_{1},\bm p_{2}\rangle$):
$$\begin{aligned}
\langle[\rho^{(1)}\,,\, V_{\ssst{PNC}}^{\pi}]\rangle & \textrm{=} & \bm k\cdot\langle\bm j_{pair}^{\pi}+\bm j_{mesonic}^{\pi}\rangle\,,\\
\langle[\rho^{(1)}\,,\, V_{\ssst{PNC}}^{\rho}]\rangle & = & \bm k\cdot
\langle\bm j_{pair+\ssst{KR}}^{\rho}+\bm j_{mesonic}^{\rho\,\ssst{(I)}}\rangle\,,\\
\langle[\rho^{(1)}\,,\, V_{\ssst{PNC}}^{\omega}]\rangle & = & \bm k\cdot\langle\bm j_{pair}^{\omega}\rangle\,,\\
\langle[\rho_{mesonic}^{\rho}\,,\, H]\rangle & = & \bm k\cdot\langle\bm j_{mesonic}^{\rho\,\ssst{(II)}}\rangle\,,\label{eq:self conserved}\\
0 & = & \bm k\cdot\langle\bm j_{mesonic}^{\rho\pi}+\bm j_{mesonic}^{\omega\pi}\rangle\,,\label{eq:NB continuity}\end{aligned}$$
where $H$ is the total Hamiltonian, which is the sum of kinetic energy ($T$) and both PC and PNC potentials ($V_{\ssst{PC}}$ and $V_{\ssst{PNC}}$); and the $\rho$ mesonic current, Eq. (\[eq:rho mesonic\]), is separated into two parts: (I) is proportional to the vector $(\bm q_{1}-\bm q_{2})$, and (II) contains the rest. The continuity equality of Eq. (\[eq:self conserved\]) indicates that $(\rho_{mesonic}^{\rho}\,,\,\bm j_{mesonic}^{\rho\,\ssst{(II)}})$ forms a conserved current not constrained by the DDH potential, which actually results from the self-conserved $\rho\rho\gamma$ vertex mentioned above, while the last equality shows the transversality of NB currents. Obviously, the total PNC EM EC operator, $(\rho_{\ssst{PNC}}^{(2)}\,,\,\bm j_{\ssst{PNC}}^{(2)})$ we construct, Eqs. (\[eq:PNC MEC first\]-\[eq:PNC MEC last\]), satisfies the total current conservation condition$$[\rho^{(1)}+\rho_{\ssst{PNC}}^{(2)}\,,\, T+V_{\ssst{PC}}+V_{\ssst{PNC}}]=\bm k\cdot(\bm j^{(1)}+\bm j_{\ssst{PC}}^{(2)}+\bm j_{\ssst{PNC}}^{(2)})\,,$$ as long as the two-body PC EC, $(\rho_{\ssst{PC}}^{(2)},\bm j_{\ssst{PC}}^{(2)})$, is conserved, i.e.,$$[\rho^{(1)}\,,\, V_{\ssst{PC}}]=\bm k\cdot\bm j_{\ssst{PC}}^{(2)}\,,$$ ($\rho_{\ssst{PC}}^{(2)}$ is higher order in $1/\mN$). Therefore, at least for the PNC part, we have every conservation condition met.
For the calculation of AM, we have to use both the DDH potential and current operators in coordinate space, these expressions could be found in Appendix \[sec:MECs in x\].
Deuteron Anapole Moment \[sec:d AM\]
====================================
Determination of the Deuteron Wave Function
-------------------------------------------
Due to the PNC $NN$ interaction, the deuteron wave function, mainly a $^{3}S_{1}$ state with some fraction of $^{3}D_{1}$ component, could have parity admixtures in $^{3}\tilde{P}_{1}$ and $^{1}\tilde{P}_{1}$ channels. The former channel, induced by the isovector part of DDH potential, is dominated by $\pi$-exchange, while the latter one, resulting from the isoscalar interaction, is purely through the heavy-meson exchange. Therefore, we express the full parity-admixed deuteron wave function as $$\begin{aligned}
\Psi(\mbfr) & = & \frac{1}{\sqrt{4\pi}r}\left[\left(u(r)+\frac{S_{12}(\hat{\bm r})}{\sqrt{8}}w(r)\right)\zeta_{00}\right.\nonumber \\
& & \left.-i\sqrt{\frac{3}{8}}\,(\vsig_{1}+\vsig_{2})\cdot\hat{\mbfr}\,\vto\,
\zeta_{10}+i\frac{\sqrt{3}}{2}\,(\vsig_{1}-\vsig_{2})\cdot\hat{\mbfr}\,\voo\,
\zeta_{00}\right]\,\chi_{1J_{z}}\,,\label{eq:d w.f.}\end{aligned}$$ where $S_{12}(\hat{\bm r})\equiv3\bm\sigma_{1}\cdot\hat{\bm r}\,\bm\sigma_{2}\cdot\hat{\bm r}-\bm\sigma_{1}\cdot\bm\sigma_{2}$ and $\chi$ and $\zeta$ represents spinor and isospinor respectively.
The Schrödinger equation in the center of mass frame is $$\begin{aligned}
H\,\Psi(\mbfr) & = & \left[-\frac{1}{\mN}\left(\frac{1}{r}\frac{\partial^{2}}{\partial\, r^{2}}\, r-\frac{l(l+1)}{r^{2}}\right)+V_{C}(r)+V_{T}(r)\, S_{12}(\hat{\bm r})+V_{\ssst{PNC}}(\mbfr)\right]\,\Psi(\mbfr)\nonumber \\
& = & E\,\Psi(\mbfr)\,,\end{aligned}$$ where $V_{C}(r)$ and $V_{T}(r)$ are the central and tensor parts of the strong potential, respectively, and $V_{\ssst{PNC}}(\mbfr)$ is the sum of the PNC potentials given above. Up to the linear order of PNC $NN$ interaction, radial wave functions of each channel satisfy the differential equations $$\begin{aligned}
&&
u''(r)+\mN\,\left(E-V_{C}(r)\right)u(r)=\sqrt{8}\mN V_{T}(r)w(r)\,,\\
&&
w''(r)-\frac{6}{r^{2}}w(r)-\mN\,(E-V_{C}(r)+2V_{T}(r))w(r)=\sqrt{8}\mN V_{T}(r)u(r)\,,\\
&&
\tilde{v}_{3p1}''(r)-\frac{2}{r^{2}}\vto+\mN(E-V_{C}(r)-2V_{T}(r))\vto=
\nonumber \\
&&
\frac{2}{\sqrt{3}}\left[\left(u(r)+\frac{1}{\sqrt{2}}w(r)\right)
\frac{\partial}{\partial r}\left(F_{\pi}^{1}(r)+\sqrt{2}F_{\rho}^{1}(r)-\sqrt{2}F_{\omega}^{1}(r)\right)
\right.
\nonumber \\
&&
\left.+2\sqrt{2}\left(F_{\rho}^{1}(r)-F_{\omega}^{1}(r)\right)\frac{\partial}
{\partial r}\left(u(r)+\frac{1}{\sqrt{2}}w(r)\right)-\frac{2\sqrt{2}}{r}
\left(F_{\rho}^{1}(r)-F_{\omega}^{1}(r)\right)\left(u(r)-\sqrt{2}w(r)
\right)\right]
\,, \label{eq:v3p1}\\
&&
\tilde{v}''_{1p1}-\frac{2}{r^{2}}\voo+\mN(E-V_{C}(r)+4V_{T}(r))\voo=
\nonumber \\
&&
\frac{2}{\sqrt{3}}\left[\left(u(r)-\sqrt{2}w(r)\right)\frac{\partial}
{\partial r}\left(3\chi_{\ssst{V}}F_{\rho}^{0}(r)-\chi_{\ssst{S}}F_{\omega}^{0}(r)\right)
\right.
\nonumber \\
&&
\left.-2\left(3F_{\rho}^{0}(r)-F_{\omega}^{0}(r)\right)\frac{\partial}{\partial r}\left(u(r)-\sqrt{2}w(r)\right)+\frac{2}{r}\left(3F_{\rho}^{0}(r)
-F_{\omega}^{0}(r)\right)\left(u(r)+2\sqrt{2}w(r)\right)\right]
\,,
\label{eq:v1p1}\end{aligned}$$ where $F_{\pi}^{1}(r)\equiv g_{\pi NN}h_{\pi}^{1}f_{\pi}(r)$, $F_{\rho}^{0}(r)\equiv g_{\rho NN}h_{\rho}^{0}f_{\rho}(r)$, $F_{\rho}^{1}(r)\equiv g_{\rho NN}h_{\rho}^{1}f_{\rho}(r)$, $F_{\omega}^{0}(r)\equiv g_{\omega NN}h_{\omega}^{0}f_{\omega}(r)$, and $F_{\omega}^{1}(r)\equiv g_{\omega NN}h_{\omega}^{1}f_{\omega}(r)$. In our numerical calculations, $\gpiNN=13.45$, $\grhoNN=2.79$, $\gomegaNN=8.37$, as well as DDH best values (in units of $10^{-7}$): $\hpiNN=4.6$, $h_{\rho}^{0}=-11.4$, $h_{\rho}^{1}=-0.2$, $h_{\rho}^{2}=-9.5$, $h_{\omega}^{0}=-1.9$, and $h_{\omega}^{1}=-1.1$, are assumed.
Anapole Moment : Expressions of Matrix Elements
-----------------------------------------------
The anapole operator we use takes the form$$\mbfa\equiv\frac{2\pi}{3}\int d\mbfx\;\mbfx\times(\mbfx\times\mbfj(\mbfx))\,,\label{eq:orig-am}$$ where $\mbfj(\mbfx)$ contains all the currents discussed in Sec. \[sec:MECs\]. Note that this form is equivalent to what have been recommended in Refs. [@HaHM89; @HLRM02], a result from implementing the extended Siegert’s theorem [@FrFa84].
With the deuteron wave function, Eq. (\[eq:d w.f.\]), we obtain the anapole moment for the spin term; $$\begin{aligned}
\mbfa_{spin} & = & -\frac{\pi}{\sqrt{6}\,\mN}\left[\mu_{\ssst{V}}\,\int dr\, r\,\left(u(r)-\sqrt{2}w(r)\right)\vto\right.\nonumber \\
& & \left.\hspace{2cm}-\sqrt{2}\,\mu_{\ssst{S}}\,\int dr\, r\,\left(u(r)+\frac{1}{\sqrt{2}}w(r)\right)\voo\right]\, e\,\mbfI\,,\end{aligned}$$ where $\bm I\equiv\frac{1}{2}\chi_{1\ssst{J_{z}}}^{\dagger}(\bm\sigma_{1}
+\bm\sigma_{2})\chi_{1\ssst{J_{z}}}$ is the intrinsic spin taken in the spinor basis, and we note that this is equivalent to the total angular momentum taken in the total angular momentum basis, i.e., $\bm I=\langle J=1,J_{z}|\bm J|J=1,J_{z}\rangle$ [^2].
The matrix element of the convection current is written as $$\begin{aligned}
\mbfj_{conv}(\mbfx) & = & \mbfj_{conv}^{+}(\mbfx)+\mbfj_{conv}^{-}(\mbfx)\,,\nonumber \\
\mbfj_{conv}^{+}(\mbfx) & \equiv & \frac{e}{4\mN}\int d\mbfr_{1}d\mbfr_{2}\frac{1}{\sqrt{4\pi}r}\chi_{1J_{z}}^{\dagger}\left(u(r)
+\frac{S_{12}(\hat{\mbfr})}{\sqrt{8}}w(r)-i\frac{\sqrt{3}}{2}(\vsig_{1}-\vsig_{2})
\cdot\hat{\mbfr}\tilde{v}_{1p1}(r)\right)\nonumber \\
& & \hspace{-0.4cm}\times(\mbfp_{1},\,\,\mbfp_{2})^{+}\left(u(r)
+\frac{S_{12}(\hat{\mbfr})}{\sqrt{8}}w(r)+i\frac{\sqrt{3}}{2}(\vsig_{1}
-\vsig_{2})\cdot\hat{\mbfr}\tilde{v}_{1p1}(r)\right)\frac{1}{\sqrt{4\pi}r}
\chi_{1J_{z}}\,,\\
\mbfj_{conv}^{-}(\mbfx) & \equiv & \frac{e}{4\mN}\int d\mbfr_{1}d\mbfr_{2}\frac{1}{\sqrt{4\pi}r}\chi_{1J_{z}}^{\dagger}\nonumber \\
& & \times\left[\left(u(r)+\frac{S_{12}(\hat{\mbfr})}{\sqrt{8}}w(r)\right)(\mbfp_{1},
\,\,\mbfp_{2})^{-}\left(-i\sqrt{\frac{3}{8}}\right)(\vsig_{1}+\vsig_{2})
\cdot\hat{\mbfr}\tilde{v}_{3p1}(r)\right.\nonumber \\
& & \left.+i\sqrt{\frac{3}{8}}(\vsig_{1}+\vsig_{2})\cdot\hat{\mbfr}\tilde{v}_{3p1}(r)
(\mbfp_{1},\,\,\mbfp_{2})^{-}\left(u(r)+\frac{S_{12}(\hat{\mbfr})}
{\sqrt{8}}w(r)\right)\right]\frac{1}{\sqrt{4\pi}r}\chi_{1J_{z}}
\,,\label{eq:conv-rel}\end{aligned}$$ where $$\begin{aligned}
(\mbfp_{1},\,\,\mbfp_{2})^{\pm}\equiv\left\{ \mbfp_{1},\,\,\delta^{3}(\mbfx-\mbfr_{1})\right\} \pm\left\{ \mbfp_{2},\,\,\delta^{3}(\mbfx-\mbfr_{2})\right\} \,.\end{aligned}$$ As been argued in [@HyDe03] $$\frac{2\pi}{3}\int d\mbfx\mbfx\times\left(\mbfx\times\mbfj_{conv}^{-}(\mbfx)\right)=\left\langle -i\frac{\pi e}{12\mN}\left[\mbfl^{2},\,\,\mbfr\right]\right\rangle \,,$$ where $\mbfl=\mbfr\times\mbfp$ [^3]. Indeed, it can be shown that the matrix element of the operator $(\mbfp_{1},\,\,\mbfp_{2})^{+}$ is proportional to $$-2i\mbfR+\mbfR\left(\mbfR\cdot\mbfP\right)+\mbfr\left(\mbfR\cdot\mbfp
+\frac{1}{4}\mbfr\cdot\mbfP\right)-\left(\mbfR^{2}+\frac{1}{4}\mbfr^{2}\right)
\mbfP-2\mbfR\cdot\mbfr\,\mbfp\,,$$ where $\mbfR=(\mbfr_{1}+\mbfr_{2})/2$ is the coordinate of the center of mass and $\mbfP=\mbfp_{1}+\mbfp_{2}$ is its conjugate momentum. In the center of mass frame where only the relative coordinate or momentum are relevant, the terms proportional to $\mbfR$ or $\mbfP$ are discarded. Consequently, the contribution from $(\mbfp_{1},\,\,\mbfp_{2})^{+}$ is zero and the magnitude from the convection current is determined by Eq. (\[eq:conv-rel\]). The anapole moment contributed by the convection term then reads $$\begin{aligned}
\mbfa_{conv} & = & \frac{1}{3}\frac{\pi}{\sqrt{6}\,\mN}\int dr\, r\,\left(u(r)-\sqrt{2}w(r)\right)\vto\, e\,\mbfI\,.\end{aligned}$$
Contributions from the PNC exchange currents are evaluated with the parity-even channels ($^{3}S_{1}$ and $^{3}D_{1}$) in the initial and final states. These channels are in spin-triplet, $S=1$, and isospin-singlet, $T=0$, state and for these states we have the spin and isospin selections rules $$\begin{aligned}
\langle S=1||\,(\vsig_{1}-\vsig_{2})\,||S=1\rangle & = & \langle S=1||\,(\vsig_{1}\times\vsig_{2})\,||S=1\rangle=0\,,\\
\langle T=0||\,(\vtau_{1}\times\vtau_{2})^{z}||T=0\rangle & = & 0\,,\end{aligned}$$ where the double bar $||$ means the reduced matrix element. It is easily seen that the matrix elements of $\mbfj_{pair+\ssst{KR}}^{\rho^{\pm}}$, $\mbfj_{mesonic}^{\rho^{\pm}}$, $\mbfj_{mesonic}^{\rho\pi}$, and the part proportional to $\vsig_{1}-\vsig_{2}$ in $\mbfj_{pair}^{\rho^{0}}$ and $\mbfj_{pair}^{\omega}$ vanish. Then we are left with $\pi$-pair, $\pi$-mesonic, and the parts proportional to $\vsig_{1}+\vsig_{2}$ in $\rho^{0}$-pair and $\omega$-pair terms. Pair and mesonic pion contributions were already calculated in [@HyDe03] which are given as $$\begin{aligned}
\mbfa_{pair}^{\pi} & = & -\frac{\sqrt{2}\pi\gpiNN}{9\mN}\int dr\, r^{2}\, f_{\pi}(r)\,\left(u(r)+\frac{1}{\sqrt{2}}w(r)\right)\left(u(r)-\sqrt{2}w(r)
\right)e\,\mbfI\,\hpiNN\,, \label{eq:pipair}\\
\mbfa_{mesonic}^{\pi} & = & \frac{\sqrt{2}\pi\gpiNN}{3\mN m_{\pi}}\int dr\, r\, f_{\pi}(r)\left(u(r)+\frac{1}{\sqrt{2}}w(r)\right)\nonumber \\
& & \times\left[u(r)\left(1-\frac{1}{3}m_{\pi}r\right)-\frac{1}{\sqrt{2}}w(r)\left(1
+\frac{1}{3}m_{\pi}r\right)\right]e\,\mbfI\,\hpiNN\,,
\label{eq:pimesonic}\end{aligned}$$ and $\rho^{0}$- and $\omega$-meson contributions read $$\begin{aligned}
\mbfa_{pair}^{\rho^{0}} & = & -\frac{2\pi\, g_{\rho\ssst{NN}}}{9\,\mN}\int dr\, r^{2}\, f_{\rho}(r)\,\left(u(r)+\frac{1}{\sqrt{2}}w(r)\right)\left(u(r)
-\sqrt{2}w(r)\right)\, e\,\mbfI\, h_{\rho}^{1}\,, \label{eq:rhopair}\\
\mbfa_{pair}^{\omega} & = & \frac{2\pi\, g_{\omega\ssst{NN}}}{9\,\mN}\int dr\, r^{2}\, f_{\omega}(r)\,\left(u(r)+\frac{1}{\sqrt{2}}w(r)\right)\left(u(r)
-\sqrt{2}w(r)\right)\, e\,\mbfI\, h_{\omega}^{1}\,.
\label{eq:omegapair}\end{aligned}$$
Numerical Results for the Argonne $v_{18}$ $NN$ Interaction Model
-----------------------------------------------------------------
With the A$v_{18}$ model, we obtain the numerical results (in units of fm$^2$) $$\begin{aligned}
\mbfa_{spin} & = & -0.547\, h_{\pi}^{1}\, e\, \bm{I}
+ (-3.7\, h_{\rho}^{1}+ 10.8\, h_{\omega}^{1}+7.3\, h_{\rho}^{0}
+ 3.7\, h_{\omega}^{0}) \times 10^{-3} \, e\,\mbfI\,\,,\\
\mbfa_{conv} & = & 0.039\, h_{\pi}^{1}\, e\, \mbfI
+ (2.7\, h_{\rho}^{1}-7.6\, h_{\omega}^{1})\times 10^{-4}\, e\,\mbfI\,,\\
\mbfa_{ex}^{\pi} & = & \mbfa_{pair}^{\pi}+\mbfa_{mesonic}^{\pi}=(-0.027+0.028)\, h_{\pi}^{1}\, e\,\mbfI\,,\\
\mbfa_{ex}^{\rho} & = & \mbfa_{pair}^{\rho^{0}}=-0.7\times10^{-4}\, h_{\rho}^{1}\, e\,\mbfI\,,\\
\mbfa_{ex}^{\omega} & = & \mbfa_{pair}^{\omega}=1.8\times10^{-4}\, h_{\omega}^{1}\, e\,\mbfI\,.\end{aligned}$$
The contribution from the nucleonic AM, $\bm a_{N}$, should also be included in the full result. Since the deuteron is an isosinglet state, only the isoscalar component of the nucleonic AM contributes, i.e.,$$\bm a_{\ssst{N}}=\bra{d}\,\sum_{i=1}^{2}\,(a_{\ssst{S}}^{(1)}+a_{\ssst{V}}^{(1)}
\,\tau_{i}^{z})\,\bm\sigma_{i}\,\ket{d}=2\, a_{\ssst{S}}^{(1)}\,(1-\frac{3}{2}P_{\ssst{D}})\,\bm I\,,$$ where $a_{\ssst{S.V}}^{(1)}$ denote the isoscalar and isovector nucleonic AM, and $P_{\ssst{D}}$ is the probability of deuteron $D$-state. Several theoretical estimates for the nucleonic AM exist [@HaHM89; @MuHo90; @MuHo91; @KaSa93; @MavK00; @Zh..00], and here we use the result of Ref. [@Zh..00] because it is the most recent one which includes the full DDH interaction at the nucleon level.
For the pion sector, Ref. [@Zh..00] gave
$$a_{\ssst{S}}^{\pi\,(1)}=
-\frac{g_{\ssst{A}}\, h^1_\pi}{12\sqrt{2}f_{\pi}m_{\pi}}
\frac{\Lambda_{\chi}^{2}}{\mN^{2}}\,(1-\frac{6}{\pi}\frac{m_{\pi}}{\mN}
\ln\frac{\Lambda_{\chi}}{m_{\pi}})\,,$$ where $\Lambda_{\chi}$ is the chiral symmetry breaking scale (the authors chose it to be $4\pi f_{\pi}$) [^4]. When setting $\Lambda_{\chi}=\mN$, the leading term is equivalent to what has been used in Refs. [@SaSp98; @SaSp01; @HyDe03] for the deuteron AM calculations, while the full result is the same as been used in Ref. [@KhKo00]. By including the heavy mesons, the numerical result is [@Zh..00] $$a_{\ssst{S}}^{(1)}=-0.274\, h_{\pi}^{1}-0.419\, h_{\rho}^{1}-0.129\, h_{\omega}^{0}\,,
\label{eq:aN}$$ and this leads to the nucleonic contribution$$\bm a_{\ssst{N}}=( -0.250\, h_{\pi}^{1}-0.383\, h_{\rho}^{1}-0.118\, h_{\omega}^{0})\, e\, \mbfI \,.$$
Although we do not have the exact PC EC corresponding to A$v_{18}$ potential, we try to approximate this by the EC coming out from one-pion exchange diagrams. Corresponding currents in the configuration space are given in the Appendix A. Pair- and mesonic-term of the PC EC contributions read $$\begin{aligned}
\mbfa^{\ssst{PC}}_{pair} &=&
\frac{1}{3 \sqrt{6}}\left(\frac{g_{\pi \ssst{NN}}}{2 \mN} \right)^2
\int dr\, {\rm e}^{-m_\pi r}\, (1 + m_\pi r)\, \tilde{v}_{3p1}(r)
\left( u(r) - \sqrt{2} w(r)\right) \, e\, \mbfI, \\
\mbfa^{\ssst{PC}}_{mesonic} &=&
- \frac{1}{3 \sqrt{6}}\left(\frac{g_{\pi\ssst{NN}}}{2 \mN} \right)^2
\int dr\, {\rm e}^{-m_\pi r}\, \tilde{v}_{3p1}(r)
\left[ 3 u(r) - m_\pi\, r
\left(u(r) + \frac{1}{\sqrt{2}} w(r)\right) \right] e\, \mbfI.\end{aligned}$$ Numerically we obtain $$\begin{aligned}
\bm a_{ex}^{\ssst{PC}} &=&
\bm{a}^{\ssst{PC}}_{pair}
+ \bm{a}^{\ssst{PC}}_{mesonic} \nonumber \\
&=& - (7.8\ h^1_\pi + 0.4\ h^1_\rho - 1.1\ h^1_\omega)
\times 10^{-4}\, e\, \bm{I}.\end{aligned}$$ In the next section, we will discuss if this is a reasonable approximation and to what extent is the breaking of current conservation due to the non-exact PC EC.
Finally, the full deuteron AM could be expressed as: $$\begin{aligned}
\bm a_{d} & = & \bm a_{spin}+\bm a_{conv}+\bm a_{ex}^{\ssst{PNC}}+\bm a_{\ssst{N}}+\bm a_{ex}^{\ssst{PC}} \nonumber \\
&=& ( -0.756\, h^1_\pi - 0.387\, h^1_\rho + 0.010\, h^1_\omega
+ 0.007\, h^0_\rho - 0.114\, h^0_\omega )\, e\, \mbfI.\end{aligned}$$
Discussions \[sec:dis\]
=======================
![The left panel shows the behavior of $r^2\, (u(r)+w(r)/\sqrt{2})(u(r) - \sqrt{2} w(r))$. The central panel compares the Yukawa function of the pion (solid line) and rho-meson (dotted line). The right panel is the behavior of the integrand for the $\pi-$ and $\rho-$ pair terms in Eq. (\[eq:pipair\]) and Eq. (\[eq:rhopair\]), respectively \[fig:suppression\] ](r2uw.eps "fig:") ![The left panel shows the behavior of $r^2\, (u(r)+w(r)/\sqrt{2})(u(r) - \sqrt{2} w(r))$. The central panel compares the Yukawa function of the pion (solid line) and rho-meson (dotted line). The right panel is the behavior of the integrand for the $\pi-$ and $\rho-$ pair terms in Eq. (\[eq:pipair\]) and Eq. (\[eq:rhopair\]), respectively \[fig:suppression\] ](fcomp.eps "fig:") ![The left panel shows the behavior of $r^2\, (u(r)+w(r)/\sqrt{2})(u(r) - \sqrt{2} w(r))$. The central panel compares the Yukawa function of the pion (solid line) and rho-meson (dotted line). The right panel is the behavior of the integrand for the $\pi-$ and $\rho-$ pair terms in Eq. (\[eq:pipair\]) and Eq. (\[eq:rhopair\]), respectively \[fig:suppression\] ](tcomp.eps "fig:")
Presenting the results, we separated the contributions of the pion and the heavy mesons so that their relative order of magnitude can be compared easily. In the result of the nuclear part, i.e. the contributions from deuteron wave function or exchange currents, which include spin, convection, PNC EC and PC EC, the magnitudes of the heavy meson contributions are commonly suppressed by an order of 2 or 3. This suppression can be understood most easily from Eqs. (\[eq:pipair\], \[eq:rhopair\], \[eq:omegapair\]), PNC pair terms of $\pi$, $\rho$ and $\omega$ mesons, respectively. Aside from the weak coupling constants, they differ only by the Yukawa functions $f_\pi(r)$ or $f_{\rho,\, \omega}(r)$ in the integrand. At small $r$, the deuteron wave function is proportional to $r$ (for the simplicity of argument, we neglect $D-$state wave function) and increases very fast up to $r \sim 2$ fm at which the maximum is reached. Afterwards, the wave function decreases very slowly, converging to zero. The factor $r^2$ multiplied by $u^2(r)$ gives rise to the suppression at short range. The left panel of Fig. \[fig:suppression\] shows this behavior. One can expect that this $r^4$ behavior at short distances makes the contribution from $r \leq 1$ fm quite small. On the contrary, the Yukawa function, which behaves like $1/r$ at small $r$ and also depends strongly on the mass of the meson, is short-range peaked. A comparison of $f_\pi(r)$ and $f_\rho(r)$ is given in the central panel of Fig. \[fig:suppression\]. The quantity $f_\rho(r)$ is non-negligible for about $r \leq 0.5$ fm but the remaining part of the integrand, approximately $r^2\, u^2(r)$ is small in this region. Consequently the heavy meson will be suppressed substantially compared to $\pi$. The right panel of Fig. \[fig:suppression\] shows the behavior of the total integrands in Eqs. (\[eq:pipair\], \[eq:rhopair\]). If we approximate the integrand as $r^3\, \exp(- m_{\ssst{X}}r)$, the maximum of the integrand is reached at $r \sim 3/m_{\ssst{X}}$. This approximation is good for $\rho$ but bad for $\pi$, however, this can provide a semi-quantitative explanation for the suppression of the heavy mesons. At the maximum, the integrand has the value proportional to $1/m^3_{\ssst{X}}$. If comparison is made for $\pi$ and $\rho$, the maximum value of $\rho$ is smaller than $\pi$ by a factor of 1/200, which can account for the suppression of order 2 of the heavy mesons.
To a lesser extent, a similar argument can be applied to the qualitative understanding of the heavy-meson suppression of spin and convection terms. The right hand side of Eqs. (\[eq:v3p1\], \[eq:v1p1\]) which are the sources of parity admixed $P-$states also contains the Yukawa functions. When the equations are solved, i.e. integrated with respect to $r$, one can expect some amount of suppression for the heavy mesons as was discussed in the analysis of pair EC’s, but without the factor $r^2$. In Tab. \[tab:sum1\], we summarize the ratio of the heavy meson contribution to the pion with the weak coupling constants given by DDH best values. The magnitudes of the heavy meson terms are suppressed by an order of two compared to the pion terms in common. As far as the nuclear part is concerned, $\pi$ contribution is by far dominant. If one can disentangle the nuclear part contribution from the total deuteron anapole moment, this can provide information to extract the magnitude of $h^1_\pi$ with high accuracy. As a side remark, it is noticed that the combination of the isovector couplings, $h^1_{\pi}$, $h^1_{\rho}$ and $h^1_{\omega}$, entering the spin and convection contributions is close to the one determining low energy PNC nucleon-nucleon scattering, roughly $3\;h^1_{\pi} +0.02\;h^1_{\rho}-0.06\;h^1_{\omega}$ [@Desp98]. This lets us think that the Danilov’s approach [@Dani65], whose application for the deuteron anapole moment was proposed later on by Savage [@Sa01], should provide a reasonable estimate for this contribution.
term spin conv. PNC EC PC EC nucleonic total
--------------------- ------ ------- --------- --------- ----------- --------
$(\rho+\omega)/\pi$ 4.0 0.6 $- 2.0$ $- 3.1$ $- 26.2$ $-5.9$
: Comparison of the heavy meson with the pion contribution to the deuteron anapole moment, term by term. The ratios are given in %. \[tab:sum1\]
Contrary to the nuclear part, the nucleonic one has sizable contribution from the heavy mesons. The relative ratio of the heavy meson is about 26% of the pion based on the nucleonic anapole estimate of Ref. [@Zh..00]. However, we should also note that these authors considered more non-DDH type couplings such as non-Yukawa type $\pi NN$ couplings and the inclusion of hyperons. So far, no detailed knowledge about these exotic couplings exist, and the theoretical uncertainty could be huge. The study of these terms and their implication for two-body nuclear contributions will be an interesting topic for further exploration.
Another issue that should be addressed is the gauge invariance of the results. We showed in Sect. \[sec:MECs\] that the PNC ECs we constructed satisfy the gauge invariance with the DDH potential. However, with the phenomenological strong interaction models like the one adopted in this work, gauge invariance may not be satisfied if the ECs are not consistent with the phenomenological potentials. For A$v_{18}$, the potential contains 18 types of operators but we took into account the dominant ECs only, which naturally leads to inconsistency, or breakdown of gauge invariance. Even though the inconsistency may only change the result slightly, the investigation of the extent to which gauge invariance is broken could help to get an estimate of the error.
Most phenomenological potentials are very complicated and the analytic analysis of gauge invariance is a formidable or practically impossible work. However as suggested in [@HyDe03], one can estimate the amount of gauge invariance breaking by comparing the anapole moment obtained from Eq. (\[eq:orig-am\]) and $$\bm{a} = - \pi \int d\bm{x}\, x^2 \bm{j}(\bm{x})\, ,
\label{eq:alt-am}$$ which can be obtained from Eq. (\[eq:orig-am\]) with the assumption $$\nabla \cdot \bm{j}(\bm{x}) = 0\, .
\label{eq:crnt-conserv}$$ In [@HyDe03], we observed that the inclusion of the PC ECs of $\pi$ removes almost all the inconsistency. In this work, we follow the same procedure to investigate the breakdown of gauge invariance.
The spin current can be shown easily to satisfy Eq. (\[eq:crnt-conserv\]). The convection, pair and mesonic anapole moment of $\pi$, $\rho-$ and $\omega-$mesons with the definition given by Eq. (\[eq:alt-am\]) can be calculated straightforwardly to give $$\begin{aligned}
\bm{a}^{\ssst{CC}}_{conv} &=& \frac{\pi}{2 \sqrt{6}\, \mN}
\int dr\, r^2\, \left[
u(r) \left(\tilde{v}'_{3p1}(r) + 2 \frac{\tilde{v}_{3p1}(r)}{r} \right)
+ \frac{w(r)}{\sqrt{2}}
\left(\tilde{v}'_{3p1}(r) - \frac{\tilde{v}_{3p1}(r)}{r} \right) \right]
\, e\, \mbfI,\\
\bm{a}^{\pi,\ssst{CC}}_{pair} &=&
- \frac{\pi\, g_{\pi\, \ssst{NN}}}{2 \sqrt{2}\, \mN}
\int dr\, r^2\, f_\pi(r) \left( u^2(r) - \frac{w^2(r)}{2} \right)\,
h^1_\pi\, e\, \mbfI,
\\
\mbfa^{\pi,\ssst{CC}}_{mesonic} &=&
\frac{g_{\pi\ssst{NN}}}{24\sqrt{2}\, \mN\, m_\pi}
\int dr\, {\rm e}^{-m_\pi r} \nonumber \\
& & \times \left[
\left( u^2(r) - \frac{w^2(r)}{2}\right) (m_\pi r + 4) -
\frac{1}{3}\left(u(r) + \frac{w(r)}{\sqrt{2}} \right)^2
m_\pi r\, (m_\pi r + 3) \right] h^1_\pi\, e\, \mbfI, \\
\bm{a}^{\rho^0,\ssst{CC}}_{pair} &=&
- \frac{\pi\, g_{\rho\, \ssst{NN}}}{2 \, \mN} \int dr\, r^2\,
f_\rho(r) \left( u^2(r) - \frac{w^2(r)}{2} \right)\, h^1_\rho\, e\, \mbfI,
\\
\bm{a}^{\omega,\ssst{CC}}_{pair} &=&
\frac{\pi\, g_{\omega\, \ssst{NN}}}{2 \, \mN} \int dr\, r^2\,
f_\omega(r) \left( u^2(r) - \frac{w^2(r)}{2} \right)\, h^1_\omega\, e\, \mbfI,
\\
\mbfa^{\ssst{PC},\ssst{CC}}_{pair} &=& \frac{1}{\sqrt{6}}
\left(\frac{g_{\pi\ssst{NN}}}{2\, \mN} \right)^2
\int dr\, {\rm e}^{-m_\pi r} (1+ m_\pi r)\,
\tilde{v}_{3p1}(r)\left(u(r) - \frac{1}{\sqrt{8}}\, w(r)\right)
\, e\, \mbfI\, , \\
\mbfa^{\ssst{PC},\ssst{CC}}_{mesonic} &=& - \frac{1}{12 \sqrt{6}}
\left(\frac{g_{\pi\ssst{NN}}}{2\, \mN} \right)^2
\int dr\, {\rm e}^{-m_\pi r}\, \tilde{v}_{3p1}(r) \nonumber \\
& & \times \left[ u(r) (18 + 2 m_\pi r - m^2_\pi r^2) -
\frac{w(r)}{\sqrt{2}}\, m_\pi r\, (4 + m_\pi r) \right]\, e\, \mbfI,\end{aligned}$$ where the superscript $CC$ denotes the quantity calculated with Eq. (\[eq:alt-am\]). Numerical results are summarized in Tab. \[tab:sum2\].
------------------- ------------ ------------ -------------- ------------ ------------ --------------
$h^1_\pi$ $h^1_\rho$ $h^1_\omega$ $h^1_\pi$ $h^1_\rho$ $h^1_\omega$
conv. 0.03925 0.00027 $-0.00076$ 0.05348 0.00024 $-0.00066$
PNC $\pi$ pair $-0.02668$ $-0.07895$
PNC $\pi$ mesonic 0.02830 0.04706
PNC $\rho$ pair $-0.00007$ $-0.00023$
PNC $\omega$ pair 0.00018 0.00061
PC $\pi$ pair 0.01400 0.00013 $-0.00038$ 0.06325 0.00062 $-0.00177 $
PC $\pi$ mesonic $-0.01478$ $-0.00017$ 0.00049 $-0.04366$ $-0.00047$ 0.00133
Total$-$PC 0.04087 0.00020 $-0.00058$ 0.02159 0.00001 $-0.00005$
Total 0.04010 0.00016 $-0.00047$ 0.04117 0.00017 $-0.00049 $
------------------- ------------ ------------ -------------- ------------ ------------ --------------
: Coefficients of the weak coupling constants for given terms with different definitions of the anapole operator, Eq. (\[eq:orig-am\]) and Eq. (\[eq:alt-am\]). \[tab:sum2\]
Judging from the “Total” column in Table II, two definitions of the anapole operator only differ in results by 3%, 6%, and 4% for the $\pi$-, $\rho$-, and $\omega$-exchange respectively; this means our result satisfy current conservation very well. This is quite surprising because only the PC one-pion exchange current—not fully consistent with the adopted strong potential, A$v_{18}$—is included in our calculation. The reason is partly because the AM is a $r^2$-weighted moment, plus the deuteron wave function peaks around 2 fm with a long tail; therefore the long-range physics, which is dominated by the one-pion exchange including the case of A$v_{18}$, becomes much more important. This explains why our conclusion differs from the one found in Ref. [@Scetal03] where large effects due to undetermined ECs were found. Their observation was concerning a different operator (E1 transition), which has a range relatively shorter than the quantity considered in this work.
Furthermore, one can observe that for the contributions from exchange currents, Eq. (\[eq:orig-am\]) always give smaller values than Eq. (\[eq:alt-am\]). This implies the calculation using Eq. (\[eq:orig-am\]) suffers less from the incomplete knowledge or uncertainty of exchange currents which causes the breaking of current conservation. For example, if the PC pion ECs are left out in our calculation, by comparing the “Total” and “Total-PC” columns in Table II, the error is 2%, 25%, and 23% for $\pi$, $\rho$, and $\omega$ component respectively when Eq. (\[eq:orig-am\]) is used; while the error is -46%, -94%, and -90% for Eq. (\[eq:alt-am\]). Perhaps, the reason for the small contribution of ECs in the former case is to be looked for in the proportionality of the dominant ECs to the position vector $\vec{x}$, which readily gives zero when inserted into the corresponding anapole operator, Eq. (\[eq:orig-am\]). Therefore the merit of Eq. (\[eq:orig-am\]) is justified, at least for the case of deuteron.
In conclusion, we constructed the PNC ECs due to one $\pi-$, $\rho-$ and $\omega-$exchange, and showed that they satisfy the current conservation; and hence are fully consistent with the adopted DDH PNC potential. An application was made to the calculation of the deuteron anapole moment. We observed that, for the nuclear part, the contribution of heavy mesons is suppressed by an order of two compared to the pion, a result consistent with the similar work by Blunden [@Blun02]. Consistency with A$v_{18}$ was also checked. We found that the approximation of using only the PC one-pion exchange current is pretty good—the breaking of current conservation only amounts to a few percent, and this should be fixed by using the PC exchange current fully consistent with A$v_{18}$. Therefore, the contribution from the nuclear part to the deuteron anapole moment can be determined with an error less than 5%, while the major uncertainty should come from the nucleonic anapole moment instead.
Acknowledgements {#acknowledgements .unnumbered}
================
Work of C.H.H. is partially supported by KOSEF (Grant No. 2000-2-11100-004-4).
PNC $NN$ Potential and Exchange Currents in Coordinate Space \[sec:MECs in x\]
==============================================================================
The DDH potential in coordinate space could easily be obtained by applying the following transformation rules to Eqs. (\[eq:vpnc-pi-p\]-\[eq:vpnc-omega-p\])$$\begin{aligned}
\bm u_{\ssst{X}} & \rightarrow & \bm u_{\ssst{X}}(\bm r)=[\bm p\,,\, f_{\ssst{X}}(r)]\,,\\
\bm v_{\ssst{X}} & \rightarrow & \bm v_{\ssst{X}}(\bm r)=\{\bm p\,,\, f_{\ssst{X}}(r)\}\,,\end{aligned}$$ where $\bm r\equiv\bm r_{1}-\bm r_{2}$; $r=|\bm r|$; $\bm p\equiv(\bm p{}_{1}-\bm p{}_{2})/2=-i\bm\nabla_{r}$; and the Yukawa functions $f_{\ssst{X}}(r)$ are defined as$$f_{\ssst{X}}(r)=\frac{{\rm e}^{-m_{\ssst{X}}r}}{4\pi r}\,.$$
For the PNC MECs, we list all the leading-order, $O(1/\mN)$, 3-currents, which are relevant for the AM calculation:
$$\begin{aligned}
\mbfj_{pair}^{\pi}(\mbfx;\,\mbfr_{1},\,\mbfr_{2}) & = & -\frac{e\;\gpiNN\; h_{\pi}^{1}}{2\,\sqrt{2}\;\mN}\;(\vtau_{1}\cdot\vtau_{2}-\tau_{1}^{z}\;
\tau_{2}^{z})\; f_{\pi}(r)\sum_{i=1}^{2}\,\delta^{(3)}(\mbfx-\mbfr_{i})\,\vsig_{i}\,,\\
\mbfj_{mesonic}^{\pi}(\mbfx;\,\mbfr_{1},\,\mbfr_{2}) & = & -\frac{e\;\gpiNN\; h_{\pi}^{1}}{2\,\sqrt{2}\;\mN}\;(\vtau_{1}\cdot\vtau_{2}-\tau_{1}^{z}\;
\tau_{2}^{z})\nonumber \\
& & \times(\bm\nabla{}_{1}-\bm\nabla_{2})\;\left[(\vsig_{1}\cdot\bm\nabla{}_{1}
-\vsig_{2}\cdot\bm\nabla{}_{2})\,,\; f_{\pi}(r_{x1})\, f_{\pi}(r_{x2})\right]\,,\\
\mbfj_{pair+\ssst{KR}}^{\rho^{\pm}}(\mbfx;\,\mbfr_{1},\,\mbfr_{2}) & = & -\frac{e\;\grhoNN}{2\,\mN}\, f_{\rho}(r)\left(h_{\rho}^{0}-\frac{1}{2\sqrt{6}}\; h_{\rho}^{2}\right)\nonumber \\
& & \hspace{-2cm}\times\Bigg((\vtau_{1}\cdot\vtau_{2}-\tau_{1}^{z}\tau_{2}^{z})
\;(\vsig_{1}-\vsig_{2})\;\Big(\delta^{(3)}(\mbfx-\mbfr_{1})-\delta^{(3)}(\mbfx
-\mbfr_{2})\Big)\nonumber \\
& & \hspace{1cm}+(1+\chi_{\ssst{V}})\,(\vtau_{1}\times\vtau_{2})^{z}\,
(\vsig_{1}\times\vsig_{2})\,\sum_{i}\delta^{(3)}(\mbfx-\mbfr_{i})\Bigg)\,,\\
\mbfj_{pair}^{\rho^{0}}(\mbfx;\,\mbfr_{1},\,\mbfr_{2}) & = & -\frac{e\;\grhoNN}{2\,\mN}\, f_{\rho}(r)\;\;\tau_{1}^{z}\;\tau_{2}^{z}\nonumber \\
& & \hspace{-2cm}\times\Bigg(\Big(h_{\rho}^{0}+\frac{1}{2}\; h_{\rho}^{1}\;(\tau_{1}^{z}+\tau_{2}^{z})+\frac{1}{\sqrt{6}}\; h_{\rho}^{2}\Big)\;(\vsig_{1}-\vsig_{2})+\frac{1}{2}\; h_{\rho}^{1}\;(\tau_{1}^{z}-\tau_{2}^{z})\;(\vsig_{1}+\vsig_{2})\Bigg)\nonumber \\
& & \times\;\Big((1+\tau_{1}^{z})\;\delta^{(3)}(\mbfx-\mbfr_{1})-(1+\tau_{2}^{z})
\;\delta^{(3)}(\mbfx-\mbfr_{2})\Big)\,,\\
\mbfj_{mesonic}^{\rho^{\pm}}(\mbfx;\,\mbfr_{1},\,\mbfr_{2}) & = & -\frac{e\;\grhoNN}{2\,\mN}\left(h_{\rho}^{0}-\frac{1}{2\sqrt{6}}\; h_{\rho}^{2}\right)\,(\vtau_{1}\times\vtau_{2})^{z}\nonumber \\
& & \hspace{-2cm}\times\Bigg\{(\bm\nabla_{1}-\bm\nabla{}_{2})\;\bigg(-i\,(\vsig_{1}
-\vsig_{2})\cdot\{(\bm\nabla_{1}-\bm\nabla{}_{2})\,,\, f_{\rho}(r_{x1})\, f_{\rho}(r_{x2})\}\nonumber \\
& & \hspace{0.8cm}+(1+\chi_{\ssst{V}})\,(\vsig_{1}\times\vsig_{2})\cdot
\left[(\bm\nabla_{1}-\bm\nabla_{2})\,,\, f_{\rho}(r_{x1})\, f_{\rho}(r_{x2})\right]\bigg)\nonumber \\
& & \hspace{-1.5cm}+2\,\nabla_{x}^{a}\;\bigg(i\,\{\nabla_{1}^{a}\,\bm\sigma_{2}
-\nabla_{2}^{a}\,\bm\sigma_{1}+\sigma_{1}^{a}\,\bm\nabla_{2}-\sigma_{2}^{a}
\,\bm\nabla_{1}\,,\, f_{\rho}(r_{x1})\, f_{\rho}(r_{x2})\}\nonumber \\
& & \hspace{0cm}-(1+\chi_{\ssst{V}})\;[(\sigma_{1}\times\nabla_{1})^{a}\,
\bm\sigma_{2}-(\sigma_{2}\times\nabla_{2})^{a}\,\bm\sigma_{1}\nonumber \\
& & \hspace{2.0cm}+\sigma_{1}^{a}\,\bm\sigma_{2}\times\bm\nabla_{2}-\sigma_{2}^{a}
\,\bm\sigma_{1}\times\bm\nabla_{1}\,,\, f_{\rho}(r_{x1})\, f_{\rho}(r_{x2})]\bigg)\nonumber \\
& & \hspace{-1.5cm}+4\, i\,\mN\,(\bm\sigma_{1}+\bm\sigma_{2})\;[H\,,\, f_{\rho}(r_{x1})\, f_{\rho}(r_{x2})]\Bigg\}\\
\mbfj_{pair}^{\omega}(\mbfx;\,\mbfr_{1},\,\mbfr_{2}) & = & -\frac{e\;\gomegaNN}{2\,\mN}\, f_{\omega}(r)\nonumber \\
& & \hspace{-2cm}\times\Bigg(\Big(h_{\omega}^{0}+\frac{1}{2}\; h_{\omega}^{1}\;(\tau_{1}^{z}+\tau_{2}^{z})\Big)(\vsig_{1}-\vsig_{2})
+\frac{1}{2}\; h_{\omega}^{1}\;(\tau_{1}^{z}-\tau_{2}^{z})\;(\vsig_{1}+\vsig_{2})
\Bigg)\nonumber \\
& & \times\Big((1+\tau_{1}^{z})\;\delta^{(3)}(\mbfx-\mbfr_{1})-(1+\tau_{2}^{z})
\;\delta^{(3)}(\mbfx-\mbfr_{2})\Big)\,,\\
\mbfj_{mesonic}^{\rho\pi}(\mbfx;\,\mbfr_{1},\,\mbfr_{2}) & = & -\frac{e\;\grhoNN g_{\rho\pi\gamma\;}h_{\pi}^{1}}{\sqrt{2}\, m_{\rho}}\,(\bm\tau_{1}\times\bm\tau_{2})^{z}\nonumber \\
& & \hspace{-2cm}\times(\bm\nabla_{1}\times\bm\nabla_{2})\,\Big(f_{\rho}(r_{x1})\, f_{\pi}(r_{x2})+f_{\pi}(r_{x1})\, f_{\rho}(r_{x2})\Big), \end{aligned}$$
where $r_{xi}\equiv\left|\bm x-\bm r_{i}\right|$; $\bm\nabla_{i}$ and $\bm\nabla_{x}$ act on the source point $\bm r_{i}$ and field point $\bm x$ respectively; and the superscript $a$ is the index to be summed from 1 to 3. Note that we separate charged and neutral $\rho$ mesons according to their isospin structure; and the last term of Eq. (A5) should be combined with a charge density corresponding to Eq. (24) in order to insure the current conservation.
PC one-pion ECs of the pair and mesonic terms that contribute to the AM read $$\begin{aligned}
\mbfj^{\ssst{PC}}_{pair}(\mbfx ;\, \mbfr_1,\, \mbfr_2) &=&
\left(\frac{g_{\pi \ssst{NN}}}{2 \mN} \right)^2 (\bm{\tau}_1 \times \bm{\tau}_2)^z
\nonumber \\ & & \times
\Big[ \bm{\sigma}_2\ \delta^{(3)}(\mbfx - \mbfr_2)\ \bm{\sigma}_1 \cdot \nabla
+ \bm{\sigma}_1 \ \delta^{(3)}(\mbfx - \mbfr_1)\ \bm{\sigma}_2 \cdot \nabla
\Big] f_\pi(r), \\
\mbfj^{\ssst{PC}}_{mesonic}(\mbfx ;\, \mbfr_1,\, \mbfr_2) &=&
- \left(\frac{g_{\pi \ssst{NN}}}{2 \mN} \right)^2 (\bm{\tau}_1 \times \bm{\tau}_2)^z
(\bm{\sigma}_1\cdot\nabla_{1})\ (\bm{\sigma}_2\cdot\nabla_{2})\
(\nabla_{1} - \nabla_{2})\ f_\pi(r_{x1})\ f_\pi(r_{x2}).\end{aligned}$$
[^1]: There are a lot of discussions in literature addressing this issue. We refer readers to more extensive reviews such as Refs. [@Chem79; @Town87; @Math89; @Risk89] (and references therein) for details.
[^2]: Since the anapole moment is a vector moment, it should be proportional to the total angular momentum which is the only intrinsic vector of a system.
[^3]: We stress that the equality holds for the matrix elements and the operators themselves do not satisfy the equality.
[^4]: Note that a factor of $4\pi/\mN^{2}$ is needed for converting the dimensionless AM defined in Lorentz-Heaviside units into our definition.
|
---
abstract: 'Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a $C^*$-bialgebra, the noncommutative counterpart of a locally compact semigroup. On locally compact quantum groups we obtain a bijective correspondence between such convolution semigroups and a class of $C_0$-semigroups of maps which we characterise. On $C^*$-bialgebras of discrete type we show that all weakly continuous convolution semigroups of states are automatically norm-continuous. As an application we deduce a known characterisation of continuous conditionally positive-definite Hermitian functions on a compact group.'
address: 'Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, U.K.'
author:
- 'J. Martin Lindsay and Adam G. Skalski'
title: '**Convolution semigroups of states**'
---
Introduction {#introduction .unnumbered}
============
Convolution semigroups of probability measures on locally compact groups, and semigroups, play an important role in both probability theory and functional analysis. They underpin all approaches to stochastic processes with independent identically distributed increments, that is Lévy processes ([@Bertoin]). The analysis of convolution semigroups is also an independent area of research, well documented in the monograph [@Heyer]. Given the recent development of a satisfactory theory of locally compact quantum groups ([@KV]) it is natural to investigate convolution semigroups of states on such objects—states being the noncommutative counterpart of probability measures.
We recently introduced the notion of quantum Lévy process on a locally compact quantum semigroup, in other words a $C^*$-bialgebra ([@qscc3]). As in the classical theory of Lévy processes, and the earlier noncommutative theory of Lévy processes on purely algebraic quantum semigroups ([@Schurmann]), the concept of a convolution semigroup of states plays a fundamental role here too. In our papers the strongest results are obtained under the assumption that the Markov semigroup of the process is norm-continuous. This class of processes may be viewed as a quantum analogue of the class of compound Poisson processes.
The present paper forms the first step in analysing general quantum Lévy processes. We initiate a theory of weakly continuous convolution semigroups of functionals on locally compact quantum semigroups. In particular we prove that on $C^*$-bialgebras enjoying a ‘residual vanishing at infinity’ property (which is satisfied by all locally compact quantum groups) every weakly continuous convolution semigroup of functionals induces a $C_0$-semigroup on the algebra. From a probabilistic viewpoint this amounts to a Feller property ([@Jacob])). We give several characterisations of the resulting class of $C_0$-semigroups. We also prove that on $C^*$-bialgebras of discrete type weak continuity for a convolution semigroup of states implies norm-continuity. Here slice maps for non-normal functionals, in the sense of Tomiyama ([@Tomiyama]), play a crucial role. Our results show that all quantum Lévy processes on a discrete quantum semigroup are automatically of the type thoroughly studied in [@qscc3]. As an application we prove a result of Guichardet, characterising the continuous conditionally positive-definite Hermitian functions on a compact group ([@Guichardet]).
Preliminaries {#prelim}
=============
Given $C^*$-algebras ${\mathsf{A}}$ and ${\mathsf{C}}$, their spatial/minimal (respectively algebraic) tensor product will be denoted ${\mathsf{A}}{\otimes}{\mathsf{C}}$ (resp. ${\mathsf{A}}\odot{\mathsf{C}}$) and the multiplier algebra of ${\mathsf{A}}$ will be denoted by $M({\mathsf{A}})$. The multiplier algebra is equipped with the strict topology, with respect to which the unit ball of ${\mathsf{A}}$ is dense in that of $M({\mathsf{A}})$. A linear map $T$ from ${\mathsf{A}}$ (or $M({\mathsf{A}})$) to $M({\mathsf{C}})$ is *strict* if it is bounded and strictly continuous on bounded subsets of ${\mathsf{A}}$ (respectively $M({\mathsf{A}})$). Standard examples of strict maps include $^*$-homomorphisms which are nondegenerate (that is $T({\mathsf{A}}){\mathsf{C}}$ is total in ${\mathsf{C}}$), continuous linear functionals on ${\mathsf{A}}$ and, more generally, slice maps of the type $\mu{\otimes}{{\textup{id}}}_{{\mathsf{C}}}: {\mathsf{A}}{\otimes}{\mathsf{C}}\to {\mathsf{C}}\subset M({\mathsf{C}})$, where $\mu \in {\mathsf{A}}^*$. The collection of strict linear functionals on $M({\mathsf{A}})$ forms a closed subspace of $M({\mathsf{A}})^*$ which we denote by $M({\mathsf{A}})^*_\beta$. An important property of strict maps $T: {\mathsf{A}}\to M({\mathsf{C}})$ is that they possess a unique strict extension ${\widetilde}{T}: M({\mathsf{A}}) \to M({\mathsf{C}})$; the extension procedure does not change the norm (or cb-norm if $T$ is completely bounded) and preserves (complete) positivity. In particular, strict extension defines an isometric isomorphism ${\mathsf{A}}^* \to M({\mathsf{A}})^*_\beta$ mapping states to states. A strict map $T$ is called *preunital* if its strict extension is unital: ${\widetilde}{T}(1)=1$. Thus all states are preunital, as are nondegenerate \*-homomorphisms. The existence and uniqueness of extensions allows *composition* of strict maps $T:{\mathsf{A}}_1 \to M({\mathsf{A}}_2)$ and $S:{\mathsf{A}}_2 \to M({\mathsf{A}}_3)$, with the resulting map ${\mathsf{A}}_1 \to M({\mathsf{A}}_3)$ being strict. Finally, for strict completely positive maps $T_i:{\mathsf{A}}_i\to{\mathsf{C}}_i$ ($i=1,2$), their tensor product map $T_1{\otimes}T_2: {\mathsf{A}}_1{\otimes}{\mathsf{A}}_2 \to M({\mathsf{C}}_1){\otimes}M({\mathsf{C}}_2)\subset M({\mathsf{C}}_1{\otimes}{\mathsf{C}}_2)$, is strict. For more information on multiplier algebras, strict maps, their tensor products and extensions, we refer to [@Lance], [@Johan] and [@qscc3].
By a *semigroup on* ${\mathsf{A}}$, or $M({\mathsf{A}})$, we mean a one-parameter family of bounded operators $(P_t)_{t\geq 0}$ satisfying $P_{s+t}=P_s P_t$ and $P_0 = {\textup{id}_{{\mathsf{A}}}}$ (no continuity in $t$ is assumed a priori). We call it a *strict/completely bounded/completely positive* semigroup if each map $P_t$ has that property. By elementary semigroup theory ([@Davies], Proposition 1.23), $(P_t)_{t\geq 0}$ is a $C_0$-semigroup, in other words it is strongly continuous in $t$, if it is weakly continuous, that is $t\mapsto \mu(P_t(a))$ is continuous for each $a\in{\mathsf{A}}$ and $\mu\in{\mathsf{A}}^*$, at $t=0$.
We shall need two general results, in Sections \[on Cstar Bialgebras\] and \[disc type\] respectively. The first concerns the preservation of pointwise convergence under strict extension.
\[contstat\] Let $(\omega_\lambda)_{\lambda \in \Lambda}$ be a net of states on a $C^*$-algebra ${\mathsf{A}}$ converging pointwise to a state $\omega$ on ${\mathsf{A}}$. Then $({\widetilde}{\omega}_\lambda)_{\lambda \in \Lambda}$ converges pointwise to ${\widetilde}{\omega}$.
Let $m \in M({\mathsf{A}})$ and let $(e_i)_{i \in {\mathcal{I}}}$ be an approximate unit for ${\mathsf{A}}$. Then, for each $i\in{\mathcal{I}}$, $${\widetilde}{\omega}_\lambda ((1- e_i)^2) =
1 + \omega_\lambda (e_i^2-2 e_i)
\stackrel{\lambda }{\longrightarrow} 1 + \omega (e_i^2-2 e_i) =
{\widetilde}{\omega}((1-e_i)^2)$$ and so, by the Cauchy-Schwarz inequality for states, $$\begin{aligned}
|({\widetilde}{\omega}_\lambda - {\widetilde}{\omega})(m)|
&\leq
|{\widetilde}{\omega}_\lambda (m(1-e_i))| + |{\widetilde}{\omega} (m(1-e_i))| +
|(\omega_\lambda - \omega)(me_i)|
\\&\leq
\|m\|({\widetilde}{\omega}_\lambda ((1-e_i)^2))^{1/2} +
\|m\| ({\widetilde}{\omega} ((1-e_i)^2))^{1/2}+
|(\omega_\lambda - \omega)(me_i)|
\\&
\stackrel{\lambda }{\longrightarrow}
2 \|m\| {\widetilde}{\omega}\big((1-e_i)^2\big)^{1/2}.\end{aligned}$$ Since $\big((1-e_i)^2\big)_{i\in{\mathcal{I}}}$ converges strictly to $0$, the result follows.
Using Kadison-Schwarz, instead of Cauchy-Schwarz, the above proof remains valid for strict completely positive preunital maps into the multiplier algebra of another $C^*$-algebra $(T_\lambda: {\mathsf{A}}\to M({\mathsf{C}}))_{\lambda\in\Lambda}$ converging pointwise to another such map $T$.
The following observation is due to Tomiyama ([@Tomiyama]); we include its elementary proof for the convenience of the reader. The notation ${\overline{{\otimes}}}$ is used for ultraweak tensor products of von Neumann algebras and ultraweakly continuous slice maps ([@KaR]).
\[new\] Let ${\mathsf{M}}= {\mathsf{M}}_1{\overline{{\otimes}}}{\mathsf{M}}_2$, for von Neumann algebras ${\mathsf{M}}_1$ and ${\mathsf{M}}_2$, and let $\nu\in ({\mathsf{M}}_1)^*$. Then there is a unique map $\nu{\otimes_{\mathrm{M}}}{{\textup{id}}}_2: {\mathsf{M}}\to{\mathsf{M}}_2$ satisfying $$\label{U}
\varphi\circ (\nu{\otimes_{\mathrm{M}}}{{\textup{id}}}_2) =
\nu\circ ({{\textup{id}}}_1\, {\overline{{\otimes}}}\, \varphi),
\quad \varphi\in ({\mathsf{M}}_2)_*,$$ where ${{\textup{id}}}_i$ denotes the identity map on ${\mathsf{M}}_i$ [(]{}$i=1,2$[)]{}. Moreover, $\nu{\otimes_{\mathrm{M}}}{{\textup{id}}}_2$ is a bounded operator of norm $\|\nu\|$.
Let $x\in{\mathsf{M}}$. Then, for $\varphi\in ({\mathsf{M}}_2)_*$, $$\big| \nu \big(({{\textup{id}}}_1\, {\overline{{\otimes}}}\, \varphi )(x)\big)\big| \leq
\|\nu\| \big\|{{\textup{id}}}_1\, {\overline{{\otimes}}}\, \varphi \big\| \| x \| =
\|\nu\| \|\varphi \| \| x \|$$ so the prescription $\varphi \mapsto \nu (({{\textup{id}}}_1\, {\overline{{\otimes}}}\, \varphi )(x))$ defines a bounded linear functional on $({\mathsf{M}}_2)_*$ of norm at most $\| \nu\| \| x \|$. Therefore, invoking the canonical identification of $(({\mathsf{M}}_2)_*)^*$ with ${\mathsf{M}}_2$, there is a bounded operator $\nu\, {\otimes_{\mathrm{M}}}\, {{\textup{id}}}_2 : {\mathsf{M}}\to {\mathsf{M}}_2$, of norm at most $\|\nu\|$, satisfying . Since $\nu\, {\otimes_{\mathrm{M}}}\, {{\textup{id}}}_2$ clearly extends the map $\nu\, {\otimes}\, {{\textup{id}}}_2 : {\mathsf{M}}_1 {\otimes}{\mathsf{M}}_2 \to {\mathsf{M}}_2$, which has norm $\|\nu \|$, it follows that $\|\nu\, {\otimes_{\mathrm{M}}}\, {{\textup{id}}}_2\| = \|\nu\|$.
By uniqueness, and the fact that $({\mathsf{M}}_2)_* = \{ \phi|_{{\mathsf{M}}_2}: \phi \in B({\mathsf{h}})_* \}$ where ${\mathsf{h}}$ is the Hilbert space on which ${\mathsf{M}}_2$ acts, it follows that $\nu {\otimes_{\mathrm{M}}}{{\textup{id}}}_2 = \nu {\otimes_{\mathrm{M}}}{{\textup{id}}}_{B({\mathsf{h}}_2)}|_{{\mathsf{M}}}$. In particular, by Remark 1.4 of [@SteveMartin], $\nu {\otimes_{\mathrm{M}}}{{\textup{id}}}_2$ is completely bounded with cb-norm equal to $\|\nu\|_{{{\text{{\textup}{cb}}}}} = \|\nu\|$; moreover the map $\nu {\otimes_{\mathrm{M}}}{{\textup{id}}}_2$ is completely positive if the functional $\nu$ is positive. In fact, the above result still holds if $\nu$ is replaced by a completely bounded map into another von Neumann algebra ${\mathsf{M}}_3$, with now reading $({{\textup{id}}}_3\, {\overline{{\otimes}}}\, \varphi) (\nu{\otimes_{\mathrm{M}}}{{\textup{id}}}_2) =
\nu\circ ({{\textup{id}}}_1\, {\overline{{\otimes}}}\, \varphi)$ ([@SteveMartin]).
If $\nu$ is ultraweakly continuous then $\nu {\otimes_{\mathrm{M}}}{{\textup{id}}}_2$ equals $\nu\, {\overline{{\otimes}}}\, {{\textup{id}}}_2$. Naturally there are also bounded operators ${{\textup{id}}}_1 {{\ }_{\mathrm{M}}\!\!\otimes}\mu : {\mathsf{M}}\to{\mathsf{M}}_1$, for each $\mu\in ({\mathsf{M}}_2)^*$, and the validity of the commutation relation $$\label{comm}
\nu_1 \circ ({{\textup{id}}}_1 {{\ }_{\mathrm{M}}\!\!\otimes}\nu_2) = \nu_2 \circ (\nu_1 {\otimes_{\mathrm{M}}}{{\textup{id}}}_2)$$ for functionals $\nu_i \in ({\mathsf{M}}_i)^*$, $i=1,2$, naturally arises. Under the assumptions that ${\mathsf{M}}_2$ is infinite dimensional and the Hilbert space on which ${\mathsf{M}}_1$ acts is separable, Tomiyama showed that holds for all $\nu_2 \in ({\mathsf{M}}_2)^*$ if and only if $\nu_1$ is ultraweakly continuous, in other words $\nu_1\in({\mathsf{M}}_1)_*$ ([@Tomiyama], Theorem 5.1). There is a corresponding result for completely bounded maps ([@Neufang], Theorem 5.4).
Convolution semigroups on $C^*$-bialgebras {#on Cstar Bialgebras}
==========================================
We first recall the definition.
A $C^*$-*bialgebra* is a $C^*$-algebra ${\mathsf{B}}$ equipped with a nondegenerate $*$-homomorphism ${\Delta}:{\mathsf{B}}\to M({\mathsf{B}}{\otimes}{\mathsf{B}})$, called the *coproduct*, and a character ${\epsilon}: {\mathsf{B}}\to {\mathbb C}$, called the *counit*, satisfying the coassociativity and counital properties: $$({\textup{id}_{{\mathsf{B}}}}{\otimes}{\Delta}) {\Delta}= ({\Delta}{\otimes}{\textup{id}_{{\mathsf{B}}}}) {\Delta}\ \text{ and } \
({\textup{id}_{{\mathsf{B}}}}{\otimes}{\epsilon}) {\Delta}= {\textup{id}_{{\mathsf{B}}}}= ({\epsilon}{\otimes}{\textup{id}_{{\mathsf{B}}}}) {\Delta}.$$ A useful neutral expression for the first two maps is ${\Delta}\!^{(2)}$. Thus ${\Delta}\!^{(2)}: {\mathsf{B}}\to M({\mathsf{B}}{\otimes}{\mathsf{B}}{\otimes}{\mathsf{B}})$. $C^*$-bialgebras are also called *locally compact quantum semigroups*; unital $C^*$-bialgebras are called *compact quantum semigroups*, or $C^*$-*bialgebras of compact type*.
*Fix now, and for the rest of the paper*, a (counital) $C^*$-bialgebra ${\mathsf{B}}$. For ${\lambda}, \mu \in {\mathsf{B}}^*$, their *convolution* is defined as the following composition of strict maps: $${\lambda}\star \mu = ({\lambda}{\otimes}\mu) {\Delta}.$$ Convolution may also be viewed as a product on ${M({\mathsf{B}})_{\beta}^*}$.
\[banisom\] Both $({\mathsf{B}}^*, \star)$ and $({M({\mathsf{B}})_{\beta}^*}, \star)$ are unital Banach algebras, with respective identities ${\epsilon}$ and ${\widetilde}{{\epsilon}}$, and strict extension defines a unital isometric algebra isomorphism from the former to the latter.
Straightforward.
A *convolution semigroup of functionals* on ${\mathsf{B}}$ is a family of functionals $(\lambda_t)_{t\geq 0}$ in ${\mathsf{B}}^*$ satisfying $${\lambda}_0 ={\epsilon}\ \text{ and } \ {\lambda}_{s+t}={\lambda}_s\star{\lambda}_t,\quad s,t \geq 0;$$ It is *weakly continuous* if $$\lim_{t \to 0^+} {\lambda}_t(a) = {\epsilon}(a), \quad a \in {\mathsf{B}}$$ and *norm-continuous* if $$\lim_{t\to 0^+} \|{\lambda}_t - {\epsilon}\| = 0.$$
Restricting to semigroups of states, Propositions \[contstat\] and \[banisom\] imply the following.
\[2.2\*\] Strict extension defines a one-to-one correspondence between weakly continuous convolution semigroups of states on ${\mathsf{B}}$ and weakly continuous convolution semigroups of strict states on $M({\mathsf{B}})$.
The norm-continuous case is summarised in the next result. In brief, norm-continuous convolution semigroups have bounded generators, from which the semigroup may be recovered by exponentiation. This also operates at the multiplier algebra level.
\[char\] Let $({\lambda}_t)_{t\geq 0}$ be a norm-continuous convolution semigroup of functionals on ${\mathsf{B}}$. Then there is a unique functional $\gamma \in {\mathsf{B}}^*$ such that $$\lim_{t \to 0^+} \big\|\frac{1}{t}({\lambda}_t - {\epsilon})-\gamma \big\|=0.$$ Moreover, $${\lambda}_t = \sum_{n=0}^{\infty}\frac{1}{n!}(t\gamma)^{\star n}, \;\;\;\; {\widetilde}{{\lambda}}_t = \sum_{n=0}^{\infty}
\frac{1}{n!}(t{\widetilde}{\gamma})^{\star n}, \quad t\geq 0,$$ and $$\lim_{t \to 0^+} \big\| \frac{1}{t}({\widetilde}{{\lambda}}_t - {\widetilde}{{\epsilon}})- {\widetilde}{\gamma} \big\|=0.$$
The functional $\gamma$ is called the *generating functional* of $({\lambda}_t)_{t \geq 0}$.
In view of Proposition \[banisom\], this follows from elementary properties of norm-continuous semigroups in a unital Banach algebra.
The class of norm-continuous convolution semigroups of states is analogous to the class of classical convolution semigroups of *compound Poisson type*, with ‘initial measure’ $\gamma + {\epsilon}$. More justification for this terminology is given in [@Uwe] and [@qscc2]. The class of functionals $\gamma\in{\mathsf{B}}^*$ which generate a convolution semigroup of states on ${\mathsf{B}}$ is characterised as follows ([@qscc3]): $\gamma$ is \*-preserving, *conditionally positive* in the sense that $\gamma (a)\geq 0$ for $a \in {\mathsf{B}}_+\cap {{\textup{Ker}}}\, {\epsilon}$ and satisfies ${\widetilde}{\gamma}(1)=0$. This is a form of Schönberg correspondence; see [@Schurmann] for its algebraic (i.e. non-analytic) counterpart. For further classical motivations we refer to [@Heyer].
To each functional $\mu \in {\mathsf{B}}^*$ there are associated two completely bounded strict maps ${\mathsf{B}}\to M({\mathsf{B}})$: $$\label{L and R}
L_{\mu} = (\mu {\otimes}{\textup{id}_{{\mathsf{B}}}}) {\Delta}\ \text{ and } \
R_{\mu} = ({\textup{id}_{{\mathsf{B}}}}{\otimes}\mu) {\Delta},$$ and their strict extensions $\widetilde{L}_\mu, \widetilde{R}_\mu: M({\mathsf{B}})\to M({\mathsf{B}})$. Note that the original functional may be recovered from either of these maps: ${\epsilon}\circ L_\mu = \mu = {\epsilon}\circ R_\mu$; from which it follows that $\|L_\mu\|_{{\text{{\textup}{cb}}}}=
\|L_\mu\| = \|\mu\| = \|R_\mu\| = \|R_\mu\|_{{\text{{\textup}{cb}}}}$. If the map $L_\mu$ is positive then the functional $\mu$ is positive, so $L_\mu$ is actually completely positive; also $\widetilde{L}_\mu$ is unital if and only if $\widetilde{\mu}$ is. Thus $\widetilde{L}_\mu$ is completely positive and unital if and only if $\mu$ is a state. The same goes for the maps $R_\mu$ and $\widetilde{R}_\mu$. The strict extensions of the $L$-map $\mu \mapsto L_\mu$ and $R$-map $\mu \mapsto R_\mu$ define completely isometric unital algebra morphisms between the Banach algebras $({\mathsf{B}}^*,\star)$ and $CB(M({\mathsf{B}}))$, sharing the same left inverse: ${\widetilde}{T} \mapsto {\epsilon}\circ T$, where $T := {\widetilde}{T}|_{\mathsf{B}}$. For any convolution semigroup of functionals $({\lambda}_t)_{t \geq 0}$ on ${\mathsf{B}}$, $({\widetilde}{P}_t:={\widetilde}{R}_{{\lambda}_t})_{t\geq 0}$ therefore defines a strict semigroup of completely bounded maps on $M({\mathsf{B}})$, which we call the *associated semigroup on the multiplier algebra*. It determines the original convolution semigroup via the identity: $$\label{recover}
{\widetilde}{{\lambda}}_t = {\widetilde}{{\epsilon}} \circ {\widetilde}{P}_t, \quad t \geq 0.$$ Moreover, the semigroup $({\widetilde}{P}_t)_{t\geq 0}$ is completely positive and unital if and only if $(\lambda_t)_{t\geq 0}$ is a convolution semigroup of states. We stress the point that, in the noncompact case (${\mathsf{B}}$ nonunital), $P_t:=R_{\lambda_t}$ need not leave ${\mathsf{B}}$ invariant and so *there may be no semigroup on ${\mathsf{B}}$ itself*. In the next section we shall see that, under a natural condition on ${\mathsf{B}}$, this obstruction is removed.
The semigroups on ${M({\mathsf{B}})}$ which are associated with convolution semigroups of functionals on ${\mathsf{B}}$ are characterised in several simple ways. This is the content of the next result. The convention on composing strict maps permits us to dispense with almost all tildes.
\[assoc\] Let $({\widetilde}{P}_t)_{t \geq 0}$ be a strict semigroup on $M({\mathsf{B}})$ and set $P_t := {\widetilde}{P}_t|_{{\mathsf{B}}}: {\mathsf{B}}\to M({\mathsf{B}})$ [(]{}$t\geq 0$[)]{}. Then the following conditions are equivalent:
$({\widetilde}{P}_t)_{t \geq 0}$ is associated to a convolution semigroup of functionals on ${\mathsf{B}}$;
$(P_t)_{t \geq 0}$ enjoys the commutativity property: $$L_{\mu} P_t = P_t L_{\mu}, \quad \mu \in {\mathsf{B}}^*, t \geq 0;$$
for each $t\geq 0$, $P_t$ is completely bounded, ${\textup{id}_{{\mathsf{B}}}}{\otimes}P_t$ is strict as a map from ${\mathsf{B}}{\otimes}{\mathsf{B}}$ to $M({\mathsf{B}}{\otimes}{\mathsf{B}})$, and the following strong invariance condition holds: $$\label{2.3}
{\Delta}P_t = ({\textup{id}_{{\mathsf{B}}}}{\otimes}P_t) {\Delta};$$
$(P_t)_{t \geq 0}$ satisfies the weak invariance condition: $$\label{invar}
P_t = \big({\textup{id}_{{\mathsf{B}}}}{\otimes}({\epsilon}\circ P_t)\big) {\Delta},
\quad t\geq 0.$$
The implication (iii)$\Rightarrow$(ii) is immediate.
(ii)$\Rightarrow$(iv): Assume that (ii) holds and let $t\geq 0$. Then, since ${\epsilon}\circ L_\mu = \mu$, $$\mu \circ \big( {\textup{id}_{{\mathsf{B}}}}{\otimes}({\epsilon}\circ P_t) \big) {\Delta}=
({\epsilon}\circ P_t) L_\mu = {\epsilon}\circ (L_\mu P_t) =
\mu \circ P_t, \quad \mu \in {\mathsf{B}}^*,$$ so (iv) holds.
(iv)$\Rightarrow$(i): Assume that (iv) holds and define $\lambda_t = {\epsilon}\circ P_t \in {\mathsf{B}}^*$ ($t\geq 0$). Then, for $s,t\geq 0$, $$\begin{aligned}
(\lambda_s {\otimes}\lambda_t){\Delta}&=
(({\epsilon}\circ P_s) {\otimes}({\epsilon}\circ P_t)) {\Delta}\\&=
\big({\epsilon}\circ P_s\big)\big({\textup{id}_{{\mathsf{B}}}}{\otimes}({\epsilon}\circ P_t)\big){\Delta}=
\big({\epsilon}\circ P_s\big)P_t = {\epsilon}\circ P_{s+t}
=
\lambda_{s+t},\end{aligned}$$ and $$R_{\lambda_t} =
({\textup{id}_{{\mathsf{B}}}}{\otimes}\lambda_t){\Delta}=
\big({\textup{id}_{{\mathsf{B}}}}{\otimes}({\epsilon}\circ P_t)\big){\Delta}= P_t.$$ Thus $(\lambda_t)_{t\geq 0}$ is a convolution semigroup of functionals on ${\mathsf{B}}$ and $({\widetilde}{P}_t)_{t\geq 0}$ is its associated semigroup on $M({\mathsf{B}})$.
(i)$\Rightarrow$(iii): Assume that $({\widetilde}{P}_t)_{t\geq 0}$ is the semigroup on $M({\mathsf{B}})$ associated with a convolution semigroup of functionals $(\lambda_t)_{t\geq 0}$ on ${\mathsf{B}}$, and let $t\geq 0$. Then $P_t$ is completely bounded, ${\textup{id}_{{\mathsf{B}}}}{\otimes}P_t$ equals the composition $({\textup{id}_{{\mathsf{B}}}}{\otimes}{\textup{id}_{{\mathsf{B}}}}{\otimes}{\lambda}_t) ({\textup{id}_{{\mathsf{B}}}}{\otimes}{\Delta})$ and so is strict, and holds by coassociativity: $${\Delta}P_t =
{\Delta}({\textup{id}_{{\mathsf{B}}}}{\otimes}{\lambda}_t) {\Delta}=
({\textup{id}_{{\mathsf{B}}}}{\otimes}{\textup{id}_{{\mathsf{B}}}}{\otimes}{\lambda}_t) {\Delta}\!^{(2)} =
({\textup{id}_{{\mathsf{B}}}}{\otimes}P_t) {\Delta}.$$
The above proof yields the following useful characterisations of the range of the $R$-map: $$\begin{aligned}
{\textup{Ran}}R &= \{ T \in B({\mathsf{B}}; M({\mathsf{B}})): T \text{ is strict and } L_\mu T = T L_\mu \text{ for all } \mu \in {\mathsf{B}}^* \}
\\ &=
\{ T \in CB({\mathsf{B}}; M({\mathsf{B}})):
({\textup{id}_{{\mathsf{B}}}}{\otimes}T) \text{ is strict and } {\Delta}T = ({\textup{id}_{{\mathsf{B}}}}{\otimes}T){\Delta}\}
\\ &=
\{ T \in B({\mathsf{B}}; M({\mathsf{B}})): T = ({\textup{id}_{{\mathsf{B}}}}{\otimes}({\epsilon}\circ T)){\Delta}\}.\end{aligned}$$ If ${\mathsf{B}}$ is a non-counital $C^*$-bialgebra then we still have $$\begin{aligned}
{\textup{Ran}}R &\subset \{ T \in CB({\mathsf{B}}; M({\mathsf{B}})): ({\textup{id}_{{\mathsf{B}}}}{\otimes}T) \text{ is strict and } L_\mu T = T L_\mu \text{ for all } \mu \in {\mathsf{B}}^* \}
\\ &=
\{ T \in CB({\mathsf{B}}; M({\mathsf{B}})):
({\textup{id}_{{\mathsf{B}}}}{\otimes}T) \text{ is strict and } {\Delta}T = ({\textup{id}_{{\mathsf{B}}}}{\otimes}T){\Delta}\}.\end{aligned}$$ The equality and inclusion above follow, in turn, from the facts that the set $\{ (\mu {\otimes}{\textup{id}_{{\mathsf{B}}}})\, {\widetilde}{\ } \, : \mu \in
{\mathsf{B}}^* \}$ separates $M({\mathsf{B}}{\otimes}{\mathsf{B}})$, and that $(\mu {\otimes}{\textup{id}_{{\mathsf{B}}}}{\otimes}{\lambda}){\Delta}\!^{(2)}$ is a common expression for $L_\mu R_{\lambda}$ and $R_{\lambda}L_\mu$ (${\lambda}, \mu \in {\mathsf{B}}^*$).
So far we have not considered the question of continuity, in $t$, for convolution semigroups and their associated semigroups. It is easily seen that one is norm-continuous if and only the other is, in which case the generating functional and the generator of the associated semigroup are related by ${{Z }_{\kern-1pt\raisebox{-1.25ex}{\mbox{\,$\tilde{\ }$}}}} = {\widetilde}{R}_\gamma$ and ${\widetilde}{\gamma} = {\widetilde}{{\epsilon}} \circ {{Z }_{\kern-1pt\raisebox{-1.25ex}{\mbox{\,$\tilde{\ }$}}}}$. The corresponding statement at the level of weak/strong continuity need not be true. For example the translation semigroup on $C_{{{\text{{\textup}{b}}}}}({\mathbb R})$ is the semigroup associated with the convolution semigroup of Dirac measures $(\delta_t)_{t\geq 0}$ on ${\mathbb R}$. Whereas the latter is weakly continuous, the former is not. However the translation semigroup leaves $C_0({\mathbb R})$ invariant and restricts to a strongly continuous semigroup there. This is a *Feller property* ([@Jacob]) which, in our framework, corresponds to the semigroup $({\widetilde}{P}_t)_{t\geq 0}$ leaving the $C^*$-algebra ${\mathsf{B}}$ invariant and restricting to a $C_0$-semigroup on ${\mathsf{B}}$. In the next section we shall see that this holds for all weakly continuous semigroups on a wide class of locally compact quantum semigroups which includes all locally compact (quantum) groups. We warn the reader that there is a variety of definitions of ‘Feller’ and ‘strong Feller’—in both the classical and noncommutative literature (e.g. [@Sau]).
Convolution semigroups on $C^*$-bialgebras satisfying\
the residual vanishing at infinity condition {#rvai}
======================================================
*Throughout this section we assume* that ${\mathsf{B}}$ is a $C^*$-bialgebra which enjoys the following ‘residual vanishing at infinity’ property: $$({\mathsf{B}}{\otimes}1) {\Delta}({\mathsf{B}}) \subset {\mathsf{B}}{\otimes}{\mathsf{B}}\ \text{ and } \
(1 {\otimes}{\mathsf{B}}) {\Delta}({\mathsf{B}}) \subset {\mathsf{B}}{\otimes}{\mathsf{B}}.$$ The $C^*$-bialgebras arising from locally compact quantum groups in the sense of Kustermans and Vaes ([@KV]) satisfy this property, as do (trivially) all unital $C^*$-bialgebras. In the classical case of a commutative quantum semigroup (see Section \[section: cases\]), ${\mathsf{B}}= C_0({H})$ for a locally compact Hausdorff semigroup ${H}$, ${\mathsf{B}}{\otimes}{\mathsf{B}}$ and $M({\mathsf{B}}{\otimes}{\mathsf{B}})$ are identified with $C_0(H \times H)$ and $C_{{\text{{\textup}{b}}}}(H \times H)$ respectively, and the coproduct is given by ${\Delta}(F)(h_1,h_2)=F(h_1 h_2)$, for $F\in{\mathsf{B}}$ and $h_1, h_2\in H$. To see how the condition applies then, let $F,F'\in
C_0({H})$. If $|F(h)|, |F'(h)| < \epsilon$ for $h\in{H}\setminus K$ then $|F(h_1)F'(h_1h_2)| < \epsilon^2$ for $(h_1,h_2)\in{H}\times{H}\setminus \phi^{-1}(K\times K)$ where $\phi$ is the continuous map $(h_1,h_2)\mapsto(h_1,h_1h_2)$. Thus if $\phi$ is proper then $(F{\otimes}1){\Delta}F' \in C_0({H}\times{H}) = {\mathsf{B}}\otimes
{\mathsf{B}}$ and a similar argument applies to $(1{\otimes}F){\Delta}F'$. Lance credits Iain Raeburn for the suggestive terminology (in [@Lance]).
The maps $R_\mu$ and $L_{\mu}$ defined in now act on ${\mathsf{B}}$, as we show next.
Let $\mu\in{\mathsf{B}}^*$. Then $$R_\mu({\mathsf{B}})\subset {\mathsf{B}}\text{ and }
L_{\mu}({\mathsf{B}}) \subset {\mathsf{B}}.$$
This follows from the fact, commonly used in topological quantum group theory, that $\mu \in {\mathsf{B}}^*$ may be factorised as $\lambda c$, for some $\lambda\in{\mathsf{B}}^*$ and $c\in{\mathsf{B}}$ ([@FactLem]); $$\begin{aligned}
L_{\mu} (a) &= (\mu {\otimes}{\textup{id}_{{\mathsf{B}}}})\, {\widetilde}{\ }\, ({\Delta}(a))
\\ &=
(\lambda {\otimes}{\textup{id}_{{\mathsf{B}}}})\, {\widetilde}{\ }\, ((c {\otimes}1) {\Delta}(a))
\in (\lambda {\otimes}{\textup{id}_{{\mathsf{B}}}})({\mathsf{B}}{\otimes}{\mathsf{B}})\subset {\mathsf{B}},
\quad a\in{\mathsf{B}},\end{aligned}$$ and similarly for $R_\mu$.
The significance of this is that each convolution semigroup of functionals $(\lambda_t)_{t\geq 0}$ on ${\mathsf{B}}$ now has an *associated semigroup on the $C^*$-algebra* ${\mathsf{B}}$ itself: $(P_t := R_{\lambda_t})_{t\geq 0}$. Reconstruction of the convolution semigroup from its associated semigroup now reads: $$\label{simplifies}
{\lambda}_t = {\epsilon}\circ P_t, \quad t \geq 0,$$ and Theorem \[assoc\] now has the following version.
\[rvaiassoc\] Let $(P_t)_{t \geq 0}$ be a strict semigroup on ${\mathsf{B}}$. Then the following conditions are equivalent:
$(P_t)_{t \geq 0}$ is associated to a convolution semigroup of functionals on ${\mathsf{B}}$[;]{}
$(P_t)_{t \geq 0}$ enjoys the commutativity property $$L_{\mu} P_t = P_t L_{\mu}, \quad \mu \in {\mathsf{B}}^*, t \geq 0;$$
for each $t\geq 0$, $P_t$ is completely bounded, ${\textup{id}_{{\mathsf{B}}}}{\otimes}P_t$ is strict and the following strong invariance condition holds: $${\Delta}P_t = ({\textup{id}_{{\mathsf{B}}}}{\otimes}P_t) {\Delta};$$
$(P_t)_{t \geq 0}$ satisfies the weak invariance condition: $$P_t = ({\textup{id}_{{\mathsf{B}}}}{\otimes}({\epsilon}\circ P_t)) {\Delta}, \quad t\geq 0.$$
Moreover we now have a satisfactory correspondence between continuity properties of the respective semigroups, showing that the semigroup on $M({\mathsf{B}})$ associated with any weakly continuous convolution semigroup of functionals on ${\mathsf{B}}$ is necessarily Feller.
\[posgen\] Let $(P_t)_{t \geq 0}$ be the semigroup on ${\mathsf{B}}$ associated with a convolution semigroup of functionals $({\lambda}_t)_{t \geq 0}$ on ${\mathsf{B}}$. Then the following are equivalent:
$({\lambda}_t)_{t \geq 0}$ is weakly continuous;
$(P_t)_{t \geq 0}$ is a $C_0$-semigroup.
This follows from the identities $$({\lambda}_t - {\epsilon}) = {\epsilon}\circ (P_t - {{\textup{id}}}_{\mathsf{B}}) \text{ and }
\mu \circ (P_t - {{\textup{id}}}_{\mathsf{B}}) = ({\lambda}_t - {\epsilon}) \circ L_\mu,$$ for $t\geq 0$ and $\mu \in {\mathsf{B}}^*$.
Under the weak continuity assumption, Proposition \[rvaiassoc\] can be formulated at the level of generators.
Let $(P_t)_{t \geq 0}$ be a $C_0$-semigroup on ${\mathsf{B}}$ with generator ${Z}$. Then the following are equivalent:
$(P_t)_{t \geq 0}$ is the semigroup associated with a weakly continuous convolution semigroup of functionals;
${Z}$ enjoys the commutativity property: $$L_\mu {Z}\subset {Z}L_\mu, \quad \mu \in {\mathsf{B}}^*,$$ that is, $L_\mu({{\textup{Dom}}}{Z}) \subset {{\textup{Dom}}}{Z}$ and $L_{\mu} {Z}(a) = {Z}L_{\mu}(a)$ for all $a\in{{\textup{Dom}}}{Z}$.
This follows from Theorem \[posgen\], the equivalence (i)$\Leftrightarrow$(ii) of Proposition \[rvaiassoc\], and Theorem 1.15 of [@Davies].
For a weakly continuous convolution semigroup of functionals $({\lambda}_t)_{t \geq 0}$ on ${\mathsf{B}}$, the functional $\gamma: {{\textup{Dom}}}\, \gamma \subset {\mathsf{B}}\to {\mathbb C}$ defined by $$\begin{aligned}
&{{\textup{Dom}}}\, \gamma:=
\big\{ a \in {\mathsf{B}}:
\lim_{t \to 0^+} \frac{{\lambda}_t (a) - {\epsilon}(a)}{t} \textrm{ exists} \big\};
\\&
\gamma(d) :=
\lim_{t \to 0^+} \frac{{\lambda}_t (d) - {\epsilon}(d)}{t}, \quad d\in {{\textup{Dom}}}\, \gamma,\end{aligned}$$ is called the *generating functional* of $({\lambda}_t)_{t \geq 0}$.
If $({\lambda}_t)_{t \geq 0}$ is norm-continuous then the functional $\gamma$ defined above is equal to that of Proposition \[char\], so our terminology is consistent. If each $\lambda_t$ is \*-preserving then ${{\textup{Dom}}}\, \gamma$ is selfadjoint and $\gamma$ is \*-preserving. If each $\lambda_t$ is a state, then $\gamma$ is conditionally positive, in the sense that $\gamma(a) \geq 0$ for all $a \in {\mathsf{B}}_+\cap {{\textup{Dom}}}\, \gamma \cap {{\textup{Ker}}}\, {\epsilon}$ (again, this is consistent with the earlier definition). The identity implies the inclusion ${\epsilon}\circ {Z}\subset \gamma$, where ${Z}$ is the generator of the associated $C_0$-semigroup on ${\mathsf{B}}$, so ${{\textup{Dom}}}\, \gamma$ is norm-dense in ${\mathsf{B}}$.
Note that $\gamma$ can be defined for a weakly continous convolution semigroup on any $C^*$-bialgebra, however, without the residual vanishing at infinity property, there is no guarantee that $\gamma$ be densely defined.
We now recall a useful construction from semigroup theory. Let $(P_t)_{t \geq 0}$ be a $C_0$-semigroup on a Banach space $X$ with generator ${Z}$. Set $$\label{Dom}
{{\textup{Dom}}}\widehat{{Z}} :=
\big\{x \in X:
\forall_{\mu \in X^*}
\lim_{t\to 0^+} \frac{\mu(P_t(x)) - \mu(x)}{t} \textrm{ exists} \big\}.$$ Clearly ${{\textup{Dom}}}\widehat{{Z}} \supset {{\textup{Dom}}}{Z}$. By the Uniform Boundedness Principle, it follows that, for all $x \in {{\textup{Dom}}}\widehat{{Z}}$, there is a unique element $l \in X^{**}$ satisfying $$\lim_{t\to 0^+} \frac{\mu(P_t(x)) - \mu(x)}{t} = l (\mu),
\quad \mu \in X^*.$$ Setting $\widehat{{Z}}(x):= l$, we have an operator $\widehat{{Z}}: X \to X^{**}$ with dense domain ${{\textup{Dom}}}\widehat{{Z}}$ (justifying the notation ). Under the canonical identification of $X$ as a subspace of $X^{**}$, clearly $\widehat{{Z}}\supset {Z}$. The precise situation is summarised next. This is a rewording of Theorem 1.24 of [@Davies], where the reader is warned of a missing hypothesis (not needed here), namely that—in the notation used there—$\mathcal{L}$ must also be assumed to be invariant under each $T_t^*$.
\[Lhat\] Let $(P_t)_{t \geq 0}$ be a $C_0$-semigroup on a Banach space $X$ with generator ${Z}$, and associated operator $\widehat{{Z}}:{{\textup{Dom}}}\widehat{{Z}} \subset X \to X^{**}$. Then $${{\textup{Dom}}}{Z}= \{x \in {{\textup{Dom}}}\widehat{{Z}}: \widehat{{Z}} (x) \in X\} \textrm{ and }
{Z}= \widehat{{Z}}|_{{{\textup{Dom}}}{Z}}.
\label{LL}$$ In particular, the operator $\widehat{{Z}}$ uniquely determines the semigroup $(P_t)_{t \geq 0}$.
The significance of the operator $\widehat{{Z}}$ for present considerations is indicated by the following result.
\[Ldet\] Let $(P_t)_{t \geq 0}$ be the semigroup on ${\mathsf{B}}$ associated with a weakly continuous convolution semigroup of functionals $({\lambda}_t)_{t \geq 0}$ on ${\mathsf{B}}$. Then its associated operator $\widehat{{Z}}: {{\textup{Dom}}}\widehat{{Z}}\subset {\mathsf{B}}\to{\mathsf{B}}^{**}$ is given by $$\begin{aligned}
&{{\textup{Dom}}}\widehat{{Z}} =
\{ a \in {\mathsf{B}}: \forall_{\mu\in{\mathsf{B}}^*}\ L_\mu(a) \in {{\textup{Dom}}}\, \gamma \};
\label{3.3}\\ &\widehat{{Z}} (a) (\mu) = \gamma(L_{\mu}(a)),
\quad a\in{{\textup{Dom}}}\widehat{{Z}}, \mu\in{\mathsf{B}}^*,
\label{form}\end{aligned}$$ where $\gamma$ is the generating functional of $(\lambda_t)_{t\geq 0}$.
If $\mu \in {\mathsf{B}}^*$, $a \in {\mathsf{B}}$ and $t \geq 0$ then $\mu (P_t(a))= {\lambda}_t (L_{\mu}(a))$; in particular $\mu(a) = {\epsilon}(L_{\mu}(a))$. The identities and now follow directly from the definitions of $\gamma$ and $\widehat{{Z}}$.
The next result, the first part of which fully justifies the term ‘generating functional’, now follows easily from Proposition \[Ldet\], Lemma \[Lhat\] and the identity .
Let $({\lambda}_t)_{t \geq 0}$ be a weakly continuous convolution semigroup of functionals on ${\mathsf{B}}$ with generating functional $\gamma$. Then
the functional $\gamma$ determines the semigroup $({\lambda}_t)_{t \geq 0}$ uniquely;
$\gamma$ is bounded if and only if $({\lambda}_t)_{t \geq 0}$ is norm-continuous, in which case $\gamma$ is everywhere defined.
Convolution semigroups on $C^*$-bialgebras of discrete type {#disc type}
===========================================================
*For this section we assume that ${\mathsf{B}}$ is of discrete type*, that is, as a $C^*$-algebra, ${\mathsf{B}}$ is a $c_0$-direct sum of matrix algebras. The existence of a counit forces ${\mathsf{B}}$ to have the form $${\mathsf{B}}= {\mathbb C}\Omega \oplus {\mathsf{B}}_0 \text{ where }
{\mathsf{B}}_0 = \bigoplus_{\lambda \in {\Lambda}} M_{n({\lambda})},
\label{disc}$$ for an index set ${\Lambda}$ and family $(n({\lambda}))_{\lambda\in\Lambda}$ in ${\mathbb N}$; the counit being given by the formula $${\epsilon}\Big(\alpha \Omega \oplus \bigoplus_{{\lambda}\in {\Lambda}} a_{{\lambda}}\Big) = \alpha.$$ We identify ${\mathsf{B}}_0$ with $0 \oplus {\mathsf{B}}_0 \subset {\mathsf{B}}$ and view ${\mathsf{B}}$ as represented on the Hilbert space ${\mathbb C}\oplus \bigoplus_{{\lambda}\in {\Lambda}} {\mathbb C}^{n({\lambda})}$. Then $M({\mathsf{B}})$ equals the $l^\infty$-direct sum $${\mathsf{M}}:= {\mathbb C}\Omega \oplus \prod_{{\lambda}\in {\Lambda}}M_{n({\lambda})},$$ which is a von Neumann algebra, coinciding with the universal enveloping von Neumann algebra of ${\mathsf{B}}$.
The examples we have in mind are the algebras of functions vanishing at infinity on a discrete semigroup, and the duals of compact (quantum) groups. In the former case $n({\lambda})=1$ for each ${\lambda}\in {\Lambda}$; in the latter case ${\Lambda}$ is the set of equivalence classes of irreducible representations of the underlying compact group – more on these in the next section.
\[str\] Let $({\lambda}_t)_{\geq 0}$ be a weakly continuous convolution semigroup of states on ${\mathsf{B}}$. Then its associated semigroup $({\widetilde}{P}_t)_{t\geq 0}$ on ${M({\mathsf{B}})}$ is strongly continuous.
Since $M({\mathsf{B}})$ is the von Neumann algebra ${\mathsf{M}}$, and by the same token $M({\mathsf{B}}{\otimes}{\mathsf{B}})$ is the von Neumann algebra ${\mathsf{M}}{\overline{{\otimes}}}{\mathsf{M}}$, Proposition \[new\] gives maps $$L_{\nu} := (\nu {\otimes_{\mathrm{M}}}{\textup{id}_{{\mathsf{M}}}}) {\widetilde}{{\Delta}} : M({\mathsf{B}})\to M({\mathsf{B}}),
\quad \nu\in
M({\mathsf{B}})^* = {\mathsf{M}}^*,$$ satisfying $${\widetilde}{\lambda}\circ L_{\nu} =
\nu \circ({\textup{id}_{{\mathsf{M}}}}\, {\overline{{\otimes}}}\, {\widetilde}{\lambda}){\widetilde}{{\Delta}} =
\nu \circ {\widetilde}{R}_\lambda,
\quad \lambda\in{\mathsf{B}}^*={\mathsf{M}}_*,\ \nu\in M({\mathsf{B}})^*={\mathsf{M}}^*.$$ In particular, $$\label{commut}
\nu \circ \big({\widetilde}{P_t} - {{\textup{id}}}_{M({\mathsf{B}})}\big) =
\big({\widetilde}{{\lambda}}_t - {\widetilde}{{\epsilon}}\big)\circ L_{\nu},
\quad t \geq 0, \ \nu \in M({\mathsf{B}})^*.$$ Therefore, by Proposition \[2.2\*\], $({\widetilde}{P}_t)_{t \geq 0}$ is weakly continuous, and so also strongly continuous.
Note how the existence of slice maps for not-necessarily-strict functionals is crucially used in the above proof (via Proposition \[new\]).
\[main\] Let $({\lambda}_t)_{t \geq 0}$ be a weakly continuous convolution semigroup of states on ${\mathsf{B}}$. Then $({\lambda}_t)_{t \geq 0}$ is norm-continuous.
Define a functional ${{\gamma }_{\kern-1pt\raisebox{-1.25ex}{\mbox{\,$\tilde{\ }$}}}} : {{\textup{Dom}}}\, {{\gamma }_{\kern-1pt\raisebox{-1.25ex}{\mbox{\,$\tilde{\ }$}}}} \subset M({\mathsf{B}}) \to {\mathbb C}$ by $$\begin{aligned}
&{{\textup{Dom}}}\, {{\gamma }_{\kern-1pt\raisebox{-1.25ex}{\mbox{\,$\tilde{\ }$}}}} :=
\big\{ m \in {M({\mathsf{B}})}:
\lim_{t \to 0^+} \frac{{\widetilde}{{\lambda}}_t (m) - {\widetilde}{{\epsilon}}(m)}{t}
\textrm{ exists} \big\};
\\&
{{\gamma }_{\kern-1pt\raisebox{-1.25ex}{\mbox{\,$\tilde{\ }$}}}} (d) =
\lim_{t \to 0^+} \frac{{\widetilde}{{\lambda}}_t (d) - {\widetilde}{{\epsilon}}(d)}{t},
\quad d\in {{\textup{Dom}}}\, {{\gamma }_{\kern-1pt\raisebox{-1.25ex}{\mbox{\,$\tilde{\ }$}}}}.\end{aligned}$$ Clearly ${{\textup{Dom}}}\, {{\gamma }_{\kern-1pt\raisebox{-1.25ex}{\mbox{\,$\tilde{\ }$}}}}$ is a selfadjoint subspace of $M({\mathsf{B}})$ containing $1$ and ${{\gamma }_{\kern-1pt\raisebox{-1.25ex}{\mbox{\,$\tilde{\ }$}}}}(1)=0$. Moreover, the identity and Lemma \[str\] imply that ${{\textup{Dom}}}\, {{\gamma }_{\kern-1pt\raisebox{-1.25ex}{\mbox{\,$\tilde{\ }$}}}}$ is norm-dense in $M({\mathsf{B}})$ (cf. the remarks in Section \[rvai\]).
Let $p$ be the orthogonal projection $1 - \Omega \in M({\mathsf{B}})$. Note that $p$ is the identity of the ideal ${{\textup{Ker}}}\, {\widetilde}{{\epsilon}}
= M({\mathsf{B}}_0)$ of $M({\mathsf{B}})$. By density and selfadjointness of ${{\textup{Dom}}}\, {{\gamma }_{\kern-1pt\raisebox{-1.25ex}{\mbox{\,$\tilde{\ }$}}}}$ there exists $z \in {{\textup{Dom}}}\, {{\gamma }_{\kern-1pt\raisebox{-1.25ex}{\mbox{\,$\tilde{\ }$}}}}$ such that $z=z^*$ and $\|p-z\| \leq \frac{1}{4}$. Note that $|{\widetilde}{{\epsilon}}(z)| = |{\widetilde}{{\epsilon}}(z-p)|\leq \frac{1}{4}$. Put $y = z - {\epsilon}(z) 1$. Then $y \in {{\textup{Dom}}}\, {{\gamma }_{\kern-1pt\raisebox{-1.25ex}{\mbox{\,$\tilde{\ }$}}}}$ and ${\widetilde}{{\epsilon}}(y)=0$, so in particular $\sup_{t>0}\frac{{\widetilde}{{\lambda}}_t(y)}{t} <\infty$. Moreover, since $y \in M({\mathsf{B}}_0)$ and $z\geq p - \frac{1}{4}1$, $$y = pyp =
(pzp-{\widetilde}{{\epsilon}}(z)p)\geq p\big( p-\frac{1}{4}1\big)p - \frac{1}{4}p = \frac{1}{2} p.$$ Therefore $$C:=\sup_{t>0} \frac{{\widetilde}{{\lambda}}_t(p)}{t} \leq 2 \sup_{t>0} \frac{{\widetilde}{{\lambda}}_t(y)}{t} <\infty.$$ Now, for each $t >0$, define a positive functional $\mu_t$ in ${M({\mathsf{B}})_{\beta}^*}$ by the formula $$\mu_t (m) = \frac{1}{t} {\widetilde}{{\lambda}}_t (pmp), \quad m \in M({\mathsf{B}}).$$ All these functionals have norm smaller than $C$: $$\|\mu_t\| = \mu_t(1) = \frac{1}{t} {\widetilde}{{\lambda}}_t (p) \leq C.
$$ Now let $\nu$ be an arbitrary functional in ${M({\mathsf{B}})}^*$. Then, since $p$ is the identity element of $M({\mathsf{B}}_0)$, it follows from that, for all $m \in M({\mathsf{B}})$, $$\begin{aligned}
\nu\big({\widetilde}{P}_t(m) - m\big) &=
{\widetilde}{\lambda}_t \big(L_{\nu}(m)- {\widetilde}{{\epsilon}}(L_{\nu}(m))1\big)
\\ &=
{\widetilde}{\lambda}_t \big( p\big( L_{\nu}(m) - {\widetilde}{{\epsilon}}(L_{\nu}(m))1\big) p\big)
=
t \,\mu_t \big( L_{\nu}(m) - {\widetilde}{{\epsilon}}(L_{\nu}(m))1\big).\end{aligned}$$ Thus, letting ${{Z }_{\kern-1pt\raisebox{-1.25ex}{\mbox{\,$\tilde{\ }$}}}}$ be the generator of the $C_0$-semigroup $({\widetilde}{P}_t)_{t\geq 0}$, $$\begin{aligned}
|\nu({{Z }_{\kern-1pt\raisebox{-1.25ex}{\mbox{\,$\tilde{\ }$}}}} (m))| = &
\lim_{t \to 0^+} \frac{1}{t}
\left|\nu\big({\widetilde}{P}_t(m) - m\big)\right|
\\ \leq &\
2 C\| L_\nu(m)\|
\leq 2 C\|\nu\| \|m\|,
\quad m \in {{\textup{Dom}}}{{Z }_{\kern-1pt\raisebox{-1.25ex}{\mbox{\,$\tilde{\ }$}}}} .\end{aligned}$$ It follows that ${{Z }_{\kern-1pt\raisebox{-1.25ex}{\mbox{\,$\tilde{\ }$}}}}$ is bounded, so $({\widetilde}{P}_t)_{t \geq 0}$ is norm-continuous, and therefore $({\lambda}_t)_{t \geq 0}$ is too.
Since the multiplier algebra $M({\mathsf{B}})$ is a von Neumann algebra, Theorem \[main\] can alternatively be deduced from Lemma \[str\] using the fact that any strongly continuous completely positive semigroup on a von Neumann algebra is automatically norm-continuous ([@George]).
We note here that the results of Sections \[on Cstar Bialgebras\], \[rvai\] and \[disc type\] remain valid for a *$C^*$-hyperbialgebra* ${\mathsf{B}}$, where the coproduct is only assumed to be completely positive, strict and preunital. Examples of $C^*$-hyperbialgebras are provided by compact quantum hypergroups ([@ChV]). The classical theory is described in [@BloomHeyer].
Commutative and cocommutative cases {#section: cases}
===================================
Let ${\mathsf{B}}$ be a commutative $C^*$-bialgebra of discrete type. Then it follows from Gelfand theory that ${\mathsf{B}}$ is of the form $C_0(\Gamma)$, where $\Gamma$ is a discrete semigroup, and the coproduct and counit are given by $${\Delta}(F) (\gamma,\gamma') = f(\gamma \gamma')
\text{ and }
{\epsilon}(F) = F(e), \quad F \in C_0(\Gamma), \gamma, \gamma' \in \Gamma,$$ under the natural identifications $M({\mathsf{B}}{\otimes}{\mathsf{B}}) = M(C_0(\Gamma \times \Gamma)) = C_{{\text{{\textup}{b}}}}(\Gamma\times \Gamma)$. If $\Gamma$ satisfies both left and right cancellation properties then the map $\phi$ introduced in the first paragraph of Section \[rvai\] is proper, and so is its ‘right’ version, so that the residual vanishing at infinity property holds. The convolution semigroups of states on ${\mathsf{B}}$ correspond to convolution semigroups of probability measures on $\Gamma$ via the Riesz Representation Theorem, and Theorem \[main\] specialises as follows.
Every convolution semigroup of probability measures on a discrete semigroup has bounded infinitesimal generator.
This extends the central conclusion of Theorem 4.1.5 of [@Heyer], which is formulated for discrete groups.
Recall that *cocommutativity* for a $C^*$-bialgebra ${\mathsf{B}}$ means $\Sigma{\Delta}= {\Delta}$ where $\Sigma$ denotes the tensor flip on ${\mathsf{B}}{\otimes}{\mathsf{B}}$, which is strict as a map ${\mathsf{B}}{\otimes}{\mathsf{B}}\to M({\mathsf{B}}{\otimes}{\mathsf{B}})$; it is equivalent to commutativity of the convolution product on ${\mathsf{B}}^*$. Now, for a locally compact group $G$, the reduced and universal group $C^*$-algebras are isomorphic via the left regular representation if and only if $G$ is amenable ([@Pedersen], Theorem 7.3.9), in particular this holds if $G$ is compact. Therefore, by the Peter-Weyl theory of unitary representations of compact groups, the group $C^*$-algebra of a compact group is isomorphic to a $c_0$-direct sum of matrix algebras.
We are not aware of any characterisation of cocommutative $C^*$-bialgebras of discrete type. However every cocommutative discrete quantum group ${\mathsf{B}}$ is the group $C^*$-algebra of a compact group $G$ where, viewing $C^*(G)$ as the $C^*$-subalgebra of $B(L^2(G, \text{Haar}))$ generated by the set of convolution operators $\lambda(F): \xi \mapsto F \star \xi$ ($F \in C(G)$), the coproduct and counit are determined by $${\widetilde}{{\Delta}} (\lambda_g) = \lambda_g {\otimes}\lambda_g
\text{ and } {\widetilde}{{\epsilon}} (\lambda_g) = 1,
\quad g \in G,$$ $\lambda_g\in M({\mathsf{B}})$ being the translation operator $(\lambda_g \xi)(g') = \xi(g^{-1}g')$. This follows from duality theory for locally compact quantum groups ([@BB], [@Kuslect]). Specifically, discrete quantum groups are naturally isomorphic to their quantum group biduals; the dual of a cocommutative discrete quantum group is a commutative compact quantum group; and the dual of a commutative compact quantum group $C(G)$ is the group $C^*$-algebra of $G$ with the quantum group structure defined above.
It is easily verified that the unital \*-algebra spanned by $\{{\lambda}_g: g \in G\}$ is strictly dense in the multiplier algebra $M({\mathsf{B}})$, and that the map $g \mapsto {\lambda}_g$ is strictly continuous. The following consequence of Proposition 7.1.9 of [@Pedersen] is noted here for convenience.
Let ${\mathsf{B}}$ be the group $C^*$-algebra of a compact group $G$. Then $$\label{corresp}
{\widetilde}{\omega}({\lambda}_g) = \phi(g), \quad g \in G,$$ describes a one-to-one correspondence between the positive linear functionals $\omega$ on ${\mathsf{B}}$ and the continuous positive-definite functions $\phi$ on $G$.
Given a positive-definite function $\phi$ on a group $G$, the function $\psi: G\to\mathbb{C}$, $g \mapsto \phi(g) - \phi(e)$ plainly enjoys the following three properties: it is *conditionally positive definite*, that is, for all $n\in\mathbb{N}$, $g_1, \ldots ,g_n \in G$ and $z_1, \ldots ,z_n \in {\mathbb C}$ satisfying $\sum_{i=1}^n z_i = 0$, $$\sum_{i,j=1}^n \overline{z_i}z_j \psi(g_i^{-1}g_j) \geq 0,$$ it is *Hermitian*, i.e. $$\psi = \psi^* \text{ where }
\psi^*(g) := \overline{\psi(g^{-1})}, \quad g\in G,$$ and it vanishes at the identity element $e$: $$\psi (e) = 0.$$ The content of the theorem below is that the converse holds if $G$ is compact and $\psi$ is continuous. We deduce this from our results.
Let $G$ be a compact group and let $\psi$ be a continuous, Hermitian, conditionally positive-definite function on $G$ vanishing at the identity element $e$. Then there is a positive-definite function $\phi$ on $G$ such that $$\label{diff}
\psi(g)= \phi(g) - \phi(e), \quad g \in G,$$ in other words there is a constant $c \in \mathbb{R}_+$ such that $\psi + c$ is positive-definite.
Let ${\mathsf{B}}$ be the discrete quantum group $C^*(G)$, as above, and let $\mathcal{B}$ be the \*-algebraic span of $\{{\lambda}_g: g \in
G\}$ in $M({\mathsf{B}})$. By the Schönberg correspondence, the continuous function $e^{t\psi}$ is positive-definite for each $t\geq
0$ ([@ParthaSch], Lemma 1.7). Let $(\mu_t)_{t\geq 0}$ be the corresponding family of states on ${\mathsf{B}}$, defined via . Then, for all $g \in G$ and $s,t\in \mathbb{R}_+$, $$({\widetilde}{\mu}_s \star {\widetilde}{\mu}_t) ({\lambda}_g) =
{\widetilde}{\mu}_s({\lambda}_g) {\widetilde}{\mu}_t ({\lambda}_g) =
e^{s\psi(g)} e^{t\psi(g)} =
e^{(s+t)\psi(g)} =
{\widetilde}{\mu}_{s+t}({\lambda}_g)$$ and $${\widetilde}{\mu}_0({\lambda}_g) = 1 = {\widetilde}{{\epsilon}}({\lambda}_g).$$ By the strict density of $\mathcal{B}$ in $M({\mathsf{B}})$, it follows that $(\mu_t)_{t\geq 0}$ is a convolution semigroup of states on ${\mathsf{B}}$. In view of the identity $$\mu_t(\lambda(F)) =
\int_G F(g) e^{t\psi(g)} d\,g,
\quad t \in \mathbb{R}_+, F \in C(G),$$ it follows from the compactness of $G$ and Lebesgue’s Dominated Convergence Theorem, that $(\mu_t)_{t\geq0}$ is weakly continuous. Therefore, by Theorem \[main\], it is norm-continuous and so, by Proposition \[char\], the strict extension of its generating functional $\gamma$ satisfies $${\widetilde}{\gamma}({\lambda}_g) =
\lim_{t \to 0^+}
t^{-1} \big( {\widetilde}{\mu}_t({\lambda}_g) - {\widetilde}{{\epsilon}}({\lambda}_g) \big) =
\lim_{t \to 0^+}
t^{-1} \big( e^{t\psi(g)} - 1 \big) = \psi(g), \quad g\in G.$$ By Theorem 7.3 of [@qscc3], there is a nondegenerate representation $(\pi,{\mathsf{h}})$ of ${\mathsf{B}}$ and vector $\eta\in{\mathsf{h}}$ such that $\gamma = \omega_\eta \circ (\pi - \iota \circ {\epsilon})$ where $\iota$ is the ampliation ${\mathbb C}\to B({\mathsf{h}})$ and $\omega_\eta$ denotes the vector functional $T \mapsto \langle \eta, T \eta \rangle$ on $B({\mathsf{h}})$. Letting $\phi: G \to {\mathbb C}$ be the positive-definite function $g \mapsto (\omega_\eta \circ {\widetilde}{\pi})({\lambda}_g)$, we have $\phi(g) - \phi(e) = {\widetilde}{\gamma}({\lambda}_g) = \psi(g)$ for all $g\in G$, and so the proof is complete.
Guichardet’s approach is to reduce the proposition to the vanishing of the first cohomology group of unitary representations of compact groups and then to appeal to Theorem 15.1 of [@ParthaSch], which delivers just that.
In the opposite direction to that presented here, Guichardet’s Theorem can be used to give an alternative proof of Theorem 7.2 of [@qscc3] in the special case of multiplier $C^*$-bialgebras of the type $C^*(G)$, where $G$ is a compact group.
*Acknowledgement.* We are grateful to Uwe Franz for useful remarks on an earlier draft, in particular for bringing Guichardet’s Theorem to our attention.
[KaR]{}
S.Baaj and G.Skandalis, Unitaires multiplicatifs commutatifs, *C.R. Math. Acad. Sci.Paris* **336** (2003), no. 4, 299–304.
J.Bertoin, “Lévy Processes,” Cambridge Tracts in Mathematics **121**, CUP, 1996.
W.R.Bloom and H.Heyer, “The Harmonic Analysis of Probability Measures on Hypergroups,” de Gruyter, Berlin, 1995.
Yu.Chapovsky and L.Vainerman, Compact quantum hypergroups, *J. Operator Theory* **41** (1999) no.2, 261–289.
E.B.Davies, “One-Parameter Semigroups,” London Mathematical Society Monographs **15**, Academic Press, London, 1980.
G.Elliott, On the convergence of a sequence of completely positive maps to the identity, *J. Austral. Math. Soc. Ser. A* **68** (2000) no.3, 340–348.
U.Franz, Lévy processes on quantum groups and dual groups, *in* “Quantum Independent Increment Processes, Vol. II: Structure of Quantum Lévy Processes, Classical Probability and Physics,” LNM **1866**, Springer, Heidelberg, 2005.
A.Guichardet, “Symmetric Hilbert Spaces and Related Topics,” Lecture Notes in Mathematics **267**, Springer, Heidelberg, 1970.
H.Heyer, “Probability Measures on Locally Compact Groups,” Springer-Verlag, Berlin, 1977.
N.Jacob, “Pseudo-differential Operators and Markov Processes, Vol.I: Fourier Analysis and Semigroups,” Imperial College Press, London, 2001.
R.V.Kadison and J.R.Ringrose, “Fundamentals of the theory of operator algebras, Vol.II: Advanced Theory,” Graduate Studies in Mathematics **16**, AMS, 1997.
J.Kustermans, One-parameter representations on $C^*$-algebras, *Preprint, Odense Universitet*, arXiv: \#funct-an/9707010.
— — , Locally compact quantum groups, *in*, “Quantum Independent Increment Processes, Vol. I: From Classical Probability to Quantum Stochastics," *eds. U. Franz & M. Schürmann*, Lecture Notes in Mathematics **1865**, Springer, Heidelberg, 2005.
J.Kustermans and S.Vaes, Locally compact quantum groups, *Ann. Sci. École Norm. Sup. (4)* **33** (2000) no.6, 837–934.
E.C.Lance, “Hilbert $C^*$-modules,” London Mathematical Society Lecture Notes Series **210**, Cambridge University Press, Cambridge, 1995.
J.M.Lindsay and A.G.Skalski, Quantum stochastic convolution cocycles II, *Comm. Math. Phys.* **280** (2008) no.3, 575–610; — — , Quantum stochastic convolution cocycles III, *Preprint*, arXiv:0905/2410 \[math.QA\].
J.M.Lindsay and S.J.Wills, Existence of Feller cocycles on a $C^*$-algebra, *Bull. London Math. Soc.* **33** (2001) no.5, 613–621.
M.Neufang, Amplification of completely bounded operators and Tomiyama’s slice maps, *J. Funct. Anal.* **207** (2004), 300–329.
K.R.Parthasarathy and K.Schmidt, “Positive Definite Kernels, Continuous Tensor Products, and Central Limit Theorems of Probability Theory,” Lecture Notes in Mathematics **272**, Springer, Berlin, 1972.
G.K. Pedersen, “$C^*$-algebras and their automorphism groups,” London Mathematical Society Monographs **14**, Academic Press, London New York, 1979.
J.-L.Sauvageot, Strong Feller semigroups on $C\sp \ast$-algebras, *J. Operator Theory* **42** (1999) no.1, 83–102.
M.Schürmann, “White Noise on Bialgebras,” Lecture Notes in Mathematics **1544**, Springer, Heidelberg, 1993.
D.C.Taylor, The strict topology for double centralizer algebras, *Trans. Amer. Math. Soc.* **150** (1970), 633–643.
J.Tomiyama, Tensor products and approximation problems of $C^*$-algebras, *Publ. RIMS, Kyoto Univ.* **11** (1975), 163–183.
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---
abstract: 'In this paper we present a locally and dimension-adaptive sparse grid method for interpolation and integration of high-dimensional functions with discontinuities. The proposed algorithm combines the strengths of the generalised sparse grid algorithm and hierarchical surplus-guided local adaptivity. A high-degree basis is used to obtain a high-order method which, given sufficient smoothness, performs significantly better than the piecewise-linear basis. The underlying generalised sparse grid algorithm greedily selects the dimensions and variable interactions that contribute most to the variability of a function. The hierarchical surplus of points within the sparse grid is used as an error criterion for local refinement with the aim of concentrating computational effort within rapidly varying or discontinuous regions. This approach limits the number of points that are invested in ‘unimportant’ dimensions and regions within the high-dimensional domain. We show the utility of the proposed method for non-smooth functions with hundreds of variables.'
author:
- 'J.D Jakeman and S.G. Roberts'
bibliography:
- 'Collocation.bib'
- 'SparseGrid.bib'
- 'Misc.bib'
- 'GPC.bib'
- 'SensitivityAnalysis.bib'
- 'EpistemicUncertainty.bib'
- 'HDMR.bib'
- 'JDJakeman.bib'
- 'DiscontinuityDetection.bib'
- 'Cubature.bib'
title: Local and Dimension Adaptive Sparse Grid Interpolation and Quadrature
---
|
---
abstract: 'In this work, operator version of Popoviciu’s inequality for positive selfadjoint operators in Hilbert spaces under positive linear maps for superquadratic functions is proved. Analogously, using the same technique operator version of Popoviciu’s inequality for convex functions is obtained. Some other related inequalities are also deduced.'
address: 'Department of Mathematics, Faculty of Science and Information Technology, Irbid National University, 2600 Irbid 21110, Jordan.'
author:
- 'M.W. Alomari'
title: 'Operator Popoviciu’s inequality for superquadratic and convex functions of selfadjoint operators in Hilbert spaces'
---
Introduction
============
Let $\mathcal{B}\left( \mathcal{H}\right) $ be the Banach algebra of all bounded linear operators defined on a complex Hilbert space $\left( \mathcal{H};\left\langle \cdot ,\cdot \right\rangle
\right)$ with the identity operator $1_{\mathcal{H}}$ in $\mathcal{B}\left(\mathcal{H} \right)$. Denotes $\mathcal{B}^+\left( \mathcal{H}\right) $ the convex cone of all positive operators on $\mathcal{H}$. A bounded linear operator $A$ defined on $\mathcal{H}$ is selfadjoint if and only if $
\left\langle {Ax,x} \right\rangle \in \mathbb{R}$ for all $x\in
\mathcal{H}$. For two selfadjoint operators $A,B\in \mathcal{H}$, we write $A\le B$ if $ \left\langle {Ax,x} \right\rangle \le
\left\langle {Bx,x} \right\rangle$ for all $x\in \mathcal{H}$. Also, we define $$\begin{aligned}
\left\| A \right\| = \mathop {\sup }\limits_{\left\| x \right\| =
1} \left| {\left\langle {Ax,x} \right\rangle } \right| = \mathop
{\sup }\limits_{\left\| x \right\| = \left\| y \right\| = 1}
\left| {\left\langle {Ax,y} \right\rangle } \right|.\end{aligned}$$
If $\varphi$ is any function defined on $\mathbb{R}$, we define $$\begin{aligned}
\left\| \varphi \right\|_A = \sup \left\{ {\left| {\varphi
\left( \lambda \right)} \right|:\lambda \in \operatorname{sp}\left( A
\right)} \right\}.\end{aligned}$$ If $\varphi$ is continuous then we write $\left\| \varphi
\right\|_A=\left\| A \right\| $.
Let $A\in \mathcal{B}\left(\mathcal{H} \right) $ be a selfadjoint linear operator on $\left( \mathcal{H};\left\langle \cdot ,\cdot
\right\rangle \right)$. Let $C\left(\operatorname{sp}\left(A\right)\right)$ be the set of all continuous functions defined on the spectrum of $A$ $\left(\operatorname{sp}\left(A\right)\right)$ and let $C^*\left(A\right)$ be the $C^*$-algebra generated by $A$ and the identity operator $1_{\mathcal{H}}$.
Let us define the map $\mathcal{G}:
C\left(\operatorname{sp}\left(A\right)\right) \to C^*\left(A\right)$ with the following properties ([@FMPS], p.3):
1. $\mathcal{G}\left(\alpha f + \beta g\right) = \alpha
\mathcal{G}\left(f\right)+\beta \mathcal{G}\left(g\right)$, for all scalars $\alpha, \beta$.
2. $\mathcal{G}\left(fg\right) = \mathcal{G}\left(f\right)
\mathcal{G}\left(g\right)$ and $\mathcal{G}\left(\overline{f}\right)=\mathcal{G}\left(f\right)^*$; where $\overline{f}$ denotes to the conjugate of $f$ and $\mathcal{G}\left(f\right)^*$ denotes to the Hermitian of $\mathcal{G}\left(f\right)$.
3. $\left\|\mathcal{G}\left(f\right)\right\|=\left\|f \right\|
= \mathop {\sup }\limits_{t \in \operatorname{sp}\left(A\right)} \left|
{f\left( t \right)} \right| $.
4. $\mathcal{G}\left( {f_0 } \right) = 1_H$ and $\mathcal{G}\left( {f_1 } \right) = A$, where $f_0\left(t\right)=1$ and $f_1\left(t\right)=t$ for all $t \in
\operatorname{sp}\left(A\right)$.
Accordingly, we define the continuous functional calculus for a selfadjoint operator $A$ by $$\begin{aligned}
f\left(A\right) = \mathcal{G}\left(f\right) \text{for all} \,f\in
C\left(\operatorname{sp}\left(A\right)\right).\end{aligned}$$ If both $f$ and $g$ are real valued functions on $\operatorname{sp}(A)$ then the following important property holds: $$\begin{aligned}
f\left( t \right) \ge g\left( t \right) \,\text{for all} \, \,t
\in \operatorname{sp}\left( A \right) \,\,\text{implies}\,\, f\left( A \right)
\ge g\left( A \right), \label{eq1.1}\end{aligned}$$ in the operator order of $\mathcal{B}\left(\mathcal{K} \right)$.
A linear map is defined to be $\Phi:\mathcal{B}\left(\mathcal{H} \right)\to \mathcal{B}\left(\mathcal{K} \right)$ which preserves additivity and homogeneity, i.e., $\Phi \left(\lambda_1 A +\lambda_2 B \right)=
\lambda_1\Phi \left( A \right)+ \lambda_2\Phi \left( B \right)$ for any $\lambda_1,\lambda_2 \in \mathbb{C}$ and $A, B \in \mathcal{B}\left(\mathcal{H} \right)$. The linear map is positive $\Phi:\mathcal{B}\left(\mathcal{H} \right)\to \mathcal{B}\left(\mathcal{K} \right)$ if it preserves the operator order, i.e., if $A\in \mathcal{B}^+\left(\mathcal{H} \right)$ then $\Phi\left(A\right)\in \mathcal{B}^+\left(\mathcal{K} \right)$, and in this case we write $\textfrak{B} [\mathcal{B}\left(\mathcal{H} \right),\mathcal{B}\left(\mathcal{K} \right)] $. Obviously, a positive linear map $\Phi$ preserves the order relation, namely $A\le B \Longrightarrow \Phi\left(A\right)\le \Phi\left(B\right)$ and preserves the adjoint operation $\Phi\left(A^*\right)=\Phi\left(A\right)^*$. Moreover, $\Phi$ is said to be normalized (unital) if it preserves the identity operator, i.e. $\Phi\left(1_{\mathcal{H}}\right) = 1_{\mathcal{K}}$, in this case we write $\textfrak{B}_n [\mathcal{B}\left(\mathcal{H}
\right),\mathcal{B}\left(\mathcal{K} \right)] $.
Superquadratic functions
------------------------
A function $f:J\to \mathbb{R}$ is called convex iff $$\begin{aligned}
f\left( {t\alpha +\left(1-t\right)\beta} \right)\le tf\left(
{\alpha} \right)+ \left(1-t\right) f\left( {\beta}
\right),\label{eq1.2}\end{aligned}$$ for all points $\alpha,\beta \in J$ and all $t\in [0,1]$. If $-f$ is convex then we say that $f$ is concave. Moreover, if $f$ is both convex and concave, then $f$ is said to be affine.
Geometrically, for two point $\left(x,f\left(x\right)\right)$ and $\left(y,f\left(y\right)\right)$ on the graph of $f$ are on or below the chord joining the endpoints for all $x,y?\in I$, $x <
y$. In symbols, we write $$\begin{aligned}
f\left(t\right)\le \frac{f\left( y \right) - f\left( x \right)
}{y-x} \left( {t-x} \right)+ f\left( x \right)\end{aligned}$$ for any $x \le t \le y$ and $x,y\in J$.
Equivalently, given a function $f : J\to \mathbb{R}$, we say that $f$ admits a support line at $x \in J $ if there exists a $\lambda
\in \mathbb{R}$ such that $$\begin{aligned}
f\left( t \right) \ge f\left( x \right) + \lambda \left( {t - x}
\right) \label{eq1.3}\end{aligned}$$ for all $t\in J$.
The set of all such $\lambda$ is called the subdifferential of $f$ at $x$, and it’s denoted by $\partial f$. Indeed, the subdifferential gives us the slopes of the supporting lines for the graph of $f$. So that if $f$ is convex then $\partial f(x) \ne
\emptyset$ at all interior points of its domain.
From this point of view Abramovich [@SJS] extend the above idea for what they called superquadratic functions. Namely, a function $f:[0,\infty)\to \mathbb{R}$ is called superquadratic provided that for all $x\ge0$ there exists a constant $C_x\in
\mathbb{R}$ such that $$\begin{aligned}
f\left( t \right) \ge f\left( x \right) + C_x \left( {t - x}
\right) + f\left( {\left| {t - x} \right|} \right)\label{eq1.4}\end{aligned}$$ for all $t\ge0$. We say that $f$ is subquadratic if $-f$ is superquadratic. Thus, for a superquadratic function we require that $f$ lie above its tangent line plus a translation of $f$ itself.
Prima facie, superquadratic function looks to be stronger than convex function itself but if $f$ takes negative values then it may be considered as a weaker function. Therefore, if $f$ is superquadratic and non-negative. Then $f$ is convex and increasing [@SJS] (see also [@S]).
Moreover, the following result holds for superquadratic function.
[@SJS] \[lemma1\]Let $f$ be superquadratic function. Then
1. $f\left(0\right)\le 0$
2. If $f$ is differentiable and $f(0)=f^{\prime}(0)=0$, then $C_x=f^{\prime}(x)$ for all $x\ge0$.
3. If $f(x)\ge0$ for all $x\ge 0$, then $f$ is convex and $f(0)=f^{\prime}(0)=0$.
The next result gives a sufficient condition when convexity (concavity) implies super(sub)quaradicity.
[@SJS] \[lemma2\]If $f^{\prime}$ is convex (concave) and $f(0)=f^{\prime}(0)=0$, then is super(sub)quadratic. The converse of is not true.
Subquadraticity does always not imply concavity; i.e., there exists a subquadratic function which is convex. For example, $f(x)=x^p$, $x\ge 0$ and $1\le p \le2$ is subquadratic and convex.
Popoviciu’s inequality
----------------------
In 1906, Jensen in [@J] proved his famous characterization of convex functions. Simply, for a continuous functions $f$ defined on a real interval $I$, $f$ is convex if and only if $$\begin{aligned}
f\left( {\frac{{x + y}}{2}} \right) \le \frac{{f\left( x \right) +
f\left( y \right)}}{2},\end{aligned}$$ for all $x,y\in I$.
In 1965, a parallel characterization of Jensen convexity was presented by Popoviciu [@P] (for more details see [@NP], p.6), where he proved his celebrated inequality, as follows:
\[thm2\] Let $f:I\to \mathbb{R}$ be continuous. Then, $f$ is convex if and only if $$\begin{aligned}
\frac{2}{3}\left[{ f\left( {\frac{{x + z}}{2}} \right)+f\left(
{\frac{{y + z}}{2}} \right)+f\left( {\frac{{x + y}}{2}}
\right)}\right]
\le f\left( {\frac{{x + y + z}}{3}} \right)
+\frac{f\left( x \right)+f\left( y \right)+f\left( z \right)}{3}
\label{eq1.5}
\end{aligned}$$ for all $x,y,z\in I$, and the equality occurred by $f(x)=x$, $x\in
I$.
In fact, Popoviciu characterization of convex function is sound and several mathematicians greatly received his work since that time and much of them considered his characterization as alternative approach to describe convex functions. For instance, the Popoviciu’s inequality can be considered as an elegant generalization of Hlawka’s inequality using convexity as a simple tool of geometry. Indeed, if $f(x)=|x|$, $x\in \mathbb{R}$, then the Popoviciu inequality reduces to the famous Hlawka inequality, which reads: $$\begin{aligned}
\label{Hlawkaineq}\left| x \right| + \left| y \right| + \left| z
\right| + \left| {x + y + z} \right| \ge \left| {x + z} \right| +
\left| {z + y} \right| + \left| {x + y} \right|.\end{aligned}$$ Geometrically, Hlawka inequality means the total length over all sums of pairs from three vectors is not greater than the perimeter of the quadrilateral defined by the three vectors. This geometric meaning was given by D. Smiley & M. Smiley [@W] (see also [@SS], p. 756). For other related results see [@MPF] and [@NF].
Also, The extended version of Hlawka’s inequality to several variables was not possible without the help of Popoviciu’s inequality, as it inspired the authors of [@BNP] to develop a higher dimensional analogue of Popoviciu’s inequality based on his characterization. Interesting generalizations and counterparts of Popoviciu inequality with some ramified consequences can be found in [@G], [@TTH], [@TTMT] and [@VS].\
Recenty, The corresponding version of Popoviciu inequality for ${\rm{GG}}$-convex (Recall that: a positive real valued function $f$ is ${\rm{GG}}$-convex if and only if $
f\left( {x^t y^{1 - y} } \right) \le \left[ {f\left( x \right)} \right]^t \left[ {f\left( y \right)} \right]^{1 - t} $ for all $t\in [0,1]$ and all $x,y \ge0$) was discussed eighteen years ago by Niculescu in [@N], where he proved that for all $x,y,z\in I \subset [0,\infty)$, the inequality $$\begin{aligned}
f^2\left( {\sqrt {xz}} \right)f^2\left( {\sqrt {yz}}
\right)f^2\left( {\sqrt {xy}} \right) \le f^3\left(
{\sqrt[3]{xyz} } \right) f\left( x \right)f\left( y \right)f\left(
z \right)\end{aligned}$$ holds for all $x,y,z\in I$.
Seeking the operator version of Popoviciu’s inequality , the expected version of for selfadjoint operators is $$\begin{gathered}
\frac{2}{3}\left[f\left( {\left\langle { \frac{{A + B}}{2} u,u} \right\rangle } \right) + f\left(
{\left\langle { \frac{{B+D}}{2} u,u}\right\rangle } \right)+ f\left( {\left\langle { \frac{{A + D}}{2} u,u} \right\rangle }
\right)\right]
\\
\le \left\langle { \frac{{f\left( A \right) + f\left(B \right) + f\left( D \right)}}{3} u,u} \right\rangle+f\left( {\left\langle { \frac{{A + B + D}}{3}u,u} \right\rangle } \right)\end{gathered}$$ for every selfadjoint operators $A,B,D\in \mathcal{B}\left(\mathcal{H} \right)$ whose spectra contained in $I$ and every convex function $f$ defined on $I$ and this is valid for each $u\in \mathcal{K}$ with $\|u\| =1$. The proof of the above inequality is obvious by taking $x=\left\langle { A u,u} \right\rangle$, $y=\left\langle { B u,u} \right\rangle$ and $z=\left\langle { D u,u} \right\rangle$ in .
In this work, we offer two operator versions of Popoviciu’s inequality for positive selfadjoint operators in Hilbert spaces under positive linear maps for both superquadratic and convex functions with some other related results.
Main Result
===========
Throughout this work and in all needed situations, $f$ is real valued continuous function defined on $\left[0,\infty\right)$. In order to prove our main result, we need the following result concerning Jensen’s inequality for superquadratic functions. Let us don’t miss the chance here to mention that the next result was proved in [@KS] and originally in [@K] for positive selfadjoint $(n \times n)$–matrices with complex entries under unital completely positive linear maps. However, let us state down this result in more general Hilbert spaces for normalized positive linear maps.
\[thm1\]Let $A\in \mathcal{B}\left(\mathcal{H} \right)$ be a positive selfadjoint operator, $\Phi:\mathcal{B}\left(\mathcal{H}
\right)\to \mathcal{B}\left(\mathcal{K} \right)$ be a normalized positive linear map. If $f:\left[0,\infty\right)\to \mathbb{R}$ is super(sub)quadratic, then we have $$\begin{aligned}
\left\langle {\Phi \left( {f\left( A \right)}
\right)x,x} \right\rangle \ge (\le) f\left( {\left\langle {\Phi
\left( {A} \right)x,x} \right\rangle } \right) + \left\langle
{\Phi \left( {f\left( {\left| {A - \left\langle
{\Phi\left(A\right)x,x} \right\rangle 1_{\mathcal{H}} } \right|}
\right)} \right)x,x} \right\rangle\end{aligned}$$ for every $x\in \mathcal{K}$ with $\|x\|=1$.
For more recent results concerning inequalities for selfadjoint operatos and other related result, we suggest [@SIP], [@AD]–[@FMPS], [@KLPP], [@MPP] and [@MP].
The operator version of Popoviciu’s inequality for superquadratic functions under positive linear maps is proved in the next result.
\[thm2.1\]Let $A,B,C\in \mathcal{B}\left(\mathcal{H} \right)$ be three positive selfadjoint operators, $\Phi:\mathcal{B}\left(\mathcal{H} \right)\to \mathcal{B}\left(\mathcal{K} \right)$ be a normalized positive linear map. If $f:\left[0,\infty\right)\to \mathbb{R}$ is superquadratic, then we have $$\begin{aligned}
&\left\langle {\Phi \left( {\frac{{f\left( A \right) + f\left(
B \right) + f\left( D \right)}}{3}} \right)x,x} \right\rangle
+f\left( {\left\langle {\Phi \left( {\frac{{A + B + D}}{3}}
\right)x,x} \right\rangle } \right)
\label{eq2.3}\\
&\ge \frac{2}{3}\left[f\left( {\left\langle {\Phi \left(
{\frac{{A + B}}{2}} \right)x,x} \right\rangle } \right) + f\left(
{\left\langle {\Phi \left( {\frac{{B+D}}{2}} \right)x,x}
\right\rangle } \right)+ f\left( {\left\langle {\Phi \left(
{\frac{{A + D}}{2}} \right)x,x} \right\rangle }
\right)\right]\nonumber
\\
&\qquad+ \frac{1}{3} \left[\left\langle {\Phi \left( {f\left(
{\left| {A -
\left\langle {\Phi\left(\frac{{B + D}}{2}\right)x,x}
\right\rangle 1_{\mathcal{H}} } \right|} \right)} \right)x,x}
\right\rangle + f\left( {\left| {\left\langle
{\Phi \left( {\frac{{2A - B - D}}{6}} \right)x,x} \right\rangle
} \right|} \right)\right.
\nonumber\\
&\qquad\qquad+\left\langle {\Phi \left( {f\left( {\left| {D -
\left\langle {\Phi\left(\frac{{A + B}}{2}\right)x,x}
\right\rangle 1_{\mathcal{H}} } \right|} \right)} \right)x,x}
\right\rangle+ f\left( {\left| {\left\langle {\Phi
\left( {\frac{{2D - A - B}}{6}} \right)x,x} \right\rangle }
\right|} \right)
\nonumber\\
&\qquad\qquad\left. + \left\langle {\Phi \left( {f\left( {\left|
{B -
\left\langle {\Phi\left(\frac{{A + D}}{2}\right)x,x}
\right\rangle 1_{\mathcal{H}} } \right|} \right)} \right)x,x}
\right\rangle + f\left( {\left| {\left\langle
{\Phi \left( {\frac{{2B - A - D}}{6}} \right)x,x} \right\rangle
} \right|} \right)\right]\nonumber
\end{aligned}$$ for each $x\in \mathcal{K}$ with $\|x\| =1$.\
Since $f$ is superquadratic on $I$, then by utilizing the continuous functional calculus for the operator $E\ge 0$ we have by the property for the inequality we have $$\begin{aligned}
f\left( E \right) \ge f\left( s \right)\cdot 1_{\mathcal{H}} +
C_s \left( {E -s \cdot 1_{\mathcal{H}}} \right) +
f\left( {\left| {E - s\cdot 1_{\mathcal{H}}} \right|} \right).
\end{aligned}$$ and since $\Phi$ is normalized positive linear map we get $$\begin{aligned}
\Phi\left(f\left( E \right)\right) \ge f\left( s \right)\cdot
1_{\mathcal{K}} + C_s \Phi\left(E -s \cdot
1_{\mathcal{H}} \right) + \Phi\left(f\left( {\left| {E - s\cdot
1_{\mathcal{H}}} \right|} \right)\right)
\end{aligned}$$ and this implies that $$\begin{aligned}
\left\langle {\Phi \left( {f\left( E \right)} \right)x,x}
\right\rangle \ge f\left( {s} \right) \left\langle { x,x}
\right\rangle + C_s \left\langle {\left[ {\Phi \left( {E - s
\cdot 1_{\mathcal{H}} } \right)} \right]x,x} \right\rangle +
\left\langle {\Phi \left( {f\left( {\left| {E - s \cdot
1_{\mathcal{H}} } \right|} \right)} \right)x,x} \right\rangle
\label{eq2.2}
\end{aligned}$$ for each vector $x\in \mathcal{K}$ with $\|x\|=1$.
Let $A,B,D$ be three positive selfadjoint operators in $
\mathcal{B}\left(\mathcal{H} \right)$. Since $f$ is superquadratic then by applying for the operator $A\ge0$ with $s_1=
\left\langle {\Phi\left(\frac{{B + D}}{2}\right)x,x}
\right\rangle$, we get $$\begin{aligned}
\left\langle {\Phi \left( {f\left( A \right)} \right)x,x}
\right\rangle &\ge f\left( {\left\langle {\Phi \left( {\frac{{B +
D}}{2}} \right)x,x} \right\rangle } \right) +C_{s_1} \left\langle {\Phi \left( {\frac{{2A
- B - D}}{2}} \right)x,x} \right\rangle \label{eq2.4}
\\
&\qquad+ \left\langle {\Phi \left( {f\left( {\left| {A -
\left\langle {\Phi \left( {\frac{{B + D}}{2}} \right)x,x}
\right\rangle 1_{\mathcal{H}} } \right|} \right)} \right)x,x}
\right\rangle \nonumber
\end{aligned}$$ for each $x\in \mathcal{K}$ with $\|x\| =1$.
Again applying for the operator $D\ge0$ with $s_2=
\left\langle {\Phi\left(\frac{{A + B}}{2}\right)x,x}
\right\rangle$ $$\begin{aligned}
\left\langle {\Phi \left( {f\left( D \right)} \right)x,x}
\right\rangle &\ge f\left( {\left\langle {\Phi \left( {\frac{{A +
B}}{2}} \right)x,x} \right\rangle } \right) + C_{s_2} \left\langle {\Phi \left(
{\frac{{2D-A-B}}{2}} \right)x,x} \right\rangle \label{eq2.5}
\\
&\qquad + \left\langle {\Phi \left( {f\left( {\left| {D -
\left\langle {\Phi \left( {\frac{{A + B}}{2}} \right)x,x}
\right\rangle 1_{\mathcal{H}} } \right|} \right)} \right)x,x}
\right\rangle \nonumber
\end{aligned}$$ for each $x\in \mathcal{K}$ with $\|x\| =1$.
Also, for the operator $B\ge0$ with $s_3= \left\langle
{\Phi\left(\frac{{A + D}}{2}\right)x,x} \right\rangle$ $$\begin{aligned}
\left\langle {\Phi \left( {f\left(B \right)} \right)x,x}
\right\rangle &\ge f\left( {\left\langle {\Phi \left( {\frac{{A +
D}}{2}} \right)x,x} \right\rangle } \right) + C_{s_3} \left\langle {\Phi \left( { \frac{{2B
- A - D}}{2} } \right)x,x} \right\rangle \label{eq2.6}
\\
&\qquad + \left\langle {\Phi \left( {f\left( {\left| {B -
\left\langle {\Phi \left( {\frac{{A + D}}{2}} \right)x,x}
\right\rangle 1_{\mathcal{H}} } \right|} \right)} \right)x,x}
\right\rangle. \nonumber
\end{aligned}$$ for each $x\in \mathcal{K}$ with $\|x\| =1$.
Adding the inequalities – and then multiplying by $\frac{1}{3}$ we get $$\begin{aligned}
&\left\langle {\Phi \left( {\frac{{f\left( A \right) + f\left( B
\right) + f\left( D \right)}}{3}} \right)x,x} \right\rangle
\nonumber\\
&\ge \frac{1}{3}\left[f\left( {\left\langle {\Phi \left( {\frac{{A
+ B}}{2}} \right)x,x} \right\rangle } \right) + f\left(
{\left\langle {\Phi \left( {\frac{{B+D}}{2}} \right)x,x}
\right\rangle } \right)+ f\left( {\left\langle {\Phi \left(
{\frac{{A + D}}{2}} \right)x,x} \right\rangle } \right)\right]
\label{eq2.7}\\
&\qquad+ \frac{1}{3}\left[C_{s_1} \left\langle {\Phi \left(
{\frac{{2A - B-D}}{2}} \right)x,x} \right\rangle +C_{s_2} \left\langle
{\Phi \left( {\frac{{2D-A-B}}{2}} \right)x,x} \right\rangle
\right.
\nonumber\\
&\qquad\qquad+ C_{ s_3} \left\langle {\Phi \left( {
\frac{{2B - A - D}}{2} } \right)x,x} \right\rangle
\nonumber\\
&\qquad+ \left\langle {\Phi \left( {f\left( {\left| {A -
\left\langle {\Phi\left(\frac{{B + D}}{2}\right)x,x} \right\rangle
1_{\mathcal{H}} } \right|} \right)} \right)x,x} \right\rangle +
\left\langle {\Phi \left( {f\left( {\left| {D - \left\langle
{\Phi\left(\frac{{A+B}}{2}\right)x,x} \right\rangle
1_{\mathcal{H}} } \right|} \right)} \right)x,x} \right\rangle
\nonumber\\
&\qquad\qquad\left. + \left\langle {\Phi \left( {f\left( {\left|
{B - \left\langle {\Phi\left(\frac{{A + D}}{2}\right)x,x}
\right\rangle 1_{\mathcal{H}} } \right|} \right)} \right)x,x}
\right\rangle\right].\nonumber
\end{aligned}$$ Setting $C:=\min\left\{C_{s_1},C_{s_2},C_{s_3}\right\}$, then reduces to $$\begin{aligned}
&\left\langle {\Phi \left( {\frac{{f\left( A \right) + f\left( B
\right) + f\left( D \right)}}{3}} \right)x,x} \right\rangle
\nonumber\\
&\ge \frac{1}{3}\left[f\left( {\left\langle {\Phi \left( {\frac{{A
+ B}}{2}} \right)x,x} \right\rangle } \right) + f\left(
{\left\langle {\Phi \left( {\frac{{B+D}}{2}} \right)x,x}
\right\rangle } \right)+ f\left( {\left\langle {\Phi \left(
{\frac{{A + D}}{2}} \right)x,x} \right\rangle } \right)\right]
\nonumber\\
&\qquad+ \frac{1}{3}C\left[ \left\langle {\Phi \left( {\frac{{2A
- B-D}}{2}} \right)x,x} \right\rangle + \left\langle {\Phi \left(
{\frac{{2D-A-B}}{2}} \right)x,x} \right\rangle
+ \left\langle {\Phi \left( { \frac{{2B - A - D}}{2} } \right)x,x} \right\rangle \right.
\nonumber\\
&\qquad+ \left\langle {\Phi \left( {f\left( {\left| {A -
\left\langle {\frac{{B + D}}{2}x,x} \right\rangle 1_{\mathcal{H}}
} \right|} \right)} \right)x,x} \right\rangle + \left\langle {\Phi
\left( {f\left( {\left| {D - \left\langle {\frac{{A+B}}{2}x,x}
\right\rangle 1_{\mathcal{H}} } \right|} \right)} \right)x,x}
\right\rangle
\nonumber\\
&\qquad\left. + \left\langle {\Phi \left( {f\left( {\left| {B -
\left\langle {\frac{{A+D}}{2}x,x} \right\rangle 1_{\mathcal{H}} }
\right|} \right)} \right)x,x} \right\rangle\right]\nonumber
\\
&=\frac{1}{3}\left[f\left( {\left\langle {\Phi \left( {\frac{{A +
B}}{2}} \right)x,x} \right\rangle } \right) + f\left(
{\left\langle {\Phi \left( {\frac{{B+D}}{2}} \right)x,x}
\right\rangle } \right)+ f\left( {\left\langle {\Phi \left(
{\frac{{A + D}}{2}} \right)x,x} \right\rangle } \right)\right]
\label{eq2.8}\\
&\qquad+\frac{1}{3} \left[ \left\langle {\Phi \left( {f\left(
{\left| {A -
\left\langle {\frac{{B + D}}{2}x,x} \right\rangle 1_{\mathcal{H}}
} \right|} \right)} \right)x,x} \right\rangle + \left\langle
{\Phi \left( {f\left( {\left| {D - \left\langle
{\frac{{A+B}}{2}x,x} \right\rangle 1_{\mathcal{H}} } \right|}
\right)} \right)x,x} \right\rangle\right.
\nonumber\\
&\qquad\left. + \left\langle {\Phi \left( {f\left( {\left| {B -
\left\langle {\frac{{A+D}}{2}x,x} \right\rangle 1_{\mathcal{H}} }
\right|} \right)} \right)x,x} \right\rangle\right].\nonumber
\end{aligned}$$ Now, applying three times for $t= {\left\langle
{\Phi \left( {\frac{{A + B + C}}{3}} \right)x,x} \right\rangle }$ with $s_1,s_2,s_3$, then for each unit vector $x\in \mathcal{K}$, we get respectively, $$\begin{aligned}
& f\left( {\left\langle {\Phi \left( {\frac{{A + B + D}}{3}}
\right)x,x} \right\rangle } \right)
\nonumber\\
&\ge f\left( {\left\langle {\Phi \left( {\frac{{A + B}}{2}}
\right)x,x} \right\rangle } \right)
+ C_{s_1} \left\langle {\Phi \left( {\frac{{A + B + D}}{3}} \right)x,x} \right\rangle - C_{s_1} \left\langle {\Phi \left( {\frac{{A + B}}{2}} \right)x,x} \right\rangle
\label{eq2.9}\\
&\qquad+ f\left( {\left| {\left\langle {\Phi \left( {\frac{{A + B
+ D}}{3}} \right)x,x} \right\rangle - \left\langle {\Phi \left(
{\frac{{A + B}}{2}} \right)x,x} \right\rangle } \right|} \right)
\nonumber
\\
&=
f\left( {\left\langle {\Phi \left( {\frac{{A + B}}{2}} \right)x,x} \right\rangle } \right) + C_{s_1} \left\langle {\Phi \left( {\frac{{2D - A - B}}{6}} \right)x,x} \right\rangle + f\left( {\left| {\left\langle {\Phi \left( {\frac{{2D - A - B}}{6}} \right)x,x} \right\rangle } \right|} \right),\nonumber
\end{aligned}$$ $$\begin{aligned}
& f\left( {\left\langle {\Phi \left( {\frac{{A + B +D}}{3}}
\right)x,x} \right\rangle } \right)
\nonumber\\
&\ge f\left( {\left\langle {\Phi \left( {\frac{{A + B}}{2}}
\right)x,x} \right\rangle } \right) + C_{s_2}\left\langle {\Phi \left( {\frac{{A + B + D}}{3}}
\right)x,x} \right\rangle - C_{s_2}\left\langle {\Phi \left( {\frac{{B+D}}{2}} \right)x,x}
\right\rangle
\label{eq2.10}\\
&\qquad+ f\left( {\left| {\left\langle {\Phi \left( {\frac{{A + B+ D}}{3}} \right)x,x} \right\rangle - \left\langle {\Phi \left(
{\frac{{B+D}}{2}} \right)x,x} \right\rangle } \right|} \right)
\nonumber
\\
&= f\left( {\left\langle {\Phi \left( {\frac{{A + B}}{2}}
\right)x,x} \right\rangle } \right) + C_{s_2} \left\langle {\Phi \left( {\frac{{2A - B -
D}}{6}} \right)x,x} \right\rangle + f\left( {\left|
{\left\langle {\Phi \left( {\frac{{2A - B - D}}{6}} \right)x,x}
\right\rangle } \right|} \right)\nonumber
\end{aligned}$$ and $$\begin{aligned}
& f\left( {\left\langle {\Phi \left( {\frac{{A + B + D}}{3}}
\right)x,x} \right\rangle } \right)
\nonumber\\
&\ge f\left( {\left\langle {\Phi \left( {\frac{{A + D}}{2}}
\right)x,x} \right\rangle } \right) + C_{s_3} \left\langle {\Phi \left( {\frac{{A + B + D}}{3}}
\right)x,x} \right\rangle - C_{s_3} \left\langle {\Phi \left( {\frac{{A+D}}{2}} \right)x,x}
\right\rangle
\label{eq2.11}\\
&\qquad+ f\left( {\left| {\left\langle {\Phi \left( {\frac{{A + B +D}}{3}} \right)x,x} \right\rangle - \left\langle {\Phi \left(
{\frac{{A+D}}{2}} \right)x,x} \right\rangle } \right|} \right)
\nonumber
\\
&= f\left( {\left\langle {\Phi \left( {\frac{{A + D}}{2}}
\right)x,x} \right\rangle } \right) + C_{s_3}\left\langle {\Phi \left( {\frac{{2B - A - D}}{6}}
\right)x,x} \right\rangle + f\left( {\left| {\left\langle {\Phi
\left( {\frac{{2B - A - D}}{6}} \right)x,x} \right\rangle }
\right|} \right).\nonumber
\end{aligned}$$ Multiplying each inequality by $\frac{1}{3}$ and summing up the inequalities –, we get $$\begin{aligned}
&f\left( {\left\langle {\Phi \left( {\frac{{A + B + D}}{3}} \right)x,x} \right\rangle } \right)
\nonumber\\
&\ge \frac{1}{3}f\left( {\left\langle {\Phi \left( {\frac{{A + B}}{2}} \right)x,x} \right\rangle } \right) + \frac{1}{3}f\left( {\left\langle {\Phi \left( {\frac{{A + D}}{2}} \right)x,x} \right\rangle } \right) + \frac{1}{3}f\left( {\left\langle {\Phi \left( {\frac{{A + B}}{2}} \right)x,x} \right\rangle } \right)\label{eq2.12}
\\
&\qquad+ \frac{1}{3} C_{s_1} \left\langle {\Phi \left( {\frac{{2D - A - B}}{6}} \right)x,x} \right\rangle + \frac{1}{3} f\left( {\left| {\left\langle {\Phi \left( {\frac{{2C - A - B}}{6}} \right)x,x} \right\rangle } \right|} \right)
\nonumber\\
&\qquad+\frac{1}{3}C_{s_2} \left\langle {\Phi \left( {\frac{{2A - B - D}}{6}} \right)x,x} \right\rangle + \frac{1}{3} f\left( {\left| {\left\langle {\Phi \left( {\frac{{2A - B - D}}{6}} \right)x,x} \right\rangle } \right|} \right)\nonumber
\\
&\qquad+ \frac{1}{3} C_{s_3} \left\langle {\Phi \left( {\frac{{2B - A - D}}{6}} \right)x,x} \right\rangle +\frac{1}{3} f\left( {\left| {\left\langle {\Phi \left( {\frac{{2B - A - D}}{6}} \right)x,x} \right\rangle } \right|} \right)\nonumber
\end{aligned}$$ But since $C:=\min\left\{C_{s_1},C_{s_2},C_{s_3}\right\}$, then becomes $$\begin{aligned}
&f\left( {\left\langle {\Phi \left( {\frac{{A + B + D}}{3}}
\right)x,x} \right\rangle } \right)
\nonumber\\
&\ge \frac{1}{3}f\left( {\left\langle {\Phi \left( {\frac{{A +
B}}{2}} \right)x,x} \right\rangle } \right) + \frac{1}{3}f\left(
{\left\langle {\Phi \left( {\frac{{A + D}}{2}} \right)x,x}
\right\rangle } \right) + \frac{1}{3}f\left( {\left\langle {\Phi
\left( {\frac{{A + B}}{2}} \right)x,x} \right\rangle } \right)
\nonumber\\
&\qquad+ \frac{1}{3} C \left[\left\langle {\Phi \left( {\frac{{2D
- A - B}}{6}} \right)x,x} \right\rangle + \left\langle {\Phi
\left( {\frac{{2A - B - D}}{6}} \right)x,x} \right\rangle +
\left\langle {\Phi \left( {\frac{{2B - A - D}}{6}} \right)x,x}
\right\rangle \right]
\nonumber\\
&\qquad+ \frac{1}{3} f\left( {\left| {\left\langle {\Phi \left(
{\frac{{2D - A - B}}{6}} \right)x,x} \right\rangle } \right|}
\right) + \frac{1}{3} f\left( {\left| {\left\langle {\Phi \left(
{\frac{{2A - B - D}}{6}} \right)x,x} \right\rangle } \right|}
\right)
\nonumber\\
&\qquad+ \frac{1}{3} f\left( {\left| {\left\langle {\Phi \left(
{\frac{{2B - A - D}}{6}} \right)x,x} \right\rangle } \right|}
\right)
\nonumber\\
&=\frac{1}{3}f\left( {\left\langle {\Phi \left( {\frac{{A +
B}}{2}} \right)x,x} \right\rangle } \right) + \frac{1}{3}f\left(
{\left\langle {\Phi \left( {\frac{{A + D}}{2}} \right)x,x}
\right\rangle } \right)\label{eq2.13}
\\
&\qquad + \frac{1}{3}f\left( {\left\langle {\Phi \left( {\frac{{A
+ B}}{2}} \right)x,x} \right\rangle } \right) + \frac{1}{3}
f\left( {\left| {\left\langle {\Phi \left( {\frac{{2D - A -
B}}{6}} \right)x,x} \right\rangle } \right|} \right)
\nonumber\\
&\qquad+\frac{1}{3} f\left( {\left| {\left\langle {\Phi \left(
{\frac{{2A - B - D}}{6}} \right)x,x} \right\rangle } \right|}
\right) +\frac{1}{3}f\left( {\left| {\left\langle {\Phi \left(
{\frac{{2B - A - D}}{6}} \right)x,x} \right\rangle } \right|}
\right).\nonumber
\end{aligned}$$ Adding the inequalities and we get that $$\begin{aligned}
&\left\langle {\Phi \left( {\frac{{f\left( A \right) + f\left( B
\right) + f\left( D \right)}}{3}} \right)x,x} \right\rangle
+f\left( {\left\langle {\Phi \left( {\frac{{A + B + D}}{3}}
\right)x,x} \right\rangle } \right)
\nonumber\\
&\ge \frac{2}{3}\left[f\left( {\left\langle {\Phi \left( {\frac{{A
+ B}}{2}} \right)x,x} \right\rangle } \right) + f\left(
{\left\langle {\Phi \left( {\frac{{B+D}}{2}} \right)x,x}
\right\rangle } \right)+ f\left( {\left\langle {\Phi \left(
{\frac{{A + D}}{2}} \right)x,x} \right\rangle }
\right)\right]\nonumber
\\
&\qquad+ \frac{1}{3} \left[\left\langle {\Phi \left( {f\left(
{\left| {A - \left\langle {\Phi\left(\frac{{B + D}}{2}\right)x,x}
\right\rangle 1_{\mathcal{H}} } \right|} \right)} \right)x,x}
\right\rangle + \left\langle {\Phi \left( {f\left( {\left| {D -
\left\langle {\Phi\left(\frac{{A + B}}{2}\right)x,x} \right\rangle
1_{\mathcal{H}} } \right|} \right)} \right)x,x}
\right\rangle\right.
\nonumber\\
&\qquad\qquad + \left\langle {\Phi \left( {f\left( {\left| {B -
\left\langle {\Phi\left(\frac{{A + D}}{2}\right)x,x} \right\rangle
1_{\mathcal{H}} } \right|} \right)} \right)x,x} \right\rangle+
f\left( {\left| {\left\langle {\Phi
\left( {\frac{{2D - A - B}}{6}} \right)x,x} \right\rangle }
\right|} \right) \nonumber
\\
&\qquad\qquad\left.+ f\left( {\left| {\left\langle {\Phi \left(
{\frac{{2A - B - D}}{6}} \right)x,x} \right\rangle } \right|}
\right) + f\left( {\left| {\left\langle {\Phi \left( {\frac{{2B -
A - D}}{6}} \right)x,x} \right\rangle } \right|}
\right)\right]\nonumber
\end{aligned}$$ for each $x\in \mathcal{K}$ with $\|x\| =1$, which gives the required inequality in .
\[cor1\]Let $A,B,C\in \mathcal{B}\left(\mathcal{H} \right)$ be three positive selfadjoint operators, $\Phi:\mathcal{B}\left(\mathcal{H} \right)\to \mathcal{B}\left(\mathcal{K} \right)$ be a normalized positive linear map. If $f:\left[0,\infty\right)\to \mathbb{R}$ is subquadratic, then we have $$\begin{aligned}
&\left\langle {\Phi \left( {\frac{{f\left( A \right) + f\left(
B \right) + f\left( D \right)}}{3}} \right)x,x} \right\rangle
+f\left( {\left\langle {\Phi \left( {\frac{{A + B + D}}{3}}
\right)x,x} \right\rangle } \right)
\label{eq2.14}\\
&\le \frac{2}{3}\left[f\left( {\left\langle {\Phi \left(
{\frac{{A + B}}{2}} \right)x,x} \right\rangle } \right) + f\left(
{\left\langle {\Phi \left( {\frac{{B+D}}{2}} \right)x,x}
\right\rangle } \right)+ f\left( {\left\langle {\Phi \left(
{\frac{{A + D}}{2}} \right)x,x} \right\rangle }
\right)\right] \nonumber
\\
&\qquad+ \frac{1}{3} \left[\left\langle {\Phi \left( {f\left(
{\left| {A -
\left\langle {\Phi\left(\frac{{B + D}}{2}\right)x,x}
\right\rangle 1_{\mathcal{H}} } \right|} \right)} \right)x,x}
\right\rangle + f\left( {\left| {\left\langle
{\Phi \left( {\frac{{2A - B - D}}{6}} \right)x,x} \right\rangle
} \right|} \right)\right.
\nonumber\\
&\qquad\qquad+\left\langle {\Phi \left( {f\left( {\left| {D -
\left\langle {\Phi\left(\frac{{A + B}}{2}\right)x,x}
\right\rangle 1_{\mathcal{H}} } \right|} \right)} \right)x,x}
\right\rangle+ f\left( {\left| {\left\langle {\Phi
\left( {\frac{{2D - A - B}}{6}} \right)x,x} \right\rangle }
\right|} \right)
\nonumber\\
&\qquad\qquad\left. + \left\langle {\Phi \left( {f\left( {\left|
{B -
\left\langle {\Phi\left(\frac{{A + D}}{2}\right)x,x}
\right\rangle 1_{\mathcal{H}} } \right|} \right)} \right)x,x}
\right\rangle + f\left( {\left| {\left\langle
{\Phi \left( {\frac{{2B - A - D}}{6}} \right)x,x} \right\rangle
} \right|} \right)\right]\nonumber
\end{aligned}$$ for each $x\in \mathcal{K}$ with $\|x\| =1$.\
Repeating the same steps in the proof of Theorem \[thm2.1\], by writing $`\le$’ instead of $`\ge$’ and in this case we consider $C:=\max\left\{C_{s_1},C_{s_2},C_{s_3}\right\}$,
A generalization of the result in Theorem \[thm1\] is deduced as follows:
\[cor2\]Let $A\in \mathcal{B}\left(\mathcal{H} \right)$ be a positive selfadjoint operator, $\Phi:\mathcal{B}\left(\mathcal{H}
\right)\to \mathcal{B}\left(\mathcal{K} \right)$ be a normalized positive linear map. If $f:\left[0,\infty\right)\to \mathbb{R}$ is super(sub)quadratic, then we have $$\begin{aligned}
f\left( {\left\langle {\Phi \left( {A} \right)x,x} \right\rangle }
\right) \le (\ge) \left\langle {\Phi \left( {f\left( A \right)}
\right)x,x} \right\rangle - \left\langle {\Phi \left( {f\left(
{\left| {A - \left\langle {\Phi\left(A\right)x,x} \right\rangle 1_{\mathcal{H}} }
\right|} \right)} \right)x,x} \right\rangle - f\left( {0} \right) \label{eq2.15}\end{aligned}$$ for each $x\in \mathcal{K}$ with $\|x\| =1$.
Setting $D=B=A$ in we get the required result.
According to Corollary \[cor2\], if $f\ge0$ $(f\le 0)$ then the above inequality refines and improves Theorem \[thm1\].
The classical Bohr inequality for scalars reads that if $a$, $b$ are complex numbers and $p,q>1$ with $\frac{1}{p}+\frac{1}{q}=1$, then $$\begin{aligned}
\left|a-b\right|^2\le p \left|a\right|^2+q\left|b\right|^2.\end{aligned}$$ The first result regarding operator version of Bohr inequality was established in [@H]. For refinements, generalizations and other related results see [@CHP], [@CP], [@FZ] and [@Z].
The following Bohr’s type inequalities for positive selfadjoint operators under positive linear maps are hold:
Let $A\in \mathcal{B}\left(\mathcal{H} \right)$ be a positive selfadjoint operator, $\Phi:\mathcal{B}\left(\mathcal{H}
\right)\to \mathcal{B}\left(\mathcal{K} \right)$ be a normalized positive linear map.
1. If $f:\left[0,\infty\right)\to \mathbb{R}$ is superquadratic, then we have $$\begin{aligned}
\left\| {\Phi \left( {f\left( {\left| {A - \left\| {\Phi \left( A
\right)} \right\|1_H } \right|} \right)} \right)} \right\| \le
\left\| {\Phi \left( {f\left( A \right)} \right)} \right\| -
f\left( {\left\| {\Phi \left( A \right)} \right\|} \right) -
f\left( 0 \right).\end{aligned}$$ In particular, let $f(t)=t^r$, $r\ge2$, $t\ge0$. $$\begin{aligned}
\left\| {\Phi \left( { \left| {A - \left\| {\Phi \left( A \right)}
\right\|1_H } \right|^r } \right)} \right\| \le \left\| {\Phi
\left( {A^r} \right)} \right\| - \left\| {\Phi \left( A \right)}
\right\|^r.\end{aligned}$$
2. If $f:\left[0,\infty\right)\to \mathbb{R}$ is subquadratic, then we have $$\begin{aligned}
\left\| {\Phi \left( {f\left( {\left| {A - \left\| {\Phi \left( A
\right)} \right\|1_H } \right|} \right)} \right)} \right\| \ge
\left\| {\Phi \left( {f\left( A \right)} \right)} \right\| -
f\left( {\left\| {\Phi \left( A \right)} \right\|} \right) -
f\left( 0 \right).\end{aligned}$$ In particular, let $f(t)=t^r$, $0<r\le2$, $t\ge0$. $$\begin{aligned}
\left\| {\Phi \left( { \left| {A - \left\| {\Phi \left( A \right)}
\right\|1_H } \right|^r } \right)} \right\| \ge \left\| {\Phi
\left( {A^r} \right)} \right\| - \left\| {\Phi \left( A \right)}
\right\|^r.\end{aligned}$$
Taking the supremum in over $x\in \mathcal{K}$ with $\|x\|=1$ we obtain the required result(s).
\[cor5\]Let $A_j,B_j,D_j\in \mathcal{B}\left(\mathcal{H} \right)$ be three positive selfadjoint operators for every $j=1,\cdots,n$. Let $\Phi_j:\mathcal{B}\left(\mathcal{H} \right)\to \mathcal{B}\left(\mathcal{K} \right)$ be positive linear maps such that $\sum\limits_{j = 1}^n {\Phi _j \left( {1_H } \right)} = 1_K$. If $f:\left[0,\infty\right)\to \mathbb{R}$ is superquadratic, then we have $$\begin{aligned}
&{f\left(\left\langle {\sum\limits_{j =
1}^n\Phi_j\left(\frac{A_j+B_j+D_j}{3}\right) u,u}
\right\rangle\right)}+ {\left\langle { \sum\limits_{j =
1}^n\Phi_j\left(\frac{{f\left( A_j \right) +f\left( B_j \right) +
f\left( D_j \right)}}{3}\right) u,u}\right\rangle}
\label{eq2.16}\\
&\ge \frac{2}{3}\left[ {f\left( {\left\langle{ \sum\limits_{j =
1}^n\Phi_j\left(\frac{{A_j + D_j}}{2}\right)u,u} \right\rangle }
\right) + f\left( {\left\langle { \sum\limits_{j =
1}^n\Phi_j\left(\frac{{B_j +D_j}}{2}\right)u,u} \right\rangle }
\right) + f\left( {\left\langle{ \sum\limits_{j =
1}^n\Phi_j\left(\frac{{A_j + B_j}}{2}\right)u,u} \right\rangle }
\right)} \right]
\nonumber\\
&\qquad+ \frac{1}{3} \left[\left\langle { \sum\limits_{j =
1}^n\Phi_j \left( {f\left( {\left| {A_j -
\left\langle { \sum\limits_{j = 1}^n\Phi_j\left(\frac{{B_j + D_j}}{2}\right)u,u}
\right\rangle 1_{\mathcal{H}} } \right|} \right)} \right)u,u}
\right\rangle + f\left( {\left| {\left\langle { \sum\limits_{j =
1}^n\Phi_j
\left( {\frac{{2A_j - B_j - D_j}}{6}} \right)u,u} \right\rangle }
\right|} \right)\right.
\nonumber \\
&\qquad + \left\langle { \sum\limits_{j = 1}^n\Phi_j \left(
{f\left( {\left| {D_j -
\left\langle { \sum\limits_{j = 1}^n\Phi_j\left(\frac{{A_j + B_j}}{2}\right)u,u}
\right\rangle 1_{\mathcal{H}} } \right|} \right)} \right)u_j,u_j} \right\rangle+ f\left( {\left| {\left\langle
{ \sum\limits_{j = 1}^n\Phi_j \left( {\frac{{2D_j - A_j- B_j}}{6}} \right)u,u} \right\rangle
} \right|} \right)
\nonumber\\
&\qquad + \left. \left\langle {\sum\limits_{j = 1}^n\Phi_j \left(
{f\left( {\left| {B_j -
\left\langle { \sum\limits_{j = 1}^n\Phi_j\left(\frac{{A_j + D_j}}{2}\right)u,u}
\right\rangle 1_{\mathcal{H}} } \right|} \right)} \right)u_j,u_j}
\right\rangle + f\left( {\left| {\left\langle
{ \sum\limits_{j = 1}^n\Phi_j \left( {\frac{{2B_j - A_j - D_j}}{6}} \right)u,u} \right\rangle
} \right|} \right)\right].\nonumber
\end{aligned}$$
Let $E$ stands for $A,B,D$. Since $ E\in
\mathcal{B}^+\left(\mathcal{H}\right)$, then there exists $E_1,\cdots, E_n \in \mathcal{B}^+\left(\mathcal{H}\right)$ (where $E_j$ stands for $A_j,B_j, D_j$ for all $j=1,\cdots, n$) such that $E= E_1 \oplus \cdots \oplus E_n \in \mathcal{B}^+\left(\mathcal{H} \oplus \cdots \oplus \mathcal{H} \right)$ for every unit vector $u=\left(u_1,\cdots,u_n\right)\in \mathcal{H} \oplus \cdots \oplus \mathcal{H}$. Let $\Phi:\mathcal{B}^+\left(\mathcal{H} \oplus \cdots \oplus \mathcal{H} \right)\to \mathcal{B}^+\left(\mathcal{K} \right)$ be a positive normalized linear map defined by $\Phi\left(E\right)=\sum\limits_{j = 1}^n {\Phi _j \left( {E_j } \right)}$. By utilizing Theorem \[thm2.1\] we get the desired result.
\[cor5\]Let $A_j\in \mathcal{B}\left(\mathcal{H} \right)$ be positive selfadjoint operators for each $j=1,\cdots,n$. Let $\Phi_j:\mathcal{B}\left(\mathcal{H} \right)\to \mathcal{B}\left(\mathcal{K} \right)$ be positive linear maps such that $\sum\limits_{j = 1}^n {\Phi _j \left( {1_H } \right)} = 1_K$. If $f:\left[0,\infty\right)\to \mathbb{R}$ is superquadratic, then we have $$\begin{gathered}
\left\langle {\sum\limits_{j = 1}^n\Phi_j \left( {f\left( A_j \right)} \right)u,u}
\right\rangle
\\
\ge (\le) f\left( {\left\langle {\sum\limits_{j = 1}^n\Phi_j \left( {A_j} \right)u,u} \right\rangle } \right) + \left\langle {\sum\limits_{j = 1}^n\Phi_j \left( {f\left( {\left| {A_j - \sum\limits_{j = 1}^n\left\langle {\Phi_j\left(A_j\right)u,u} \right\rangle 1_{\mathcal{H}} } \right|} \right)} \right)u,u} \right\rangle + f\left( {0} \right) \label{eq2.17}
\end{gathered}$$ for each $x\in \mathcal{K}$ with $\|x\| =1$.
Setting $D_j=B_j=A_j$ for each $j=1,\cdots,n$, in we get the required result.
As a direct consequence of Theorem \[thm2.1\], the expected operator version Popoviciu’s inequality for convex functions would be as follows:
\[prp1\]Let $A,B,C\in \mathcal{B}\left(\mathcal{H} \right)$ be three positive selfadjoint operators, $\Phi:\mathcal{B}\left(\mathcal{H} \right)\to \mathcal{B}\left(\mathcal{K} \right)$ be a normalized positive linear map. If $f:\left[0,\infty\right)\to \mathbb{R}$ is non-negative and superquadratic, then $f$ is convex and $$\begin{aligned}
&\left\langle {\Phi \left( {\frac{{f\left( A \right) + f\left( B \right) + f\left( D \right)}}{3}} \right)x,x} \right\rangle +f\left( {\left\langle {\Phi \left( {\frac{{A + B + D}}{3}} \right)x,x} \right\rangle } \right)
\label{eq2.18}\\
&\ge \frac{2}{3}\left[f\left( {\left\langle {\Phi \left( {\frac{{A + B}}{2}} \right)x,x} \right\rangle } \right) + f\left( {\left\langle {\Phi \left( {\frac{{B+D}}{2}} \right)x,x} \right\rangle } \right)+ f\left( {\left\langle {\Phi \left( {\frac{{A + D}}{2}} \right)x,x} \right\rangle } \right)\right]\nonumber
\end{aligned}$$ for each $x\in \mathcal{K}$ with $\|x\| =1$.
Since $f$ is non-negative superquadratic then by Lemma \[lemma1\] $f$ is convex and so that from , we get $$\begin{aligned}
&\left\langle {\Phi \left( {\frac{{f\left( A \right) + f\left( B
\right) + f\left( D \right)}}{3}} \right)x,x} \right\rangle
+f\left( {\left\langle {\Phi \left( {\frac{{A + B + D}}{3}}
\right)x,x} \right\rangle } \right)
\\
&\ge \frac{2}{3}\left[f\left( {\left\langle {\Phi \left(
{\frac{{A + B}}{2}} \right)x,x} \right\rangle } \right) + f\left(
{\left\langle {\Phi \left( {\frac{{B+D}}{2}} \right)x,x}
\right\rangle } \right)+ f\left( {\left\langle {\Phi \left(
{\frac{{A + D}}{2}} \right)x,x} \right\rangle }
\right)\right]\nonumber
\\
&\qquad+ \frac{1}{3} \left[\left\langle {\Phi \left( {f\left(
{\left| {A -
\left\langle {\Phi\left(\frac{{B + D}}{2}\right)x,x}
\right\rangle 1_{\mathcal{H}} } \right|} \right)} \right)x,x}
\right\rangle + f\left( {\left| {\left\langle
{\Phi \left( {\frac{{2A - B - D}}{6}} \right)x,x} \right\rangle
} \right|} \right)\right.
\nonumber\\
&\qquad\qquad+\left\langle {\Phi \left( {f\left( {\left| {D -
\left\langle {\Phi\left(\frac{{A + B}}{2}\right)x,x}
\right\rangle 1_{\mathcal{H}} } \right|} \right)} \right)x,x}
\right\rangle+ f\left( {\left| {\left\langle {\Phi
\left( {\frac{{2D - A - B}}{6}} \right)x,x} \right\rangle }
\right|} \right)
\nonumber\\
&\qquad\qquad\left. + \left\langle {\Phi \left( {f\left( {\left|
{B -
\left\langle {\Phi\left(\frac{{A + D}}{2}\right)x,x}
\right\rangle 1_{\mathcal{H}} } \right|} \right)} \right)x,x}
\right\rangle + f\left( {\left| {\left\langle
{\Phi \left( {\frac{{2B - A - D}}{6}} \right)x,x} \right\rangle
} \right|} \right)\right]\nonumber
\\
&\ge \frac{2}{3}\left[f\left( {\left\langle {\Phi \left(
{\frac{{A + B}}{2}} \right)x,x} \right\rangle } \right) + f\left(
{\left\langle {\Phi \left( {\frac{{B+D}}{2}} \right)x,x}
\right\rangle } \right)+ f\left( {\left\langle {\Phi \left(
{\frac{{A + D}}{2}} \right)x,x} \right\rangle }
\right)\right]\nonumber
\end{aligned}$$ which gives .
\[prp2\]Let $A,B,D\in \mathcal{B}\left(\mathcal{H} \right)$ be three selfadjoint operators, $\Phi:\mathcal{B}\left(\mathcal{H} \right)\to \mathcal{B}\left(\mathcal{K} \right)$ be a normalized positive linear map and $f:\left[0,\infty\right)\to \mathbb{R}$ be a differentiable function with $f(0)=f^{\prime}(0)=0$. If $f^{\prime}$ is convex (concave), then $f$ is super(sub)quadratic and $$\begin{aligned}
&\left\langle {\Phi \left( {\frac{{f^{\prime}\left( A \right) + f^{\prime}\left(
B \right) + f^{\prime}\left( D \right)}}{3}} \right)x,x} \right\rangle
+f^{\prime}\left( {\left\langle {\Phi \left( {\frac{{A + B + D}}{3}}
\right)x,x} \right\rangle } \right)
\label{eq2.19}\\
&\ge (\le) \frac{2}{3}\left[f^{\prime}\left( {\left\langle {\Phi \left(
{\frac{{A + B}}{2}} \right)x,x} \right\rangle } \right) + f^{\prime}\left(
{\left\langle {\Phi \left( {\frac{{B+D}}{2}} \right)x,x}
\right\rangle } \right)+ f^{\prime}\left( {\left\langle {\Phi \left(
{\frac{{A + D}}{2}} \right)x,x} \right\rangle }
\right)\right]\nonumber
\end{aligned}$$ for each $x\in \mathcal{K}$ with $\|x\| =1$.
The superquadratic of $f$ follows from Lemma \[lemma2\]. To obtain the inequality we apply the same technique considered in the proof of Theorem \[thm2.1\], by applying for $f^{\prime}$ instead of for $f$, so that we get the required result.
\[prp3\]Let $A,B,D\in \mathcal{B}\left(\mathcal{H} \right)$ be three selfadjoint operators, $\Phi:\mathcal{B}\left(\mathcal{H} \right)\to \mathcal{B}\left(\mathcal{K} \right)$ be a normalized positive linear map and $g:\left[0,\infty\right)\to \mathbb{R}$ be a continuous function. If $g$ is convex (concave) and $g(0)=0$, then $$\begin{aligned}
&\left\langle {\Phi \left( {\frac{{g\left( A \right) + g\left(
B \right) + g\left( D \right)}}{3}} \right)x,x} \right\rangle
+g\left( {\left\langle {\Phi \left( {\frac{{A + B + D}}{3}}
\right)x,x} \right\rangle } \right)
\label{eq2.20}\\
&\ge (\le) \frac{2}{3}\left[g\left( {\left\langle {\Phi \left(
{\frac{{A + B}}{2}} \right)x,x} \right\rangle } \right) + g\left(
{\left\langle {\Phi \left( {\frac{{B+D}}{2}} \right)x,x}
\right\rangle } \right)+ g\left( {\left\langle {\Phi \left(
{\frac{{A + D}}{2}} \right)x,x} \right\rangle }
\right)\right]\nonumber
\end{aligned}$$ for each $x\in \mathcal{K}$ with $\|x\| =1$.
Applying Corollary \[cor5\] for $G\left( t \right) = \int_0^t
{g\left( s \right)ds}$, $t\in \left[0,\infty\right)$, then it’s easy to observe that $G\left(0\right)=G^{\prime}\left(0\right)=0$ and $G^{\prime}\left(t\right)=g\left(t\right)$ is convex (concave) for all $t\in \left[0,\infty\right)$.
Inequality holds with more weaker conditions, indeed neither continuity assumption nor the image of $0$ is needed, it is hold just with convexity assumption, as follows:
\[thm3\]Let $A,B,D\in \mathcal{B}\left(\mathcal{H} \right)$ be three selfadjoint operators with $\operatorname{sp}\left(A\right),\operatorname{sp}\left(B\right),\operatorname{sp}\left(D\right)\subset
\left[\gamma,\Gamma\right]$ for some real numbers $\gamma,\Gamma$ with $\gamma<\Gamma$. Let $\Phi:\mathcal{B}\left(\mathcal{H} \right)\to \mathcal{B}\left(\mathcal{K} \right)$ be a normalized positive linear map. If $f:
\left[\gamma,\Gamma\right]\to \mathbb{R}$ is convex (concave) function, then holds for each $x\in \mathcal{K}$ with $\|x\| =1$. The inequality is satisfied with $f(t)=t$.
Applying the same technique considered in the proof of Theorem \[thm2.1\], by applying for $f$ instead of for $f$.
Employing for $g(x)=|x|$, $x\in \mathbb{R}$ then we observe that $$\begin{aligned}
& \left| {\left\langle { \Phi \left( A+C \right) x ,x }
\right\rangle } \right| + \left|\left\langle { \Phi \left( B+C
\right)x ,x } \right\rangle \right| + \left|\left\langle { \Phi
\left(A+B\right)x,x} \right\rangle \right|
\nonumber\\
&\le \left| { \left\langle { \Phi \left( A+B+C \right) x
,x } \right\rangle }\right| + \left\langle {\Phi \left( { \left|
A \right| + \left| B \right| + \left| C \right|} \right)x ,x}
\right\rangle, \label{eq2.21}
\end{aligned}$$ which gives the operator version of Hlawka’s inequality for positive linear maps of selfadjoint operators in Hilbert space. Furthermore, by taking the supremum in over $x\in
\mathcal{K}$ with $\|x\|=1$, we obtain the following Hlawka’s norm inequality $$\begin{aligned}
&\left| {\left\| { \Phi \left( A+C \right) } \right\|} \right| +
\left|\left\| { \Phi \left( B+C \right) } \right\| \right| +
\left|\left\langle { \Phi \left(A+B\right) } \right\| \right|
\\
&\le \left| { \left\| { \Phi \left( A+B+C \right) }
\right\| }\right| + \left\| {\Phi \left( { \left| A \right| +
\left| B \right| + \left| C \right|} \right) } \right\|.
\end{aligned}$$ Generally, the Popoviciu’s extension of Hlawka’s norm inequality can be presented in the form: $$\begin{aligned}
&\frac{2}{3}\left[g\left(\left\| { \Phi \left( \frac{A+C}{2}
\right) } \right\| \right) +g\left(\left\|{ \Phi \left(
\frac{B+C}{2} \right) } \right\| \right) +g\left(\left\| { \Phi
\left( \frac{A+B}{2} \right) } \right\| \right) \right]
\nonumber\\
&\le g\left( { \left\| { \Phi \left( \frac{A+B+C}{3}
\right) } \right\| }\right) + \left\| {\Phi \left(
{\frac{{g\left( A \right) + g\left( B \right) + g\left( C
\right)}}{3}} \right) } \right\|
\end{aligned}$$ for every positive linear map $\Phi$ and convex increasing function $g$.
Let $A_j,B_j,D_j\in \mathcal{B}\left(\mathcal{H} \right)$ be three selfadjoint operators with $\operatorname{sp}\left(A_j\right),\operatorname{sp}\left(B_j\right),\operatorname{sp}\left(D_j\right)\subset
\left[\gamma,\Gamma\right]$ for some real numbers $\gamma,\Gamma$ with $\gamma<\Gamma$ and for every $j=1,\cdots,n$. Let $\Phi_j:\mathcal{B}\left(\mathcal{H} \right)\to \mathcal{B}\left(\mathcal{K} \right)$ be a positive linear map. such that $\sum\limits_{j = 1}^n {\Phi _j \left( {1_H } \right)} = 1_K$. If $ f:
\left[\gamma,\Gamma\right]\to \mathbb{R}$ is convex (concave) function, then $$\begin{gathered}
{f\left(\left\langle {\sum\limits_{j = 1}^n\Phi_j\left(\frac{A_j+B_j+D_j}{3}\right)
u,u} \right\rangle\right)}+
{\left\langle { \sum\limits_{j = 1}^n\Phi_j\left(\frac{{f\left( A_j \right) +f\left( B_j \right) +
f\left( D_j \right)}}{3}\right) u,u}\right\rangle}
\\
\ge (\le) \frac{2}{3}\left[ {f\left(
{\left\langle{ \sum\limits_{j = 1}^n\Phi_j\left(\frac{{A_j + D_j}}{2}\right)u,u} \right\rangle }
\right) + f\left( {\left\langle { \sum\limits_{j = 1}^n\Phi_j\left(\frac{{B_j +D_j}}{2}\right)u,u}
\right\rangle } \right) + f\left( {\left\langle{ \sum\limits_{j = 1}^n\Phi_j\left(\frac{{A_j +
B_j}}{2}\right)u,u} \right\rangle } \right)} \right]
\end{gathered}$$ for each $u\in \mathcal{K}$ with $\|u\| =1$.
[5]{}
S. Abramovich, On superquadraticity, [*J. Math. Inequal.* ]{}, [**3**]{} (3) (2009), 329–339.
S. Abramovich, S. Ivelić and J. Pečarić, Improvement of Jensen-Steffensen’s inequality for superquadratic functions, [*Banach J. Math. Anal.*]{}, [**4**]{} (1) (2010), 146-158.
S. Abramovich, G. Jameson and G. Sinnamon, Refining Jensen’s inequality, [*Bull. Math. Soc. Sci. Math. Roumanie*]{}, [**47**]{} (2004), 3–14.
R.P. Agarwal and S.S. Dragomir, A survey of Jensen type inequalities for functions of selfadjoint operators in Hilbert spaces, [*Comput. Math. Appl.*]{}, [**59**]{} (2010), 3785–3812.
J. Barić, A. Matković and J. Pečarić, A variant of the Jensen–Mercer operator inequality for superquadratic functions, [*Math. Comput. Modelling*]{}, [**51**]{} (2010) 1230–1239.
S. Banić, J. Pečarić and S. Varošanec, Superquadratic functions and refinements of some classical inequalities, [*J. Korean Math. Soc.*]{}, [**45**]{}, (2) (2008), 513–525.
M. Bencze, C.P. Niculescu and F. Popovici, Popoviciu’s inequality for functions of several variables, [*J. Math. Anal. Appl.,*]{} [**365**]{} (2010), 399–409.
P. Chansangiam P. Hemchote and P. Pantaragphong, Generalizations of Bohr inequality for Hilbert space operators, [*J. Math. Anal. Appl.*]{}, [**356**]{} (2009), 525–536.
W.–S. Cheung and J. Pečarić, Bohr’s inequalities for Hilbert space operators, [*J. Math. Anal. Appl.*]{}, [**323**]{} (2006), 403–412.
S.S. Dragomir, Operator inequalities of the Jensen, Čebyšev and Grüss type, Springer, New York, 2012.
M.Fuji and H. ZUO, Matrix order in Bohr inequality for operators, [*Banach J. Math. Anal.*]{}, [**4**]{} (2010), 21–27.
T. Furuta, J. Mićić, J. Pečarić and Y. Seo, Mond-Pečarić method in operator inequalities. Inequalities for bounded selfadjoint operators on a Hilbert space, Element, Zagreb, 2005.
D. Grinberg, Generalizations of Popoviciu’s inequality, (2008), arXiv:0803.2958v1.
O. Hirzallah, Non-commutative operator Bohr inequality, [*J. Math. Anal. Appl.*]{}, [**282**]{} (2003), 578–583.
J. Jensen, Sur les fonctions convexes et les inégalités entre les valeurs moyennes, [*Acta Math.*]{}, [**30**]{} (1906), 175–193.
M. Krnić, N. Lovrinčević, J. Pečarić, J. Perić, Superadditivity and monotonicity of the Jensen-type functionals: New Methods for improving the Jensen-type Inequalities in Real and in Operator Cases, Element, Zagreb, 2016.
M. Kian, Operator Jensen inequality for superquadratic functions, [*Linear Algebra and its Applications*]{}, [**456**]{}, (2014), 82–87.
M. Kian and S.S. Dragomir, Inequalities involving superquadratic functions and operators, [*Mediterr. J. Math.*]{}, [**11**]{} (4) (2014), 1205–1214.
J. Mićić, J. Pečarić and J. Perić, Extension of the refined Jensen’s operator inequality with condition on spectra, [*Ann. Funct. Anal.*]{}, [**3**]{} (1) (2012), 67–85.
D.S. Mitrinović, J. Pečarić and A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic, Dordrecht, 1993.
F.C. Mitroi and N. Minculete, On the Jensen functional and superquadraticity, [*Aequat. Math.*]{}, [**90**]{} (4) (2016), 705–718.
B. Mond and Pečarić, Convex inequalities in Hilbert space, [*Houston J. Math.*]{}, [**19**]{} (1993), 405–420.
M.S. Moslehian and R. Rajć Generalizations of Bohr’s inequality in Hilbert $C^*$-modules, [*Linear and Multilinear Algebra*]{}, [**58**]{} (2010), 323–331.
C.P. Niculescu, Convexity according to the geometric mean, *Math. Inequal. Appl.*, [**3**]{} (2) (2000), 155–167.
C.P. Niculescu, F. Popoviciu, A refinement of Popoviciu’s inequality, [*Bull. Math. Soc. Sci. Math. Roumanie*]{}, Tome [ **49**]{} (3) (2006), 285–290.
C.P. Niculescu, L.E. Persson, Convex Functions and Their Applications. A Contemporary Approach, CMS Books Math., vol. 23, Springer-Verlag, New York, 2006.
T. Popoviciu, Sur certaines inégalités qui caractérisent les fonctions convexes, Analele Stiintifice Univ. Al. I. Cuza, Iasi, Sectia Mat., 11 (1965), 155–164.
D.M. Smiley and M.F. Smiley, The polygonal inequalities, [*Amer. Math. Mon.*]{}, [**71**]{} (7) (1964), 755–760.
S.-E. Takahasi, Y. Takahashi and A. Honda, A new interpretation of Djoković’s inequality, [*J. Nonlinear Convex Anal.*]{}, [ **1**]{} (3) (2000), 343–350.
S.-E. Takahasi, Y. Takahashi, S. Miyajima and H. Takagi, Convex sets and inequalities, [*J. Inequal. Appl.*]{}, [**2**]{} (205), 107–117.
P.M. Vasić and L.R. Stanković, Some inequalities for convex functions, [*Math. Balkanica*]{}, [**6**]{} (1976), 281–288.
R. Whitty, A generalised Hlawka inequality by D. Smiley & M. Smiley, Theorem of The day.
F. Zhang, On the Bohr inequality of operators, [*J. Math. Anal. Appl.*]{}, [**333**]{} (2007), 1264–1271.
|
---
author:
- |
[^1]\
Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India\
E-mail:
- |
M. Padmanath\
Institut fur Theoretische Physik, Universitat Regensburg, Universitatsstrase 31, 93053 Regensburg, Germany\
E-mail:
- |
Randy Lewis\
Department of Physics & Astronomy, York University, Toronto, ON M3J 1P3, Canada\
E-mail:
title: 'Charmed-Bottom mesons from Lattice QCD'
---
Introduction
============
Lattice QCD methods provide a unique opportunity to study hadronic physics, particularly the energy spectra of hadrons. Substantial progress has been made to extract the ground and the excited states of charmed hadrons, particularly for charmonia. However, the study of bottom hadrons with relativistic actions and controlled discretizations is still prohibitively computer intensive though recent progress in relativistic heavy quark actions is promising. Most studies involving bottom quarks are based on the non-relativistic QCD (NRQCD) formulation, heavy quark effective theory (HQET) and the static quark formulation.
Study of heavy mesons plays a very important role in understanding the nature of strong interactions. Though heavy quarkonia and heavy light mesons have been investigated in great detail, not much is known about the heavy mesons containing only charm and bottom quarks. It is expected that the physics of charmed-bottom mesons involves multiple scales : $1/m_b$ ($v_b = 0.05$), $1/m_c$ ($v_c = 0.4-0.5$), and $\Lambda_{QCD}$. It is interesting to investigate whether charmed-bottom mesons behave like heavy-light mesons or like quarkonia states. The information about hyperfine splittings and other spin splittings, in addition to the results on decay constants, can shed light on the structure of these states.
Experimentally only one state, $B_c(0^{-})$, is established with mass at 6275(1) MeV [@PDG]. Recently ATLAS observed another $B_c$ meson with mass $6842\pm 4 \pm 5$ MeV, which was interpreted as the excited state, $B_c^{0^{-}}(2S)$ [@Aad:2014laa]. However, this excitation has not been confirmed by other experiments yet. On the theoretical side, potential model predictions for these states vary widely. For example, the prediction for $1S$ hyperfine splitting varies in the range 40-90 MeV [@q1; @q2; @Kiselev:1994rc; @Ebert:2002pp; @Godfrey:2004ya]. This is due to various ways of tuning the heavy quark potentials with spin dependent terms and less clarity of the wavefunction at the origin [@q1; @q2; @Kiselev:1994rc; @Ebert:2002pp; @Godfrey:2004ya]. On the other hand, using lattice QCD methods, though heavy-light or heavy-heavy mesons were studied extensively not many calculations were carried out for charmed-bottom mesons and baryons. Only two collaborations, HPQCD [@Gregory:2009hq; @Dowdall:2012ab] and Wurtz [*et al.*]{} [@Wurtz:2015mqa], have studied charmed-bottom mesons recently. Similarly, there are very few recent results for charmed-bottom baryons [@Francis:2016; @Brown:2014ena].
In this report, we present our preliminary results on charmed-bottom mesons and baryons. For the bottom quark we use an NRQCD action with non-perturbatively tuned coefficients with terms up to $\mathcal{O}(v^4)$, whereas the overlap action is utilized for the valence charm quark. Hyperfine splitting between $1S$ $B_c$ states as well as masses for other $B_c$ mesons are predicted. Moreover, ground state energy spectra of baryons with charm and bottom quarks are also predicted.
Simulation Details
==================
We use three sets of dynamical 2+1+1 flavours HISQ gauge field ensambles generated by the MILC collaboration : $24^3 \times 64$, $32^3 \times
96$ and $48^3 \times 96$ lattices at gauge couplings $10/g^2 = 6.00,
6.30$ and $6.72$, respectively. The details of these gauge configurations are summarized in Ref. [@Bazavov:2012xda]. We use the unphysical $\bar{s} s$ pseudoscalar mass equal to 685 MeV to tune the strange quark mass while the $\Omega_{sss}$ baryon mass is used for calculating the lattice spacings as mentioned in Refs. [@Basak:2012py; @Basak:2013oya]. The measured lattice spacings, 0.1192(14), 0.0877(10) and 0.0582(5) [*[fm]{}*]{}, are consistent with 0.1207(11) 0.0888(8) and 0.0582(5) [*[fm]{}*]{}, respectively, measured by MILC collaboration with this set of ensembles using the $r_1$ parameter.
We have adopted an NRQCD formulation for the bottom quark. The non-relativistic action that we use is discussed in Ref. [@Lepage:1992tx]. We have considered all terms up to $1/M_{0}^2$ and the leading term of the order of $1/M_{0}^3$, where $M_{0} = am_b $ is the bare mass for bottom quark in lattice units. The NRQCD Hamiltonian is given by, $$H = H_{0} + \delta H,$$ where $H_{0}$ is the kinetic term and is defined by, $$H_{0} = - {{\Delta^{(2)}}\over{2M_0}},$$ whereas, $\Delta H$ contains interaction terms and is given by, $$\begin{aligned}
\Delta H &=& - c_1 {{(\Delta^{(2)})^2}\over{8M_0^3}} +
c_2 {{i}\over {8M_0^2}} {\left(\tilde{\nabla} \cdot {\bf\tilde{E}} - {\bf\tilde{E}} \cdot \tilde{\nabla}\right)}
- c_3 {{1}\over{8M_0^2}} \bf{\sigma} \cdot {\left(\tilde{\nabla} \times {\tilde{\bf{E}}} - {\tilde{\bf{E}}} \times \tilde{\nabla}\right)} \nonumber \\
&& - c_4 {{1}\over{2M_0}} {\bf{\sigma}} \cdot {\tilde{\bf{B}}} \,+ c_5 {{\Delta^{(4)}}\over{24M_0}} \, - c_6 {{(\Delta^{(2)})^2}\over{8nM_0^2}} .\end{aligned}$$ The quantities with a tilde assume an $\mathcal{O}(a)$ corrected form of discretization. $\nabla$ is the symmetric lattice derivative while $\Delta^{(2)}$ and $\Delta^{(4)}$ represent the lattice discretized versions of $\sum_{i}D^2$ and $\sum_{i}D^4$, respectively. More details of this Hamiltonian can be found in Ref. [@Lewis:2008fu]. This Hamiltonian is improved by including spin-independent terms through $\mathcal{O}(v^4)$. For the coarser two ensembles, we use the values of the improvement coefficients, $c_1$ to $c_6$, as estimated non-perturbatively by the HPQCD collaboration [@Dowdall:2011wh]. For the finer lattice, we use tree level coefficients. The NRQCD quark propagators are obtained by usual time evolution $$G({\bf{x}},t+1) = \left(1 - {\delta H\over 2}\right)\left(1 - {\delta H_0\over 2n}\right)^nU^{\dagger}_t(x)
\left(1 - {\delta H_0\over 2n}\right)^n\left(1 - {\delta H\over 2}\right) G({\bf{x}},t).$$ A wall source is utilized as smearing function for calculating these quark propagators.
The bottom quark mass is tuned by equating the lattice spin-average mass of $1S$ bottomonium to its experimental value. The lattice spin-average mass is obtained from the kinetic mass relation : $$\bar{M}_{kin}(1S) = {3\over 4} aM_{kin}(\Upsilon) + {1\over 4} aM_{kin}(\eta_b),$$ where the kinetic mass is calculated from the relativistic energy-momentum dispersion relation : $$aM_{kin} = {{a^2p^2 - (a\Delta E)^2}\over{2a\Delta E}}.$$ Here $a\Delta E$s are extracted from the energy difference between the mesons with momenta $pa$ and zero. A momentum induced wall-source is utilized to obtain energy values from the correlators with finite momenta. This method was found to be very efficient compared to point or smeared sources [@Basak:2012py] and helps to obtain kinetic masses precisely with significantly little statistics.
For valence charm quark propagators we use the overlap action as described in Refs. [@Basak:2012py; @Basak:2013oya]. The overlap action does not have $\mathcal{O}(ma)$ errors, and it is chirally symmetric at finite lattice spacings. The charm mass is tuned by equating the spin-averaged kinetic mass of the $1S$ charmonia states to its physical value. The tuned bare charm quark masses ($am_c$) are found to be 0.528, 0.425 and 0.29 on coarser to finer lattices respectively. Details of the charm tuning was described in Ref. [@Basak:2013oya].
Results
=======
Charmed-bottom mesons
---------------------
Lattice methods suffer from discretization errors, particularly for hadrons with heavy quarks. A good agreement of hyperfine splitting of the $1S$ mesons on the lattice with the respective experimental value ensures good control over the discretization errors and hence a reliable estimation of the heavy meson spectra using lattice methods. Figure 1 shows our results for the hyperfine splittings in $1S$ quarkonium for three different ensembles we use. Figure 1(a) is for $1S$ charmonium showing our results with filled circles and experimental value with star symbol. An extrapolation with a form $1/a^3$, including systematics, gives a fit value 115(3) MeV for this splitting which is consistent with the experimental value 113.9(6) MeV. We have excluded the disconnected diagrams which is shown to reduce the splittings by a few MeV [@Levkova:2010ft]. Figure 1(b) is for $1S$ bottomonium. Black circles are results from tree-level coefficients and blue are with improved coefficients in the NRQCD action (this color coding will be followed throughout). A naive fit combining the results from improved coefficients for coarser lattices and the result from unimproved coefficients at the finer lattice gives $1S$ bottomonium hyperfine splitting as 64(3) MeV, whereas the experimental value is $62.3\pm 3$ MeV. These results show that the discretization errors in our calculation for quarkonia are well within control.
In figure 2(a), we show our main results on the hyperfine splittings of $B_c$ mesons : $M_{B^*_c}-M_{B_c}$. Results from other lattice calculations are shown by red squares. A combined fit with results from improved and tree-level coefficients yields an estimate for this splitting as $56^{+4}_{-3}$ MeV (shown by blue star) which is consistent with predictions from other lattice calculations [@Dowdall:2012ab; @Wurtz:2015mqa]. With an NRQCD action the hyperfine splitting of quarkonia is proportional to the $c^{2}_4$ term, which is $\mathcal{O}(\alpha^2_{s})$ in our case, and it also depends on higher order ($\mathcal{O}(v^6)$) operators. These higher order corrections would also be present in the $B_c$ mesons, which could be reduced in the ratio of such hyperfine splittings, e.g. between $B_s$ and $B_c$ mesons [@Dowdall:2012ab]. Similar to HPQCD [@Dowdall:2012ab] we have constructed the following ratio, $$R_{B_c} = {{\Delta^{hyp}_{B_c}}\over {\Delta^{hyp}_{B_s}}} = {{E_0(B_c^*) - E_0(B_c)}\over {E_0(B_s^*) - E_0(B_s)}}.$$ In figure \[fig\_hyp\_bc\](b) we show this ratio at three lattice spacings.
We also calculate the ground state masses of the axial-vector and scalar $B_c$ mesons. In figure \[fig\_Bc\_ax\_sc\](a) we plot the splitting between scalar and pseudoscalar states ($0^{+}-0^{-}$) while in figure 3(b) the splitting between axial vector and vector states ($1^{+}-1^{-}$) is shown. HPQCD results are shown with open square. After extrapolation with a form $1/a^2$ to the data from unimproved coefficients we obtain $\Delta_{B_c}^{0^{+}-0^{-}} = 414(16)$ MeV and $\Delta_{B_c}^{1^{+}-1^{-}} = 395(15)$ MeV (shown by blue star). Taking the experimental value of $B_c(0^{-})$, the ground state masses for the scalar and axial-vector $B_c$ mesons are $M_{B_c(0^{+})}$ = 6690(17) MeV and $M_{B_c(1^{+})}$ = 6726(16) MeV respectively.
Charmed-bottom baryons
----------------------
Baryons with three heavy quarks are interesting systems as they can provide important information about the potential between three heavy quarks and they can also be a study ground for effective field theories as well as perturbative QCD. Here we present the ground state masses of baryons containing only charm and bottom quarks, namely $\Omega_{ccb}({1\over 2}^+),
\Omega^*_{ccb}({3\over 2}^+), \Omega_{cbb}({1\over 2}^+)$ and $\Omega^*_{cbb}({3\over 2}^+)$. Very similar to mesons, the hyperfine splittings between these baryons can provide important information about the spin dependent interactions in heavy quark systems. In figure \[fig\_bc\_baryons\] we plot these splittings along with other lattice results [@Francis:2016; @Brown:2014ena] and quark model predictions [@Roberts:2007ni].
Finally, we present results for the triply bottom baryon $\Omega_{bbb}({3\over
2}^+)$, which may be viewed as baryonic analogues of bottomonium. Being heavier with three bottom quarks it provides a good arena to test the discretization error in a lattice calculation with bottom quarks. For the relative removal of discretization error due to the bottom quark we calculate the subtracted energy defined as $E_{sub}^{\Omega_{bbb}} =
M(\Omega_{bbb}) - {3\over 2} {\bar{M}}(\bar bb)$, where ${\bar{M}}(\bar bb)$ is the lattice estimate of the spin average $1S$ bottomonium mass. Figure \[fig\_triply\_bottom\] shows our results for this subtracted energy at different lattice spacings with improved and tree-level coefficients along with the result from another lattice calculation [@Brown:2014ena]. Flatness of the plot shows that discretization errors due to bottom quarks after subtraction are well within control. Combining data from improved coefficients, the prediction for the ground state spin-3/2 triply bottom baryon is following : $E^{\Omega_{bbb}}_{sub} = 194^{+4}_{-3}$ MeV, and taking the PDG spin average mass for $1S$ bottomonium as 9445(3) MeV, $M_{\Omega_{bbb}} = 14362^{+5}_{-4}$ MeV.
![The ground state energy of the spin-3/2 $\Omega_{bbb}$ baryon : The subtracted energy $E_{sub} = M(\Omega_{bbb}) - {3\over 2}{\bar M}(\bar bb)$, where ${\bar M}(\bar bb)$ is the $1S$ spin average bottomonium mass, is plotted at different lattice spacings. \[fig\_triply\_bottom\]](figs/bbb_sub.pdf){width="45.00000%"}
Conclusions
===========
We report the ground state energy spectra of charmed-bottom mesons. An NRQCD Hamiltonian with non-perturbatively improved coefficients is used for bottom quarks while a relativistic overlap action is adopted for charm quarks on a background of 2+1+1 flavours HISQ gauge configurations generated by the MILC collaboration. Results are obtained at three lattice spacings with physical volume of about 3 [*fm*]{}. We tune the bottom and the charm quark masses by equating the lattice spin-averaged kinetic masses of $1S$ states in bottomonia and charmonia with their experimental values. The hyperfine splittings between these $1S$ states are found to be 64(3) and $115(3)$ MeV for bottomonia and charmonia, respectively, which are quite consistent with their experimental values. Our result on hyperfine splitting between $1S$ states of $B_c$ mesons is $56^{+4}_{-3}$ MeV. This result is consistent with HPQCD [@Dowdall:2012ab] as well as Wurtz [*et al.*]{} [@Wurtz:2015mqa] who found this splitting to be 54(3) and 57.5(3) MeV, respectively. It is interesting to note that all lattice results are inconsistent with many quark model results which vary in the range between 40-90 MeV [@q1; @q2; @Kiselev:1994rc; @Ebert:2002pp; @Godfrey:2004ya]. Due to this rather small splitting, detection of $B_c^{*}(1^-)$ from its $B_c\gamma$ decay may turn out to be challenging in experiments. Taking the PDG value of $B_c(0^-)$ state as 6275(1) MeV, our prediction for the ground state mass of $B^{*}_c$ is $6331^{+4}_{-3}$ MeV. We also calculate the ground state energy of other $B_c$ mesons, namely, $B_c(0^+)$ and $B_c(1^+)$. Our preliminary results on the splittings between $(0^+ - 0^-)$ and $(1^+ - 1^-)$ states are found to be 411(16) MeV and 395(15) MeV, respectively. The ground state masses of baryons containing only charm and bottom quarks are also extracted. The hyperfine splittings between spin-3/2 and spin-1/2 $\Omega(ccb)$ and $\Omega(cbb)$ states are found to be 20(5) MeV and 21(5) MeV, respectively. Our prediction for the ground state spin-3/2 triply bottom baryon is $M_{\Omega_{bbb}} = 14362^{+5}_{-4}$ MeV.
Acknowledgement
===============
Computations were carried out on the Blue Gene/P of the Indian Lattice Gauge Theory Initiative, and on the Gaggle cluster of the Department of Theoretical Physics, TIFR. We thank A. Salve and K. Ghadiali for technical support. M. P. acknowledges support from the Austrian Science Fund FWF:I1313-N27 and the Deutsche and Forschungsgemeinschaft Grant No. SFB/TRR 55. R.L. is supported in part by NSERC of Canada. We are grateful to the MILC collaboration and in particular to S. Gottlieb for providing us with the HISQ lattices.
[99]{} C. Patrignani [*et al.*]{} \[Particle Data Group\], Chin. Phys. C, 40, 100001 (2016). G. Aad [*et al.*]{} \[ATLAS Collaboration\], Phys. Rev. Lett. [**113**]{}, no. 21, 212004 (2014).
W. Kwong and J. Rosner, Phys. Rev. D 44, 212 (1991). E. Eichten and C. Quigg, Phys. Rev. D 49, 5845 (1994).
V. V. Kiselev [*et al.*]{}, Phys. Rev. D [**51**]{}, 3613 (1995). D. Ebert [*et al.*]{}, Phys. Rev. D [**67**]{}, 014027 (2003). S. Godfrey, Phys. Rev. D [**70**]{}, 054017 (2004). E. B. Gregory [*et al.*]{} \[HPQCD\], Phys. Rev. Lett. [**104**]{}, 022001 (2010). R. J. Dowdall [*et al.*]{} \[HPQCD\], Phys. Rev. D [**86**]{}, 094510 (2012). M. Wurtz, R. Lewis and R. M. Woloshyn, Phys. Rev. D [**92**]{}, no. 5, 054504 (2015). A. Francis [*et al.*]{}, PoS LATTICE 2016, 133 (2016).
Z. S. Brown [*et al.*]{}, Phys. Rev. D [**90**]{}, no. 9, 094507 (2014).
A. Bazavov [*et al.*]{} \[MILC Collaboration\], Phys. Rev. D [**87**]{}, no. 5, 054505 (2013).
S. Basak [*et al.*]{}, PoS LATTICE [**2012**]{}, 141 (2012). S. Basak [*et al.*]{}, PoS LATTICE [**2013**]{}, 243 (2014). G. P. Lepage [*et al.*]{}, Phys. Rev. D [**46**]{}, 4052 (1992). R. Lewis and R. M. Woloshyn, Phys. Rev. D [**79**]{}, 014502 (2009). R. J. Dowdall [*et al.*]{} \[HPQCD Collaboration\], Phys. Rev. D [**85**]{}, 054509 (2012). L. Levkova and C. DeTar, Phys. Rev. D [**83**]{}, 074504 (2011).
W. Roberts and M. Pervin, Int. J. Mod. Phys. A [**23**]{}, 2817 (2008).
[^1]: Indian Lattice Gauge Theory Initiative
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---
bibliography:
- 'supplementv2.bib'
title: 'Supplement to “Single quasiparticle trapping in aluminum nanobridge Josephson junctions"'
---
Fabrication Details
===================
The device is fabricated using conventional double-angle shadow-mask evaporation and a liftoff process. First, a two-layer PMMA/copolymer resist stack is spun onto a silicon substrate. The nanobridge resonator structure is then defined lithographically in an SEM running at 30 keV. The SQUID pattern is a 750 nm wide, $2\times2 \ \mu$m square washer, with two 100 nm gaps spanned by the 25 nm wide nanobridges. After development in a solution of 1 part methyl isobutyl ketone to 3 parts isopropanol at $-15^\circ$ C, the sample is put in an electron-beam evaporator with a base pressure of $\sim 10^{-8}$ Torr. The evaporation is performed in two stages. First, 8 nm of aluminum are evaporated at normal incidence, depositing metal on the substrate everywhere exposed by the SEM beam. Next, the sample is tilted in-situ to an angle of $36^\circ$. This causes the nanobridge region to be occluded, as there is no longer a line-of-sight from the substrate to the evaporation source through the upper resist layer due to the steep aspect ratio of the nanobridge resist window. We evaporate 80 nm of aluminum in this configuration, defining banks which are $8+80\cos(36^\circ) = 73$ nm thick, connected by the 8 nm bridge. Finally, the resist is lifted off by soaking in acetone, leaving only the resonator with variable-thickness bridge defined.
Extracting $q_0$ and $L_J$
==========================
We model the nanobridge SQUID as two identical nanobridge junctions in parallel. The resonant frequency of the device is given by $\omega_0 = (C(L+L_J/2))^{-1/2}$, where $C$ is the total capacitance, $L_J$ is the inductance of a single nanobridge, and $L$ is the linear inductance of the circuit. Defining the junction participation ratio $q$ by $$\begin{aligned}
q(\phi) &\equiv\frac{ L_J(\phi)/2}{L + L_J(\phi)/2}\\
q_0 &\equiv q(\phi=0) ,\end{aligned}$$ one can write $$\omega_0(\phi) = \omega_0(0)[1 + q_0\frac{L_J(\phi) - L_J(0)}{L_J(0)}]^{-1/2}\,.
\label{eq:omega0}$$ In order to fit $q_0$, we assume a KO-1 current-phase relation (CPR) for the nanobridges[@KO1_cpr], thus giving $$\omega_0(\phi) = \omega_0(0)[1 + q_0\frac{\sin\frac{\delta}{2}\tanh^{-1}\sin\frac{\delta}{2}}{1 - \sin\frac{\delta}{2}\tanh^{-1}\sin\frac{\delta}{2}}]^{-1/2}\,.$$ Here, we use $\delta = \pi \phi$. Fitting the flux tuning of the resonant frequency with this equation gives $q_0 = 0.015$. We performed EM simulations of the oscillator structure in Microwave Office using the AXIEM solver and found good agreement with experimentally measured quantities. These simulations gave $L = 1.2$ nH, and so we find $L_J(0) = 2 L q_0 / (1-q_0) = 36$ pH.
Theoretical Model
=================
We consider a weak link with $N_e$ channels, i.e. $N_e$ Andreev states. Let $n_i =0,1$ denote the number of quasiparticles (QPs) trapped in the $i$-th channel. For a given configuration $\{n_i \}$ of QPs, we can find the resonant frequency of the circuit, $\omega_0(\{ n_i\})$. We denote the frequency-dependent response function as $S(\omega,\omega_0(\{n_i \}))$, since it depends on both $\omega$ and resonant frequency, which is a function of QP configuration. In the experiment we measure the average value of the response function, $\bar S (\omega)$: $$\bar S (\omega) = \sum_{\{ n_i \}} p(\{n_i\}) S(\omega, \omega_0(\{ n_i \}))\,. \label{barS}$$ where $p(\{n_i\})$ is the probability to find the system in the configuration $\{ n_i \}$. We assume it can be written as a product $p(\{n_i\}) = \prod_i p_i(n_i)$. The average number of QPs is then $\bar n_{trap} = \sum_i p_i$, where $p_i=p_i(n_i=1)$. Assuming $p_i \ll 1$, the probability to have zero QPs is $e^{-\bar n_{trap}}$. The expression for $\bar S (\omega)$ can be written as a convolution: $$\bar S (\omega) = \int d\Omega F(\Omega) S(\omega, \Omega)\,, \label{Sconv}$$ where $$F(\Omega) = \sum_{\{ n_i \}} p(\{ n_i \}) \delta(\Omega - \omega_0 (\{ n_i \}))\,. \label{Fnew}$$ We distinguish two different cases – large average number of QPs ($\bar n_{trap} \gg 1$) and small average number of QPs ($\bar n_{trap} \lesssim 1$). In the case of large average number of QPs, we evaluate $F(\Omega)$ using the saddle point approximation as follows: $$\begin{aligned}
F(\Omega) = \int\frac{d\alpha}{2 \pi} \sum_{ \{ n_i \} } p(\{ n_i \}) e^{i \alpha (\Omega - \omega_0(\{ n_i \}))} = \int\frac{d\alpha}{2 \pi} \sum_{\{ n_i \}} p(\{ n_i \}) \exp[i \alpha ( \Omega - \omega_0^{(0)} - \sum_i \frac{\partial \omega_0}{\partial n_i} n_i ) ]\,.\end{aligned}$$ Here we have expanded $\omega_0(\{ n_i \})$ around $\omega_0^{(0)}$, corresponding to no QPs, since each QP only slightly shifts the resonant frequency. Writing the probability $p(\{ n_i \})$ as a product of $p_i$’s we get: $$\begin{aligned}
F(\Omega) = \int \frac{d\alpha}{2 \pi} \exp[ i\alpha (\Omega - \omega_0^{(0)}) + \sum_i \ln (1 - p_i + p_i e^{-i\alpha \partial \omega_0/\partial n_i} ) ]\,.\end{aligned}$$ The saddle point approximation of this integral then gives a Gaussian for $F(\Omega)$: $$\begin{aligned}
F(\Omega) \propto \exp\left[- \frac{(\Omega - \omega_0^{(0)} -\sum_i p_i \partial \omega_0/\partial n_i)^2}{2 \sum_i p_i (\partial \omega_0/\partial n_i)^2}\right] \,,\label{Flarge}\end{aligned}$$ yielding the Gaussian referred to in the main text with center frequency $\omega_0 = \omega_0^{(0)} + \sum_i p_i \frac{\partial \omega_0}{\partial n_i}$ and width $\delta \omega_0^2 = \sum_i p_i (\frac{\partial \omega_0}{\partial n_i})^2 $. For $S(\omega, \Omega)$ we assume a Lorentzian lineshape: $$\begin{aligned}
S(\omega, \Omega) = \frac{\Gamma/\pi}{(\omega- \Omega)^2 + \Gamma^2}\,, \label{Snew}\end{aligned}$$ where the width $\Gamma$ is easily measured from the resonance at zero flux. Thus, in the case of large number of QPs, the resonance is a convolved Gaussian-Lorentzian with its maximum shifted from the 0-QP peak. The 0-QP peak is suppressed by a factor $e^{-\bar n_{trap}}$ and thus is too small to be observed. In the case of a diffusive nanobridge we can find the exact expression for the inductance $L_J$ with a general distribution of QPs. The channel transmittivities $\tau$ in a diffusive nanobridge are given by the Dorokhov distribution: $$\rho(\tau) = \frac{\pi \hbar G}{2 e^2} \frac{1}{\tau \sqrt{1-\tau}} \,,$$ where $G$ is the normal-state conductance of the nanobridge. The number of channels is not well defined for this distribution, but we can define the effective number of channels as $$N_e = \frac{\left(\int_0^1d\tau \rho(\tau) \tau\right)^2}{\int_0^1d\tau \rho(\tau) \tau^2} = 3\pi \hbar G/2e^2.$$ Assuming that the quasiparticles in the nanobridge are distributed according to a given distribution $f(E)$, the inductance is given by: $$\begin{aligned}
1/L_J = \int_0^1 d\tau \rho(\tau) \frac{e^2 \Delta \tau}{\hbar^2} \frac{\cos \delta + \tau \sin^4\frac{\delta}{2}}{\left(1 - \tau\sin^2 \frac{\delta}{2} \right)^{3/2}} [1-2f(E_A(\delta,\tau))]\,,
\label{eq:LJ}\end{aligned}$$ where $E_A(\delta,\tau)$ is the Andreev energy of a channel with transmittivity $\tau$ (see Eq. (1) in the main text). It is clear from Eq. (\[eq:LJ\]) that the presence of QPs changes the expectation value of the inductance. Similarly, fluctuations of the number or distribution of trapped QPs cause fluctuations of $L_J$. Since the resonant frequency depends on the inductance through Eq. (\[eq:omega0\]), QPs therefore cause a resonant frequency shift and broaden the resonance. Assuming $f(E)$ is small and given by the Gibbs distribution with some low temperature $k_B T\ll \Delta-\Delta|\cos\frac{\delta}{2}|$, we get the following expressions: $$\begin{aligned}
\omega_0^{(0)} - \omega_0 = \omega_0^{(0)} g(\delta) \frac{\bar{n}_{trap}}{N_e}\,, &\qquad
\delta \omega_0 ^2 = \frac{1}{4} (\omega_0^{(0)})^2 g^2(\delta)\frac{\bar{n}_{trap}}{N_e^2}\,, \label{eq:shiftwidth} \\
g(\delta) \equiv \frac{3q(\delta)}{2} & \frac{|\cos \frac{\delta}{2}|}{1- \sin\frac{\delta}{2}\tanh^{-1}\sin\frac{\delta}{2}}\,.\end{aligned}$$ All quantities appearing in the above expressions are measured except $\bar{n}_{trap}$. Therefore, we may fit the resonance to the convolved Gaussian-Lorentzian described above with $\bar{n}_{trap}$ as the only fitting parameter. However, we find out that such a procedure does not fit the data well. Thus, we take the frequency shift and the width as independent fitting parameters, and extract $\bar{n}_{trap}$ from the best fit value of each of the parameters using Eq. (\[eq:shiftwidth\]). As explained in the main text, we find that the value of $\bar{n}_{trap}$ extracted from the width of the resonance is about a factor of 3 greater than the value $\bar{n}_{trap}$ extracted from the shift of the resonance.
In the case of small number of QPs ($\bar n_{trap} \lesssim 1$) the main resonance peak comes from the 0-QP resonance, which is suppressed by a factor $e^{-\bar n_{trap}} \lesssim 1$. Additional contributions to the response function come from the 1-QP and 2-QP resonances, while the multi-QP contributions are further suppressed. We can thus keep only terms corresponding to 0, 1, and 2 QPs in the expression for $F(\Omega)$, Eq. (\[Fnew\]): $$\begin{aligned}
F(\Omega) = e^{-\bar n_{trap}}[ \delta(\Omega - \omega_0^{(0)}) + \sum_i p_i \delta(\Omega - \omega_0^{(i)}) + \sum_{i,j} p_i p_j \delta(\Omega - \omega_0^{(i,j)})]\,, \label{Fsmall}\end{aligned}$$ where $\omega_0^{(0)}$, $\omega_0^{(i)}$, and $\omega_0^{(i,j)}$ are the resonant frequencies at zero, one, and two trapped QPs, respectively; $i$ and $j$ label channels.
We write the resonant frequency with a QP trapped in a channel with transmittivity $\tau^{(i)}$ and energy $E_A^{(i)}$ as $$\begin{aligned}
\omega_0^{(i)} \equiv \omega_0(\tau^{(i)}) = \omega_0^{(0)} - \frac{q}{2} \omega_0^{(0)} \cdot L_J^{(0)}\Delta\frac{1}{L_J^{(i)}}\, , \quad \Delta\frac{1}{L_J^{(i)}} = \frac{2 e^2\tau_i\Delta\left( \cos\delta + \tau_i \sin^4\frac{\delta}{2} \right)}{\hbar^2(E_A^{(i)}/\Delta)^3}\,. \label{omegai}\end{aligned}$$ Here $\tau_i$ and $\Delta\frac{1}{L_J^{(i)}}$ are the transmittivity of the $i$th channel and its contribution to the inductance, respectively. Using Eq. (\[omegai\]) in expression (\[Fsmall\]) and transforming the sums into integrals by averaging over the distribution of transmission coefficients, we find $$\begin{aligned}
F(\Omega) = e^{-\bar n_{trap}}[ \delta (\Omega -\omega_0^{(0)}) + \int\limits_0^1 d\tau \rho(\tau) p(E_A(\tau)) \delta(\Omega - \omega_0(\tau)) \notag\\
+\int\limits_0^1 d\tau_1 \int\limits_0^1 d\tau_2 \rho(\tau_1)\rho(\tau_2 )p(E_A(\tau_1) )p(E_A(\tau_2)) \delta(\Omega - \omega(\tau_1,\tau_2))].
\label{Fsmall2}\end{aligned}$$ We assume a Gibbs distribution for the QPs: $$\begin{aligned}
p(E_A(\tau)) \equiv \bar n_{trap} f(E_A(\tau)) = \bar n_{trap} \ \mathcal{N}e^{-E_A(\tau)/k_B T}\,,\end{aligned}$$ where the normalization factor $\mathcal{N}$ is determined from the condition $\int d\tau \rho(\tau)p(E_A(\tau))=\bar n_{trap}$. Finally, convolving the expression for $F(\Omega)$ with the Lorentzian lineshape given in Eq. (\[Snew\]), we find the response function (see Eq. (2) in the main text): $$\begin{aligned}
\bar S (\omega) = P_0 S(\omega, \omega_0^{(0)}) + P_1 \int\limits_0^1 d\tau \rho(\tau) f(E_A(\tau)) S(\omega, \omega_0(\tau)) \notag\\
+ P_2 \int\limits_0^1 d\tau_1\int\limits_0^1 d\tau_2 \ \rho(\tau_1) \rho(\tau_2) f(E_A(\tau_1))f(E_A(\tau_2)) S(\omega, \omega_0(\tau_1,\tau_2))\,,
\label{Sfinal2}\end{aligned}$$ where $P_k$ is the probability of having exactly $k$ trapped QPs. Our data typically do not agree with the Poisson-distributed values of $P_k = \bar{n}_{trap}^k \exp(-\bar n_{trap})/k! $ that are expected for rare, independent trapping events. Because of this, we allow the $P_k$ to vary independently, improving the fits. We may also use temperature as an additional fitting parameter, but even allowing $T$ to vary freely gives $P_k$ which are not Poisson-distributed.
Retrapping Time Measurements
============================
We probe the resonator at its 0-QP resonant frequency with a constant tone. We mix the reflected tone down to dc with an IQ demodulation setup and measure the two quadratures with a fast digitizer circuit. We then pulse the QP excitation tone at a frequency which is significantly above $\Delta_A$. This removes some percentage of the QPs out of the junction trap states. We test the pulse’s efficacy by increasing its power, and determine that at the power at which the response saturates there are no trapped QPs. Fig. S1(a) shows a sample trace at 10 mK monitoring one quadrature of the measurement as the pulse is turned on and off, averaged over many iterations. The resonance shifts to a new steady state with the pulse on and decays back to its old value after it is turned off, with an exponential envelope for both processes. The first exponential corresponds to the clearing of QPs from the junction. It has a time constant which depends linearly on the excitation pulse power, and is roughly independent of $\phi$ as shown in Fig. S1(b). The second exponential corresponds to the retrapping of QPs as the system returns to steady state. It has a time constant which is independent of excitation pulse amplitude and duration, and which decreases as a function of $\phi$ from $60$ to $20 \ \mu$s as $\phi$ increases from 0.35 to 0.55; see Fig. S1(b). Below $\phi=0.35$ the pulse response signal was too small to accurately measure.
![\[figs1\] (Color online) (a) Representative measurement of QP excitation and retrapping at $\phi = 0.476$. The shaded region indicates the time during which the excitation pulse is on. We fit the exponential rise and fall to extract excitation and retrapping time constants, respectively. (b) QP excitation (top) and retrapping (bottom) time constants as a function of flux. The excitation time is roughly constant as a function of flux for a given power, while the retrapping time decreases with $\phi$, in qualitative agreement with the theoretical prediction of electron-phonon relaxation (solid blue line).](figs1v1.png)
We attempt to fit the retrapping time by assuming electron-phonon relaxation as the dominant mechanism. The crude model we use disregards the influence of disorder on the electron-phonon interaction, and the effect of bridge geometry. The electron-phonon interaction Hamiltonian then takes the form: $$H_{e-ph} = \frac{1}{\sqrt{N}}\sum_{\bf{k},\bf{q},\sigma}\alpha q^{1/2}(a^\dag_{\bf{k}+\bf{q},\sigma}a_{\bf{k},\sigma}b_{\bf{q}} + a^\dag_{\bf{k},\sigma}a_{\bf{k}+\bf{q},\sigma}b^\dag_{\bf{q}})\,,$$ where $a$ and $b$ are the annihilation operators and $\bf{k}$ and $\bf{q}$ are the momenta for electrons and phonons, respectively. We want to relate the retrapping time in our experiment, $\tau_T$, to the QP recombination time $\tau_R$ in a bulk superconductor that was measured in earlier experiments [@Prober2004]. The recombination rate of a QP with momentum $\mathbf{p}$ in the superconductor is given by Fermi’s Golden Rule as: $$\begin{aligned}
w_R(\mathbf{p}) &= \frac{2\pi}{N}\sum_{\mathbf{p_2}}\sum_{\mathbf{q}}\alpha^2 q |u_p v_{p_2} + u_{p_2}v_p|^2 (N_q+1) f_{p_2} \delta_{\mathbf{p}+\mathbf{p_2},\mathbf{q}} \delta(E_p + E_{p_2}-\omega_q)\,,\end{aligned}$$ where $u_p$ and $v_p$ are Bogoliubov amplitudes of a QP with momentum $\mathbf{p}$ and energy $E_p$, and $N_q$, $f_p$ are distribution functions of phonons and QPs, respectively. In the case where the QP energy is close to the gap, $E_p \approx \Delta$, and the QPs are Gibbs-distributed at a low temperature $k_B T \ll \Delta$, the integrals in the above expression for the recombination rate can be evaluated to give[@Scalapino1976] $$w_R = 4\pi x_{qp}b\Delta^3 \,,
\label{eq:wR}$$ where $b = 2V\alpha^2/\pi^2Ns^3v_F$, with $s$ and $v_F$ representing the speed of sound and Fermi velocity, respectively. The recombination time is defined as $\tau_R = 1/w_R$.
Next we consider the single-channel point contact with one Andreev state $\alpha_{A\sigma}^\dagger|0\rangle$, which can be expressed in terms of the electron operators as: $$\alpha_{A\sigma} = \sum_{\mathbf{k}} \left( u_{A\mathbf{k}}^{L*} a_{\mathbf{k}\sigma}^{L} + v_{A\mathbf{k}}^{L*} a_{\mathbf{k}-\sigma}^{L\dagger} + u_{A\mathbf{k}}^{R*} a_{\mathbf{k}\sigma}^{R} + v_{A\mathbf{k}}^{R*} a_{\mathbf{k}-\sigma}^{R\dagger}\right) \,.$$ Indices $L$ and $R$ denote left and right lead, respectively. The retrapping rate $w_T$ can be written as $w_T = \sum_\mu w_{\mu \to A} f(E_\mu)$, where the sum goes over all single-QP states and $f(E_\mu)$ is the initial distribution function of QPs. The transition rate from a given QP state $\mu$ to the Andreev level is given by Fermi’s Golden Rule as: $$\begin{aligned}
w_{\mu\to A} = 2\pi \sum_{\mathbf{q}} | \langle 0 | \alpha_{A\sigma_A}b_q H_{e-ph} \alpha_{\mu \sigma}^\dagger |0\rangle |^2 \delta (E_\mu - E_A -\omega_q)\,.\end{aligned}$$ The retrapping rate can then be expressed in the form: $$\begin{aligned}
w_T = \frac{-i}{N} &\sum_{\mathbf{q}} \alpha^2 q f(E_A+\omega_q) \sum_{i,j} \sum_{\mathbf{k}_1,\mathbf{k}_4} \big[ u^{i*}_{A\mathbf{k}_1} u^j_{A\mathbf{k}_4} \mathcal{G}_{ij}(\mathbf{k}_1-\mathbf{q},\mathbf{k}_4-\mathbf{q};E_A+\omega_q) \notag\\
+& v^{i*}_{A\mathbf{k}_1} v^j_{A\mathbf{k}_4} \mathcal{G}_{ij}(\mathbf{k}_4 + \mathbf{q}, \mathbf{k}_1 + \mathbf{q}; -E_A-\omega_q) + v^{i*}_{A\mathbf{k}_1} u^j_{A\mathbf{k}_4} \mathcal{F}^+_{ij}(\mathbf{k}_1+\mathbf{q},\mathbf{k}_4-\mathbf{q};E_A+\omega_q) \notag\\
+& u^{i*}_{A\mathbf{k}_1} v^j_{A\mathbf{k}_4} \mathcal{F}_{ij}(\mathbf{k}_1-\mathbf{q},\mathbf{k}_4+\mathbf{q};E_A+\omega_q) \big]\,.
\label{eq:wA}\end{aligned}$$ Here $\mathcal{G}_{ij} =G_{ij}^R - G_{ij}^A$ and $\mathcal{F^\pm}_{ij} =F_{ij}^{\pm R} - F_{ij}^{\pm A}$ are electron-electron Green’s functions in the point contact. The indices $i, j$ in the above sum are going over the values $L, R$. Describing the point contact with transmittivity $\tau$ using the tunneling Hamiltonian[@Kos2013], it is possible to find the exact expressions for the Green’s functions and the Bogoliubov amplitudes of the Andreev states. However, the integrals in Eq. (\[eq:wA\]) are difficult to evaluate for arbitrary $\tau$, so we only consider the limit of a shallow Andreev level, in which case the amplitudes are given by: $$\begin{aligned}
|u^{L,R}_{A\mathbf{k}}|^2 = \frac{(\Delta^2-E_A^2)^{3/2}}{2\pi\nu_0(E_k^2-\Delta^2)^2} \,,\\
v^L_{A\mathbf{k}} = e^{i\delta/2}u^L_{A\mathbf{k}} = e^{-i\delta/2}u^{R*}_{A\mathbf{k}}=v^{R*}_{A\mathbf{k}}\,,\end{aligned}$$ where $\nu_0$ is the normal-state density of states at Fermi level. For the crude estimate of the retrapping time, we use the bulk superconductor Green’s functions. Taking as the initial distribution of QPs a Gibbs distribution at a low temperature $k_BT \ll \Delta-E_A$, we can evaluate Eq. (\[eq:wA\]): $$w_T = \pi x_{qp} b \Delta (\Delta - E_A)^2\,.$$ Defining the retrapping time as $\tau_T = 1/w_T$ and comparing with Eq. (\[eq:wR\]), we get: $$\tau_T = \left( \frac{2\Delta}{\Delta-E_A} \right)^2 \tau_R \label{retrapping}\,.$$ A fit to the relaxation time measurements using this functional form, with $E_A(\phi) =\Delta |\cos\frac{\phi \pi}{2}| $, is shown in Fig. S1(b). The functional form derived for $\tau_T(\phi)$ agrees reasonably with the measurements. However, the value $\tau_R=0.3\mu$s which we extract with the help of Eq. (\[retrapping\]) is much lower than the value $\tau_R = 100\mu$s reported in earlier experiments at 250mK[@Prober2004]. We note, however, that our derivation of Eq. (\[retrapping\]) was very crude; we assumed the clean superconductor limit, did not take into account the geometry of the junction, and considered only the limit of a shallow Andreev level. A more complete theoretical treatment may be able to resolve the discrepancy.
|
---
abstract: 'We introduce a new class of Monte Carlo methods, which we call exact estimation algorithms. Such algorithms provide unbiased estimators for equilibrium expectations associated with real-valued functionals defined on a Markov chain. We provide easily implemented algorithms for the class of positive Harris recurrent Markov chains, and for chains that are contracting on average. We further argue that exact estimation in the Markov chain setting provides a significant theoretical relaxation relative to exact simulation methods.'
author:
- |
Peter W. Glynn\
Stanford University\
Stanford, CA, 94305, USA\
\
Chang-han Rhee\
Georgia Institute of Technology\
Atlanta, GA, 30332, USA\
\
bibliography:
- 'References-ExactEstimation.bib'
title: Exact Estimation for Markov Chain Equilibrium Expectations
---
Introduction {#sec:1}
============
A key advance in the development of Monte Carlo algorithms for Markov chains has been the introduction of what are known as *exact simulation* or *perfect simulation* algorithms for equilibrium distributions of such stochastic processes. By assuming suitable structure on the underlying Markov chain, one can construct algorithms that draw samples perfectly from the equilibrium distribution of the Markov chain, based only on an ability to generate sample paths of the Markov chain from an arbitrary state. In particular, such algorithms have been developed for finite state Markov chains [@Propp1996], uniformly recurrent Markov chains [@Asmussen1992], certain stochastically monotone Markov chains [@Propp1996; @Corcoran2001], some queueing models [@Ensor2000; @Blanchet2014], and various sub-classes of Harris recurrent Markov chains [@Kendall2004; @Connor2007]. While the idea is powerful, it apparently requires exploiting significant structure within the Markov chain itself. In particular, no universal and practically implementable such perfect sampler has been constructed for Harris recurrent Markov chains, or even for countable state positive recurrent Markov chains. In fact, we shall argue below in Section \[sec:2\] that exact simulation is inherently restrictive, in the sense that such algorithms can typically be constructed only for Markov chains that are $\phi$-irreducible.
In this paper, we relax the algorithmic formulation so as to require only that the algorithm output unbiased estimators for equilibrium expectations, rather than to demand (as in exact simulation) that such an unbiased estimator necessarily is constructed from an exact sample from the equilibrium distribution. To differentiate this new class of algorithms from exact simulation algorithms, we shall refer to them as *exact estimation* algorithms. Our exact estimation algorithms exploit a recent idea of [@Rhee2013] that shows how unbiased estimators can often be constructed from a sequence of biased approximations; see also [@McLeish2011] and [@Rhee2012]. As we shall see below, this new class of algorithms can provide unbiased estimators for equilibrium expectations for any positive recurrent Harris chain. In fact, we shall see that exact estimation algorithms can even be developed for non $\phi$-irreducible Markov chains, provided that the Markov chain is “contractive” in a certain sense and the equilibrium expectation involves a suitably Lipschitz functional (Theorem 1). This makes clear that exact estimation is indeed a significant relaxation of exact simulation.
One key additional feature of our exact estimation algorithms is that unlike most existing exact simulation methods, our algorithms do not involve explicitly simulating paths from multiple initial conditions, nor do they require monotonicity. Furthermore, our exact estimation procedures can easily be implemented with a minimal need to store sample paths. As a consequence, our proposed exact estimation methods are relatively straightforward to implement.
Our paper is organized as follows. In Section \[sec:2\], we illustrate exact estimation in the setting of contractive Markov chains, and exploit the fact that Markov chains can, in great generality, be viewed as a sequence of random iterated functions. Section \[sec:3\] develops exact estimation algorithms in the context of Harris recurrent Markov chains, thereby establishing that our proposed relaxation is indeed a generalization of exact simulation. Section \[sec:4\] proves a variant of the Glivenko-Cantelli theorem for our newly developed estimator, and Section \[sec:5\] provides a brief computational discussion.
Exact Estimation for Contracting Markov Chains {#sec:2}
==============================================
Given an $S$-valued Markov chain $X=(X_n: n \geq 0)$, we wish to develop exact estimation algorithms for computing ${\mathbf{E}}f(X_\infty)$, where $f$ is real-valued and $X_\infty$ has the equilibrium distribution $\pi$ of $X$ (assumed to exist uniquely). We start by briefly describing and (slightly) generalizing the algorithms and analysis presented in [@Rhee2012; @Rhee2013] and [@McLeish2011], following related earlier work by [@Rychlik1990; @Rychlik1995].
Suppose that we wish to compute ${\mathbf{E}}Y$. We assume that we have available to us a sequence $(Y_k: k\geq 0)$ of approximations for which $$\label{eq:2.1}
{\mathbf{E}}Y_k \to {\mathbf{E}}Y$$ as $k\to \infty$. If $(\Delta_k: k\geq 0)$ is a sequence of random variables (rv’s) for which ${\mathbf{E}}\Delta_k = {\mathbf{E}}(Y_k - Y_{k-1})$ for $k\geq 0$ (with $Y_{-1} \triangleq 0$) and $$\label{eq:2.2}
{\mathbf{E}}\sum_{k=0}^\infty |\Delta_k| < \infty,$$ then it is easy to verify that $$\label{eq:2.3}
Z = \sum_{k=0}^N \frac{\Delta_k}{{\mathbf{P}}(N\geq k)}$$ is an unbiased estimator for ${\mathbf{E}}Y$, provided that $N$ is a ${\mathbb{Z}}_+$-valued rv independent of $(\Delta_k: k \geq 0)$. Of course, one implication of (\[eq:2.2\]) is that $\Delta_k \Rightarrow 0$ as $k\to \infty$, where $\Rightarrow$ denotes weak convergence. Note that in our Markov chain setting, the most natural choice of approximating sequence $(Y_k: k\geq 0)$ is to choose $Y_k = f(X_k)$ (with $Y = f(X_\infty)$). However, the obvious choice for $\Delta_k$, namely $\Delta_k = f(X_k ) - f(X_{k-1})$, then fails to satisfy $\Delta_k \Rightarrow 0$ as $k\to \infty$. Thus, the key to the development of an exact estimation algorithm for $X$ is the construction of a computationally implementable “coupling” between $X_{k-1}$ and $X_k$ that forces $\Delta_k$ to converge to zero (hopefully rapidly).
We now illustrate one potential coupling that can be applied in the setting of “contractive” Markov chains. We presume (for the purpose of this section) that $S$ is a complete separable metric space with metric $\rho:S\times S\to {\mathbb{R}}_+$ and that $X$ can be represented in terms of a sequence of independent and identically distributed (iid) random functions $(\varphi_i: i\geq 1)$, independent of $X_0$, so that $$X_i = \varphi_i(X_{i-1})$$ for $i \geq 1$. In particular, conditional on $X_0 = x \in S$, $X_n = (\varphi_n \circ \varphi_{n-1}\circ \cdots \circ \varphi_1)(x)$. An obvious means of coupling $X_{n-1}$ and $X_n$ is then to set $$\tilde X_{n-1} = (\varphi_n\circ \varphi_{n-1}\circ \cdots \circ \varphi_2)(x)$$ for $n\geq 2$, with $\tilde X_0 \triangleq x$. Clearly, $\tilde X_{n-1} \stackrel{\mathcal D}{=} X_{n-1}$ for $n\geq 1$, where $\stackrel{\mathcal D}{=}$ denotes “equality in distribution.”
For $y, z \in S$, set $$r(y,z) = {\mathbf{E}}\rho^2(\varphi_1(y), \varphi_1(z))$$ and assume that there exists $b<1$ for which $$\label{eq:2.4}
r(y,z) \leq b\rho^2(y,z)$$ for $y, z \in S$, so that $X$ is “contractive on average.” Suppose further that $f$ is Lipschitz with respect to the metric $\rho$, so that there exists $\kappa<\infty$ for which $$\label{eq:2.5}
|f(y) - f(z)| \leq \kappa\rho(y,z)$$ for $y,z \in S$. Evidently, $$\begin{aligned}
{\mathbf{E}}\Delta_k^2
&\leq \kappa^2 {\mathbf{E}}\rho^2(X_k, \tilde X_{k-1})\\
&= \kappa^2 {\mathbf{E}}\rho^2(\varphi_k(X_{k-1}), \varphi_k(\tilde X_{k-2}))\\
&\leq \kappa^2 b {\mathbf{E}}\rho^2(X_{k-1}, \tilde X_{k-2})\\
&\leq \ldots \leq \kappa^2 b^{k-1} {\mathbf{E}}\rho^2(X_1, x).
\end{aligned}$$ Hence, if $$\label{eq:2.6}
{\mathbf{E}}\rho^2(\varphi_1(x), x) < \infty$$ for $x\in S$, it follows that ${\mathbf{E}}\Delta_k^2 \to 0$ geometrically fast, so that (\[eq:2.2\]) holds. Also, Theorem \[thm:1\] of [@Diaconis1999] applies, in the presence of (\[eq:2.4\]) and (\[eq:2.6\]), so that $X$ has a unique stationary distribution $\pi$. In fact, their proof makes clear that ${\mathbf{E}}f(X_n)\to{\mathbf{E}}f(X_\infty)$ geometrically fast (where $X_\infty$ has distribution $\pi$) when $f$ is Lipschitz. Consequently, (\[eq:2.1\]) is valid, thereby proving that $Z$ is an unbiased estimator for ${\mathbf{E}}f(X_\infty)$.
But more can be said. Note that $$\label{eq:*}
{\mathbf{E}}Z^2 = \sum_{k=0}^\infty \frac{{\mathbf{E}}\Delta_k^2 + 2\sum_{j=k+1}^\infty {\mathbf{E}}\Delta_k \Delta_j}{{\mathbf{P}}(N\geq k)}.$$ By virtue of the Cauchy-Schwarz inequality, it follows that ${\mathbf{var\,}}Z < \infty$, provided that we choose the distribution for $N$ so that $$\label{eq:2.7}
\sum_{k=0}^\infty b^k / P(N\geq k) < \infty$$ Under condition (\[eq:2.7\]), the central limit theorem (CLT) asserts that if we generate iid copies $Z_1, Z_2, \ldots$ of $Z$ and form the sample mean $\bar Z_n = (Z_1 + \cdots + Z_n)/n$, then $\bar Z_n$ converges weakly to ${\mathbf{E}}f(X_\infty)$ at rate $n^{-1/2}$ in the number of samples $n$ that are generated.
Of course, the amount of computer time needed to generate each $Z_i$ could be excessive. To take this effect into account, we let $\xi_i$ be the computer time needed to generate $Z_i$. In view of the fact that computing $Z$ requires generating $\varphi_1, \varphi_2, \ldots, \varphi_N$, it seems natural to assess the computer time as being equal to $N$. Hence, we put $\xi_i = N_i$, where $N_i$ is the corresponding “randomization” rv $N$ associated with $Z_i$. If $\Gamma(c)$ is the number of $Z_i$’s generated in $c$ units of computer time, $\Gamma(c) = \max\{k\geq0: \xi_1 + \cdots + \xi_k \leq c\}$ and the estimator for ${\mathbf{E}}f(X_\infty)$ available after expending $c$ units of computer time is $\bar Z_{\Gamma(c)}$. If $$\label{eq:2.9}
{\mathbf{E}}\xi_i = {\mathbf{E}}N = \sum_{k=0}^\infty {\mathbf{P}}(N\geq k) < \infty$$ and ${\mathbf{var\,}}Z<\infty$, it is well known (see, for example, [@Glynn1992]) that $$c^{1/2} (\bar Z_{\Gamma(c)} - {\mathbf{E}}f(X_\infty)) \Rightarrow \sqrt{{\mathbf{E}}N \cdot {\mathbf{var\,}}Z} \ N(0,1)$$ as $c\to\infty$, where $N(0,1)$ denotes a normal rv with mean $0$ and variance $1$.
We summarize our discussion with the following result, which establishes that exact estimation algorithm exhibiting “square root” convergence rates can be obtained for any suitably contractive chain and Lipschitz function $f$.
\[thm:1\] Assume (\[eq:2.4\]) - (\[eq:2.6\]). If the distribution of $N$ is chosen so that (\[eq:2.7\]) and (\[eq:2.8\]) hold (e.g. ${\mathbf{P}}(N\geq k) = ck^{-\alpha}$ for $\alpha > 1$), then $$c^{1/2} (\bar Z_{\Gamma(c)} - {\mathbf{E}}f(X_\infty))\Rightarrow\sqrt{{\mathbf{E}}N \cdot {\mathbf{var\,}}Z} \ N(0,1)$$ as $c \to \infty$.
In view of Theorem \[thm:1\], a natural question arises as to the optimal choice for the distribution of $N$. According to Proposition 1 of [@Rhee2013] the optimal choice is to set $$\label{eq:2.8}
{\mathbf{P}}(N\geq k) = \left(\frac{{\mathbf{E}}\Delta_k^2 + 2\sum_{j=k+1}^\infty {\mathbf{E}}\Delta_k \Delta_j}{{\mathbf{E}}\Delta_0^2 + 2\sum_{j=1}^\infty {\mathbf{E}}\Delta_0 \Delta_j}\right)^{1/2}$$ for $k\geq 0$, provided that the right-hand side is non-[increasing]{}; see Theorem 3 of [@Rhee2013] for details of the form of the optimal distribution when the right-hand side fails to be non-[increasing]{}. Given this result and the geometric decay of the ${\mathbf{E}}\Delta_k^2$, it therefore seems reasonable to expect that requiring the tail of $N$ to be geometric will often be a good choice in this setting.
We turn next to a slightly different implementation of our coupling idea in the $S$-valued metric space contractive setting. Given the independence of $N$ from the $\varphi_i$’s, an alternative coupling for $(X_{i-1}, X_i)$ is to set $$X_j^* = (\varphi_N \circ \varphi_{N-1}\circ \cdots \circ \varphi_{N-j+1}) (x)$$ for $0 \leq j \leq N$. Clearly, conditional on $N$, $X_j^* \stackrel{\mathcal D}{=} X_j$ and $\Delta_j^* \triangleq X_j^* - X_{j-1}^* \stackrel{\mathcal D}{=} \Delta_j$ for $j\geq 0$. Because ${\mathbf{E}}(\Delta_j^*)^2 = {\mathbf{E}}\Delta_j^2$ for $j\geq 0$, the same argument as for $Z$ shows that $$Z^* = \sum_{j=0}^N \frac{\Delta_j^*}{{\mathbf{P}}(N\geq j)}$$ is unbiased for ${\mathbf{E}}f(X_\infty)$. Furthermore, the estimator for ${\mathbf{E}}f(X_\infty)$ corresponding to computing a sample average of iid copies of $Z^*$ satisfies Theorem \[thm:1\], under the conditions (\[eq:2.4\])-(\[eq:2.8\]).
The estimator based on $Z^*$ is slightly more complicated to implement, because $X_i^*$ can not be recursively computed from $X_{i-1}^*$ in this setting (whereas $(X_i, \tilde X_i)$ can be recursively computed from $(X_{i-1}, \tilde X_{i-1})$). Of course, the estimator based on $Z^*$ will have a different variance than does $Z$, because ${\mathbf{E}}\Delta_k \Delta_j \neq {\mathbf{E}}\Delta_k^*\Delta_j^*$ for $k<j$. In particular, while all four of the quantities $f(X_k^*)$, $f(X_{k-1}^*)$, $f(X_j^*)$, $f(X_{j-1}^*)$ appearing in ${\mathbf{E}}\Delta_k^*\Delta_j^*$ will be close to one another when $k$ is large, $f(X_j)- f(X_k)$ will exhibit significant variability, regardless of the magnitude of $k$. We close this section by noting that construction of exact simulation algorithm typically requires that the underlying Markov chain be $\phi$-irreducible. Recall that an $S$-valued Markov chain $X=(X_n: n\geq 0)$ is *$\phi$-irreducible* if there exists a $\sigma$-finite (non-negative) measure $\phi$ such that whenever $\phi(A)>0$ for some (measurable) $A$, $$R(x,A) \triangleq \sum_{n=0}^\infty 2^{-n} {\mathbf{P}}_x(X_n\in A) > 0$$ for all $x\in S$, where ${\mathbf{P}}_x(\cdot) \triangleq {\mathbf{P}}(\cdot|X_0 = x)$. Equivalently, $$\phi(\cdot) \ll R(x,\cdot)$$ for each $x\in S$, where $\ll$ denotes “is absolutely continuous with respect to.”
A typical exact simulation algorithm involves simulating $X$ from multiple initial states $x_1, x_2, \cdots$, thereby yielding a family of random variables (rv’s) $(X_{ij}: i \geq 1, j\geq 0)$ such that $${\mathbf{P}}((X_{ij}: j \geq 0) \in \cdot) =
{\mathbf{P}}((X_j: j\geq 0)\in \cdot | X_0 = x_i)$$ for $i\geq 1$; the exact simulation algorithm then outputs $X_{IJ}$ for some appropriately chosen pair of rv’s $(I,J).$ Exact simulation demands that if $X$ has an equilibrium distribution $\pi(\cdot)$, then $${\mathbf{P}}(X_{IJ} \in \cdot) = \pi(\cdot).$$ The probability ${\mathbf{P}}(X_{IJ}\in \cdot)$ is mutually absolutely continuous with respect to ${\mathbf{E}}2^{-I-J} {\mathbb{I}}(X_{IJ} \in \cdot).$ But $$\begin{aligned}
{\mathbf{E}}2^{-I-J} {\mathbb{I}}(X_{IJ} \in \cdot)
&= \sum_{i=1}^\infty \sum_{j=0}^\infty 2^{-i-j} {\mathbf{P}}(X_{ij}\in \cdot, I=i, J=j)\\
&\leq \sum_{i=1}^\infty \sum_{j=0}^\infty 2^{-i-j} {\mathbf{P}}(X_{ij}\in \cdot)\\
&= \sum_{i=1}^\infty 2^{-i}R(x_i, \cdot).\end{aligned}$$ Hence, it follows that $$\label{eq:2.99}
\pi(\cdot) \ll \sum_{i=1}^\infty 2^{-i} R(x_i, \cdot).$$ Thus, the existence of an exact simulation algorithm requires that one have a priori knowledge of a set of states $x_1, x_2, \cdots$ satisfying (\[eq:2.99\]). Without additional structure on the chain, the only way to guarantee this is to require that $$\pi(\cdot) \ll R(x,\cdot)$$ for each $x\in S$. In other words, $X$ must be $\phi$-irreducible, with $\phi = \pi$. (Note that when a $\phi$-irreducible Markov chain has a stationary distribution, one choice for $\phi$ is always $\phi = \pi$.) Consequently, $\phi$-irreducibility and exact simulation are tightly connected concepts.
Exact Estimation for Harris Recurrent Markov Chains {#sec:3}
===================================================
In this section, we will establish that exact estimation algorithms can be constructed for any positive recurrent Harris chain, under the assumption that $S$ is a separable metric space. In the presence of such separability, it is well known that there exists a so-called small set $A$ i.e., there exists $m\geq 1$, a probability $\nu$, and $\lambda > 0$, for which $$\label{eq:3.1}
{\mathbf{P}}(X_m \in \cdot | X_0 = x) \geq \lambda \nu(\cdot)$$ for all $x \in A$. Given (\[eq:3.1\]), we can express the $m$-step transition probability on $A$ via the mixture representation $$\label{eq:3.2}
{\mathbf{P}}(X_m\in \cdot |X_0 = x) = \lambda \nu(\cdot) + (1-\lambda)Q(x,\cdot),$$ where $Q(x,\cdot)$ is a probability on $S$ for each $x\in A$. In view of (\[eq:3.2\]), we can construct regeneration times $T(1),T(2),\ldots$ for $X$ by first running the chain until it hits $A$. Once it hits $A$ at time $T$ (say), we distribute the chain at time $T+m$ according to $\nu$ with probability $\lambda$ and according to $Q(X_T, \cdot)$ with probability $1-\lambda$; we then “condition in” the values of $X_{T+1}$, …, $X_{T+m-1}$, conditional on $X_T$ and $X_{T+m}$. If we succeed in distributing $X$ according to $\nu$ at time $T+m$, we set $T(1) = T+m$. Otherwise, we continue simulating $X$ forward from time $T+m$, and continue attempting to distribute $X$ according to $\nu$ at successive visits to $A$ until we are successful, thereby defining the first regeneration time $T(1)$. We then successively follow the same procedure from time $T(1)$ forward to construct $T(2),T(3),\ldots$. The Markov chain $X$ is *wide-sense regenerative* with respect to the sequence of random times $T(1),T(2),\ldots$. In particular, the random element $((X_{T(i)+j}, T(i+j+1)-T(i)): j\geq 0)$ is identically distributed and independent of $T(i)$ for $i\geq 1$; see [@Meyn2009] and [@Thorisson2000] for details. As noted in [@Asmussen2007], one can implement the above algorithm using acceptance / rejection so that explicit generation from the conditional distribution (given $X_T$ and $X_{T+m}$) can be avoided.
We now explain our exact estimation algorithm in the special case that $X$ is aperiodic, and we later generalize to the periodic case. As in Section \[sec:2\], the key is to construct a coupling of $(X_{n-1}, X_n)$ that makes $\Delta_n$ small. Specifically, we will construct alongside $(X_n:n\geq 0)$ another sequence $(X_n': n\geq 0)$ such that $(X_n': n\geq 0)\stackrel{\mathcal D}{=} (X_n : n \geq 0)$, and then attempt to (distributionally) couple the $X_n'$’s to the $X_{n+1}$’s so that $X_{\tau} \stackrel{\mathcal D}{=}X'_{\tau-1}$.
We start by drawing $X_0$ from the distribution $\nu$ and set $X_0' = X_0$. We have already discussed the simulation of $X$ and the construction of the associated $T(n)$’s. Conditional on $X_0$, we simulate $(X_n': n \geq 1)$ independently of $(X_n: n\geq 1)$, thereby producing an associated sequence of regeneration times $0 = T'(0) < T'(1) < T'(2) < \ldots$. We then let the (distributional) coupling time $\tau$ be the first time at which one of the $(T(j)-1)$’s coincides with one of the $T'(i)$’s, specifically $\tau = \inf\{T(n): n\geq 1, \text{ there exists } m \geq 0 \text{ such that } T'(m) = T(n)-1\}$. With this definition of $\tau$, $(X_{\tau+j}: j\geq 0) \stackrel{\mathcal D}{=} (X'_{\tau+j-1} : j \geq 0)$ and $((X_{\tau+j}, X'_{\tau+j-1}): j \geq 0)$ is independent of $\tau$. As a consequence, if we set $\Delta_k = (f(X_k) - f(X_{k-1}')){\mathbb{I}}(\tau > k)$ (with $f$ bounded), it follows that $$\begin{aligned}
{\mathbf{E}}\Delta_k
&= {\mathbf{E}}\left(f(X_k)-f(X_{k-1}')\right){\mathbb{I}}(\tau>k)\\
&= {\mathbf{E}}\left(f(X_k)-f(X_{k-1}')\right){\mathbb{I}}(\tau>k)\\
&\quad + \sum_{j=1}^k {\mathbf{E}}\left(f(X_{\tau+k-j})-f(X_{\tau+k-j-1}')\right){\mathbf{P}}(\tau = j)\\
&= {\mathbf{E}}\left(f(X_k)-f(X_{k-1}')\right){\mathbb{I}}(\tau>k)\\
&\quad + \sum_{j=1}^k {\mathbf{E}}\left(f(X_{\tau+k-j})-f(X_{\tau+k-j-1}')\right){\mathbb{I}}(\tau = j)\\
&= {\mathbf{E}}\left(f(X_k)-f(X_{k-1}')\right) = {\mathbf{E}}\big(f(X_k)-f(X_{k-1})\big).
\end{aligned}$$ Furthermore, the aperiodicity of $X$ and the boundedness of $f$ imply that ${\mathbf{E}}f(X_k) \to {\mathbf{E}}f(X_\infty)$, where $X_\infty$ has the distribution of $\pi$, with $\pi$ being the unique stationary distribution of the Harris chain.
It remains to establish condition (\[eq:2.2\]). Observe that the $(T(n)-1)$’s are the regeneration times for the sequence $(X_{n+1}: n \geq 0)$, in which $X$ is initialized with the distribution ${\mathbf{P}}(X_1 \in \cdot)$. Equivalently, the $(T(n)-1)$’s are renewal times for the delayed renewal process where the inter-renewal times share the same inter-renewal distribution as for the $T'(n)$’s, but in which the probability mass function for the initial delay is given by $(q_j: j \geq 0)$, where $q_j = {\mathbf{P}}(T'(2) - T'(1) = j+1)$. Thus, $\tau$ is the first time that two independent aperiodic renewal processes couple, in which one is non-delayed (corresponding to $(X_n': n \geq 0)$) and the other is delayed with initial delay $(q_j: j \geq 0)$. According to [@Lindvall2002], p.27, ${\mathbf{E}}\tau^r<\infty$ for $r\geq 1$, provided that ${\mathbf{E}}(T(2) - T(1))^r < \infty$. (Note that $\sum_{j=0}^\infty j^r q_j < \infty = {\mathbf{E}}(T(2) - T(1))^r$ for the specific delay distribution that arises here.) Of course, the positive recurrence of $X$ implies that ${\mathbf{E}}(T(2)-T(1))<\infty$ (see [@Athreya1978]), implying that ${\mathbf{E}}\tau < \infty$. Hence $${\mathbf{E}}\sum_{k=0}^\infty |\Delta_j| \leq 2\,\|f\|\, {\mathbf{E}}\sum_{k=0}^\infty {\mathbb{I}}(\tau>k) = 2 \|f \|\, {\mathbf{E}}\tau < \infty,$$ where $\|f\| = \sup\{|f(x)|: x \in S\}<\infty$, validating (\[eq:2.2\]). It follows that $Z$ is an unbiased estimator for ${\mathbf{E}}f(X_\infty)$.
For the periodic case (with period $p$), we can apply the above algorithm to $(X_{pn}: n \geq 0)$, and apply the coupling $\tau$ to coupling the $X_{pn}$’s to the $X_{p(n-1)}$’s. (Note that by setting $X_0 = X_0'$, we guarantee that both $(X_n: n \geq 0)$ and $(X_n': n \geq 0)$ start off in the same periodic sub-class, so that $(X_{p(n+1)}: n \geq 0)$ can successfully couple with $(X'_{pn}: n \geq 0)$.) We summarize our discussion thus far with the following result.
\[thm:2\] If $X$ is a positive recurrent Harris chain and $f$ is bounded, the estimator $Z$ described above is unbiased for ${\mathbf{E}}f(X_\infty)$.
Of course, this estimator may fail to exhibit a “square root convergence rate,” because $Z$ may not have finite variance and the expected computation time to generate $Z$ may be infinite. However, we note that because $\Delta_k = 0$ for $k > \tau$ in this setting, the number of time steps of $((X_j, X_j'): j \geq 0)$ that need to be simulated in order to compute $Z$ is bounded by $2\min(\tau, N)$. (The factor of 2 appears because we need to simulate both the $X_j$’s and $X_j'$’s.) Hence, if $\xi$ is a measure of the computational effort required to generate $Z$, ${\mathbf{E}}\xi$ is automatically finite because ${\mathbf{E}}\tau < \infty$, regardless of the distribution of $N$. (In fact, we may set $N=\infty$ a.s. in this setting, if we so wish.)
Turning now to the variance of $Z$, we note that if $f$ is bounded, ${\mathbf{E}}\Delta_k^2 = O({\mathbf{P}}(\tau>k))$ as $k\to \infty$. Furthermore, if ${\mathbf{E}}(T(2)-T(1))^r < \infty$ for $r>1$, then ${\mathbf{E}}\tau^r < \infty$, so that ${\mathbf{E}}\Delta_k^2 = O(k^{-r})$ by virtue of Markov inequality. Also, for $k<j$, $\Delta_k \Delta_j = 0$ unless $\tau> j$, so ${\mathbf{E}}\Delta_k \Delta_j = O(j^{-r})$ as $j \to \infty$, uniformly in $k$. Thus $$\sum_{k=j+1}^\infty {\mathbf{E}}\Delta_k \Delta_j = O(k^{1-r})$$ as $k\to\infty$. In order that there exist a probability distribution $N$ so that ${\mathbf{E}}Z^2 < \infty$, (\[eq:\*\]) implies that it is therefore sufficient that $\alpha > 2$ (in which case we can, for example, choose $N$ so that ${\mathbf{P}}(N\geq k)$ is of order $k^{1-\alpha/2}$ for $k$ large).
We have therefore proved the following theorem, establishing a “square root convergence” (in the computational effort $c$) for our estimator.
\[thm:3.2\] If $X$ is Harris chain with ${\mathbf{E}}(T(2)-T(1))^r < \infty$ for $r>2$ and $f$ is bounded, then $$c^{1/2}(\bar Z_{\Gamma(c)} - {\mathbf{E}}f(X(\infty))) \Rightarrow \sqrt{{\mathbf{E}}\xi\cdot {\mathbf{var\,}}Z} \ N(0,1)$$ as $c\to \infty$.
An improvement to the above coupling can be easily implemented. In the above algorithm, $\tau$ occurs whenever $X$ and $X'$ $m$ time steps earlier were in $A$, and both $X$ and $X'$ independently chose at that time to distribute themselves according to $\nu$ $m$ time units later. But an alternative coupling is to generate $(X_{T+m}, X'_{T+m-1})$ as follows, whenever $(X_T, X'_{T-1})\in A\times A$. As in the previous algorithm, we distribute $X_{T+m}$ according to $\nu$ with probability $\lambda$, and according to $Q(X_T, \cdot)$ with probability $1-\lambda$. However, we now modify the dynamics for $X'$. Whenever $X_{T+m}$ is distributed according to $\nu$, set $X'_{T+m-1} = X_{T+m}$. On the other hand, whenever $X_{T+m}$ is distributed according to $Q(X_T, \cdot)$, independently generate $X'_{T+m-1}$ according to $Q(X'_{T-1}, \cdot)$. This coupling preserves the marginal distribution of $X$ and $X'$, but the time $\tau'$ at which $X$ and $X'$ couple (so that $X_{\tau'}=X'_{\tau'-1}$) is a.s. smaller than under the previous “independent coupling.” Consequently, ${\mathbf{P}}(\tau' \geq k) \leq {\mathbf{P}}(\tau\geq k)$ for $k\geq 0$, so ${\mathbf{E}}(\tau')^r \leq {\mathbf{E}}\tau^r$ for $r>0$, thereby establishing that this coupling can be used in place of $\tau$ in proving Theorem \[thm:3.2\]. Given that $\tau' \leq \tau$, this coupling is computationally preferable to $\tau$.
A Glivenko-Cantelli Result {#sec:4}
==========================
In some settings, one may be interested in computing the equilibrium distribution of some real-valued functional $f$ of the Markov chain, rather then merely its expected value ${\mathbf{E}}f(X_\infty)$. In this section, we study the behavior of our unbiased estimator for the equilibrium probability ${\mathbf{P}}(f(X_\infty) \leq x)$ ($= {\mathbf{E}}{\mathbb{I}}(f(X_\infty)\leq x)$) as a function of $x$. Because the mapping ${\mathbb{I}}(f(\cdot)\leq x)$ is not Lipschitz, Section \[sec:2\]’s theory does not apply. As a consequence, we focus here on the case where $X$ is a positive recurrent Harris chain.
Set $Y_k = f(X_k)$ and $Y_k' = f(X_k')$. Let $((Y_{k,j}, Y_{k,j}'): 0 \leq k \leq \min(\tau_j, N_j)): j \geq 1)$ be a sequence of iid copies of $((Y_k, Y_k'): 0 \leq k \leq \min(\tau, N))$, where the $Y_k$’s and $Y_k'$’s are constructed as in Section \[sec:3\]. The empirical measure (intended to estimate $F_\infty(\cdot)\triangleq {\mathbf{P}}(f(X_\infty)\leq \cdot)$) associated with sample size $n$ is then given by the random signed measure $$\pi_n(\cdot) = \frac{1}{n}\sum_{j=1}^n \sum_{k=0}^{\tau_j \wedge N_j} \frac{\delta_{Y_{k,j}}(\cdot) - \delta_{Y_{k-1,j}'}(\cdot)}{{\mathbf{P}}(N\geq k)},$$ where $a\wedge b \triangleq \min(a,b)$, and $\delta_y(\cdot)$ is a unit point mass measure at $y$. Observe that $$\int_S y \pi_n(dy)$$ is Section \[sec:3\]’s unbiased estimator for ${\mathbf{E}}f(X_\infty))$, and $$F_n(x) \triangleq \int_S {\mathbb{I}}(y \leq x) \pi_n(dy)$$ is Section \[sec:3\]’s unbiased estimator for $F_\infty(x)$. We may re-write $F_n(\cdot)$ as $$F_n(x) = \sum_{k=0}^\infty \frac{1}{n} \sum_{j=1}^n ({\mathbb{I}}(Y_{k,j} \leq x) - {\mathbb{I}}(Y'_{k-1,j} \leq x)) \frac{{\mathbb{I}}(\tau_j \wedge N_j \geq k)}{{\mathbf{P}}(N\geq k)}.$$ Because the sample functions $$\frac{1}{n} \sum_{j=1}^n {\mathbb{I}}(Y_{k,j} \leq x) {\mathbb{I}}(\tau_j \wedge N_j \geq k)$$ and $$\frac{1}{n} \sum_{j=1}^n {\mathbb{I}}(Y_{k-1,j}' \leq x){\mathbb{I}}(\tau_j \wedge N_j \geq k)$$ are monotone in $x$, a proof identical to that of the standard Glivenko-Cantelli theorem (see, for example, [@Chung2001]) establishes that $$\sup_x \left| \sum_{j=1}^n{\mathbb{I}}(Y_{kj}\leq x) {\mathbb{I}}(\tau_j\wedge N_j \geq k) - {\mathbf{E}}{\mathbb{I}}(Y_k \leq x) {\mathbb{I}}(\tau\geq k) {\mathbf{P}}(N_j \geq k) \right| \to 0 \quad a.s.$$ and $$\sup_x \left| \sum_{j=1}^n{\mathbb{I}}(Y_{k-1,j}'\leq x) {\mathbb{I}}(\tau_j\wedge N_j \geq k) - {\mathbf{E}}{\mathbb{I}}(Y'_{k-1} \leq x) {\mathbb{I}}(\tau\geq k) {\mathbf{P}}(N_j \geq k) \right| \to 0 \quad a.s.$$ as $n \to \infty$, for each fixed $k\geq 0$. Since we proved in Section \[sec:3\] that $${\mathbf{E}}({\mathbb{I}}(Y_k \leq x) - {\mathbb{I}}(Y_{k-1}' \leq x)) {\mathbb{I}}(\tau\geq k) = {\mathbf{P}}(Y_k \leq x) - {\mathbf{P}}(Y_{k-1} \leq x),$$ it follows that for any $m\geq 1$, $$\label{eq:4.1}
\sup_x \left| \sum_{k=0}^m \frac{1}{n} \sum_{j=1}^n ({\mathbb{I}}(Y_{kj}\leq x) - {\mathbb{I}}(Y_{k-1,j}'\leq x))\frac{{\mathbb{I}}(\tau_j \wedge N_j \geq k)}{{\mathbf{P}}(N\geq k)} - {\mathbf{P}}(Y_m \leq x) \right| \to 0 \quad a.s.$$ as $n\to \infty$. If $X$ is any aperiodic positive recurrent Harris chain, $Y_m$ converges to $Y_\infty$ in total variation, and hence $$\label{eq:4.2}
\sup_x |{\mathbf{P}}(Y_m \leq x) - F_\infty(x) | \to 0$$ as $m \to \infty$. (If $X$ is periodic, we adapt (\[eq:4.2\]) by restricting $m$ to multiples of $p$, and (\[eq:4.2\]) is replaced by $$\sup_x \left|\frac{1}{p}\sum_{i=0}^{p-1}{\mathbf{P}}(Y_{m+i} \leq x) - F_\infty(x) \right| \to 0$$ as $m \to \infty$.)
Suppose now that ${\mathbf{P}}(N\geq k ) \sim ck^{-\alpha}$ as $k\to \infty$ (where $a_k \sim b_k$ means that $a_k / b_k \to 1$ as $k \to \infty$). Note that for $m$ sufficiently large, $$\begin{aligned}
\label{eq:4.3}
&\sum_{k > m} \left| \frac{1}{n} \sum_{j=1}^n ({\mathbb{I}}(Y_{kj} \leq x) - {\mathbb{I}}(Y_{k-1,j}' \leq x) ) \frac{{\mathbb{I}}(\tau_j \wedge N_j \geq k)}{{\mathbf{P}}(N\geq k)}\right| \nonumber\\
&\leq \sum_{k > m} \frac{1}{n} \sum_{j=1}^n \frac{{\mathbb{I}}(\tau_j \wedge N_j \geq k)}{{\mathbf{P}}(N\geq k)} \nonumber\\
&\leq \frac{2}{c} \sum_{k > m} k^\alpha \frac{1}{n} \sum_{j=1}^n {\mathbb{I}}(\tau_j \wedge N_j \geq k) \nonumber\\
&\leq \frac{3}{c(\alpha+1)} \cdot \frac{1}{n} \sum_{j=1}^n (\tau_j \wedge N_j )^{\alpha+1} {\mathbb{I}}(\tau_j \wedge N_j \geq m) \nonumber\\
&\to \frac{3}{c(\alpha+1)} {\mathbf{E}}(\tau \wedge N)^{\alpha+1} {\mathbb{I}}(\tau \wedge N \geq m) \qquad a.s.
\end{aligned}$$ as $n \to \infty$. If ${\mathbf{E}}(\tau\wedge N)^{\alpha + 1} < \infty$, it follows that $$\label{eq:4.4}
{\mathbf{E}}(\tau\wedge N)^{\alpha+1} {\mathbb{I}}(\tau\wedge N \geq m) \to 0$$ as $m \to \infty$.
By first fixing $m$, then letting $n \to \infty$, and finally sending $m \to \infty$, (\[eq:4.1\])-(\[eq:4.4\]) therefore prove that $$\sup_x |F_n (x) - F_\infty(x) | \to 0$$ a.s. as $n\to \infty$. It remains to consider the finiteness of ${\mathbf{E}}(\tau\wedge N)^{\alpha+1}$. Observe that $${\mathbf{E}}(\tau\wedge N)^{\alpha+1} \leq (\alpha+1)\sum_{k=0}^\infty k^\alpha {\mathbf{P}}(\tau\geq k) {\mathbf{P}}(N\geq k).$$ But $\sum_{k=0}^\infty {\mathbf{P}}(\tau\geq k) = {\mathbf{E}}(\tau + 1) < \infty $ and $k^{\alpha} {\mathbf{P}}(N\geq k) \to c$ as $k \to \infty$, thereby proving that ${\mathbf{E}}(\tau\wedge N)^{\alpha+1}$ is automatically finite.
This proves the following Glivenko-Cantelli type theorem for the estimator of Section \[sec:3\].
\[thm:4.1\] Suppose ${\mathbf{P}}(N\geq k)\sim ck^{-\alpha}$ as $k \to \infty$ for $\alpha > 0$. If $X$ is a positive recurrent Harris chain, then $F_{n}(x)$ is an unbiased estimator for $F_\infty(x)$ for each $x\in {\mathbb{R}}$, and $$\sup_x|F_n(x) - F_\infty(x)| \to 0\qquad a.s.$$ as $n \to \infty$.
Numerical Results {#sec:5}
=================
We present here a brief account of the numerical performance of our exact estimation algorithms. For our contracting chain example, we consider the non $\phi$-irreducible Markov chain $X=(X_n: n\geq 0)$ given by $$X_{n+1} = \frac{1}{2} X_n + V_{n+1},$$ where the $V_i$’s are iid with ${\mathbf{P}}(V_n = 0) = 1/2 = {\mathbf{P}}(V_n = 1)$, with corresponding Lipschitz functions $f_1(x) = x$, $f_2(x) = \min(1, x)$, and $f_3 (x) = x^2$. For this example, $\pi$ is uniform on $[0,2]$, ${\mathbf{E}}f_1(X_\infty) =1$, ${\mathbf{E}}f_2(X_\infty) = 3/4$, and ${\mathbf{E}}f_3(X_\infty) = 4/3$. Tables \[table:contracting\] and \[table:contracting\_large\_mean\] report results for two different distributions for $N$. As expected, the algorithm based on $Z$ becomes more attractive when $N$ has a heavier tail, because the computational effort for $Z^*$ increases quadratically in $N$ (because of the non-recursive computation of the $\Delta_i$’s), whereas the effort for $Z$ increases linearly in $N$.
$f(x)$ Estimator 90% Confidence Interval \#Samples
------------- ----------- ---------------------------------- --------------------
$x$ $Z$ 1.013 $\pm$ 1.1$\times 10^{-2}$ 6.7$\times 10^{4}$
$Z^*$ 0.9974 $\pm$ 7.3$\times 10^{-3}$ 5.0$\times 10^{4}$
$\min(1,x)$ $Z$ 0.7531 $\pm$ 6.2$\times 10^{-3}$ 6.7$\times 10^{4}$
$Z^*$ 0.7552 $\pm$ 4.7$\times 10^{-3}$ 5.1$\times 10^{4}$
$x^2$ $Z$ 1.344 $\pm$ 2.3$\times 10^{-2}$ 6.7$\times 10^{4}$
$Z^*$ 1.334 $\pm$ 1.6$\times 10^{-2}$ 5.0$\times 10^{4}$
: Contracting chain: with $10^6$ time steps, $X_0 = 1$, ${\mathbf{P}}(N\geq n) = 0.95^{n-1}$ \[table:contracting\_large\_mean\]
$f(x)$ Estimator 90% Confidence Interval \#Samples
------------- ----------- --------------------------------- --------------------
$x$ $Z$ 1.009 $\pm$ 3.3$\times 10^{-2}$ 2.7$\times 10^{3}$
$Z^*$ 1.006 $\pm$ 6.1$\times 10^{-2}$ 2.4$\times 10^{2}$
$\min(1,x)$ $Z$ 0.743 $\pm$ 1.7$\times 10^{-2}$ 2.7$\times 10^{3}$
$Z^*$ 0.764 $\pm$ 3.6$\times 10^{-2}$ 2.5$\times 10^{2}$
$x^2$ $Z$ 1.356 $\pm$ 6.7$\times 10^{-2}$ 2.7$\times 10^{3}$
$Z^*$ 1.39 $\pm$ 1.3$\times 10^{-1}$ 2.5$\times 10^{2}$
: Contracting chain: with $10^6$ time steps, $X_0 = 1$, ${\mathbf{P}}(N\geq n) = 0.95^{n-1}$ \[table:contracting\_large\_mean\]
We turn next to the Harris chain algorithm, implemented with the coupling $\tau'$ of Section \[sec:3\]. We consider the Markov chain $W=(W_n: n\geq 0)$ on ${\mathbb{R}}_+$ corresponding to the waiting time sequence for the $M/M/1$ queue, with arrival rate 1/2 and unit service rate. The equilibrium distribution $\pi$ here is a mixture of a unit point mass on 0 and an exponential distribution with rate parameter $1/2$, with equal probability 1/2 on each mixture component. We let the function $f$ be given by $f(x) = {\mathbb{I}}(x > 1)$, so that ${\mathbf{P}}( W_\infty > 1) = \frac{1}{2} e^{-1/2} \approx 0.303.$
As for $N$, we note that the same proof technique as for Proposition 1 of [@Rhee2013] establishes that the optimal choice for the distribution of $N$ is to choose ${\mathbf{P}}(N\geq k)$ proportional to $$\sqrt{\frac{{\mathbf{E}}\Delta_k^2 + 2\sum_{j = k+1}^\infty {\mathbf{E}}\Delta_k \Delta_j}{{\mathbf{P}}(\tau \geq k)}},$$ provided that this sequence is non-[increasing]{}. Since it seems likely that ${\mathbf{E}}\Delta_k^2$ will frequently be of roughly the same order as ${\mathbf{P}}(\tau \geq k)$ for large $k$, this suggests that the optimal distribution will often have positive mass at infinity. In view of this observation, we have chosen to use a very heavy-tailed specification for $N$, namely ${\mathbf{P}}(N\geq k)= 1/k$ for $k\geq 0$. Table \[table:HarrisChain\] reports the results of our computations with $\lambda=1$ and with small set $A=\{0\}$; the results show the “square root” decrease in the width of the confidence interval that is to be expected.
\#Steps Simulated 90% Confidence Interval \#Samples
-------------------- ---------------------------------- --------------------
1.0$\times 10^{5}$ 0.283 $\pm$ 8.3$\times 10^{-2}$ 3.2$\times 10^{4}$
2.0$\times 10^{5}$ 0.279 $\pm$ 5.8$\times 10^{-2}$ 6.5$\times 10^{4}$
5.0$\times 10^{5}$ 0.296 $\pm$ 3.4$\times 10^{-2}$ 1.6$\times 10^{5}$
1.0$\times 10^{6}$ 0.329 $\pm$ 2.7$\times 10^{-2}$ 3.2$\times 10^{5}$
2.0$\times 10^{6}$ 0.294 $\pm$ 1.8$\times 10^{-2}$ 6.5$\times 10^{5}$
5.0$\times 10^{6}$ 0.308 $\pm$ 1.2$\times 10^{-2}$ 1.6$\times 10^{6}$
1.0$\times 10^{7}$ 0.2992 $\pm$ 8.4$\times 10^{-3}$ 3.2$\times 10^{6}$
2.0$\times 10^{7}$ 0.3089 $\pm$ 5.8$\times 10^{-3}$ 6.5$\times 10^{6}$
5.0$\times 10^{7}$ 0.2995 $\pm$ 3.7$\times 10^{-3}$ 1.6$\times 10^{7}$
1.0$\times 10^{8}$ 0.3041 $\pm$ 2.6$\times 10^{-3}$ 3.3$\times 10^{7}$
2.0$\times 10^{8}$ 0.3024 $\pm$ 1.9$\times 10^{-3}$ 6.5$\times 10^{7}$
5.0$\times 10^{8}$ 0.3036 $\pm$ 1.2$\times 10^{-3}$ 1.6$\times 10^{8}$
: Harris chain \[table:HarrisChain\]
Acknowledgements {#acknowledgements .unnumbered}
================
The authors gratefully acknowledge the support of National Science Foundation Grant DMS-1320158 and a Samsung Scholarship.
|
---
author:
- |
Shin [Mineshige]{} and Atsunori [Yonehara]{} [^1]\
[*Department of Astronomy, Faculty of Science, Kyoto University, Sakyo-ku, Kyoto 606-8502*]{}\
[*E-mail(SM): [email protected]*]{}
title: Gravitational Microlens Mapping of a Quasar Accretion Disk
---
Introduction
============
In many cases of astrophysical objects, their direct images are too small to resolve with the usual telescopes. In the case of a quasar accretion disk, for example, its angular size is, typically, on the order of $\theta_{\rm d} \sim 0.01{\rm pc}/1{\rm Gpc}
\!=\! 10^{-11} {\rm rad}
\!\approx\! 1\mu{\rm as}$. In the case of a binary accretion disk, on the other hand, we estimate $\theta_{\rm d} \sim 3\times 10^{10}{\rm cm}/1{\rm kpc}
\!=\! 10^{-11} {\rm rad}
\!\approx\! 1\mu{\rm as}$. These are both considerably below the angular resolution of any present-day telescopes. What is usually attempted is, therefore, to construct theoretical models based on the basic equations of (magneto)hydronamics and to infer their structure from the limited number of information, such as the total radiation output originating from their entire surface at some wavelength bands.
In some special cases, however, we can ‘resolve’ the spatial structure of the disk by using a ‘natural’ telescope. A good example is the technique of eclipse mapping (Horne 1985). Since a certain fraction of an accretion disk is shadowed by a companion star in an eclipsing close binary and since the shadowed part varies with time in accordance with the orbital phase, we can map the disk emissivity distribution from the eclipsing light curves. A variety of disk luminosity profiles has been revealed with this technique and our knowledge about the disk structure has been remarkably enriched. Unfortunately, however, this technique is irrelevant to probing a quasar accretion-disk structure because there are no eclipsing objects known to date (see, however, McKernan, Yaqoob 1998).
An alternative and potentially powerful method using a microlensing phenomenon has been considered. Chang and Refsdal (1979, 1984) considered flux changes that might occur due to gravitational lensing by a single star in an extended gravitational lens galaxy. Now, a stimulating discussion has started that such a microlensing phenomenon can be used to investigate the quasar central structure (Grieger et al. 1988; Rauch, Blandford 1991) as a ‘gravitational telescope’ (Blandford, Hogg 1995). The Einstein Cross, Q2237+0305 (e.g., Huchra et al. 1985), is the first object in which quasar microlensing events were detected (Corrigan et al. 1991, Houde, Racine 1994, see also Ostensen et al. 1996). These observations suggest that microlensing events seem to take place almost every year. This rather high frequency is consistent with the microlens optical depth of $\tau \simeq 0.2 - 0.8$ obtained by a realistic simulation of the lensing galaxy (i.e., Wambsganss, Paczyński 1994).
We, here, specifically consider the microlensing event of this source caused by the so-called ‘caustic crossings.’ Several authors have already calculated and inspired this ‘caustic’ case based on simple disk models (e.g., Wambsganss, Paczyński 1991; Jaroszyński et al. 1992) or recently on the realistic disk models (Yonehara et al. 1998, 1999). All of these calculations correspond to the so-called forward problem; i.e., they calculated microlens light curves based on disk models prescribed [*a priori*]{}. In contrast, we are, in the present study, concerned with a distinct approach called the inverse problem; i.e., we consider how to reconstruct the disk image (or more precisely, the emissivity distribution) of a quasar accretion disk from microlensing light curves.
For this purpose, we first simply apply the regularization method, one of the most well-known non-classical inversion techniques. The basic methodology of this method was already given by Grieger et al. (1991, hereafter GKS). Indeed, their proposed procedure has been very successful, but they have assumed rather idealized situations and, thus, there remain some problems requiring further investigation before realistic applications are obtained.
1. GKS assumed no radiation from any part outside a caustic. As a result, they could precisely determine the contribution from a small outer portion of the disk to the total light from the shape of the light curve at the beginning of a microlens event. If this were the case, we would not be able to see the disk emission in the absence of a microlensing, contrary to the observations.
2. The goal of GKS was to reconstruct a one-dimensional luminosity profile \[$P(\xi$) in their notation\] which is the luminosity integrated over the disk plane in the direction parallel to that of a caustic (see figure 1 for the definition of $\xi$).
To make a direct comparison with accretion disk models, however, it will be more convenient to express the emissivity distribution as a function of the distance from the center, $Q(r)$, since the central part of the disk can be reasonably assumed to be axisymmetric. It is thus needed to transform $P(\xi)$ to $Q(r)$.
3. GKS assumed a rather smooth emissivity distribution around the center, which certainly makes the analysis easier than otherwise. According to realistic disk models, however, it seems more likely that the emissivity has a power-law dependence on the radius, thus being sharply peaked around the center. We wish to know what fraction of the total energy output originates from a compact region in the disk.
The aim of the present study was to improve the GKS method so that it can be applied to more realistic situations. Our version of the inverse technique will be described in section 2. We then present the results of disk mapping for the calculated microlensing light curve in section 3. The final section will be devoted to a summary and discussion.
Inversion Procedures
====================
Setting Matrix Forms
--------------------
We first describe the regularization method adopted by GKS with some modifications.
Chang and Refsdal (1979, 1984) have shown that the flux from the bright double image of a point source close to a caustic can be amplified approximately according to $(x-\xi)^{-1/2}$, where $x$ and $\xi$ are, respectively, the radial distances of the caustic and that of a part of the disk in question from a line which is parallel to the caustic and which crosses the disk center (see figure 1). Note that $x$ is time-dependent; i.e., $$\label{vcaus}
x = x(t) = V_{\rm caus}t/r_0,$$ where the unit of length, $r_0$, is taken to be a typical disk dimension, $t=0$ corresponds to the time when a caustic crosses the center of the disk, and $V_{\rm caus}$ is the transverse velocity of the caustic, also including that of the peculiar motion of the foreground galaxy relative to the source and the observer, on the disk plane. The total observed flux will then be approximated by $$\label{fx}
F(t) = F_0\int_{-1}^1 A(x-\xi) P(\xi)d\xi.$$ Here, $F_0$ is a constant representing the total disk flux outside the caustic ($x < \xi$); i.e., $\int P(\xi)d\xi = 1$, and the amplification factor is given by $$\label{ax}
A(x-\xi) = \left\{ \begin{array}{lcl}
1+\frac{\displaystyle k}{\displaystyle\sqrt{x-\xi}}
& \mbox{for} & x > \xi \\
1 & \mbox{for} & x \leq \xi, \\
\end{array} \right.$$ where $k$ denotes the amplification factor inside the caustic ($x>\xi$) and depends on the spatial distribution of lensing stars and the number distribution of lens masses. Note that GKS set $A= 0$ outside the caustics and, hence, $k$ is arbitrary, which grossly simplified the analysis. Also note that inclination effects are included in the expression of $F_0$ and $V_{\rm caus}$ (discussed later).
We assume that $P(\xi)$ is represented by a continuous, piecewise linear function, $$\begin{aligned}
P(\xi) = \frac{\xi_j-\xi}{\xi_j-\xi_{j-1}} P_{j-1}
+ \frac{\xi-\xi_{j-1}}{\xi_j-\xi_{j-1}} P_{j} \\
\quad{\rm for}\quad\xi_{j-1}\leq\xi\leq\xi_j. \nonumber\end{aligned}$$ For simplicity, we take equal-distant mesh points normalized by the disk radius, $$\begin{aligned}
\xi_j = -1+ \left(j-\frac 12\right)\Delta\xi
= \frac{2j-J-1}{J} \\
\quad(j=1, 2, \cdots, J), \nonumber\end{aligned}$$ with $\Delta\xi\equiv 2/J$, and we take the total number of mesh points ($J$) to be equal to that of the observed time sequences ($t_i$, $i=1, 2, \cdots, J$). If we set $F_i = F(t_i)$, equation (\[fx\]) leads $$\begin{aligned}
\label{fkp}
\vec{F} = F_0{\bf K} \vec{P}
\quad{\rm or}\quad
F_i = F_0\sum_{j=1}^JK_{ij}P_j \\
\quad (i = 1, 2, \cdots, J) \nonumber\end{aligned}$$ with $$\vec{F} = (F_1, F_2, \cdots, F_J)^{\rm T}, \quad
\vec{P} = (P_1, P_2, \cdots, P_J)^{\rm T},$$ and $\bf{K}$ being a $J \times J$ matrix given in Appendix.
Regularization Technique
------------------------
It seems straightforward to derive the emissivity distribution, $P(\xi)$ or $P_j$ ($j=1, 2, \cdots, J$), from equation (\[fkp\]) by calculating the inverse matrix of ${\bf K}$, $$\label{soln1}
\vec{P} = {\bf K}^{-1}(\vec{F}/F_0).$$ However, this does not work efficiently, since the observational data usually contain measuring errors, which will be greatly amplified when calculating the inverse matrix (GKS).
To resolve this issue, a regularization technique was proposed. First, we express the observed flux as $$\label{fkpd}
\vec{F} = F_0{\bf K} \vec{P} + \vec{\delta}
\quad{\rm or}\quad
F_i = F_0\sum_{j=1}^J K_{ij}P_j + \delta_i,$$ with $\delta_i$ being the error in measuring the flux, $F_i$. Next, as a measure to evaluate how the $P(\xi)$ profile is smooth we introduce a badness function, $$L(P) = \int \left[\frac{d^2 P(\xi)}{d\xi^2}\right]^2d\xi.$$ (Note that $L(P)=0$ if $P(\xi)$ is a linear function of $\xi$.)
Then, the problem is reduced to determining the functional form of $P(\xi)$ which gives a minimum value of $$\label{min}
[(\vec{F}/F_0) - {\bf K}\vec{P}]^2 + \lambda L(\vec{P}).$$ Namely, there are two important factors to be considered: fitting to the given light curves ($\vec{F}$, the first term) and smoothing the emissivity profile ($\vec{P}$, the second term), where $L(\vec{P})$ is the matrix form of $L(P)$, $$L(\vec{P}) = \sum_j \left[\frac{P_{j+1}-2P_j+P_{j-1}}
{(\Delta\xi)^2}\right]^2\Delta\xi.$$ The controlling parameter, $\lambda$, is called the smoothing parameter. In the limit of vanishing $\lambda$, the solution, $P(\xi)$, is given by equation (\[soln1\]) which can precisely reproduce the observed light curves but is not always very smooth, especially in cases with large measuring errors (GKS). In the limit of very large $\lambda$, on the other hand, the solution is smoothest, since it gives a straight line on the \[$\xi,P(\xi)$\] plane, but may not give an excellent fit to the observed flux variations. We thus need to choose a moderate value of $\lambda$ which satisfies $$\label{chi2}
{\frac 1J}\sum_{i=1}^J
\left[\frac{1}{\delta_i}\left(
{F_i}-F_0\sum_{j=1}^JK_{ij}P_j
\right)\right]^2 = 1.
$$ In other words, the final solutions should be [*as smooth as possible under the constraint that a reproduced flux variation agrees with the observed variation within the error bars.*]{}
After some algebra, minimizing equation (\[min\]) is equivalent to setting $$\label{soln2}
({\bf K}^{\rm T}{\bf K} + \lambda {\bf H}) \vec{P}
= {\bf K}^{\rm T}(\vec{F}/F_0),$$ where $\bf{H}$ is a matrix (see Appendix of GKS). The final solution, $P(\xi)$, is obtained successively by solving equation (\[soln2\]) for a given $\lambda$ satisfying equation (\[chi2\]).
These procedures are performed for a specific value of $k$, say, $k=0.5$ (which gives a variation with an amplitude of $\sim$ 0.5 mag for $Q(r) \propto 1/r$). We repeat the same procedures with different values of $k$ and determine the $k$-value so that $L(P)$ reaches the minimum, yielding the smoothest $P(\xi)$.
Transformation from $P(\xi)$ to $Q(r)$
--------------------------------------
As stated in Introduction, our goal is to determine the emissivity distribution of the disk as a function of $r$ (not $\xi$), where $r$ is the radial distance from the center of the accretion disk in the unit of $r_0$ ($r=1$ at the outer rim). Thus, we need to transform $P(\xi)$ to $Q(r)$.
Note that the dimension of $Q(r)$ is flux; i.e., the energy emitted per unit time from a unit surface area. Then, $P(\xi)$ can be expressed as an integral of $Q(r)$ as $$\label{pxi}
P(\xi) = \int_{-1}^1 Q(r) d\eta
= 2\int_{|\xi|}^1 Q(r)\frac{rdr}{\sqrt{r^2-\xi^2}}.$$
It is of great importance to note that equation (\[pxi\]) takes the form of Abel’s integral; that is, the inverse transformation is straightforward (see, e.g., Binney, Tremaine 1987). Since we have two independent sets of $P(\xi)$, one at $\xi < 0$ and the other at $\xi > 0$, we can separately obtain two sets of $Q(r)$, $Q^-(r)$ and $Q^+(r)$; namely, $$\begin{aligned}
\label{qr}
Q^-(r)=\frac{1}{\pi}\int_{-\infty}^{-r}
\frac{dP(\xi)}{d\xi} \frac{d\xi}{\sqrt{\xi^2-r^2}}
\quad{\rm and}\quad \nonumber \\
Q^+(r)=\frac{1}{\pi}\int_r^\infty \left[
-\frac{dP(\xi)}{d\xi}\right] \frac{d\xi}{\sqrt{\xi^2-r^2}}.\end{aligned}$$ Note that quite generally $dP/d\xi > 0$ for $\xi < 0$ and $dP/d\xi < 0$ for $\xi > 0.$
The next procedure is used to derive an expression for $Q^\pm_k = Q^\pm(r_k)$ in terms of $P_j = P(\xi_j)$ ($j=1,2, \cdots, J$) with $r_k \equiv (2k-1)/J$ for $k=1, 2, \cdots, J/2$. We have $$\begin{aligned}
Q^-(r_k)\!\!&=&\!\!\frac{1}{\pi}\sum_{j=1}^{(J/2)-k}
\frac{P_{j+1}-P_j}{\sqrt{\xi_{j+(1/2)}^2-r_k^2}} \nonumber \\
& &+ \frac{1}{\pi}\frac{P_1}{\sqrt{1-r_k^2}} \end{aligned}$$ and $$\begin{aligned}
Q^+(r_k)\!\!&=&\!\!\frac{1}{\pi}\sum_{j=(J/2)+k}^{J-1}
\frac{P_j-P_{j+1}}{\sqrt{{\xi}_{j+(1/2)}^2-r_k^2}} \nonumber \\
& &+ \frac{1}{\pi}\frac{P_J}{\sqrt{1-r_k^2}} \end{aligned}$$ with ${\xi}_{j+(1/2)} \equiv |(\xi_j+\xi_{j+1})/2| = (2j-J)/J$, and we used ${\xi}_{1/2} = -1$ and ${\xi}_{J+(1/2)}= +1$. We can successively find $Q^\pm(r_k)$’s for $k=1,2,\cdots, (J/2)-1$ from $P(\xi_j)$’s ($j=1,2,\cdots,J$).
Producing Light Curves
----------------------
To test how the above procedures work, we calculate the expected flux variations based on specific models for $Q(r)$ \[hereafter denoted as $Q_{\rm model}(r)$ to distinguish from the reconstructed values, $Q^\pm(r)$\]. It is important to note that accretion disk structure can well be described as being self-similar (or more precisely, each physical quantity is expressed as a power-law function of the radius, see, e.g., Shakura, Sunyaev, 1973; Narayan, Yi 1995). We, hence, prescribe the emissivity distribution as $$Q_{\rm model}(r) \propto r^{-a} \quad{\rm for~}
0.01 \leq r \leq 1.0$$ with $a$ being a positive constant. Note that $a=3$ for the standard-type disks, whereas $Q_{\rm model}(r)$ is much flatter, $a \lsim 1$, in optically-thin, advection-dominated disks (Manmoto et al. 1997).
Light curves are calculated by $$\begin{aligned}
\label{ft}
F(t_i)\!\!&=&\!\!F_0\int_0^1 rdr\int_0^{2\pi} d\theta
A(x-r\cos\theta) Q_{\rm model}(r) \nonumber \\
& & + \delta_i
$$ with the amplification factor \[$A(x-r\cos\theta)$\] being given by equation (\[ax\]). We continuously change the angular separation between the caustic and the center of the accretion disk (see Eq.\[\[vcaus\]\]). The unit of time ($t_0$) is taken to be the crossing time over which the caustic moves on the quasar disk plane from $\xi = 0$ to $+1$. That is, $t_0 = r_0/V_{\rm caus}$, and, hence, $x(t) = t/t_0$ from equation (\[vcaus\]). On the intrinsic light curve, we add measuring errors, $\delta_i$. We calculate three models for each $Q_{\rm model}(r)$ prescription; $F(t)$ at each time is randomly fluctuated around the mean value in a Gaussian way with assumed standard deviations, $\Delta m$ (in magnitude).
We plot in figure 2 how different microlens light curves are produced by changes of the $Q_{\rm model}(r)$ prescription.
The normalizations are taken so as to give $F=1.5F_0$ at $t=0.1t_0$.
Importantly, the peak luminosities are reached at the time of caustic crossing over the disk center ($t=0$) only for $a\geq 1$, while a disk with a flat emissivity profile (with $a=0$) yields the peak at a later time, $t\sim 0.7t_0$. This is because for the flat disk the amplified area (which reaches its maximum at $t=t_0$) is also an important factor. For $a\geq 1$, in contrast, the emergent flux is rather insensitive to the extent of the amplified area, since a large fraction of radiation comes from the very center of the disk.
To summarize, those cases with $a=0$ and $a\geq 1$ give distinct flux variations. It is thus important to distinguish these two critical cases in disk mapping.
Results of Image Reconstruction
===============================
The results of the reconstruction are displayed in figure 3 for the case with $Q_{\rm model}(r) \propto 1/r$. The prescribed emissivity profile is illustrated by the dashed lines in the lower three panels. To calculate the flux variation, we set a constant time interval, $\Delta t= 0.2t_0$, and the total number of observations is $J=24$. The calculated (i.e., ‘observed’) flux at each time is displayed together with error bars in the upper three panels by the dashed lines for different magnitudes of the mean errors, $\Delta m$ = 0.1 (left), 0.05 (middle), and 0.02 (right), respectively. The dashed lines in the middle ones show the one-dimensional emissivity, $P(\xi)$, calculated from $Q_{\rm model}(r)$.
Similarly, the reconstructed $F(t)$ and $P(\xi)$ are shown in the upper and middle panels with the solid lines. The two solid lines in the lower panels display $Q^-(r)$ at $r < 0$ and $Q^+(r)$ at $r > 0$, respectively. Note that the reconstructed $k$-value does not always reproduce the original value of $k=0.50$; we find $k=0.61$, 0.51, and 0.47 for $\Delta m = 0.10$, 0.05, and 0.02, respectively. The reconstructed flux variations and emissivity profiles are not very sensitive to these small deviations in the $k$-values, however.
For $a=1$, large errors ($\Delta m=0.1$ mag) tend to produce a rather flat $P(\xi)$ profile and thus a smooth $Q(r)$. This is because a flat model \[$P(\xi) =$ const.\] is compatible with the light curves within the error bars so that the technique prefers a flatter $P(\xi)$ profile (see figure 2). Note again that the peak shifts from $t=0$ in the original light curve to $t\sim 0.7 t_0$ in the reconstructed one. Such a problem does not arise for those cases with smaller errors. To reproduce a steep $Q(r)$ profile up to the inner parts, $\Delta m = 0.02$ mag is necessary. It depends on the observing intervals how close to the origin the mapping technique can reproduce the original image. Surely, frequent observations, especially at around the peak flux, are preferable (GKS).
Figures 4 and 5 display the results of those cases with $Q_{\rm model}\propto 1/r^2$, and $\propto 1/r^3$, respectively. In each figure, we omit the cases with $\Delta m$ = 0.05 and $P(\xi)$ plots to avoid any complications.
The reconstructed $k$-values are 0.63 ($\Delta m = 0.10$) and 0.74 ($\Delta m=0.02$) for $a=2$, and 0.76 ($\Delta m = 0.10$) and 0.78 ($\Delta m = 0.02$) for $a=3$, respectively. These all deviate greatly from the original value of $k=0.5$, particularly when $a$ is large or $\Delta m$ is small. Nevertheless, the inner steep rise parts are reasonably well reproduced by the mapping. The wing parts are poorly reproduced for a steep $Q(r)$ profile. In fact, $Q(r)$ sometimes goes below zero. This tendency is rather enhanced for small $\Delta m$’s.
To conclude, in cases with sharply peaked emissivity profiles we can still obtain reliable information regarding the extent of the substantially emitting region, although we cannot trust the results about the outer zones surrounding the central bright zone.
Summary and Discussion
======================
Let us consider specifically the case of Einstein Cross and thus insert the model parameters relevant to this source. The Einstein-ring radius on the source plane is $$\begin{aligned}
r_{\rm E} \equiv \theta_{\rm E}D_{\rm os}
\!\!&=&\!\! \left[\left(\frac{4GM_{\rm lens}}{c^2}\right)
\left(\frac{D_{\rm ls}D_{\rm os}}
{D_{\rm ol}}\right)\right]^{1/2} \nonumber \\
\!\!&\sim&\!\!1.5\times 10^{17}
\left(\frac{M_{\rm lens}}{M_\odot}\right)^{1/2}{\rm cm},\end{aligned}$$ where $M_{\rm lens}$ is the typical mass of a lens star, and $D_{\rm ls}$, $D_{\rm os}$, and $D_{\rm ol}$ represent the angular diameter distances from lens to source, from observer to source, and from observer to lens, respectively. To evaluate these distances, we assume the redshifts corresponding to the distances from the observer to the quasar and from the observer to the lens of, $z_{\rm os} = 1.675$ and $z_{\rm ol} = 0.039$, respectively (see Irwin et al. 1989), and also assumed an Einstein-de Sitter universe and Hubble’s constant to be $H_{\rm 0} \sim 60 {\rm km~s^{-1} Mpc^{-1}}$, according to Kundić et al. (1997).
Another important length is $r_{\rm cross}$, the caustic crossing length over the quasar image plane per observational time interval, $\Delta t$; $$\begin{aligned}
\label{cross}
r_{\rm cross}\!\!&=&\!\!
v_{\rm t}\Delta t \frac{D_{\rm os}}{D_{\rm ol}} \nonumber \\
\!\!&\sim&\!\!2.0\times 10^{13}
\left(\frac{v_{\rm t}}{300~{\rm km~s}^{-1}}\right)
\left(\frac{\Delta t}{1~{\rm d}}\right) {\rm cm},
$$ where $v_{\rm t}$ is the transverse velocity of the lens on the lens plane ($V_{\rm caus} \equiv v_{\rm t}D_{\rm os}/D_{\rm ol}$). Surprisingly, this is comparable to the Schwarzschild radius, $r_{\rm g}\simeq 3\times 10^{13}(M_8)^{-1}$cm ($\sim 2$AU) for a $10^8M_8$ black hole and is much smaller than $r_{\rm E}$. Thus, by weekly observations can one determine the disk emissivity distributions on length scales of $\sim$ 10AU or $\sim 5 r_{\rm g}(M_8)^{-1}$ for $v_{\rm t} \sim$ 300 km s$^{-1}$. To summarize, we have improved the reconstruction technique previously developed by GKS in such a way that a direct comparison with accretion disk models is possible. We have found that for deriving the emissivity distribution as a function of $r$ on scales down to several to ten AUs, we need an accuracy of $\Delta m \lsim 0.02$mag and a sampling interval within one week. In cases in which the emissivity profile is rather centrally peaked (i.e., if $a > 1$), we can still reproduce the inner bright zone reasonably well, but cannot trust the results of the outer zone.
It might be noted that the problem treated here seems to be closely related to that of reconstructing stellar brightness profile (e.g., limb darkening) from photometry of Galactic microlensing events (see Gaudi, Gould, 1999, and references therein). The present technique may be used to analyze such data.
The inclination of the disk has two important effects. If the disk plane is tilted by an angle of $i$ with respect to the line of sight, the apparent disk flux will be reduced by a factor of $(\cos i)^{-1}$. This affects the normalization constant, $F_0$. Further, if the disk is tilted by an angle of $i_\parallel$ with respect to the direction of the motion of a caustic, the apparent transverse velocity of the caustic will be increased by a factor of $(\cos i_\parallel)^{-1}$. This affects the caustic crossing, $V_{\rm caus}$, and thus the time and length scales, $t_0$ and $r_0$. In other words, the normalization constants both in the ordinate and abscissa in the $(t, F)$ diagram and thus in the $(r, Q)$ diagram are subject to the inclination angles. There remain ambiguities in the normalizations of $r_0$ and $Q_0$. Since it is difficult to evaluate these inclination angles and other uncertain factors, we rather focused our effort on the shape of the non-dimensional emissivity distribution, $Q/Q_0$, as a function of $r/r_0$.
We must consider the fact that real photometric light curves will be unevenly sampled due to gaps in the observing schedule caused by various factors, including weather. Therefore, an even more realistic case could be made by making a model of such gaps. To see such an effect, we made a light curve in which a certain fraction, $\sim 25$%, of the regularly spaced light curve samples is removed stochastically. The results for the cases with $Q_{\rm model} \propto 1/r$ are displayed in figure 6. The reconstructed $k$-values are 0.48 ($\Delta m = 0.10$) and 0.415 ($\Delta m=0.02$), respectively. Because there are no significant changes, as long as frequent observations are made around the peak, a lack of data (uneven sampling) should not cause any serious problem. Obviously, however, poor sampling rates lead to a poor spatial resolution in the disk mapping (see equation \[cross\]). The results would sensitively depend on whether or not there are enough observation runs at the times around the peak.
HST and AXAF observations of the Einstein Cross are scheduled. If observed with multi-wavelength bands, a microlensing event should first clearly resolve the multi-wavelength radiation properties of a disk in a distant quasar on length scales down to several AUs.
The authors would like to express their thanks to Joachim Wambsgan[ß]{} for valuable suggestions and an anonymous referee for useful comments, which helped in making the revised version. This work was supported in part by the Grants-in Aid of the Ministry of Education, Science, and Culture of Japan (10640228, SM) and by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists, 9852 (AY).
Appendix {#appendix .unnumbered}
========
Equation (\[fx\]) can be rewritten as $$F_i = \sum_{j=1}^J f_{ij},$$ where $f_{ij}$ is non-zero only for $x_i > \xi_{j-1}$ and is $$\begin{aligned}
f_{ij}
\!\!&=&\!\!\int_{\xi_{j-1}}^{\min(\xi_j,x_i)}A(x-\xi)P(\xi)d\xi \nonumber \\
\!\!&=&\!\!\int_{\xi_{j-1}}^{\min(\xi_j,x_i)}
\left(1+\frac{k}{\sqrt{x_i-\xi}}\right) \nonumber \\
& & \times \left(\frac{\xi_j-\xi}{\Delta\xi} P_{j-1}
+ \frac{\xi-\xi_{j-1}}{\Delta\xi} P_{j}\right)d\xi,\end{aligned}$$ with $\Delta\xi\equiv \xi_{j}-\xi_{j-1}$. We introduce matrices $E_{ij}$ and $D_{ij}$ such that $$\label{ed}
f_{ij} = E_{ij}P_{j-1} + D_{ij}P_j.$$ We then have (for $x_i > \xi_{j-1}$) $$\begin{aligned}
\label{ee}
E_{ij}\!\!&=&\!\!\frac{1}{\Delta\xi} \int_{\xi_{j-1}}^{\min(\xi_j,x_i)}
\left(\xi_j-\xi+k\frac{\xi_j-\xi}{\sqrt{x_i-\xi}}\right)d\xi \nonumber \\
\!\!&=&\!\!\frac{1}{\Delta\xi}\int_{\Delta'\xi}^{\Delta\xi}
\left(s+k\frac{s}{\sqrt{x_i+s-\xi_j}}\right)ds \nonumber \\
\!\!&=&\!\!\frac{1}{\Delta\xi}
\left[\frac{s^2}{2}\right.+2ks\sqrt{x_i+s-\xi_j} \nonumber \\
& & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ -\left.\frac{4k}{3}(x_i+s-\xi_j)^{3/2}
\right]_{\Delta'\xi}^{\Delta\xi} \nonumber \\
\!\!&=&\!\!\frac{(\Delta\xi)^2-[\rho(\xi_j-x_i)]^2}{2\Delta\xi}
+ 2k {\sqrt{x_i-\xi_{j-1}}} \nonumber \\
& & -\frac{4k}{3} \frac{(x_i-\xi_{j-1})^{3/2}-[\rho(x_i-\xi_j)]^{3/2}}
{\Delta\xi},\end{aligned}$$ with $s\equiv \xi_j-\xi$, $\Delta'\xi \equiv \rho(\xi_j-x_i)$ and $$\rho(x_i-\xi_j) \equiv \left\{ \begin{array}{lcl}
x_i-\xi_{j} & \mbox{for} & x_i > \xi_j \\
0 & \mbox{for} & x_i \leq \xi_j. \\
\end{array} \right.$$ Similarly, we find (for $x_i > \xi_{j-1}$) $$\begin{aligned}
\label{dd}
D_{ij}\!\!&=&\!\!\frac{1}{\Delta\xi} \int_{\xi_{j-1}}^{\min(\xi_j,x_i)}
\left(\xi-\xi_{j-1}
+k\frac{\xi-\xi_{j-1}}{\sqrt{x_i-\xi}}\right)d\xi \nonumber \\
\!\!&=&\!\!\frac{1}{\Delta\xi}\int_0^{\Delta''\xi}
\left(u+k\frac{u}{\sqrt{x_i-u-\xi_{j-1}}}\right)du \nonumber \\
\!\!&=&\!\!\frac{1}{\Delta\xi}
\left[\frac{u^2}{2}\right.-2ku\sqrt{x_i-u-\xi_{j-1}} \nonumber \\
& & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ -\left.\frac{4k}{3}(x_i-u-\xi_{j-1})^{3/2}
\right]_0^{\Delta''\xi} \nonumber\\
\!\!&=&\!\!\frac{(\Delta''\xi)^2}{2\Delta\xi}
-2k\frac{\Delta''\xi}{\Delta\xi} \sqrt{\rho(x_i-\xi_j)} \nonumber \\
& & +\frac{4k}{3} \frac{(x_i-\xi_{j-1})^{3/2}-[\rho(x_i-\xi_j)]^{3/2}}
{\Delta\xi},\end{aligned}$$ with $u\equiv \xi-\xi_{j-1}$ and $\Delta''\xi \equiv\min(\Delta\xi, x_i-\xi_{j-1})$. For $x_i \leq \xi_{j-1}$ both are zero, $E_{ij} = D_{ij} = 0$. Since we have from equation (\[ed\]) $$f_{ij} = \left(E_{i,j+1} + D_{ij}\right)P_j \equiv K_{ij}P_j,$$ for $(i,j)=(1,1), (1,2), \cdots, (J,J)$, it is straightforward to derive expression for $\bf{K}$, $$K_{ij} = E_{i,j+1} + D_{ij},$$ by inserting equations (\[ee\]) and (\[dd\]).
References {#references .unnumbered}
==========
Binney J., Tremaine S. 1987, Galactic Dynamics (Princeton Univ. Press, Princeton) p651
Blandford R.D., Hogg D.W. 1995, in IAU Sympo. 173, Astrophysical Application of Gravitational Lensing, ed C.S. Kochanek, J.N. Hewitt (Kluwer, Dordrecht) p355
Chang K., Refsdal S. 1979, Nature 282 561
Chang K., Refsdal S. 1984, A&A 132, 168
Corrigan R.T. et al. 1991, AJ 102, 34
Gaudi B.S., Gould A. 1999, ApJ 513, 619
Grieger B., Kayser R., Refsdal S. 1988, A&A 194, 54
Grieger B., Kayser R., Schramm T. 1991, A&A 252, 508 (GKS)
Horne K., 1985, MNRAS 213, 129
Houde M., Racine R. 1994, AJ 107, 466
Huchra J. et al. 1985, AJ 90, 691
Irwin M.J., Webster R.L., Hewett P.C., Corrigan R.T., Jedrzejewski R.I. 1989, AJ 98, 1989
Jaroszyński M., Wambsganss J., Paczyński B. 1992, ApJ 396, 65L
Kundić T. et al. 1997, ApJ 482, 75
Manmoto T., Mineshige S., Kusunose M., 1997, ApJ 489, 791
McKernan B., Yaqoob T. 1998, ApJ 501, L29
Narayan R., Yi I. 1995, ApJ 452, 710
Ostensen R., et al. 1996, A&A 309, 59
Rauch K.P., Blandford R.D. 1991, ApJ, 381, L39
Shakura N.I., Sunyaev R.A. 1973, A&A 24, 337
Wambsganss J., Paczyński B. 1991, AJ 102, 864
Wambsganss J., Paczyński B. 1994, AJ 108, 1156
Yonehara A., Mineshige S., Fukue J., Umemura M., Turner E.L. 1999, A&A 343, 41
Yonehara A., Mineshige S., Manmoto T., Fukue J., Umemura M., Turner E.L. 1998, ApJ 501, L41; Erratum 511, L65
\[last\]
[^1]: Research Fellow of the Japan Society for the Promotion of Science.
|
---
author:
- 'F. Hofmann'
- 'J.S. Sanders'
- 'K. Nandra'
- 'N. Clerc'
- 'M. Gaspari'
bibliography:
- 'auto.bib'
- 'general.bib'
title: 'Thermodynamic perturbations in the X-ray halo of 33 clusters of galaxies observed with *Chandra* ACIS [^1]'
---
Introduction {#sec:introduction}
============
In the current picture of the evolution of the universe, clusters of galaxies have formed in the deep potential wells created by clumping of dark matter (DM) around remnant density fluctuations after the big bang. The majority of the mass in clusters is made of DM, which is only observed indirectly through its gravitational effects. The second component of clusters is baryonic matter consisting mainly of very thin hot plasma (the intra cluster medium, ICM), heated by the gravitational accretion into the potential wells, and emitting strongest in X-ray wavelength due to its high temperatures. The smallest fraction of the mass is in the stellar content of the clustered galaxies, which is observable in visible light. The behaviour of DM in the cluster potential is believed to be well understood from cosmological simulations [e.g. @2005Natur.435..629S] and observations of gravitational lensing effects on visible light . The member galaxies of a cluster are well described as collisionless particles moving in the cluster potential by measuring the line of sight velocity dispersion in the optical . The complex dynamic and thermodynamic processes in the hot ICM can be studied with X-ray observations. Other phases of the ICM have been studied at different wavelength with UV and $\rm{H{\alpha}}$ [e.g. @2011ApJ...734...95M] or radio observations .
first related fluctuations in the projected pressure maps of the hot, X-ray emitting, ICM of the Coma cluster to turbulence. Turbulence has been discussed as a significant heating mechanism [see @2005ApJ...622..205D; @2010ApJ...713.1332R; @2012MNRAS.424..190G], which is important to understand the heating and cooling balance in clusters and estimate the amount of non-thermal pressure support [see e.g. simulations by @2014ApJ...792...25N]. @2014Natur.515...85Z recently studied turbulence in the Perseus and Virgo cluster by analysing fluctuations in the surface brightness of the cluster emission. Asymmetries and fluctuations within thermodynamic properties of the hot plasma can be used to estimate the amount of turbulence . This has been studied in the PKS0745-191 galaxy cluster by @2014MNRAS.444.1497S. We applied a similar technique to the current sample of 33 clusters and compared our results to the findings of cluster simulations .
The amount of turbulence in the hot ICM is hard to directly measure. Simulations of galaxy clusters predict turbulent motions of several hundreds of km/s . @2011MNRAS.410.1797S, @2013MNRAS.429.2727S, and were able to obtain upper limits on the velocity broadening of spectra for a large sample of clusters. However current X-ray instruments do not provide the spectral resolution needed to detect significant broadening due to turbulence in all but a few possible cases.
The basis for this study was the X-ray all-sky survey of the *ROSAT* mission [1990 to 1999, see @1982AdSpR...2..241T]. The clusters identified in this survey were followed up with the current generation X-ray telescopes *Chandra*, *XMM-Newton*, and *Suzaku*. For the substructure study we used observations of *ROSAT*-clusters with the X-ray observatory on board the *Chandra* satellite which delivers the best spatial and very good spectral resolution. Since its launch in 1999 *Chandra* has frequently been used to study clusters of galaxies as individual systems and as cosmological probes [e.g. @2004MNRAS.353..457A; @2008MNRAS.383..879A; @2009ApJ...692.1060V; @2009ApJ...692.1033V]. Its high resolution showed an unexpected complexity of the ICM structure in many cases [e.g. @2000MNRAS.318L..65F; @2000ApJ...534L.135M; @2000ApJ...541..542M; @2002ApJ...567L..27M; @2005MNRAS.356.1022S; @2006MNRAS.366..417F; @2007ApJ...665.1057F]. Based on the large archive of deep cluster observations we analysed a sample of clusters and mapped their thermodynamic properties. We specifically investigated perturbations in the thermodynamic parameters of the ICM, which according to recent high-resolution simulations can be used to trace turbulence in the ICM via the normalisation of the ICM power spectrum . By measuring the slope of the power spectrum we can constrain the main transport processes in the hot ICM, such as thermal conductivity .
The paper is structured as follows: Sect. 2 describes the sample selection and data reduction, Sect. 3 describes the analysis of perturbations, in Sect. 4 the main results are presented, Sect. 5 contains results for individual systems, in Sect. 6 the findings are discussed, and Sect. 7 contains the conclusions.
For all our analysis we used a standard $\rm{\Lambda CDM}$ cosmology with $\rm{H_0=71~km~s^{-1}~Mpc^{-1}}$, $\rm{\Omega_M=0.27}$ and $\rm{\Omega_{\Lambda}=0.73}$ and relative solar abundances as given by @1989GeCoA..53..197A.
Observations and data reduction {#sec:observations}
===============================
[llrrrrrr]{} & & & & & & &\
& & & & & & &\
RX J1347-114 & rxj1347 & 0.046 & 14.7$\pm$0.125 & 0.48 & 1.57$\pm$0.05 & 17.7$\pm$1.8 & 0.31\
1E 0657-56 & 1e0657 & 0.049 & 12.6$\pm$0.112 & 0.98 & 1.58$\pm$0.06 & 15.3$\pm$1.9 & 0.62\
A 2390 & a2390 & 0.062 & 10.8$\pm$0.104 & 0.55 & 1.51$\pm$0.07 & 12.4$\pm$1.8 & 0.36\
A 1689 & a1689 & 0.018 & 10.4$\pm$0.115 & 0.41 & 1.52$\pm$0.07 & 12.1$\pm$1.8 & 0.27\
A 401 & a401 & 0.099 & 8.6$\pm$0.071 & 0.36 & 1.45$\pm$0.09 & 9.5$\pm$1.7 & 0.25\
A 2204 & a2204 & 0.057 & 8.5$\pm$0.035 & 0.42 & 1.39$\pm$0.08 & 9.0$\pm$1.6 & 0.30\
A 2034 & a2034 & 0.015 & 8.3$\pm$0.107 & 0.31 & 1.40$\pm$0.09 & 8.8$\pm$1.6 & 0.22\
A 1413 & a1413 & 0.018 & 8.3$\pm$0.107 & 0.36 & 1.38$\pm$0.08 & 8.7$\pm$1.6 & 0.26\
A 2744 & a2744 & 0.014 & 8.7$\pm$0.229 & 1.09 & 1.30$\pm$0.08 & 8.6$\pm$1.5 & 0.84\
A 1835 & a1835 & 0.020 & 8.5$\pm$0.040 & 0.42 & 1.32$\pm$0.08 & 8.5$\pm$1.5 & 0.32\
PKS 0745-191 & pks0745 & 0.373 & 7.9$\pm$0.031 & 0.40 & 1.37$\pm$0.09 & 8.2$\pm$1.6 & 0.29\
A 665 & a665 & 0.043 & 7.3$\pm$0.137 & 0.48 & 1.26$\pm$0.09 & 6.9$\pm$1.5 & 0.38\
CYGNUS A & cygnusa & 0.272 & 6.9$\pm$0.022 & 0.29 & 1.30$\pm$0.10 & 6.8$\pm$1.5 & 0.22\
ZW 3146 & zw3146 & 0.025 & 7.0$\pm$0.064 & 0.39 & 1.18$\pm$0.09 & 6.3$\pm$1.4 & 0.33\
A 520 & a520 & 0.057 & 6.7$\pm$0.073 & 0.79 & 1.20$\pm$0.09 & 6.1$\pm$1.4 & 0.66\
A 1795 & a1795 & 0.012 & 6.2$\pm$0.008 & 0.30 & 1.23$\pm$0.10 & 5.8$\pm$1.4 & 0.24\
A 1650 & a1650 & 0.013 & 6.0$\pm$0.036 & 0.28 & 1.20$\pm$0.10 & 5.5$\pm$1.4 & 0.23\
A 3667 & a3667 & 0.044 & 5.8$\pm$0.025 & 0.37 & 1.20$\pm$0.10 & 5.3$\pm$1.4 & 0.31\
A 907 & a907 & 0.054 & 5.8$\pm$0.061 & 0.28 & 1.14$\pm$0.10 & 5.0$\pm$1.3 & 0.24\
A 521 & a521 & 0.049 & 5.9$\pm$0.177 & 0.54 & 1.10$\pm$0.10 & 4.8$\pm$1.3 & 0.49\
A 1995 & a1995 & 0.012 & 5.9$\pm$0.173 & 0.31 & 1.06$\pm$0.09 & 4.7$\pm$1.2 & 0.29\
A 2146 & a2146 & 0.030 & 5.7$\pm$0.031 & 0.50 & 1.08$\pm$0.10 & 4.6$\pm$1.2 & 0.46\
MS0735.6+7421 & ms0735 & 0.033 & 5.5$\pm$0.030 & 0.30 & 1.08$\pm$0.10 & 4.5$\pm$1.2 & 0.28\
MS 1455.0+2232 & ms1455 & 0.032 & 5.1$\pm$0.055 & 0.31 & 1.01$\pm$0.10 & 3.9$\pm$1.2 & 0.30\
A 2199 & a2199 & 0.009 & 4.4$\pm$0.010 & 0.19 & 1.05$\pm$0.12 & 3.5$\pm$1.2 & 0.18\
A 496 & a496 & 0.038 & 4.3$\pm$0.012 & 0.20 & 1.04$\pm$0.12 & 3.4$\pm$1.2 & 0.19\
A 2597 & a2597 & 0.025 & 4.0$\pm$0.014 & 0.21 & 0.98$\pm$0.12 & 2.9$\pm$1.1 & 0.21\
3C348 (HERCULES A) & 3c348 & 0.062 & 3.9$\pm$0.032 & 0.23 & 0.94$\pm$0.12 & 2.7$\pm$1.1 & 0.25\
A 1775 & a1775 & 0.010 & 3.7$\pm$0.047 & 0.22 & 0.94$\pm$0.13 & 2.6$\pm$1.1 & 0.23\
HYDRA A & hydraa & 0.047 & 3.5$\pm$0.009 & 0.22 & 0.93$\pm$0.13 & 2.4$\pm$1.1 & 0.23\
2A 0335+096 & 2a0335 & 0.175 & 2.9$\pm$0.005 & 0.18 & 0.84$\pm$0.15 & 1.8$\pm$1.0 & 0.21\
SERSIC 159-03 & sersic159 & 0.011 & 2.8$\pm$0.012 & 0.17 & 0.82$\pm$0.15 & 1.7$\pm$0.9 & 0.20\
A 2052 & a2052 & 0.027 & 2.3$\pm$0.002 & 0.15 & 0.76$\pm$0.17 & 1.3$\pm$0.9 & 0.19\
\[tab:sampro\]
We used archival observations with the *Chandra* Advanced CCD Imaging Spectrometer [ACIS, @2003SPIE.4851...28G] using the imaging (-I) or spectral (-S) CCD array (about 0.1 to 10keV energy range). This instrument provides high spatial ($\rm{\sim1\arcsec}$) and spectral resolution ($\rm{\sim100~eV}$ full width half maximum, FWHM). The field of view (FOV) is limited, so that only the inner $\rm{5-10\arcmin}$ of any cluster are homogeneously covered. See Sect. \[sec:aobs\] for a list of all analysed clusters and their individual exposure times.
Sample selection {#sec:sample}
----------------
We based our sample selection on the NORAS [378 sources, see @2000ApJS..129..435B], REFLEX , and CIZA [73 sources, see @2002ApJ...580..774E] catalogues. They all have been derived from *ROSAT* observations, which deliver the only true imaging all-sky X-ray survey to date. NORAS and REFLEX cover the regions north and south of the galactic plane ($\rm{\pm 20^{\circ}}$), excluding the Magellanic Clouds. The CIZA sample covers the Galactic plane and thus adds some interesting clusters to our sample. However in the Galactic plane we have to deal with higher foreground absorption of X-rays (due to the column density $\rm{n_{H}}$ of the Galaxy).
Not all of these clusters have been observed with *Chandra*, but predominantly X-ray bright clusters have, where the structure of the ICM could be well studied. We matched all *Chandra* ACIS observations available from the *Chandra* Data Archive (CDA[^2] on 2013-10-09) with cluster positions in the *ROSAT* catalogues mentioned above.
We set a luminosity cut on the $\rm{\sim300}$ clusters correlated with *Chandra* observations, and only added clusters to the sample which had a luminosity of more than $\rm{2.0x10^{44} ~erg/s}$ in the 0.1-2.4keV *ROSAT* energy band. We only accepted clusters and groups of galaxies with a redshift of $\rm{z \gtrsim 0.025}$ to ensure all clusters fit reasonably well into the *Chandra* ACIS FOV. This excludes some nearby extended systems where larger radii are not homogeneously covered [e.g. the Coma cluster, see @2009ApJ...692.1060V]. After these selections we used all clusters with $\rm{\gtrsim 100~ks}$ raw *Chandra* exposure time. Our final sample consists of 33 X-ray bright, massive, nearby clusters of galaxies. The velocity dispersion of galaxies in the clusters is around $\rm{1000~km/s}$ for most systems. The cluster halo masses within the overdensity radius $\rm{r_{500}}$ range from $\rm{1\times10^{14}~M_{\odot}}$ to $\rm{2\times10^{15}~M_{\odot}}$ (Tab. \[tab:sampro\]). At $\rm{r_{500}}$ the average density of the cluster is 500 times the critical density of the universe at the cluster redshift. The luminosity range is $\rm{(2-63)x10^{44}~erg/s}$ (0.1-2.4keV X-ray luminosity, see Tabs. \[tab:cat1\], \[tab:cat2\], and \[tab:cat3\]), the redshift ranges from 0.025 to 0.45 (see Fig. \[fig:zhist\]) and the total exposure analysed in this work is $\rm{\sim8~Ms}$ (corresponding to more than 90 days of observations). For a list of observations used in this study see Tab. \[tab:obsids\].
Cluster maps {#sec:maps}
------------
For the *Chandra* data reduction we used a pipeline of various *Python* and *C++* scripts to accurately map the structure of the ICM. The pipeline downloads all relevant datasets from the *Chandra* data archive (CDA) and reprocesses them using the *Chandra* standard data processing (SDP) with the *Chandra* Interactive Analysis of Observations software package [CIAO, @2006SPIE.6270E..1VF] version 4.5 and the *Chandra* Calibration Database [CalDB, @2007ChNew..14...33G] version 4.5.9.
A background light-curve for each observation is created and times of high background are removed from the event files, by iteratively removing times where the count rate is more than $\rm{3\sigma}$ from the median of the light-curve. Using the CIAO tool `acis_bkgrnd_lookup`, we find a suitable blank-sky background file, which is provided by the *Chandra* X-ray centre [see e.g. @2003ApJ...583...70M] and derive the background in each of the cluster observations. We correct residual spatial offsets between the individual observations if necessary by detecting point sources in each image with `wavdetect` and correlating the individual detection lists. Using the deepest observation as a reference, offsets in other event files are corrected by updating their aspect solution. With this procedure we ensure the best resolution of small-scale ICM variations. We create images from the event file of each dataset using an energy range of 0.5keV to 7.0keV and binning the image by a factor of two ($\rm{1pix\sim0.98\arcsec}$). For each image an exposure-map is created for an energy of 1.5keV. After the analysis of the individual observations, all images, background-, and exposure-maps are merged. A second point source detection with `wavdetect` is run on the merged images and after carefully screening the detection list, the point sources are masked from the image. In the following steps the image is adaptively smoothed with $snsmooth = 15$ and then binned into regions of equal S/N ratio of 50 ($\rm{>}$2500 counts per bin) using the contour binning technique `contbin` [see @2006MNRAS.371..829S].
For the asymmetry (i.e. spread) analysis (Sect. \[sec:asym\]) we generated maps with $\rm{S/N = 25}$ ($\rm{>}$625 counts per bin) for clusters where we obtained less than 50 independent spatial-spectral bins from the $\rm{S/N = 50}$ analysis. For each of the bins a detector response is calculated and the count spectrum extracted. Spectra of the same detectors are added together (ACIS-I and ACIS-S separately) and fitted using C-Statistics in `XSPEC version 6.11` [@1979ApJ...228..939C; @1996ASPC...99..409A] with the `apec` model for collisionally-ionised diffuse gas, which is based on the ATOMDB code v2.0.2 [@2012ApJ...756..128F]. The fit was done using a fixed foreground column density ($\rm{n_H~[cm^{-2}]}$, see Tab. \[tab:sampro\]), which is determined from the Leiden/Argentine/Bonn (LAB) survey of Galactic HI , and a fixed redshift from the *ROSAT* catalogues.
Only in the special case of PKS0745-191 was the fit done with $\rm{n_H}$ as a free parameter due to its location behind the Galactic plane and strong $\rm{n_H}$ variation within the FOV.
The free parameters of the fit to the count spectrum are temperature $\rm{T~[keV]}$, the metal abundance Z as a fraction of solar abundances [reference solar abundance $\rm{Z_\odot}$ from @1989GeCoA..53..197A] and the normalisation $\rm{\eta~[cm^{-5}~arcsec^{-2}]}$ of the fit, which is defined as,
$$\rm{\eta = 10^{-14}/(4\pi~{D_A}^2~(1+z)^2) ~\int{n_e~n_p~dV}}$$
with $\rm{D_A}$ the angular diameter distance to the source, $\rm{n_e}$ and $\rm{n_p}$ the electron and hydrogen densities being integrated over the volume V. Assuming a spherical source, uniform density, and full ionisation with 10 per cent helium, 90 per cent hydrogen abundance (i.e. $\rm{n_e \sim 1.2 ~n_p}$) the hydrogen density can be calculated as
$$\rm{n_p = \sqrt{\frac{(1+z)^2~10^{14}~\eta}{1.2\Theta^3~D_A}} \equiv \xi \cdot \sqrt{\eta}}$$
where $\rm{\Theta}$ is the angular size of the source (see the ATOMDB webpage[^3] for these equations). The factor $\rm{\xi}$ changes for different clusters and assumed geometries. We assumed $\rm{n_p \sim \sqrt{\eta}}$ within any given cluster.
The fit parameters of each bin are translated into images to obtain maps of the temperature, metal abundance and normalisation of the ICM.
From the 2D maps of the spectral fitting, we calculated a projected pseudo density, pressure, and entropy in each spatial bin of the ICM emission. We assumed a constant line-of-sight depth for all spectral regions calculating pseudo density n as square root of the fit normalisation, normalised by region size,
$$\rm{n \equiv \sqrt{\eta}~[cm^{-5/2}~arcsec^{-1}]}$$
pseudo pressure as,
$$\rm{P \equiv n \times temperature~[keV~cm^{-5/2}~arcsec^{-1}]}$$
and pseudo entropy as,
$$\rm{S \equiv n^{-2/3} \times temperature~[keV~cm^{5/3}~arcsec^{2/3}]}$$
We adopted a common definition of entropy for galaxy cluster studies, which is related to the standard definition of thermodynamic entropy s through
$$\rm{s = k_B~ln(S^{3/2}~(\mu~m_p)^{5/2})+s_0}$$
with mean particle mass $\rm{\mu}$ and proton mass $\rm{m_p}$ [see e.g. @2005RvMP...77..207V]. Relative cooling times of the ICM are proportional to $\rm{n^{-1} \times temperature^{1/2}~[keV^{1/2}~cm^{5/2}~arcsec]}$ where Bremsstrahlung emission dominates [see e.g. @1986RvMP...58....1S and references therein].
All distances were calculated using the redshift given in the *ROSAT* cluster catalogues. All uncertainties are $\rm{1\sigma}$ confidence intervals unless stated otherwise. All further analysis only includes regions where the area-normalised normalisation of the fit was above $\rm{10^{-7}~cm^{-5}~arcsec^{-2}}$. This corresponds roughly to a surface brightness cut below which there were insufficient counts for detailed spectral analysis. For an example refer to the maps of Abell1795 in Fig. \[fig:2dmaps\].
Analysis of perturbations {#sec:analyses}
=========================
Based on the very detailed spatial-spectral analysis of this sample of clusters with deep *Chandra* observations we were able to study the ICM in great detail for a large sample of clusters.
Asymmetry measurement {#sec:asym}
---------------------
One of the main goals of this study was to better characterise the thermodynamic state of the ICM in individual clusters and identify general trends in the whole sample. Important indicators of the state of the hot gas are fluctuations in thermodynamic properties. and , in recent high-resolution simulations, have shown a connection between such fluctuations and the Mach number of gas motion in the ICM.
We examined the asymmetry (i.e. the spread) of thermodynamic properties in concentric, circular annuli of radius r around the peak X-ray emission. As input we used the S/N 50 ICM-maps (see Sect. \[sec:maps\]). For nine clusters (a907, ms1455, a521, a665, a2744, a1775, a1995, 3c348, zw3146) we used S/N 25 instead to obtain at least five radial spread-bins with a minimum of five data points per bin for all clusters. Assuming the data points are scattered statistically due to their uncertainties $\rm{\sigma_{stat}}$ we tested for intrinsic spread $\rm{\sigma_{spread}}$ in the radial profiles (see e.g. Fig. \[fig:a1795prof\]).
To avoid contamination of the spread-measurement by the slope, we modelled the radial profile by interpolating between several nodes. This modelled average profile can be described by a function $\rm{\mu(r,\mu_1,...,\mu_N)}$ with N=7. The nodes divide the cluster profile into bins with equal number of data points. We used a minimum of five data points per model-node and if this criterion was not met we reduced the number of nodes. The intrinsic fractional spread is given by a function of the form $\rm{\sigma_{spread}(r,\sigma_{spread,1},...,\sigma_{spread,M})}$. We performed three independent spread measurements, splitting the profiles into one, two, and five radial bins (M=1,2,5) to measure the spread in the clusters with different radial resolutions.
For every data point we obtained the mean value $\rm{\mu(r_i)}$ of the thermodynamic profile at its radius $\rm{r_i}$ by interpolating between the model-nodes (see grey lines in Fig. \[fig:a1795prof\]). The individual spread values of each data point $\rm{\sigma_{spread}(r_i)}$ are constant within a given radial bin (see spread profile in Fig. \[fig:a1795prof\]).
The intrinsic spread of cluster properties was estimated using a Markov Chain Monte Carlo (MCMC) method implemented using `emcee` [see @2013PASP..125..306F] with 100 walkers, 1000 iterations, and a burn-in length of 1000. The total scatter of data values (i) was assumed to follow a Gaussian distribution with standard deviation of $\rm{\sigma_{tot}}$ calculated as,
$$\rm{\sigma_{tot,i} = \sqrt{\sigma_{stat,i}^2 + \sigma_{spread}^2(r_i)}}$$
For each iteration of the MCMC $\chi^2$ was calculated as,
$$\rm{\chi^2 = \sum\limits_{i=1}^n \frac{(D_i-\mu(r_i))^2}{\sigma_{tot,i}^2}}$$
where $\rm{D_i}$ are the individually measured values of the n data points (see Fig. \[fig:a1795prof\]). The mean-model and spread-values have $\rm{N + M}$ free parameters. Using the MCMC method these parameters were varied, giving as probability value for each iteration, the logarithm of the likelihood (see e.g. @2010arXiv1008.4686H),
$$\rm{log~L = -\frac{1}{2}~\chi^{2} -\frac{1}{2}~\sum\limits_{i=1}^n \log(2\pi~\sigma_{tot,i}^2})$$
From this procedure we obtained a distribution of mean and spread values. We selected the best fit value for each parameter as the maximum of the distribution and estimated the uncertainty by giving the range containing 34 per cent of the obtained values on each side of the maximum. If the distribution was consistent with zero, we give the 68 per cent range as an upper limit.
The spread measurements performed with just one radial spread-bin (M=1) were used to compare the overall fractional spread dn, dT, dP, and dS among clusters (see e.g. Figs. \[fig:dsdp\], \[fig:dtdn\], \[fig:dst\], and Tab. \[tab:persum\]) and constrain the general ICM properties. For additional comparison of larger and smaller physical radii we measured the spread inside and outside of 100kpc from the X-ray peak (M=2, see e.g. $\rm{dP_{cen}}$ and $\rm{dP_{out}}$ Sect. \[sec:sam\_inout\]). From the spread analysis with five radial spread-bins (M=5) we obtained profiles of the intrinsic fractional standard deviation of thermodynamic properties dn/n, dT/T, dP/P, and dS/S (see Fig. \[fig:Proall\]) in individual systems. The spread measurements are consistent with the results of a second Monte Carlo simulation based technique, which was not based on Markov Chains [similar to @2014MNRAS.444.1497S].
Surface brightness substructures {#sec:sb}
--------------------------------
The emissivity of the ICM and thus its surface brightness in X-rays is proportional to the plasma density squared [for a review, see e.g. @1986RvMP...58....1S]. From the data reduction pipeline we obtained merged count and exposure images of the clusters. To remove any structure due to inhomogeneous exposure we divide the count image by the exposure image and obtain an image of the count rate. To identify small surface brightness fluctuations we enhanced the contrast of those images by unsharp-masking, a method commonly used in image analysis. This method is implemented by subtracting two versions of an image, smoothed by two different Gaussian filters, from each other. We subtracted an image smoothed with a Gaussian function of $\rm{\sigma = 2 pixel}$ width from an image smoothed with a Gaussian function of $\rm{\sigma = 5 pixel}$ width (see Fig. \[fig:umall\]). The obtained unsharp-masked images enhance surface brightness features, complementing the analysis of substructure in thermodynamic properties, and highlighting disturbed systems.
Average cluster temperatures {#sec:temp}
----------------------------
We calculated average cluster temperatures $\rm{<T~map>}$ as the area- and error-weighted mean value of all measurements. We computed the average of the bin temperature ($\rm{<T~prof>}$, see data points in Fig. \[fig:a1795prof\]) which is usually lower than $\rm{<T~map>}$, because $\rm{<T~prof>}$ is weighted more on emission than area. Hotter regions in the outskirts are generally larger with lower emission and the relatively colder, X-ray bright, central regions cover a smaller area.
We compared our approach for estimating overall cluster temperatures with previous studies like V09 (using *Chandra*) or HiFLUGS (using *ROSAT*) [@2009ApJ...692.1033V; @2002ApJ...567..716R] as shown in Fig. \[fig:v09comp\] and Fig. \[fig:Hicomp\]. Fig. \[fig:v09comp\] shows significant scatter of up to 0.5keV between V09 and this study. This is expected since we use a different approach by averaging many independently fitted bins weighting by area and error-bars rather than by counts. The averaging of many different bins results in small error bars on our average temperature. Fig. \[fig:Hicomp\] shows similar scatter when accounting for the larger error bars on the *ROSAT* measurements. Note that we also use different extraction radii than the studies we compared to which influences the measured temperature.
To estimate the mass range of the cluster sample (see Tab. \[tab:sampro\]) we used the mass-temperature scaling relation from @2009ApJ...692.1033V with the $\rm{<T~map>}$ temperatures as input to calculate the overdensity radius $\rm{r_{500}}$ and the total mass $\rm{M_{500}}$ included by this radius. We accounted for uncertainties in the scaling by assuming a systematic temperature uncertainty of 0.5keV (see scatter in Fig. \[fig:v09comp\]). The average map temperature (area weighted) is comparable to the core-excised average temperature used for the scaling by @2009ApJ...692.1033V.
Table description {#sec:tab}
-----------------
The main results of the analysis of this study are summarized in separate tables for each cluster. We created 2D maps from the merged observations of every cluster and measured the asymmetries in thermodynamic parameters in circular concentric annuli around the centre.
### Map tables {#sec:tab:1}
The primary data products of this analysis are 2D maps of the thermodynamic properties of the ICM based on the merged observations of ACIS-S and ACIS-I for every cluster in the sample (see Sect. \[sec:maps\]).
[**Tables 2-34**]{} contain the 2D map information (one table per cluster). Each table lists the properties for every pixel in the cluster maps ($\rm{1pix\sim0.98\arcsec}$). The map tables contain X (east-west direction) and Y (south-north direction) position (x, y, Cols. 1, 2) of the pixel, the index of the independently-fitted spatial-spectral bin it belongs to (binnum, Col. 3), the photon counts (cts, Col. 4), the background counts (bgcts, Col. 5), the effective exposure time (exp, Col. 6), the temperature and its upper and lower limits (T, Tup, Tlo, Cols. 7, 8, 9), the relative metallicity and limits (Z, Zup, Zlo, Cols. 10, 11, 12), the fit normalisation and limits (norm, normup, normlo, Cols. 13, 14, 15), the redshift (redshift, Col. 16), the foreground column density $\rm{n_H}$ (NH, Col 17), distance from the centre in pixels, arc seconds, and kpc (cen\_dist, rad\_arcsec, rad\_kpc, Cols. 18, 19, 20), the angle with respect to the west direction (cen\_angl, Col. 21), the calculated projected pressure and symmetric uncertainty (P, P\_err, Cols. 22, 23), projected entropy and uncertainty (S, S\_err, Cols. 24, 25), and density and uncertainty (n, n\_err, Cols. 26, 27). All uncertainties are on $\rm{1\sigma}$ confidence level.
### Asymmetry tables {#sec:tab:2}
The secondary data products are based on the 2D maps and contain the measured spread (i.e. asymmetry, deviation from radial symmetry of the thermodynamic parameters, see M=5 in Sect. \[sec:asym\]).
[**Tables 35-67**]{} provide the measured properties in the concentric annuli for one cluster each. They contain the average radius of the annulus (rr, Col. 1), average bin-temperature $\rm{<T>}$ and uncertainty (T, Te, Cols. 2, 3), and average bin-metallicity and uncertainty (Z, Ze, Cols. 4, 5). For the intrinsic fractional spread values in projected properties best fit value and $\rm{1\sigma}$ upper and lower confidence limits are provided. Spread measurements contain the intrinsic spread in pressure with upper and lower limits (dP, dP\_eu, dP\_el, Cols. 6, 7, 8), entropy with limits (dS, dS\_eu, dS\_el, Cols. 9, 10 ,11), density with limits (dn, dn\_eu, dn\_el, Cols. 12, 13, 14), and temperature with limits (dT, dT\_eu, dT\_el, Cols. 15, 16, 17).
Results {#sec:results}
=======
The detailed analysis of this sample of clusters enabled us to derive information on the thermodynamic state of the ICM in individual clusters and the whole sample in general.
Perturbations in thermodynamic properties {#sec:sam_turbobs}
-----------------------------------------
We studied the average fluctuations in thermodynamic properties of all clusters in the sample using the M=1 spread calculations from Sect. \[sec:asym\]. Comparing the average measurements for all systems in the sample enabled us to find general trends.
Fig. \[fig:dsdp\] indicates that, on average, entropy fluctuations (fractional spread, dS) dominate pressure fluctuations (fractional spread, dP). For some clusters we only obtained upper limits (e.g. A1689). There are some outliers (e.g. A2146 and A3667), which are heavily disturbed systems (see Sect. \[sec:sam\_turbcav\]). Overall most clusters lie off a one-to-one correlation with an average of 16 per cent fluctuations in entropy and 9 per cent in pressure (see Tab. \[tab:samper\]).
Fig. \[fig:dtdn\] shows a smaller offset between the average density fluctuations (fractional spread, dn) and temperature fluctuations (fractional spread, dT). We measure significant spread in dn for every cluster in the sample but only obtain upper limits on the dT spread for some. There are some outliers which show very strong density fluctuations (e.g. A2146, 1E0657-56, and CygnusA). Overall most clusters are close to a one-to-one correlation with an average of 11 per cent fluctuations in density and 9 per cent in temperature (see Tab. \[tab:samper\]).
[rrrr]{} & & &\
0.09 $\pm$ 0.06 & 0.16 $\pm$ 0.07 & 0.11 $\pm$ 0.05 & 0.09 $\pm$ 0.04\
\[tab:samper\]
Perturbations on different scales {#sec:sam_inout}
---------------------------------
We found evidence for larger pressure perturbations dP at larger scales. Fig. \[fig:inout\] shows a clear separation of the distribution for central regions $\rm{\lesssim100~kpc}$ (cen) from the centre and regions beyond (out). To compare the distribution of spread values on different scales we used the asymmetry measurements from Sect. \[sec:asym\] where the cluster profile is divided at $\rm{\sim100~kpc}$. The division at 100kpc was chosen, after visual inspection, as a robust separation radius between central substructure and outer more homogeneous regions. We did not choose the regions relative to $\rm{r_{500}}$ to ensure we are testing the same physical scale of the ICM fluctuations.
To estimate the mean of the distributions we used a bootstrapping re-sampling technique, calculating the mean value for 1000 permutations with repetition. The contours are at 15, 50, and 85 per cent of the maximum of the obtained distribution of mean values. In addition Fig. \[fig:inout\] contains the output of a Kolmogorov-Smirnov (KS) test to quantify the difference between the cen and out distributions. The D value states the maximum fractional offset between the cumulative distribution graphs and the p value is the probability of the null hypothesis. This means the probability that the dP distributions are different is 96 per cent and thus just above the $\rm{2\sigma}$ level. It should be noted that the offset in dP is dominated by low dP data points and decreases to about $\rm{1\sigma}$ level when only including data points above 0.05dP. For the thermodynamic properties dS, dn, and dT there is no significant difference between inner and outer radii.
Metallicity correlations {#sec:sam_met}
------------------------
We found an anti-correlation between the average temperatures ($\rm{<T~map>}$, see Sect. \[sec:temp\]) of the clusters and their average metallicity. The best fit linear correlation is $\rm{Z/Z_\odot = -(1.0\pm0.7)~T/100keV + (0.34\pm0.06)}$. The average metallicity of the clusters have been weighted by area and error in the same way as the average map temperatures in Sect. \[sec:temp\]. A similar correlation can be found within the individual clusters (see maps in Sect. \[sec:tab:1\]). By repeating the same slope analysis for the inner ($\rm{\lesssim100~kpc}$) and outer ($\rm{\gtrsim100~kpc}$) regions of the cluster we found that the average metallicity is higher in the central regions and the slope of the T-Z anti-correlation is steeper. Testing different weighting methods for the average temperature and metallicity we obtained consistent correlations with some scatter. Temperature is also correlated to the redshift of the clusters in this sample (more luminous and massive systems at higher redshift due to selection bias).
We investigated the redshift-metallicity (z-Z) relation and found the correlation with redshift to be tighter than with temperature (see Fig. \[fig:metall\]). The best fit linear correlation of metallicity and redshift is $\rm{Z/Z_\odot = -(0.6\pm0.2)~z + (0.36\pm0.04)}$ (see Fig. \[fig:metall\]). There was no evidence that the z-Z anti-correlation is steeper in the central regions. The average metallicity of the sample is $\rm{Z\approx0.3\pm0.1~Z_\odot}$. The significance of a correlation was estimated using a re-sampling technique, performing a linear fit on random sub-samples of 17 clusters and using the mean value and $\rm{1\sigma}$ width as best fit and error range of slope and normalisation.
Individual clusters {#sec:indi}
===================
In addition to the sample properties the data contain important information on the properties of individual systems. The sample consists of clusters with a wide range of different structures from more relaxed systems like A496 and A2199 to disturbed clusters like the 1E0657-56 and A2146. We highlight some special cases below. Temperature maps, unsharp-masked images, and radial profiles of thermodynamic properties can be found in Sect. \[sec:amaps\], \[sec:amask\], and \[sec:arad\].\
\
[**Abell1795**]{} has the deepest exposure in our sample and thus the most detailed spatial-spectral information on the ICM could be derived in this system. @1994AJ....107..857O found that its central cD galaxy has a peculiar radial velocity of $\rm{150~km~s^{-1}}$ within the cluster, while they measured the velocity dispersion of cluster members at $\rm{\sigma=920~km~s^{-1}}$. According to @2001MNRAS.321L..33F the simplest explanation for the visible soft X-ray filament would be a cooling-wake behind the cD galaxy (approximate position at the cross in Fig. \[fig:2dmaps\], filament extending to the south), which is oscillating in the DM potential of the galaxy cluster. The centre is surrounded by linear surface-brightness features which might be remnants of past mergers with subhalos or created by the outburst of a strong AGN [see e.g. @2001ApJ...562L.153M; @2002MNRAS.331..635E; @2014MNRAS.445.3444W; @2015ApJ...799..174E].
The deep unsharp-masked count images in this study show the central surface-brightness features at higher significance than previous studies (see Fig. \[fig:umall\]). The maps (Fig. \[fig:2dmaps\]) contain 1563 bins with a S/N count ratio of 50. The detailed radial profiles of the projected thermodynamic properties (Fig. \[fig:a1795prof\]) were the basis for the radial asymmetry measurements. Entropy perturbations seem to dominate throughout the cluster, suggesting that the 1D Mach number is comparable to the variance of entropy (see below, Sect. \[sec:sam\_turbturb\]). The perturbation measurements are influenced by the presence of a cooler X-ray filament (inner $\rm{\sim40~kpc}$) and projection-effects (see Sect. \[sec:sam\_turbcav\]). The central filament seems to increase the spread in density, temperature, and entropy, but pressure seems unaffected (see scatter in Fig. \[fig:a1795prof\]). The profiles follow the average trend of higher pressure perturbations in the outskirts (see Sect. \[sec:sam\_inout\]). The findings confirm and improve the detection of many surface-brightness features at the centre of the cluster. The detailed analysis of thermodynamic perturbations support a model where isobaric processes dominate the central ICM.\
\
[**1E0657-56**]{} (the Bullet Cluster) offers an almost edge-on view of two massive merging subclusters [@2002ApJ...567L..27M]. It was possible to find a significant offset between the total mass profile from weak-lensing and the X-ray emission of the hot ICM and thus make a very convincing case for the existence of DM [@2004ApJ...606..819M; @2004ApJ...604..596C; @2006ApJ...648L.109C; @2008ApJ...679.1173R].
We found the strongest surface-brightness fluctuations around the prominent Mach cone from the impact of the smaller subcluster (see Fig. \[fig:umall\]). The density fluctuations in the cluster are among the highest in the observed sample (see Fig. \[fig:dtdn\]). Pressure fluctuations around the “Bullet” are significantly weaker than at larger radii (see Fig. \[fig:Proall\]). It should be noted that major merger shocks and AGN feedback are not modelled in the simulations and thus in those cases the direct connection between Mach number and fluctuations in thermodynamic parameters might change (see Sect. \[sec:sam\_turbcav\]). 1E0657-56 has the second highest $\rm{<T~map>}$ in the sample (after RXJ1347-114) and the lowest average metallicity (see Fig. \[fig:metall\]). The Bullet Cluster has a particularly flat pressure profile with large scatter when compared to other clusters. In the most disturbed systems the pressure profile is flat and does not drop with radius.\
\
[**Abell2052**]{} hosts an extended region of colder ICM at its centre caused by rising colder gas due to strong AGN feedback from the central cD galaxy. The radio source connected to the central AGN and its effect on the surrounding ICM have been studied in detail by @2001ApJ...558L..15B [@2003ApJ...585..227B; @2011ApJ...737...99B]. Feedback from the radio source pushes the X-ray emitting gas away from the centre and creates a sphere of enhanced pressure around the central region. @2015MNRAS.447.2915M investigated different merger scenarios in recent simulations of the cluster.
In our measurements the cluster emission shows a large scale ellipticity which could be an indicator of a past merger. The cluster has the lowest average temperature in the sample ($\rm{<T~map>\sim2.3\,keV}$, see Tab. \[tab:sampro\]) and one of the largest drops in entropy asymmetry from the centre to the outskirts (see Fig. \[fig:inout\]). At about 20kpc from the central AGN we detect an enhancement in projected pressure of more than a factor of two compared to the enclosed ICM (see Fig. \[fig:Proall\]). On larger scales the cluster shows a spiral structure in surface brightness which is most likely caused by sloshing of gas due to a past merger (see Fig. \[fig:umall\]). In A2052 all thermodynamic profiles are heavily influenced by the AGN feedback at the center. Only the strongest feedback cases in our sample show a deviation from a radially decreasing pressure profile.\
\
[**Abell2146**]{} is a major merger viewed almost edge-on. @2010MNRAS.406.1721R [@2011MNRAS.417L...1R; @2012MNRAS.423..236R] detected two opposing shock fronts in the cluster and investigated transport processes in the ICM. They find the system to be less massive and thus colder than the Bullet Cluster merger (see also Fig. \[fig:dst\]). Unlike in the Bullet Cluster, the secondary BCG in A2146 seems to be slightly lagging behind the shockfront [@2012MNRAS.420.2956C].
The cluster shows the highest entropy perturbations in the sample (see Fig. \[fig:dsdp\]). Asymmetries are especially high around the cluster centre which we choose on the smaller merging subcluster (see the cross in Fig. \[fig:Tall\]). Like in 1E0657-56, pressure fluctuations around the merging subcluster are rather low but increase at larger radii (see Fig. \[fig:inout\]). The difference in dP/dS between outer and inner radii is among the highest in the sample (see Fig. \[fig:inout\]). The extreme entropy perturbations in A2146 indicate the highest average Mach number in our sample (see Sect. \[sec:sam\_turbturb\]), almost twice as high as in the Bullet Cluster. If the two merging systems have similar impact velocities, the lower temperature of A2146 could cause the Mach number of the turbulence induced by the merger to be twice as high as in the Bullet Cluster.\
\
[**CygnusA and HydraA**]{} are two similar systems with strong AGN feedback. Both sources have strong radio jets emerging from the central AGN causing complex structures in the X-ray emitting ICM around the nucleus [see e.g. @2000ApJ...534L.135M; @2002ApJ...565..195S for HydraA and CygnusA respectively]. @2002ApJ...568..163N [@2005ApJ...628..629N] found AGN feedback to influence the ICM on large scales in HydraA.
The unsharp-masked analysis of the surface-brightness images clearly shows the strong feedback structures around the central AGN (Fig. \[fig:umall\]). Average perturbations and temperature in CygnusA are significantly higher than in HydraA (see Figs. \[fig:dsdp\], \[fig:dst\]). CygnusA shows enhanced density and temperature at larger radii to the north-west (see Fig. \[fig:Tall\]). HydraA has a very asymmetric temperature distribution which is mainly caused by continuous radial structures of colder gas extending from the centre (see Fig. \[fig:Tall\]). The average metallicity of CygnusA is significantly higher than for HydraA (see Fig. \[fig:metall\]). The thermodynamic profiles of CygnusA are more disturbed which would indicate a stronger or more recent AGN outburst.\
\
[**Abell2199**]{} is a typical relaxed cluster with a cool core and AGN feedback structures at its centre [@1999ApJ...527..545M; @2002MNRAS.336..299J]. @2013ApJ...775..117N found various substructures in deep *Chandra* observations of the cluster, including evidence for a minor merger $\rm{\sim400~Myr}$ ago. @2006MNRAS.371L..65S found a weak isothermal shock ($\rm{\sim100kpc}$ from the centre to the south-east) likely caused by the supersonic inflation of radio lobes by jets from the central AGN (3C338).
Our maps indicate a weak temperature jump in the area where the shock has been detected (see Fig. \[fig:Tall\]) and show a large scale asymmetry in the temperature distribution between north and south (also visible in the scatter of the radial profile, see Fig. \[fig:Proall\]). The surface brightness unsharp-masked image shows some of the structures at the centre and a weak indication of the surface brightness jump due to the shock to the south-east. The perturbations in entropy, pressure, temperature, and density are in the average regime of our sample as expected for an overall relaxed system (see Figs. \[fig:dsdp\], \[fig:dtdn\]). The radial profiles show a prominent discontinuity around 50kpc from the center, related AGN feedback. Just outside this jump there is a region where the fit uncertainties are larger due to overlapping chip gaps of many observations, which could bias our results (see faint spurious linear structures in Fig. \[fig:Tall\]).\
\
[**Abell496**]{} is a relaxed cluster with relatively high metallicity around the cool core and non-uniform temperature distribution on large scales . By comparing dedicated simulations of the cluster to deep *Chandra* observations @2012MNRAS.420.3632R concluded that the cluster was most likely perturbed by a merging subcluster 0.6-0.8Gyr ago.
The spiral surface brightness excess structure and the northern cold front is clearly visible in our new images of temperature and unsharp-masked count images (see Fig. \[fig:Tall\], \[fig:umall\]). The radial profiles and asymmetry measurements show relatively large spread in entropy (see Fig. \[fig:Proall\]). Figs. \[fig:dsdp\] and \[fig:dtdn\] show that the cluster has a large average temperature spread (due to the large scale asymmetry caused by the sloshing of colder gas from the centre) and thus also larger spread in entropy. A496 has a flat pressure profile similar to the strongest merging systems in the sample. The average metallicity is the highest we measured.\
\
[**PKS0745-191**]{} is a relaxed cluster at large scales out to the virial radius [@2009MNRAS.395..657G]. @2014MNRAS.444.1497S found AGN feedback and sloshing structures in deep *Chandra* observations.
The AGN feedback features are most prominent in the unsharp-masked image (Fig. \[fig:umall\]) and indication of weak asymmetry in temperature on large scales can be seen in Fig. \[fig:Tall\]. The radial profiles of the cluster follow the expected trend for relaxed systems but show remarkably low intrinsic scatter in pressure and relatively high scatter in entropy (see Fig. \[fig:Proall\]). The scatter measurements of @2014MNRAS.444.1497S are consistent with our method for dn, dT, dP, and dS. This is the only cluster in the sample where $\rm{n_H}$ was set as a free parameter in the spectral fits, because the foreground column density of this system is knows to vary significantly.
Discussion {#sec:discussion}
==========
The perturbation measurements for this cluster sample constrain the average Mach number of the systems. The sample covers a wide range of dynamic ICM-states providing insight to the influence of different perturbation events.
Caveats of perturbation measurements {#sec:sam_turbcav}
------------------------------------
It has been shown by @2014MNRAS.444.1497S that in the PKS0745-191 cluster projection effects in the measured parameters caused the calculated projected spread to be only about half of the real spread in the ICM. Therefore we expect projection effects to strongly affect the absolute values of our spread parameters. But since the effect seems to be similar on all parameters in the PKS0745-191 system the comparison of ratios between different spreads should not be affected. All asymmetry measurements are based on circular extraction regions, which means ellipticity of the cluster emission, as in PKS0745-191, will add to the spread values. Thus strong ellipticity could also influence the measured ratios between perturbations. @2014MNRAS.444.1497S showed for PKS0745-191 that different bin sizes within a certain range lead to consistent results in the spread analysis. However if the bins are too large like for some more distant clusters in our sample, we generally obtain larger absolute spread-values (e.g. zw3146 in Fig. \[fig:dsdp\]). Overall the comparison of spread measurements on maps with a S/N of 25 and 50 were in good agreement.
In real clusters there are many factors influencing perturbation measurements. The clusters that lie outside the relations expected from simulations seem to be dominated by processes that have not been included in the simulations (like mergers, AGN feedback bubbles, shocks, or uplifted cold gas). Also projection effects in our measurements and the overall geometry of a system could cause deviations. Note that the parameters dS and dP are not independent, since they are derived from the independent fit parameters T and $\rm{\eta}$ (see Sect. \[sec:maps\]). Uncertainties in the spectral fits are larger at higher temperatures. The size of the error bars of individual measurements limit the sensitivity for finding additional spread in the MCMC calculations (see Sect. \[sec:asym\]). In the case of Abell1689 the absolute spread measurements are very low and we only obtain upper limits for this system, which is caused by very low fluctuations and high temperatures, lowering the sensitivity for detecting additional spread.
With the caveats described above our measurements allow for a rough estimate of the average Mach number and thus turbulence trends in the ICM for a large sample of massive clusters of galaxies. Future simulations will help to better quantify the influence of the above described caveats on the Mach number estimates.
Relating perturbations to turbulence {#sec:sam_turbturb}
------------------------------------
Recent high-resolution 3D hydrodynamic simulations by show that entropy and pressure drive the main perturbations in the ICM depending on the Mach number in the medium. In the low Mach number case ($\rm{Mach_{3D} < 0.5}$) perturbations are mainly isobaric, implying:\
\
dP/P is negligible, $\rm{dS/S\sim Mach_{1D}}$, and $\rm{dn/n\sim \mid dT/T \mid}$\
\
For large Mach numbers ($\rm{Mach_{3D} > 0.5}$) perturbations shift to the adiabatic regime (because turbulence becomes more violent, overcoming the cluster stratification), implying:\
\
dS/S becomes negligible, $\rm{dP/P\sim Mach_{1D}}$,\
and $\rm{dn/n\sim 1/(\gamma -1)~dT/T\sim 1.5~dT/T}$\
\
The measurements of the average perturbations in the sample suggest a mixture of the two states (see Figs. \[fig:dsdp\] and \[fig:dtdn\]). We observed that entropy perturbations are slightly dominating pressure fluctuations (Fig. \[fig:dsdp\]) and density fluctuations are comparable to temperature fluctuations with a possible tendency of slightly dominating dn (Fig. \[fig:dtdn\]). Enhanced temperature asymmetry can be caused by displaced cold gas from the central region due to strong feedback like e.g. in CygnusA or strong cold fronts as observed in Abell3667. Additional conduction in the ICM can weaken temperature perturbations .
Assuming entropy perturbations dominate in the sample would it follows that $\rm{dS/S\sim Mach_{1D}}$. The dS/S value is related to the normalisation of the power spectrum, which is related to the peak of the power spectrum, that occurs at large physical radii (low wavenumber k), typically $\rm{\gtrsim100~kpc}$ or $\rm{\gtrsim0.1~r_{500}}$. As an approximation we used the overall average of perturbations (M=1, see Sect. \[sec:asym\]) as turbulence indicator since the assumption that dS dominates does not hold in the outer regions $\rm{\gtrsim100~kpc}$ of the clusters (see Sect. \[sec:sam\_inout\]). The measurements on average suggest $\rm{Mach_{1D}\approx0.16\pm0.07}$ (see Fig. \[fig:dst\]).\
Hydrodynamical simulations of galaxy clusters usually find the ratio between turbulence energy $\rm{E_{turb}}$ and the thermal energy $\rm{E_{therm}}$ to be $\rm{E_{turb}\approx3-30\%~E_{therm}}$, from relaxed to unrelaxed clusters . Since,
$$\rm{E_{turb}=0.5~\gamma~(\gamma-1)~Mach_{3D}^2~E_{therm}\simeq0.56~Mach_{3D}^2~E_{therm}}$$
where the adiabatic index $\rm{\gamma=C_P/C_V=5/3}$ ($\rm{C_P, C_V}$ heat capacity of the medium at constant pressure and volume respectively), it follows that $\rm{Mach_{3D}\simeq0.23-0.73}$ and thereby,
$$\rm{Mach_{1D}=Mach_{3D}/\sqrt{3} \simeq 0.13-0.42}$$
which is consistent with the measured average of $\rm{Mach_{1D}\approx0.16\pm0.07}$. This would suggest that turbulence energy in this sample of clusters is on average about four per cent of the thermal energy in the systems. Fig. \[fig:dst\] indicates a weak anti-correlation with average cluster temperature. In the best fit linear regression $\rm{Mach_{1D}\approx-(0.01\pm0.01)~<T_{prof}>/keV + (0.20\pm0.05)}$, the slope is consistent with zero within the errors. The significance of the slope was estimated using the re-sampling technique of Sect. \[sec:sam\_met\] by investigating the distribution of fit functions in 1000 fits to random sub-samples of half the size of the original. Note that the absolute values used as Mach number indicator are subject to some uncertainty like projection- and feedback-effects that have not been taken into account in the simulations we compare to (see Sect. \[sec:sam\_turbcav\]).
Difference between core and outskirts {#sec:diss_inout}
-------------------------------------
If the enhanced pressure perturbations in the outer regions of the clusters in this sample can be confirmed this could be interpreted as a change in the thermodynamic state of the ICM from the center to the outer regions. With increasing pressure perturbations dP we expect more turbulent, pressure-wave driven, motions in the ICM.
This fits the expectation of less relaxed outer regions in clusters where the ICM is not yet virialised and still being accreted into the cluster potential [e.g. @2009ApJ...705.1129L]. The change of the dP/dS ratio could also be due to the dependence of the ratio on the probed scales . In annuli at larger radii we probe larger scales of ICM fluctuations. A change in dP/dS with scale would also show a radial dependence in the sample.
ICM metallicity {#sec:diss_met}
---------------
The metallicity of the ICM and its local distribution are of great interest when studying the processes which are enriching the ICM with heavier nuclei. They are thought to be mainly produced by supernova explosions in the galaxies (mainly in the brightest clusters galaxy, BCG) within the clusters and then transported into the ICM .
The observed anti-correlation between average cluster temperature and metallicity (Fig. \[fig:metall\]) could have many different causes. @2004MNRAS.349..952S found a similar relation between metallicity and temperature in the Perseus cluster. After testing for various systematic effects they found the correlation to be real but no definite explanation for the effect has been found so far. Sloshing structures of colder gas have been found to coincide with higher metallicity [see e.g. @2011MNRAS.413.2057R], which could partially explain an anti-correlation.
The temperatures of the clusters in our sample are correlated to their redshift and the metallicity anti-correlates with redshift as well as temperature. probed a sample of clusters in the redshift range $\rm{0.3 < z < 1.3}$ and found significant evolution in metallicity between their higher and lower redshift clusters by more than a factor of two. This trend was confirmed by @2008ApJS..174..117M. Both studies also found a correlation between cluster temperature and metallicity. However e.g. found no significant evolution of abundance with redshift. It is not clear whether to expect any evolution within the narrow redshift range ($\rm{0.025 \leq z \leq 0.45}$) we probed. This corresponds to a time span of $\rm{\Delta t \approx 4~Gyr}$ in the standard cosmology assumed in this study.
In individual clusters there is a general trend of lower metallicity at larger radii which means the average metallicity will be lower when we cover larger radial ranges for clusters at higher redshift. The metallicity drops from the centres ($\rm{Z_{solar}\approx0.4-0.8}$) with typically lower temperature gas out to $\rm{\sim300~kpc}$ where the profile flattens around 0.2 solar metallicity fraction. Uniform metallicity distribution at larger radii has been observed in great detail in the Perseus cluster [@2013Natur.502..656W].
The colder gas generally resides in the cluster centres, often in the vicinity of large cD galaxies, that might cause enhanced enrichment. More massive (higher temperature) systems will also have undergone more mergers and generally have stronger AGN feedback, which contributes to the dilution of metals in the X-ray halo [e.g. @2011MNRAS.411..349G] and affects the average area- and error-weighted metallicity calculated in this study. Also multiphasedness of the hot ICM along the line of sight could influence the correlation as we typically probe larger volumes for higher temperatures. Multiphase gas has been found to bias the ICM metallicity measurements from X-ray spectra in some clusters [e.g. @2013MNRAS.433.3290P]. Unresolved 2D structure of the ICM could bias the measured metallicity to higher values for systems with intermediate temperatures . This study reduced multiphase effects by resolving the spatial structure and measuring the average metallicity of many spatial-spectral bins.
Another factor of influence could be that star formation efficiency decreases with cluster mass and temperature . There have been many studies in the infrared wavelength which show quenching of star formation in massive DM halos of galaxy clusters and groups. Our sample predominantly consists of relatively massive systems (see Tab. \[tab:sampro\]) at low to intermediate redshifts and in different stages of evolution. Note that all metallicity measurements are based on a fixed solar abundance model [@1989GeCoA..53..197A].
Summary and conclusions {#sec:conclusion}
=======================
We presented a very large sample of detailed cluster maps with application to understand the thermodynamic processes in clusters of galaxies. The deep observations of the individual clusters helped to identify structures in the ICM caused by mergers or AGN feedback. By comparing to recent high-resolution simulations of perturbations in the ICM we constrained the average 1D Mach number regime in the sample to $\rm{Mach_{1D}\approx0.16\pm0.07}$ with some caveats (see Sect. \[sec:discussion\]). Comparing to simulations this would suggest $\rm{E_{turb}\approx0.04~E_{therm}}$ (see Sect. \[sec:sam\_turbturb\]). By comparing perturbations in the central regions ($\rm{\lesssim100~kpc}$) and in the outer regions ($\rm{\gtrsim100~kpc}$) we found an indication for a change in the thermodynamic state from mainly isobaric to a more adiabatic regime.
In addition the sample shows a tight correlation between the average cluster metallicity, average temperature and redshift. The best fit linear correlation between metallicity and redshift is $\rm{Z/Z_\odot=-(0.6\pm0.2)~z+(0.36\pm0.04)}$ and between metallicity and temperature the best fit is $\rm{Z/Z_\odot = -(1.0\pm0.7)~T/100keV + (0.34\pm0.06)}$. The average metallicity of the sample is $\rm{Z\approx0.3\pm0.1~Z_\odot}$.
Future X-ray missions like Astro-H [@2014arXiv1412.1176K] and Athena [@2013arXiv1306.2307N] will help to further investigate turbulent velocities and chemical enrichment in the ICM. The eROSITA observatory [@2012arXiv1209.3114M] will detect a large X-ray cluster sample for cosmological studies and our detailed cluster mapping can be used to make predictions on the scatter in scaling relations due to unresolved structures in temperature. To encourage further analysis based on this unique sample of cluster observations all maps and asymmetry measurements used in this study are made publicly available in electronic form.
We thank the anonymous referee for constructive comments that helped to improve the clarity of the paper. We thank J. Buchner for helpful discussions. This research has made use of data obtained from the *Chandra* Data Archive and the *Chandra* Source Catalog, and software provided by the *Chandra* X-ray Center (CXC) in the application packages CIAO, ChIPS, and Sherpa. This research has made use of NASA’s Astrophysics Data System. This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France. This research has made use of SAOImage DS9, developed by Smithsonian Astrophysical Observatory. This research has made use of data and software provided by the High Energy Astrophysics Science Archive Research Center (HEASARC), which is a service of the Astrophysics Science Division at NASA/GSFC and the High Energy Astrophysics Division of the Smithsonian Astrophysical Observatory. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. This research has made use of the tools Veusz, the matplotlib library for Python, and TOPCAT. M.G. is supported by the National Aeronautics and Space Administration through Einstein Postdoctoral Fellowship Award Number PF-160137 issued by the *Chandra* X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060.
Cluster sample {#sec:aobs}
==============
[lrrrrrlr]{} & & & & & & &\
& & & & & & &\
CYGNUS A & 232 & 299.877 & 40.741 & 52.82 & 0.0561 & J1959.5+4044 & 7.08\
PKS 0745-191 & 174 & 116.883 & -19.290 & 45.73 & 0.1028 & J0747.5-1917 & 20.36\
\[tab:cat1\]
[lrrrrrlr]{} & & & & & & &\
& & & & & & &\
A 2052 & 651 & 229.182 & 7.013 & 47.94 & 0.0353 & RXC J1516.7+0701 & 2.58\
A 1775 & 99 & 205.448 & 26.352 & 12.58 & 0.0724 & RXC J1341.8+2622 & 2.83\
A 2199 & 156 & 247.188 & 39.553 & 97.92 & 0.0299 & RXC J1628.6+3932 & 3.77\
2A 0335+096 & 102 & 54.665 & 10.007 & 80.91 & 0.0347 & RXC J0338.6+0958 & 4.21\
3C348 (HERCULES A) & 112 & 252.778 & 4.985 & 5.39 & 0.154 & RXC J1651.1+0459 & 5.49\
A 2034 & 255 & 227.532 & 33.515 & 11.94 & 0.113 & RXC J1510.1+3330 & 6.49\
MS0735.6+7421 & 520 & 115.421 & 74.266 & 4.06 & 0.2149 & RXC J0741.7+7414 & 7.94\
A 2146 & 418 & 239.006 & 66.352 & 3.99 & 0.2339 & RXC J1556.1+6621 & 9.31\
A 1795 & 958 & 207.221 & 26.596 & 59.29 & 0.0622 & RXC J1348.8+2635 & 9.93\
A 1413 & 136 & 178.769 & 23.369 & 12.61 & 0.1427 & RXC J1155.3+2324 & 10.91\
A 401 & 163 & 44.727 & 13.579 & 50.29 & 0.0739 & RXC J0258.9+1334 & 11.76\
A 1995 & 100 & 223.168 & 58.049 & 3.18 & 0.3179 & RXC J1452.9+5802 & 13.42\
MS 1455.0+2232 & 108 & 224.253 & 22.33 & 4.89 & 0.2579 & RXC J1457.2+2220 & 13.73\
A 520 & 527 & 73.546 & 2.977 & 8.33 & 0.203 & RXC J0454.1+0255 & 14.52\
A 665 & 140 & 127.637 & 65.89 & 11.18 & 0.1818 & RXC J0830.9+6551 & 15.69\
A 2204 & 97 & 248.186 & 5.557 & 24.11 & 0.1514 & RXC J1632.7+0534 & 23.43\
A 2390 & 110 & 328.403 & 17.683 & 11.01 & 0.2329 & RXC J2153.5+1741 & 25.15\
ZW 3146 & 84 & 155.906 & 4.167 & 8.77 & 0.285 & RXC J1023.6+0411 & 29.91\
A 1835 & 223 & 210.271 & 2.895 & 12.12 & 0.2528 & RXC J1401.0+0252 & 32.56\
\[tab:cat2\]
[lrrrrrlr]{} & & & & & & &\
& & & & & & &\
SERSIC 159-03 & 106 & 348.515 & -42.713 & 23.412 & 0.0564 & J2313.9-4244 & 3.74\
A 496 & 88 & 68.403 & -13.25 & 72.075 & 0.0326 & J0433.6-1315 & 3.89\
HYDRA A & 224 & 139.527 & -12.092 & 39.461 & 0.0539 & J0918.1-1205 & 5.61\
A 1650 & 229 & 194.664 & -1.781 & 20.909 & 0.0845 & J1258.6-0145 & 6.99\
A 2597 & 146 & 351.337 & -12.136 & 20.558 & 0.0852 & J2325.3-1207 & 7.22\
A 3667 & 528 & 303.211 & -56.855 & 70.892 & 0.0556 & J2012.5-5649 & 10.02\
A 907 & 103 & 149.528 & -11.086 & 7.833 & 0.1669 & J0958.3-1103 & 10.13\
A 521 & 165 & 73.558 & -10.273 & 4.944 & 0.2475 & J0454.1-1014 & 12.97\
A 2744 & 124 & 3.586 & -30.352 & 4.964 & 0.3066 & J0014.3-3023 & 19.79\
A 1689 & 197 & 197.808 & -1.337 & 15.332 & 0.1832 & J1311.4-0120 & 23.59\
1E 0657-56 & 566 & 104.751 & -55.904 & 9.079 & 0.2965 & J0658.5-5556 & 35.55\
RX J1347-114 & 232 & 206.889 & -11.734 & 6.468 & 0.4516 & J1347.5-1144 & 63.43\
\[tab:cat3\]
*Chandra* datasets {#sec:cobs}
==================
[ll]{} &\
1e0657 & 554,3184,4984,4985,4986,5355,5356,5357,5358,5361\
2a0335 & 919,7939,9792\
3c348 & 1625,5796,6257\
a1413 & 537,1661,5002,5003,7696\
a1650 & 4178,5822,5823,6356,6357,6358,7242,7691\
a1689 & 540,1663,5004,6930,7289,7701\
a1775 & 12891,13510\
a1795 & 493,494,3666,5286,5287,5288,5289,5290,6159,6160,6161,6162,6163,\
&10432,10433,10898,10899,10900,10901,12026,12027,12028,12029,13106,13107,\
&13108,13109,13110,13111,13112,13113,13412,13413,13414,13415,13416,13417,\
&14268,14269,14270,14271,14272,14273,14274,14275,15485,15486,15487,15488,15489,\
&15490,15491,15492,16432,16433,16434,16435,16436,16437,16438,16439,16465,16466,\
&16467,16468,16469,16470,16471,16472\
a1835 & 495,496,6880,6881,7370\
a1995 & 906,7021,7713\
a2034 & 2204,7695,12885,12886,13192,13193\
a2052 & 890,5807,10477,10478,10479,10480,10879,10914,10915,10916,10917\
a2146 & 10464,10888,12245,12246,12247,13020,13021,13023,13120,13138\
a2199 & 497,498,10748,10803,10804,10805\
a2204 & 499,6104,7940\
a2390 & 500,501,4193\
a2597 & 922,6934,7329,15144\
a2744 & 2212,7712,7915,8477,8557\
a3667 & 513,889,5751,5752,5753,6292,6295,6296,7686\
a401 & 518,2309,14024\
a496 & 931,3361,4976\
a520 & 528,4215,7703,9424,9425,9426,9430\
a521 & 430,901,12880,13190\
a665 & 531,3586,7700,12286,13201,15148\
a907 & 535,3185,3205\
cygnusa & 360,5830,5831,6225,6226,6228,6229,6250,6252\
hydraa & 575,576,4969,4970\
ms0735 & 4197,10468,10469,10470,10471,10822,10918,10922\
ms1455 & 543,4192,7709\
pks0745 & 508,2427,6103,7694,12881\
rxj1347 & 506,507,3592,13516,13999,14407\
sersic159 & 1668,11758\
zw3146 & 909,9371\
\[tab:obsids\]
Perturbation table {#sec:amaps}
==================
[lrrrr]{} & & & &\
& & & &\
1e0657 & $\rm{20.4^{+1.6}_{-1.4}}$ & $\rm{17.3^{+2.1}_{-1.5}}$ & $\rm{8.8^{+1.5}_{-1.4}}$ & $\rm{20.9^{+1.2}_{-0.9}}$\
2a0335 & $\rm{11.7^{+0.6}_{-0.5}}$ & $\rm{19.8^{+0.9}_{-0.6}}$ & $\rm{9.3^{+0.5}_{-0.4}}$ & $\rm{13.5^{+0.6}_{-0.4}}$\
3c348 & $\rm{1.3^{+2.9}_{-1.3}}$ & $\rm{12.0^{+1.9}_{-1.7}}$ & $\rm{3.6^{+2.0}_{-2.8}}$ & $\rm{8.5^{+0.9}_{-0.6}}$\
a1413 & $\rm{5.4^{+1.9}_{-4.0}}$ & $\rm{6.4^{+2.8}_{-2.6}}$ & $\rm{0.3^{+2.4}_{-0.3}}$ & $\rm{8.7^{+1.0}_{-0.7}}$\
a1650 & $\rm{5.5^{+0.9}_{-1.0}}$ & $\rm{14.0^{+1.0}_{-1.0}}$ & $\rm{4.0^{+1.0}_{-1.0}}$ & $\rm{10.0^{+0.7}_{-0.5}}$\
a1689 & $\rm{0.6^{+2.7}_{-0.6}}$ & $\rm{1.1^{+2.8}_{-1.1}}$ & $\rm{0.8^{+2.1}_{-0.8}}$ & $\rm{3.8^{+0.4}_{-0.2}}$\
a1775 & $\rm{0.6^{+3.5}_{-0.6}}$ & $\rm{20.7^{+2.6}_{-2.3}}$ & $\rm{12.6^{+1.9}_{-2.1}}$ & $\rm{8.0^{+0.8}_{-0.8}}$\
a1795 & $\rm{5.3^{+0.2}_{-0.2}}$ & $\rm{22.3^{+0.5}_{-0.2}}$ & $\rm{11.7^{+0.4}_{-0.1}}$ & $\rm{12.4^{+0.3}_{-0.1}}$\
a1835 & $\rm{6.0^{+1.1}_{-1.1}}$ & $\rm{15.8^{+1.5}_{-1.1}}$ & $\rm{9.3^{+1.0}_{-0.8}}$ & $\rm{8.8^{+0.6}_{-0.5}}$\
a1995 & $\rm{5.7^{+5.5}_{-5.7}}$ & $\rm{13.6^{+4.9}_{-3.5}}$ & $\rm{11.4^{+3.7}_{-5.8}}$ & $\rm{7.0^{+1.6}_{-0.9}}$\
a2034 & $\rm{12.8^{+2.2}_{-1.7}}$ & $\rm{6.5^{+2.1}_{-2.4}}$ & $\rm{6.1^{+1.6}_{-3.4}}$ & $\rm{9.1^{+1.0}_{-0.7}}$\
a2052 & $\rm{7.5^{+0.2}_{-0.2}}$ & $\rm{18.3^{+0.6}_{-0.4}}$ & $\rm{11.6^{+0.4}_{-0.2}}$ & $\rm{10.1^{+0.3}_{-0.2}}$\
a2146 & $\rm{20.3^{+1.9}_{-1.9}}$ & $\rm{37.1^{+4.1}_{-3.2}}$ & $\rm{12.2^{+1.7}_{-1.4}}$ & $\rm{24.3^{+2.7}_{-1.5}}$\
a2199 & $\rm{7.6^{+0.4}_{-0.3}}$ & $\rm{13.2^{+0.6}_{-0.4}}$ & $\rm{7.3^{+0.4}_{-0.3}}$ & $\rm{8.6^{+0.3}_{-0.3}}$\
a2204 & $\rm{6.7^{+1.2}_{-1.3}}$ & $\rm{13.7^{+1.7}_{-1.0}}$ & $\rm{8.7^{+1.1}_{-1.0}}$ & $\rm{7.6^{+0.5}_{-0.5}}$\
a2390 & $\rm{11.4^{+2.0}_{-1.8}}$ & $\rm{19.0^{+2.6}_{-1.9}}$ & $\rm{12.4^{+2.1}_{-2.0}}$ & $\rm{11.2^{+1.1}_{-0.8}}$\
a2597 & $\rm{8.1^{+0.8}_{-0.6}}$ & $\rm{8.9^{+0.8}_{-0.6}}$ & $\rm{5.5^{+0.6}_{-0.6}}$ & $\rm{7.2^{+0.5}_{-0.4}}$\
a2744 & $\rm{9.8^{+4.2}_{-9.8}}$ & $\rm{13.4^{+3.1}_{-2.7}}$ & $\rm{7.6^{+3.5}_{-7.6}}$ & $\rm{15.1^{+1.4}_{-1.1}}$\
a3667 & $\rm{24.6^{+1.1}_{-1.1}}$ & $\rm{12.4^{+0.7}_{-0.5}}$ & $\rm{16.1^{+0.9}_{-0.7}}$ & $\rm{11.4^{+0.5}_{-0.4}}$\
a401 & $\rm{6.1^{+1.3}_{-1.5}}$ & $\rm{8.1^{+1.0}_{-1.3}}$ & $\rm{2.8^{+1.3}_{-2.8}}$ & $\rm{7.5^{+0.5}_{-0.3}}$\
a496 & $\rm{9.0^{+0.6}_{-0.6}}$ & $\rm{21.7^{+1.2}_{-1.0}}$ & $\rm{14.0^{+0.9}_{-0.6}}$ & $\rm{8.9^{+0.5}_{-0.4}}$\
a520 & $\rm{12.6^{+2.0}_{-1.8}}$ & $\rm{15.0^{+2.6}_{-1.4}}$ & $\rm{7.6^{+1.9}_{-1.7}}$ & $\rm{11.5^{+1.4}_{-0.9}}$\
a521 & $\rm{12.1^{+5.3}_{-7.8}}$ & $\rm{16.7^{+8.2}_{-7.2}}$ & $\rm{10.0^{+6.4}_{-8.0}}$ & $\rm{10.7^{+2.3}_{-1.4}}$\
a665 & $\rm{11.8^{+2.4}_{-2.4}}$ & $\rm{8.2^{+2.6}_{-4.6}}$ & $\rm{6.5^{+2.6}_{-3.9}}$ & $\rm{11.0^{+0.9}_{-0.8}}$\
a907 & $\rm{0.2^{+2.4}_{-0.2}}$ & $\rm{12.1^{+1.9}_{-1.7}}$ & $\rm{1.7^{+2.9}_{-1.7}}$ & $\rm{9.5^{+0.9}_{-0.6}}$\
cygnusa & $\rm{18.2^{+1.0}_{-0.8}}$ & $\rm{23.4^{+1.3}_{-1.0}}$ & $\rm{14.1^{+0.8}_{-0.6}}$ & $\rm{27.3^{+1.2}_{-0.9}}$\
hydraa & $\rm{9.8^{+0.4}_{-0.3}}$ & $\rm{14.4^{+0.6}_{-0.5}}$ & $\rm{9.7^{+0.4}_{-0.3}}$ & $\rm{7.8^{+0.3}_{-0.2}}$\
ms0735 & $\rm{10.6^{+1.5}_{-0.9}}$ & $\rm{14.0^{+1.7}_{-1.2}}$ & $\rm{7.8^{+1.2}_{-0.8}}$ & $\rm{10.3^{+1.0}_{-0.7}}$\
ms1455 & $\rm{4.1^{+2.0}_{-3.5}}$ & $\rm{19.0^{+2.6}_{-1.9}}$ & $\rm{9.0^{+2.0}_{-1.5}}$ & $\rm{11.1^{+1.0}_{-0.8}}$\
pks0745 & $\rm{3.4^{+1.0}_{-1.9}}$ & $\rm{19.0^{+1.0}_{-0.8}}$ & $\rm{9.4^{+0.8}_{-0.6}}$ & $\rm{11.6^{+0.4}_{-0.5}}$\
rxj1347 & $\rm{13.4^{+2.3}_{-2.3}}$ & $\rm{23.4^{+3.3}_{-2.1}}$ & $\rm{16.0^{+2.3}_{-2.0}}$ & $\rm{13.8^{+1.5}_{-0.8}}$\
sersic159 & $\rm{9.7^{+1.1}_{-0.7}}$ & $\rm{11.4^{+1.1}_{-1.1}}$ & $\rm{5.2^{+0.6}_{-0.6}}$ & $\rm{9.2^{+0.9}_{-0.5}}$\
zw3146 & $\rm{0.4^{+2.3}_{-0.4}}$ & $\rm{25.9^{+2.5}_{-2.2}}$ & $\rm{10.9^{+1.9}_{-1.9}}$ & $\rm{15.9^{+1.3}_{-0.9}}$\
\[tab:persum\]
2D Maps {#sec:amaps}
=======
Unsharp-masked count images {#sec:amask}
===========================
Projected radial profiles {#sec:arad}
=========================
[^1]: Tables 2-67 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/
[^2]: http://cxc.harvard.edu/cda/
[^3]: http://atomdb.org/faq.php
|
---
abstract: 'We study the temporal aspects of laser-assisted extreme ultraviolet (XUV) photoionization using attosecond pulses of harmonic radiation. The aim of this paper is to establish the general form of the [*phase*]{} of the relevant transition amplitudes and to make the connection with the time-delays that have been recently measured in experiments. We find that the overall phase contains two distinct types of contributions: one is expressed in terms of the phase-shifts of the photoelectron continuum wavefunction while the other is linked to continuum–continuum transitions induced by the infrared (IR) laser probe. Our formalism applies to both kinds of measurements reported so far, namely the ones using attosecond pulse trains of XUV harmonics and the others based on the use of isolated attosecond pulses (streaking). The connection between the phases and the time-delays is established with the help of finite difference approximations to the energy derivatives of the phases. The observed time-delay is a sum of two components: a one-photon Wigner-like delay and an [*universal*]{} delay that originates from the probing process itself.'
address:
- 'Department of Physics, Lund University, P.O. Box 118, 22100 Lund, Sweden'
- 'Atomic Physics, Fysikum, Stockholm University, AlbaNova University Center, SE-106 91 Stockholm, Sweden'
- |
UPMC Université Paris 6, UMR 7614, Laboratoire de Chimie Physique-Matière et Rayonnement,\
11 rue Pierre et Marie Curie, 75231 Paris Cedex 05, France
- 'CNRS, UMR 7614, LCPMR, Paris, France'
author:
- 'J. M. Dahlström'
- 'D. Guénot'
- 'K. Kl[ü]{}nder'
- 'M. Gisselbrecht'
- 'J. Mauritsson'
- 'A. L’Huillier'
- 'A. Maquet'
- 'R. Taïeb'
title: 'Theory of attosecond delays in laser-assisted photoionization'
---
Introduction {#sec:intro}
============
The dynamics of photoionization can now be explored with unprecedented time resolution thanks to high-order harmonic-based sources that deliver pulses of XUV radiation with duration in the attosecond range. Recent measurements performed with single attosecond pulses have shown the existence of an unexpected time-delay between the single-photon ionization from the 2s and the 2p sub-shells of Ne atoms in gas phase [@Schultze2010]. The “streak camera” technique used in these experiments [@Goulielmakis2004] implied nontrivial ejection times of the photoelectrons, depending on the sub-shell from which they originate. Similar delays between the ejection times from the 3s and 3p sub-shells in Ar have been measured also using trains of attosecond pulses [@Klunder2011], with the help of another technique based on interferometry called RABBIT (Reconstruction of Attosecond Beating By Interference of Two-photon transitions) [@Veniard95; @Veniard96; @Muller2002]. In both cases, delays of several tens of attoseconds have been measured. As photoionization is one of the most fundamental processes in light-matter interactions, these results have motivated a large number of theoretical investigations [@Yakovlev2010; @Kheifets2010; @Ivanov2011; @Zhang2011; @Nagele2011; @IvanovM2011; @Pazourek2011].
{width="90.00000%"}
The two kinds of measurements share many similarities since they involve a laser-assisted single-photon ionization process and they rely on a phase-locked IR laser field to probe the temporal aspects of the XUV photoionization. However, they differ in the analysis used to determine the time-delays and in the range of IR laser intensity.
The motivation of the present paper is to present an unified theoretical analysis of these processes. To achieve this goal, we shall expose first the theoretical background which has conducted us to conclude in [@Klunder2011], that in interferometric measurements, the measured delays arise from the combination of two distinct contributions: One is related to the electronic structure of the atomic target while the other is induced by the measurement process itself. The first one can be identified as a “Wigner time-delay” [@Wigner1955; @Nussensveig2002] that is directly related to the energy dependence of the different phase-shifts experienced by the photoelectrons ionized from distinct sub-shells in atoms. The other contribution is induced by the IR laser field that is used to probe the photoionization process. This latter contribution results from the continuum–continuum transitions induced by the probe IR laser field in the presence of the Coulomb potential of the ionic core. When simplifying the analysis to the cases when the process is dominated by the asymptotic form of the relevant second-order matrix elements, a characteristic measurement-induced delay can be identified, that is independent from the details of the electronic structure of the ionic core. This shows how the experimental signal can be related to the temporal dynamics of one-photon ionization.
Regarding the streaking measurements realized with a single attosecond pulse of XUV radiation [@Schultze2010], the experimental data were obtained for IR field intensities significantly higher than those obtained with attosecond pulse trains [@Klunder2011]. Understandably, the questions related to the role of the probe IR field on the photoelectron dynamics in streaking measurements have motivated several theoretical studies [@Yakovlev2010; @Kheifets2010; @Ivanov2011; @Zhang2011; @Nagele2011; @IvanovM2011; @Pazourek2011]; see also the earlier papers: [@Smirnova2006; @Smirnova2007; @Zhang2010; @Baggesen2010]. Then, a natural issue arises which is to determine to what extent the “streaking delays” so obtained differ from those derived from the interferometric data. Although both the experimental techniques and the theory treatments differ, it is of interest to compare the two approaches. Indeed, as we shall show below, a link can be found when reducing the laser intensity of the streaking field so that one reaches the domain of applicability of the recently developed Phase-Retrieval by Omega Oscillation Filtering (PROOF) scheme, [@Chini2010]. An interesting outcome of our analysis is to show the importance of the long-range Coulomb potential for understanding the absolute time-delays in the streaking experiments as well.
The interpretation of the attosecond delays in photoionization relies on our ability to determine the [*phases*]{} of the relevant transition amplitudes. Thus, before going into the details of the derivation of such phases, we shall outline the main features of the two techniques in Section \[sec:ATI\]. Then, Section \[sec:ATIphases\] is devoted to the presentation of the general expressions for two-color, two-photon, complex transition amplitudes that are relevant for Above-Threshold Ionization (ATI) in single-active electron systems. The theoretical background is based on a perturbative approach and the emphasis will be on the derivation of a closed-form approximate expression that is of interest for evaluating the phase of the amplitudes. The basis of exact computations in hydrogen will be outlined, and a simplified classical treatment will be presented. Applications to the determination of the relation between the phases and the time-delays is presented in Section \[sec:ATIapplications\]. Here we consider first ionization by an attosecond pulse train and then by a single attosecond pulse, in the presence of a relatively weak IR field. This discussion provides an interesting connection between the two types of measurements. Section \[sec:results\] contains a comparison of the results extracted from the approximate evaluation of the delays to the ones deduced from exact calculations performed in hydrogen from different initial states. Also, we present our conclusions and perspectives.
Laser-Assisted XUV Photoionization: Attosecond Pulse Train vs. Single Attosecond Pulse {#sec:ATI}
======================================================================================
The principle of the measurements of the delays using an attosecond pulse train is illustrated in Fig. \[fig1\] (a), which represents schematically the ionization of an atom in the simultaneous presence of a set of several XUV (odd) harmonics and of the IR field, used to generate the harmonics (atomic units will be used throughout the paper, unless otherwise stated). In the time domain, both pulses are “long”, [*i.e.*]{} the IR laser pulse is multi-cycle, with typical duration of a few tens of femtoseconds, and the XUV harmonic field is constituted of a train of attosecond pulses (or equivalently of a comb of coherent odd harmonic frequencies $(2q+1)\omega$: $H_{2q+1}$). Under these conditions, the photoelectron spectrum consists of equidistant lines separated by $2 \omega$ that are associated to one-photon ionization of the target by each harmonic. In-between these lines are sidebands associated to two-photon transitions involving the absorption of one harmonic and the exchange of one IR photon. The signal intensities, $S_{2q}$, of the sidebands labelled $2q$ vary periodically with the delay $\tau$ between the IR and the harmonic pulses, according to a generic expression that involves the phases of the fields together with atom-dependent contributions: $$\begin{aligned}
S_{2q} = {} & \alpha + \beta \cos[2 \omega \tau - \Delta \phi_{2q} - \Delta \theta_{2q} ],
\label{S2q}\end{aligned}$$ where $\Delta \phi_{2q} = (\phi_{2q + 1} -\phi_{2q-1})$ is the phase difference between the consecutive harmonics $H_{2q+1}$ and $H_{2q-1}$ and $\Delta \theta_{2q} $ is an intrinsic atomic quantity, associated to the difference of the phases of the transition amplitudes associated to the distinct quantum paths leading to the sideband [@Veniard96].
To make clearer the connection between the above phase differences and the time-delays we shall discuss here, it is convenient to rewrite the formula in Eq. (\[S2q\]) under the form: $$\begin{aligned}
\label{S2qt}
S_{2q} = {} & \alpha + \beta \cos [2\omega (\tau - \tau_{2q} - \tau_\theta) ], \end{aligned}$$ where $\tau_{2q} = \Delta \phi_{2q}/2\omega$ is a finite difference approximation to the group delay $GD = \partial \phi /\partial \Omega$ of the harmonic radiation at the considered frequency, $\Omega\approx 2q \omega $, as presented in refs. [@Paul2001; @Mairesse2003]. On the other hand, $\tau_\theta= \Delta \theta_{2q}/2\omega$ is an intrinsic time-delay associated to the atomic phase difference, $\Delta \theta_{2q}$. As reported in [@Klunder2011] and as we shall describe in more details here, the determination of $\tau_\theta$ gives access to the temporal dynamics of atomic photoionization. Before closing this brief presentation of the RABBIT scheme, we stress that the intensities of both fields must be kept moderate, so that the phases of the transition amplitudes associated to the sidebands can be derived from a standard time-dependent perturbation theory calculation, limited to second-order.
As represented schematically in Fig. \[fig1\] (b), streaking relies on the ionization of the atom by a single attosecond pulse, in the presence of a few-cycle IR pulse. One requirement to realize streaking is that the effective duration of the attosecond pulse has to be significantly shorter than the IR pulse cycle [@Goulielmakis2004] (more rigorously it is actually the spectral bandwidth of the attosecond pulse that must be larger than the probe photon frequency). The measurement consists then in recording the momentum, ${\vec k}_f (t)$, of the ejected photoelectron, as deflected by the instantaneous IR probe vector potential, ${\vec A}_\omega (t)$, so that its wave vector is given approximately by: $$\begin{aligned}
\label{Streak}
{\vec k}_f \approx {\vec k} - {\vec A}_\omega (t),\end{aligned}$$ where ${\vec k}$ is the field-free momentum. We note that this simple relation is being derived by assuming that the photoelectron does not experience the effects of the residual Coulomb potential of the ionic core [@Goulielmakis2004]. In this article, we will recover this streaking phenomenon using an interferometric interpretation, which includes the full effect of the ionic core, thereby, obtaining the correct absolute delay of the momentum modulation relative to the probe field.
![\[fig2\] Sketch of an attosecond photoionization time-delay experiment between two different initial states with energies, $\epsilon_i$ and $\epsilon_j$. The attosecond XUV field, $\Omega$, ionizes the electron in the presence of a phase-locked IR laser field, $\omega$. Since the same single attosecond pulse (or attosecond pulse train) is used to promote the electrons from either state, the observed delays of the modulation of the central momenta (or sidebands), $\Delta \tau_{ij}(\Omega)$, are directly related to the atomic delay difference $\tau_{\theta}^{(i)}(\Omega)-\tau_{\theta}^{(j)}(\Omega)$ defined in Eq. (\[S2qt\]), which, in turn, is related to the difference of the corresponding two-photon matrix element phases.](fig2.pdf){width="40.00000%"}
If the XUV field frequency is high enough to ionize electrons from either one of two valence sub-shells of an atom (typically the 2s and 2p states in Ne atoms) one can record two distinct streaking traces corresponding to the two photoelectron lines associated to each sub-shell. This situation is schematically displayed in Fig. 2. A delay between the ejection times of the photoelectrons from the different atomic sub-shells can be determined by comparing the corresponding streaking traces, corrected from possible biases introduced by the experimental procedure [@Schultze2010]. The same idea can be applied to attosecond pulse trains, by using the interferometric set-up [@Klunder2011]. We turn now to the presentation of the theoretical background that is common to the two kinds of techniques, in the limit of weak IR probe fields.
Phases of Laser-Assisted Photoionization Transition Amplitudes {#sec:ATIphases}
==============================================================
Two-photon Above-Threshold Ionization
--------------------------------------
A representative laser-assisted photoionization transition is depicted in Fig. \[fig1\] (c): It displays the sequential absorption of one XUV harmonic photon with frequency $\Omega$, followed by the absorption of one IR laser photon with frequency $\omega$. It corresponds to the lowest-order perturbative amplitude for an Above-Threshold Ionization (ATI) process observed when the XUV frequency is larger than the ionization energy of the system: $\Omega > I_p$. Obviously, other quantum paths are contributing to this type of two-color ionization process, [*e.g.*]{} the IR photon can be absorbed before the XUV photon, but ATI amplitudes of the former type are dominant in the class of experiments considered here.
For two fields with the same linear polarization ${\vec \epsilon}$, it is natural to choose this direction as the quantization axis $\hat z$, and the matrix element associated to the path shown in Fig. \[fig1\] (c), is of the form: $$\begin{aligned}
M ({\vec k};\epsilon_i +\Omega) = {} & \frac{1}{i} E_\omega E_\Omega \lim_{ \varepsilon \to 0^+} \int \!\!\!\!\!\!\!\! \sum_{\nu} {\langle \: \vec k \: | \: z \: | \: \nu \: \rangle \langle \: \nu \: | \: z \: | \: i \: \rangle\over \epsilon_i +\Omega - \epsilon_\nu +i \varepsilon} , \end{aligned}$$ where $E_\Omega$ and $E_\omega$ are the complex amplitudes of the harmonic and IR laser fields, respectively; $\varphi_{n_i, \ell_i,m_i} ({\vec r})=\langle \vec r | i \rangle$ is the initial state wavefunction, with negative energy $\epsilon_i$ and $\varphi_{\vec k} ({\vec r})=\langle \vec r | \vec k \rangle$ is the final state wavefunction with positive energy $\epsilon_k = {k^2 / 2} = \epsilon_i +\Omega +\omega$. The sum over the index $\nu$ runs over the whole spectrum (discrete plus continuous) of the atom. The partial wave expansion of the final state wavefunction is: $$\begin{aligned}
\varphi_{\vec k}({\vec r}) = {} & (8 \pi)^{3/2} \sum_{L,M} i^{L} e^{ -i \eta_L (k)} Y_{L,M}^*({\hat k}) Y_{L,M}({\hat r}) R_{k,L}(r),\end{aligned}$$ where the $Y_{L,M}$ are spherical harmonics, the $R_{k,L}(r)$ are (real) radial wavefunctions normalized on the energy scale and $\eta_{L} (k)$ are the phase-shifts. We note that this wavefunction behaves asymptotically as the superposition of a plane wave plus an [*ingoing*]{} spherical wave, as required to treat photoionization [@Landau]. Thus, the phase-shifts $\eta_{L} (k)$ account for the phase difference between the free motion of a plane wave and that of a photoelectron wave ejected from an atomic bound state.
The angular dependence of the matrix element can be factorized out for an initial state $$\begin{aligned}
\varphi_{n_i,\ell_i,m_i} ({\vec r })= {} & Y_{\ell_i,m_i}({\hat r}) R_{n_i,\ell_i}(r)\end{aligned}$$ and with $z = \sqrt{4 \pi/3} \: r \:Y_{1,0}({\hat r})$, it becomes: $$\begin{aligned}
\label{Ma}
M({\vec k},\epsilon_i+\Omega) = {} &
\:{4 \pi \over 3i} ~ (8 \pi)^{3/2} ~ E_\omega E_\Omega ~ \nonumber \\
\times ~ & \sum_{L,M} (-i)^{L} e^{ i \eta_{L}(k)} Y_{L,M}({\hat k}) \nonumber \\
\times ~ & \sum_{\lambda,\mu} \langle Y_{L,M}| Y_{1,0}|Y_{\lambda,\mu} \rangle \langle Y_{\lambda,\mu}| Y_{1,0}| Y_{\ell_i,m_i}\rangle \nonumber \\
\times ~ & T_{L,\lambda,\ell_i}( k, \epsilon_i +\Omega), \end{aligned}$$ where the angular momentum components of the intermediate states are labelled $(\lambda, \mu)$ and the quantity, $T_{L,\lambda,\ell_i}(k; \epsilon_i +\Omega)$, is the radial part of the amplitude. The span of accessible angular momentum states in the intermediate and final states is governed by the dipole selection rules: One has $\lambda = \ell_i \pm 1; L = \ell_i, \ell_i\pm 2$ and $M = \mu =m_i$ respectively. The explicit form of the radial amplitude $T_{L,\lambda,\ell_i}( k;\epsilon_i +\Omega)$ is: $$\begin{aligned}
T_{L,\lambda,\ell_i}(k; \epsilon_i +\Omega)
= {} \sum_{\nu:\epsilon_\nu<0} {\langle R_{k,L}| r |R_{\nu,\lambda}\rangle \langle R_{\nu,\lambda}| r |R_{n_i,\ell_i} \rangle \over \epsilon_i+ \Omega - \epsilon_\nu } \notag \\
+ \lim_{ \varepsilon \to 0^+} \int_0^{+\infty} d\epsilon_{\kappa'} {\langle R_{k,L}| r |R_{\kappa',\lambda} \rangle \langle R_{\kappa',\lambda}| r |R_{n_i,\ell_i} \rangle \over \epsilon_i +\Omega - \epsilon_{\kappa'} +i \varepsilon},
\label{Tell} \end{aligned}$$ where we have separated the contributions of the discrete and continuous spectra.
Since the frequency of the XUV harmonic is larger than the ionization potential of the atom, $\Omega > I_p = |\epsilon_i|$, it is also larger than the excitation energies of the atom, $\Omega > \epsilon_\nu - \epsilon_i$. Accordingly, the denominators of the terms in the sum over the discrete states are positive and relatively large, which makes the overall contribution of these terms significantly smaller than that of the continuous spectrum. In the integral running on the continuous spectrum of energies $\epsilon_{\kappa'} = \kappa'^2 /2$, the denominator becomes zero at the energy $\epsilon_\kappa = \kappa^2 /2 = \epsilon_i +\Omega$. Taking the limit $\varepsilon \to 0^+$, the integral becomes: $$\begin{aligned}
\lim_{ \varepsilon \to 0^+} \int_0^{+\infty} d\epsilon_{\kappa'} {\langle R_{k,L}| r |R_{\kappa',\lambda} \rangle \langle R_{\kappa',\lambda}| r |R_{n_i,\ell_i} \rangle \over \epsilon_i +\Omega - \epsilon_{\kappa'} +i \varepsilon} \nonumber \\
= {\mathcal P} \int_0^{+\infty} d\epsilon_{\kappa'} {\langle R_{k,L}| r |R_{\kappa',\lambda} \rangle \langle R_{\kappa',\lambda}| r |R_{n_i,\ell_i} \rangle \over \epsilon_i +\Omega - \epsilon_{\kappa'} } \nonumber \\
-i\pi \langle R_{k,L}| r |R_{\kappa,\lambda} \rangle \langle R_{\kappa,\lambda}| r |R_{n_i,\ell_i} \rangle.
\label{Cauchy}
\end{aligned}$$ where the first term is the Cauchy principal value of the integral, which turns out to be real, and the second term is purely imaginary. The latter is associated with a two-step transition as it contains the product of the one-photon ionization amplitude towards the state of energy $\epsilon_{\kappa} = \epsilon_i + \Omega$, times the continuum–continuum transition amplitude from $\epsilon_{\kappa}$ towards the final state of energy $k^2 /2 = \epsilon_{\kappa} + \omega$ that is reached upon the absorption of the IR photon $\omega$.
The overall phase of the radial matrix element, Eq. (\[Tell\]), is thus governed by the ratio of the imaginary term in Eq. (\[Cauchy\]) to the sum of the integral principal part in the same equation plus the contribution of the discrete spectrum contained in Eq. (\[Tell\]). Accurate computations of such amplitudes and phases represent a formidable task for most atomic systems. This entails to rely on approximate representations of the atomic potential for each angular momentum dependent state [@Toma; @Kennedy] or to use many-electron techniques [@LHuillier1986; @KheifetsComment]. There is however the notable exception of hydrogenic systems, where “exact” calculations of these amplitudes are feasible [@Zernik; @Klarsfeld79a; @Klarsfeld79b; @Aymar], see below. It is thus of importance to derive an approximate treatment which should allow to get correct estimates of the phases of interest to address the questions of the time-delays.
Asymptotic approximation for 2-photon ATI matrix elements {#sec:asympapp}
---------------------------------------------------------
Let us re-express the radial amplitude in terms of the first-order [*perturbed wavefunction*]{} denoted $\rho_{\kappa,\lambda}(r)$: $$\begin{aligned}
\label{Trho}
T_{L,\lambda,\ell_i}(k; \epsilon_i +\Omega) =
{\langleR_{k,L} |r| \rho_{\kappa,\lambda} \rangle}.\end{aligned}$$ The function $\rho_{\kappa,\lambda}(r)$ solves the inhomogeneous differential equation: $$\begin{aligned}
\label{eqdiffrho}
[H_\lambda - \epsilon_{\kappa}]\rho_{\kappa,\lambda}(r) =
- r R_{n_i,\ell_i} (r),\end{aligned}$$ where $H_\lambda$ is the radial atomic Hamiltonian for angular momentum $\lambda$. The fully developed eigenfunction expansion of $\rho_{\kappa,\lambda}(r)$ can be identified using Eqs. (\[Tell\]) and (\[Cauchy\]): $$\begin{aligned}
\label{rho}
\rho_{\kappa,\lambda}(r) ={}&
\sum_{\nu:\epsilon_\nu<0} {R_{\nu,\lambda}(r) \langle R_{\nu,\lambda}| r |R_{n_i,\ell_i} \rangle \over \epsilon_{\kappa} - \epsilon_\nu }
\nonumber \\
+~& {\mathcal P} \int_0^{+\infty} d\epsilon_{\kappa'} {R_{\kappa',\lambda}(r) \langle R_{\kappa',\lambda}| r |R_{n_i,\ell_i} \rangle \over \epsilon_{\kappa} - \epsilon_{\kappa'} }
\nonumber \\
-~&i\pi R_{\kappa,\lambda}(r) \langle R_{\kappa,\lambda}| r |R_{n_i,\ell_i} \rangle.\end{aligned}$$ We note that it describes the radial part of the *intermediate* photoelectron wave packet created upon absorption of the XUV photon ${\Omega}$, before absorbing the IR laser photon $\omega$.
The essence of the approximate treatment that we have implemented to get an estimate of $T_{L,\lambda,\ell_i}(k; \epsilon_i +\Omega) $, is based on using the asymptotic forms of both the final state function $R_{k,L}(r)$ and of the perturbed wavefunction $\rho_{\kappa,\lambda}(r)$ for large values of their radial coordinate. This is [*a priori*]{} justified by the fact that we are interested in the [*phases*]{} of the amplitudes which are governed by the asymptotic behavior of these functions. As an additional verification, we will compare the predictions of the model to those derived from an exact treatment in Hydrogen.
The asymptotic limit of the radial continuum wavefunction of the final state with angular momentum $L$ is of the generic form [@Landau]: $$\begin{aligned}
\lim_{r\rightarrow \infty} R_{k,L}(r) \ = \ {N_{k} \over r} \sin[k r+\Phi_{k,L}(r)],
\label{RkL}\end{aligned}$$ where $N_{k} = \sqrt{2/(\pi k)}$ is the normalization constant in the energy scale and the phase has the general form: $$\begin{aligned}
\Phi_{k,L}(r) = Z \ln(2 k r)/k +\eta_L(k) - \pi L /2.
\label{Phase}\end{aligned}$$ We note that this phase includes the logarithmic divergence characteristic of the Coulomb potential of the ionic core with charge $Z$, in the asymptotic region. The Coulomb potential influences also the scattering phase-shift $\eta_L (k)$, which can be rewritten under the form: $\eta_L (k) = \sigma_L (k)+\delta_L (k)$ where $\sigma_L=\arg[\Gamma(L+1-iZ/k)]$ is the Coulomb phase-shift and where the correction $\delta_L (k)$ originates from the short range deviation of the ionic potential from a pure Coulomb potential, see for instance [@Kennedy; @KheifetsComment]. Obviously, in the case of an hydrogenic system, one has $\delta_L (k) =0$.
To derive the asymptotic form of the perturbed wavefunction $\rho_{\kappa,\lambda}(r)$, it is in principle enough to establish the limiting structure of the differential equation it verifies. From Eq. (\[eqdiffrho\]), one observes that, in the asymptotic limit $r \to \infty$, the second member vanishes, as a result of the exponential decay of the bound state wavefunction $R_{n_i,\ell_i}(r)$. One is left with a standard Schrödinger equation for positive energy $\epsilon_{\kappa}$ which is solved by imposing outgoing wave boundary conditions to the solutions [@Aymar]: $$\begin{aligned}
\lim_{r\rightarrow\infty} \rho_{\kappa,\lambda}(r) \propto
{N_{\kappa}\over r}\ \exp\left[ i(\kappa r+\Phi_{\kappa,\lambda}(r))
\right] .
\label{outgoing}\end{aligned}$$ It is also of interest to derive explicitly the limiting forms of the terms entering the expression of $\rho_{\kappa,\lambda}(r)$ in Eq. (\[rho\]). Regarding its real part, the sum over the discrete states $\nu$ can be neglected, as each term goes asymptotically to zero. Thus, for large $r$, it reduces to a principal part integral: $$\begin{aligned}
\label{Rerho1}
\Re[\rho_{\kappa,\lambda}(r)] \approx
{\mathcal P} \int_0^{+\infty} d\epsilon_{\kappa'} {R_{\kappa',\lambda}(r) \langle R_{\kappa',\lambda}| r |R_{n_i,\ell_i} \rangle \over \epsilon_{\kappa} - \epsilon_{\kappa'} }. \end{aligned}$$ This integral can be estimated by extending the integration range $\epsilon_{\kappa'} \to -\infty$ and replacing the radial continuum function $R_{\kappa',\lambda}(r)$ by its asymptotic limit according to the prescription in Eq. (\[RkL\]). Then, writing the sine function under its exponential form and performing contour integrations, with semi-circles around the pole at $ \epsilon_{\kappa}$, one gets: $$\begin{aligned}
\lim_{r\rightarrow\infty} \Re[\rho_{\kappa,\lambda}(r)] \approx - {\pi N_{\kappa}\over r}
\cos[\kappa r + \Phi_{\kappa,\lambda}(r)] {\langleR_{\kappa,\lambda} |r| R_{n_i,\ell_i} \rangle}.
\label{rerho}\end{aligned}$$ Regarding the imaginary part given by the last term in Eq. (\[rho\]), it is enough to substitute again the asymptotic form of $R_{\kappa,\lambda}(r)$, so that: $$\begin{aligned}
\lim_{r\rightarrow\infty} \Im[\rho_{\kappa,\lambda}(r)] \approx - {\pi N_{\kappa} \over r}
\sin[\kappa r + \Phi_{\kappa,\lambda}(r)] {\langleR_{\kappa,\lambda} |r| R_{n_i,\ell_i} \rangle}.
\label{imrho}\end{aligned}$$ Then by regrouping the real and imaginary parts, one gets the final expression: $$\begin{aligned}
\lim_{r\rightarrow\infty} [\rho_{\kappa,\lambda}(r)] \approx - {\pi N_{\kappa} \over r}
\exp\left[
i\kappa r + i\Phi_{\kappa,\lambda}(r)\right] {\langleR_{\kappa,\lambda} |r| R_{n_i,\ell_i} \rangle},
\label{asymprho}\end{aligned}$$ which corresponds to a complex outgoing wave [@Aymar], as expected from Eq. (\[outgoing\]), weighted by the dipole matrix element associated to the one-photon transition from the initial state. We note that adopting the so-called “pole-approximation”, which consists in neglecting the off-shell part ([*i.e*]{}. the real part given in Eq. (\[rerho\])), would lead to a loss of the phase information of the process since the perturbed wavefunction then would be a standing wave rather than an outgoing wave.
The corresponding asymptotic approximation for the second-order radial matrix element, Eq. (\[Tell\]), is obtained by substituting in Eq. (\[Trho\]) the asymptotic expressions, Eqs. (\[RkL\]) and (\[asymprho\]), for the radial wavefunctions of the final state and of the intermediate state, respectively. One has explicitly: $$\begin{aligned}
T_{L,\lambda,\ell}(k;\epsilon_{\kappa}) \approx {} &
- \pi \ {\langleR_{\kappa,\lambda} |r| R_{n_i,\ell_i} \rangle}N_{k} N_{\kappa} \nonumber \\
\times~&\int_0^{\infty} dr
\sin[kr+\Phi_{k,L}(r)] \ r \ e^{\left[ i(\kappa r+\Phi_{\kappa ,\lambda}(r)) \right] }.\
\label{Tas1}\end{aligned}$$ To introduce the next step in our approximate treatment, one rewrites the sine in its exponential form and develop the expressions of the phases $\Phi_{\kappa,\lambda}(r)$ as given in Eq. (\[Phase\]). One is left with two distinct contributions containing integrals either of the type $J_+$ or $J_-$ that are defined as follows: $$\begin{aligned}
J_\pm = {}& \pm\frac{1}{2i} \int_0^{\infty} dr ~ r^{1+ iZ(1/ \kappa \pm 1/ k )} \exp\left[i(\kappa \pm k)r\right]
\nonumber \\
= {}& \pm\frac{1}{2i} \left( \frac{i}{\kappa \pm k} \right) ^ {2+iZ(1/\kappa \pm1/k)} \Gamma[2+iZ(1/\kappa \pm1/k)],
\label{Jint}\end{aligned}$$ where we have used an integral representation of a Gamma function $\Gamma(z)$ with complex argument. In our case, the contribution of the $J_+$ integral is vanishingly small as compared to that of $J_-$. This is due to the IR photon energy being small compared to the final kinetic energy of the electron, $\omega = k^2 /2 - \kappa^2/2 \ll k^2 /2$, so that the difference $|\kappa - k| \approx \omega/ k$ is much smaller than the sum $\kappa + k \approx 2 k \pm \omega /k$. As a result, the fast oscillations of $\exp [i(\kappa + k)r]$ lead to a relative cancellation of the corresponding integral, as compared to the one containing the factor $\exp [i(\kappa - k)r]$. Neglecting the $J_+$ contribution, the asymptotic expression reduces to: $$\begin{aligned}
T_{L,\lambda,\ell_i}(k;\epsilon_{\kappa}) \approx {}&
\frac{\pi}{2}~
N_{k} N_{\kappa}~
{\langleR_{\kappa,\lambda} |r| R_{n_i,\ell_i} \rangle} \nonumber \\ \times~&
\frac{ 1 }{ |\kappa-k|^2 } ~ \exp\left[-\frac{\pi Z}{2} \left( \frac{1}{\kappa}-\frac{1}{k}\right)\right]
\nonumber \\
\times ~ &
i^{L-\lambda-1}\exp[i(\eta_{\lambda}(\kappa)-\eta_{L}(k))]
\nonumber \\
\times ~ &
\frac{(2\kappa)^{iZ/\kappa}}{(2k)^{iZ/k}}
\frac{\Gamma[2+iZ(1/\kappa-1/k)]}{(\kappa-k)^{iZ(1/\kappa-1/k)}},
\label{Tas1}\end{aligned}$$ which was used by us [@Klunder2011] to obtain estimates of the phases occurring in two-photon transitions entering RABBIT transition amplitudes. The first two lines in Eq. (\[Tas1\]) are real, they contain a one-photon matrix element from the bound state into the continuum, but also an exponential factor describing the strength of the continuum–continuum transition from $\kappa$ to $k$. The exponential factor decreases with the probe photon energy, $\omega = k^2/2-\kappa^2/2$, which indicates that large energy leaps in the continuum are strongly suppressed. At a given laser probe energy, however, the exponential factor increases with the final momentum, $k$, which indicates that it becomes easier for the photoelectron to interact with the probe field. The third line is a simple phase factor containing the scattering phases of the continuum states. Finally, the fourth line is a complex factor that depends on three quantities: final momentum, $k$; the laser probe frequency, $\omega$; and the charge of the ion, $Z$.
A more formal derivation of this result, based on a closed-form representation of the Coulomb Green’s function is given in the Appendix A. \[We have found a typo in Eq. (7) in ref. [@Klunder2011]: The ratio, $(2k)^{i/k}/(2k_a)^{i/k_a}$, should be inverted, as is evident from Eq. (\[Tas1\]) in the present work\].
Thus, in the asymptotic limit, the [*phase*]{} of the radial component takes the form: $$\begin{aligned}
\arg {[T_{L,\lambda,\ell_i}(k;\epsilon_\kappa)] }
\approx {}&
{\pi\over 2} (L-\lambda - 1)
\nonumber \\
+~& \eta_\lambda (\kappa) - \eta_L(k)
+ \phi_{cc}(k,\kappa),
\label{argTas1}\end{aligned}$$ where $$\begin{aligned}
\phi_{cc}(k,\kappa)~ =~ \arg \left[ \frac{(2\kappa)^{iZ/\kappa}}{(2k)^{iZ/k}}~
{\Gamma[2+iZ(1/\kappa-1/k)] \over ( \kappa-k)^{iZ(1/\kappa-1/k)}}\right],
\label{phicc}\end{aligned}$$ is the phase associated to a continuum–continuum radiative transition resulting from the absorption of $\omega$, in the presence of the Coulomb potential, $Z$. It is important to note that it is [*independent*]{} from the characteristics of the initial atomic state, as well as from the amplitude of the field. It is illustrative to study the continuum–continuum phase in the limit of a small photon energy, $\omega\approx k(k-\kappa)$, which yields a simplified expression: $$\begin{aligned}
\phi_{cc}^{(soft)}(k;\omega)~=~\arg\left[
\left( \frac{2k^2}{\omega} \right)^{iZ\omega/k^3} ~
\Gamma[2+iZ\omega/k^3]
\right],
\label{phiccsoft}\end{aligned}$$ where it becomes clear that it is the product: $Z\omega/k^3$, which determines the size of the continuum–continuum phase. This expression is expected to be valid in the so-called “soft-photon” limit, $k^2/2 \gg \omega $, where the exchange of energy $\omega$ and the corresponding momentum transfer $\Delta k = \omega/c$ do not significantly modify the electron state [@Maquet]. The substitution, $\omega\rightarrow-\omega$, yields the (Fourier) component corresponding to stimulated emission of light. We note that the phases corresponding to absorption and emission have opposite signs, but that they are otherwise identical in the soft-photon limit.
Replacing the formula obtained in Eq. (\[Tas1\]) for the radial component in the expression of the full transition amplitude $M ({\vec k},\epsilon_i+\Omega)$ given in Eq. (\[Ma\]), one gets its general form in the asymptotic limit: $$\begin{aligned}
M({\vec k};\epsilon_\kappa)
\approx {}&
-\frac{2\pi^2}{3}(8\pi)^{3/2}E_{\omega} E_{\Omega} N_k N_{\kappa}
\nonumber \\ \times~&
\frac{1}{|k-\kappa|^2}~\exp\left[-\frac{\pi Z}{2} \left( \frac{1}{\kappa}-\frac{1}{k}\right)\right]
\nonumber \\ \times~&
\frac{(2\kappa)^{iZ/\kappa}}{(2k)^{iZ/k}}
\frac{\Gamma[2+iZ(1/\kappa-1/k)]}{(\kappa-k)^{iZ(1/\kappa-1/k)}}
\nonumber \\ \times~&
\sum_{L=\ell_i,\ell_i\pm 2} Y_{L,m_i}({\hat k})
\sum_{\lambda=\ell_i\pm 1}\langle Y_{L,m_i}| Y_{1,0}|Y_{\lambda,m_i} \rangle
\nonumber \\ \times~&
\langle Y_{\lambda,m_i}| Y_{1,0}|Y_{\ell_i,m_i}\rangle
{\langleR_{\kappa,\lambda} |r| R_{n_i,\ell_i} \rangle} i^{-\lambda} e^{i\eta_\lambda (\kappa)}
\label{Mas1}
\end{aligned}$$ To address the question of its phase, one notices that besides a trivial contribution from the spherical harmonic in the final state, $Y_{L,m_i}(\hat k)$, it contains only phase-shifts that are governed by the angular momentum $\lambda$ of the intermediate state, [*i.e.*]{} a state that can be reached via [*single*]{}-photon ionization. More precisely, for a given transition channel characterized by the angular momenta of the intermediate and final state $\ell_i \rightarrow \lambda \rightarrow L$ , the phase of the matrix element reduces to: $$\begin{aligned}
\arg{ [M_{L,\lambda,\ell_i}({\vec k},\epsilon_\kappa)}] ={}&
\pi+\arg[Y_{L,m_i}({\hat k})] + \phi_\Omega + \phi_\omega
\nonumber \\ -~&
{\pi \lambda \over 2} + \eta_\lambda (\kappa) + \phi_{cc}(k,\kappa) ,
\label{phiMas1}\end{aligned}$$ where $\phi_\Omega$ and $\phi_\omega$ are the phases of the XUV field $\Omega$ and of the IR laser $\omega$, respectively. We stress that the final state scattering phase, $\eta_L(k)$, cancels out and that it enters neither in Eq. (\[Mas1\]) nor in Eq. (\[phiMas1\]).
Eq. (\[phiMas1\]) represents one of the major results of our theoretical analysis. It shows that, within the asymptotic approximation and besides trivial spherical harmonic contributions and the phases of the fields \[line 1 in Eq. (\[phiMas1\])\], the phase of a two-color ATI transition amplitude has two components: i) One is directly linked to the quantum-mechanical phase-shift of the one-photon XUV ionization amplitude, here $-\pi\lambda/2+\eta_\lambda (\kappa)$; ii) The other, denoted $\phi_{cc}(k,\kappa)$, is in some sense “universal”, it describes the phase brought by the absorption of the probe photon $\omega$, in the presence of the Coulomb potential with charge $Z$. Then, as shown below, when comparing laser-assisted ionization originating from distinct atomic states, the energy derivative of the phase-shifts in the first term contribute to a Wigner-like time-delay. On the other hand, the difference between the “universal” terms $\phi_{cc}$ gives rise to a measurement-induced delay, associated to continuum–continuum stimulated radiative transitions in the presence of the Coulomb potential of the ionic core.
In Fig. \[results1a\], we present the continuum–continuum phases, associated with absorption (red) and emission (blue) of a probe photon leading to the same final energy, calculated using the asymptotic approximation, Eq. (\[phicc\]).
![\[results1a\] Continuum–continuum phases for absorption (red, upper curves) and emission (blue, lower curves) calculated using the asymptotic approximation (dashed curves) \[Eq. \[phicc\]\] and the long-range amplitude-corrected asymptotic approximation (full curves) \[Eq. \[phicctilde\]\]. These approximate phases are compared with the exact calculations (black symbols) from the $1s$ state in hydrogen. The exact results for final angular momentum $L=0$ ($+$ symbol) and $L=2$ ($\times$ symbol) are computed by subtracting the one-photon scattering phase from that of the exact two-photon matrix elements. The data correspond to $Z=1$ and to a laser probe with $\omega=1.55$ eV. It is interpolated between the discrete harmonic orders. ](results1a.pdf){width="45.00000%"}
The continuum–continuum phases for probe photon absorption are positive while those for stimulated emission are negative, but approximately equal in absolute value. In the next subsection, we show that, still in a single-active electron picture, it is feasible to improve the accuracy of our approximate treatment with the help of semi-classical arguments.
Long-range amplitude effects
----------------------------
In order to go to the next level of our asymptotic approximation, we must include not only long-range phase variations of the continuum states, but also long-range variations of the amplitudes. Indeed, the normalization constants contained in the asymptotic forms of the radial functions $R_{k,L}(r)$ and $\rho_{\kappa,\lambda}( r)$ can be modified to account for the long-range influence of the Coulomb potential. For instance, the modified final state normalization constant is: $$\begin{aligned}
N_k(r)=\sqrt{\frac{2}{\pi p(r)}},
\label{NWKB}\end{aligned}$$ where $$\begin{aligned}
p(r)=\sqrt{2(\epsilon-V(r))}\approx k - V(r)/k,
\label{pWKB}\end{aligned}$$ is the local momentum from Wentzel–Kramers–Brillouin (WKB) theory [@HaraldFriedrich]. The same remark applies to $N_{\kappa}(r)$ for the perturbed radial function. Long-range amplitude effects can be approximated by expanding the quantities $N_k(r)$ and $N_{\kappa}(r)$, $$\begin{aligned}
N_k(r)N_{\kappa}(r) \approx
\sqrt{\frac{4}{\pi^2 k \kappa}}
\left[
1-\frac{1}{2}
\left(\frac{1}{\kappa^2} + \frac{1}{k^2} \right) \frac{Z}{r}
\right],
\label{normalizationOfr}\end{aligned}$$ to first-order in the Coulomb potential. The second term within the brackets in Eq. (\[normalizationOfr\]) contains the first-order amplitude correction to the matrix element. Evaluation of the long-range amplitude contribution leads to a correction to the continuum–continuum phase: $$\begin{aligned}
\alpha_{cc}(k,\kappa) ={}&
\arg\left[ 1 +
\frac{iZ}{2}
\left( \frac{1}{\kappa^2} + \frac{1}{k^2} \right )
\frac { \kappa - k }{1+iZ(1/\kappa-1/k)}\right].
\label{ampCorrFactor}\end{aligned}$$ The final continuum–continuum transition phase is: $$\begin{aligned}
\tilde {\phi}_{cc}(k,\kappa) = \alpha_{cc}(k,\kappa) + \phi_{cc}(k,\kappa)
\label{phicctilde}\end{aligned}$$ where $\phi_{cc}(k,\kappa)$ is given in Eq. (\[phicc\]). In Fig. \[results1a\], we show that including such long-range effects improves the accuracy of the approximation greatly, leading to accurate quantitative result already at relatively low energies in the continuum. The accuracy in the lower energy range can be further improved by using a regularization method designed to remove the effect of the singularity in Eq. (\[normalizationOfr\]) as $r$, $k$ and $\kappa$ go to zero.
Going beyond the approximations given here, namely performing [*exact ab initio*]{} computations of the matrix elements $M({\vec k}, \epsilon_i+\Omega)$ in polyelectronic systems, is out of reach of present computational capabilities. It is only in the special case of hydrogenic systems, that such 2-photon amplitudes can be computed with arbitrary precision. Thus, with objective to delineate the range of validity of our approximation, we turn now to a brief presentation of the “exact” calculations in hydrogen.
Exact calculations of 2-photon ATI matrix elements in hydrogenic systems
------------------------------------------------------------------------
The principle of the calculation is outlined here for $s$-states. Numerical data for other states will be given below. We first express the transition amplitude given in Eq. (\[Ma\]) for the case $ \ell_i = 0, m_i = 0$ which implies $\lambda = 1$ so that the angular momentum of the photoelectron is either $L= 0,2$. Accordingly, two distinct amplitudes contribute to ATI transitions like the one depicted in Fig. \[fig1\] (c): $$\begin{aligned}
\label{Mas}
\left. M({\vec k},\epsilon_\kappa)\right|_{\ell_i=0} ={}&
\:{1 \over 3i} (8 \pi)^{3/2} E_\omega E_\Omega
\nonumber \\ \times~&
\left[ e^{ i \sigma_{0}(k)} Y_{0,0}({\hat k}) T_{0,1,0}(k; \epsilon_\kappa) \right.
\nonumber \\ -~&
\left. {2 \over \sqrt{5} }e^{ i \sigma_{2}(k)} Y_{2,0}({\hat k}) T_{2,1,0} (k; \epsilon_\kappa)\right] ,\end{aligned}$$ where the radial components for $s-$states are of the form: $$\begin{aligned}
\left. T_{L,1,0} (k;\epsilon_\kappa)\right |_{\ell_i=0}={}&
\langle R_{k,L}| r |G_1 (\epsilon_\kappa)| r |R_{n_i,0} \rangle ,
\label{TGreen}
\end{aligned}$$ with $L = 0,2$; $n_i$ labels the initial atomic $s-$state and $G_{\lambda = 1} (\epsilon_\kappa)$ is the radial component of the Coulomb Green’s function for angular momentum $\lambda =1$. The general form of the Green’s function with energy argument $\epsilon_\kappa$ is: $$\begin{aligned}
\label{Green}
G_\lambda (r',r;\epsilon_\kappa) =
\lim_{ \varepsilon \to 0^+} \int \!\!\!\!\!\!\!\! \sum_{\nu}
{R_{\nu,\lambda}(r') R_{\nu,\lambda} (r) \over \epsilon_\kappa -\epsilon_\nu +i \varepsilon} . \end{aligned}$$ As already mentioned, the infinite sum over the index $\nu$ runs over the whole (discrete + continuous) spectrum of the hydrogenic system. Closed form expressions for $G_\lambda$ are known, see for instance [@Maquet98]. Here, we have used the expression given as an expansion over a discrete Sturmian basis: $$\begin{aligned}
\label{GreenSturm}
G_\lambda (r',r;\epsilon_\kappa)= \sum_{\nu=\lambda +1} {S_{\nu,\lambda,x}(r') S_{\nu,\lambda,x} (r) \over 1 - \nu x} , \end{aligned}$$ where $x = \sqrt{-2\epsilon_\kappa}$ and the so-called Sturmian functions $S_{\nu,\lambda,x} (r)$ have a structure similar to the bound-state hydrogenic radial functions [@Rotenberg; @Maquet1977]: $$\begin{aligned}
\label{Sturm}
S_{\nu,\lambda,x}(r) ={}& 2x\sqrt{(\nu -\lambda -1)! \over(\nu +\lambda)!}
\nonumber \\ \times ~ &
e^{-xr} (2xr)^\lambda L_{\nu - \lambda-1}^{2\lambda +1} (2xr),\end{aligned}$$ where $L_{\nu - \lambda-1}^{2\lambda +1} (z)$ are associated Laguerre polynomials. In the amplitudes for ATI transitions, $\epsilon_\kappa = \epsilon_i +\Omega >0$, and the quantity, $x =i \sqrt{2|\epsilon_\kappa|}$, is pure imaginary. It is then convenient to use Padé-like resummation techniques to compute the infinite sum over the index $\nu$, see, for instance ref. [@Klarsfeld79b].
In Fig. \[results1a\], we present the exact continuum–continuum phases from the $1s$ state in hydrogen. These phases are defined as the total phase of the exact matrix element, $M_{L,1,0}(\vec k;\epsilon_\kappa)$, [*minus*]{} the one-photon phase \[see line 2 of Eq. (\[phiMas1\])\]. Our approximate calculation including long-range amplitude effects, Eq. (\[phicctilde\]), is in excellent agreement with the exact calculations except at low energy.
Phase of the classical dipole {#sec:classical}
-----------------------------
Finally, we present a simplified derivation of the continuum–continuum phase, $\phi_{cc}(k,\kappa)$, using a classical approach. The dipole associated with the absorption of radiation at frequency $\omega$ by a free electron in the presence of a Coulomb potential, can be calculated using Larmor’s formula, $$\begin{aligned}
d_C(k;\omega)=\int_0^{\infty} dt ~ r_k(t) \exp[- i\omega t],
\label{cl_dipole}\end{aligned}$$ where it is assumed that the electron follows a [*field-free*]{} trajectory, $r_k(t)$, that starts close to the ion, $r_k(0) \approx 0$, and then moves out away from the ion with an asymptotic velocity, $k$. The integral can be cast from time to space using the $r-$dependence of the velocity: $$\begin{aligned}
v_k(r)~ =~ \sqrt{k^2 - 2 V(r)},
\label{cl_time2space}\end{aligned}$$ where $k^2/2$ is the final kinetic energy of the electron at large distance from the ion. Using the differential $dt = dr/v(r)$, the time can be written as $t(k;r) = \int^r dr'/v_k (r')+C$, where $C$ is an integration constant. In the case of the Coulomb potential, $V(r)=-Z/r$, the integral becomes $$\begin{aligned}
t(k;r) ={}&
\int^{r}dr' ~ \frac{1}{ \sqrt{k^2+2 Z/r'} } + C
\nonumber \\ \approx{}&
\frac{r}{k}-\frac{Z}{k^3}\ln(r)+C,
\label{CoulombTimeApprox}\end{aligned}$$ in the asymptotic limit, [*i.e.*]{} when $k^2 /2 \gg Z/r$. This provides an approximate time–position relation valid at large distances from the origin. In the special case where the electron starts from the origin $[t,r]=[0,0]$, the exact integration in Eq. (\[CoulombTimeApprox\]) leads to $C=-Z\ln[2k^2/Z]/k^3$. Keeping for the moment this value of the constant and replacing the asymptotic form of the time in the expression of the dipole Eq. (\[cl\_dipole\]), one gets: $$\begin{aligned}
d_C(k;\omega) \approx{}&
\frac{1}{k}\int_0^{\infty}dr ~ r\exp\left[ - i\frac{\omega}{k}\left(r-\frac{Z}{k^2}\log[2k^2r/Z]
\right) \right]
\nonumber \\ {}=&
\frac{1}{k}\left(\frac{2k^2}{Z}\right)^{ iZ\omega/k^3} \int_0^{\infty}dr ~
r^{1 + iZ\omega/k^3}\exp\left[- i\frac{\omega}{k} r\right]
\nonumber \\ ={}&
-\frac{k}{\omega^2}\exp\left[-\frac{3 \pi Z \omega}{ 2k^3}\right]
\nonumber \\ \times~&
\left( \frac{2k^3}{Z\omega} \right)^{iZ\omega/k^3}
\Gamma(2+iZ\omega/k^3),
\label{C_dipole}\end{aligned}$$ where the next-to-last line is real, with an exponential factor that decreases with $\omega$, but increases with $k$, in excellent agreement with the quantum mechanical result, Eq. (\[Mas1\]). Furthermore, the last line contains the complex gamma function times an algebraic factor, also in close connection to the quantum counterpart. Clearly, the dipole corresponding to absorption is a complex quantity with phase: $$\begin{aligned}
\phi_C(k;\omega) ={}&
\arg\left[d_C(k;\omega)\right]
\nonumber \\ \approx{}&
\arg\left[-\left( \frac{2k^3}{Z\omega} \right)^{iZ\omega/k^3}~\Gamma(2+iZ\omega/k^3) \right],
\label{C_phase}\end{aligned}$$ which is closely related, but [*not*]{} identical to its soft-photon quantum counter part $\phi^{(soft)}_{cc}$, given in Eq. (\[phiccsoft\]).
In the quantum mechanical case, the electron starts in a bound state with some spatial extent and not exactly from $r=0$. In order to account for this uncertainty on the initial position, we may choose a different value of the integration constant $C$, introduced in Eq. (\[CoulombTimeApprox\]), in order to come closer to the quantum mechanical dipole. This matching-procedure is reminiscent of the method used in ref. [@IvanovM2011], to determine the “best” initial radial position for the electron within the eikonal Volkov approximation. Within the lowest-order approximation of Eq. (\[CoulombTimeApprox\]), we find a simple relation between the initial position and the integration constant: $$\begin{aligned}
r_0 \approx \exp\left[\frac{Ck^3}{Z}\right],
\label{r2C}\end{aligned}$$ which is valid at high-energy. It is convenient to set $C=-Z\ln[2k]/k^3$, a choice corresponding to an initial radial position $r_0\approx 1/(2k)$, as was identified in ref. [@IvanovM2011]. Clearly, when the first-order amplitude correction is included, Eq. (\[phicctilde\]), the initial position is adjusted accordingly. The continuum–continuum phase being only one part of the total quantum mechanical phase, we have also to include the scattering phase if we want to deduce the “true” initial position. Interestingly, in our approach this exact inital position is not critical. In fact, our results are stable with respect to rather substantial modifications of the wavefunctions close to the core. It is the behavior of the wavefunctions far away from the core that must be described accurately, using the asymptotic approximation. We now turn our attention to the applications of the complex ATI matrix elements, to the determination of attosecond delays in photoionization.
Attosecond Time-delays {#sec:ATIapplications}
======================
The complex amplitude $M({\vec k},\epsilon_i+\Omega)$ (denoted $M^{(a)}$ for conciseness in the following) for the joint absorption of an XUV photon $\Omega$ and of an IR laser photon $\omega$, is of course [*not*]{} a direct observable in any experiment. Only the square modulus of a complex transition amplitude can be measured. Thus, if $M^{(a)}$ is the only amplitude leading to a given final state, there is no way to determine its absolute phase. However, in the cases of interest here, with attosecond XUV pulses in the presence of a probe IR laser field, several other channels are open which can lead to the same final state, thus making it feasible to observe phase-dependent interference patterns. This property is exploited in the following schemes.
Attosecond delay measurements using pulse trains
------------------------------------------------
In the RABBIT scheme there are two dominant complex amplitudes (quantum paths), $M^{(a)}$ and $M^{(e)}$, associated with the absorption of harmonic $H_{2q-1}$ or $H_{2q+1}$ plus absorption or emission of a laser photon with phase, $\pm\omega\tau\equiv\pm\phi_\omega$, leading to the same final sideband, $S_{2q}$. The photoelectron will transit via different intermediate states: $\kappa_<$ and $\kappa_>$, corresponding to different intermediate energies: $\epsilon_<=\epsilon_i+2q\omega-\omega=\kappa_<^2/2$ and $\epsilon_>=\epsilon_i+2q\omega+\omega=\kappa_>^2/2$. Clearly, the energy of the corresponding intermediate states are one photon below and above the sideband, as is illustrated in Fig. \[fig1\] (a). This implies that the measured intensity of the sideband, $S_{2q}$, will depend on the [*phase difference*]{} between the two quantum paths: $$\begin{aligned}
P_{2q} \propto | M^{(a)}+M^{(e)} |^2 = |M^{(a)}|^2 + |M^{(e)}|^2 \nonumber \\
+ ~ 2 |M^{(a)} | | M^{(e)}| \ \cos\left[\arg\left(M^{(a)*} M^{(e)}\right)\right]
\label{SBprob}\end{aligned}$$ which corresponds to a standard interference phenomenon, as summarized in Eqs. (\[S2q\]) or, equivalently in (\[S2qt\]) displayed in the introduction (Sec. \[sec:intro\]). If we now apply Eq. (\[phiMas1\]), and assume that the total contribution to the complex amplitudes can be approximated by a single intermediate angular momentum component, $\lambda$, the phase difference is $$\begin{aligned}
\arg\left(M^{(a)*}_{L,\lambda,\ell_i} M^{(e)}_{L,\lambda,\ell_i}\right) \ \approx \
-2\omega\tau ~+~
\overbrace{
\phi_{2q+1}-\phi_{2q-1}
}^{\Delta\phi_{2q}}
\nonumber \\
~+~
\underbrace{
\eta_{\lambda}(\kappa_>) -\eta_{\lambda}(\kappa_< )
~+~ \phi_{cc}(k,\kappa_>) -\phi_{cc}(k,\kappa_<)
}_{\Delta\theta_{2q}}.
\label{phaseDiff}\end{aligned}$$ Here the first line contains the phases of the fields, while the second line is $\Delta\theta_{2q}$, in terms of scattering and continuum–continuum phases. The corresponding measurement delay, $\tau_\theta$ as defined in Eq. (\[S2qt\]), is the sum of finite-difference approximations to a Wigner-like time-delay: $$\begin{aligned}
{\tau}_\lambda(k) \approx \frac{ \eta_\lambda (\kappa_>)- \eta_\lambda (\kappa_< )}{2\omega}
\label{Wigner}\end{aligned}$$ and to a continuum-continuum delay: $$\begin{aligned}
\tau_{cc}(k;\omega) \approx \frac{\phi_{cc}(k,\kappa_>) - \phi_{cc}(k,\kappa_<)}{2\omega},
\label{Taucc}\end{aligned}$$ so that $\tau_\theta \approx { \tau}_{\lambda} + \tau_{cc}$.
In a more general case, the interference will result from complex amplitudes with two terms for absorption and for emission, $M^{(a/e)}=M^{(a/e,+)}+M^{(a/e,-)}$, corresponding to different intermediate angular momenta, $\lambda=\ell_i \pm 1$. The interesting point is that the $\tau_{cc}$ will not be affected, since the phases $\phi_{cc}$ are independent of $\lambda$ as well as of the final angular momenta $L=\ell_i, \ell_i \pm 2$, see Eq. (\[phicc\]). The remaining part can be interpreted as an [*effective*]{} Wigner-like delay, $\tilde {\tau}_{\lambda = \ell_i \pm 1}$, which has to be computed taking into account the relative weights of the [*four*]{} components entering the expression of the transition probability amplitude.
Attosecond delay measurements using single pulses
--------------------------------------------------
In this section we apply the perturbative treatement to laser-assisted photoionization by a single-attosecond pulse (SAP) and a probing laser field. The ionizing attosecond field is $$\begin{aligned}
\tilde E_{SAP}(t)=\int d\Omega ~ E_\Omega \exp[-i\Omega t]/2\pi,
\label{SAP}\end{aligned}$$ with $E_\Omega=|E_\Omega|~\exp[i\phi_\Omega]$ being complex Fourier coefficients. While our approach is very general, it can be illustrated by considering a Gaussian frequency distribution centered on the frequency $\Omega_0$, as depicted in the left panel of Fig. \[fig4\]. The probing laser field is assumed to be monochromatic and real, $\tilde E(t)=2|E_\omega|\cos[\omega (t - \tau)]$, so that it can account for both absorption and emission processes. Our perturbative approach cannot be used to fully account for the large momentum shifts that are typical of streaking spectrograms, but it does allow us to study the *onset* of streaking. As we shall see below, streaking-like behavior is clear already in the perturbative regime. In the right panel of Fig. \[fig4\], we sketch the interfering processes to a certain energy $\epsilon$: (d), (a) and (e) from lowest-order perturbation theory. The dominant contribution (d) corresponds to absorption of a single XUV photon, while the upshifted (a) and downshifted (e) photoelectron spectra corresponds to absorption and emission of an additional laser photon $\omega$, respectively.
![\[fig4\] Sketch of the quantum paths describing the onset of “streaking” for a photoelectron ionized by a single attosecond pulse and probed by a monochromatic laser field. The first-order photoelectron wave packet is centered at $\epsilon_0=\epsilon_i+\Omega_0$. Within the bandwidth of the attosecond pulse, any energy, $\epsilon$, can be reached by path (d), where a single XUV photon with frequency $\Omega$ is absorbed. Alternatively, the same energy can reached by path (a) by absorbing a less energetic photon, $\Omega_{<}$, and a laser photon, $\omega$; or by path (e) by absorbing a more energetic photon, $\Omega_{>}$, and then emitting a laser photon, $-\omega$. At the high-energy end of the photoelectron distribution (indicated by $\epsilon$), the dominant contributions to the photoelectron wave packet are (d) and (a), due to $\Omega_{>}$ being far in the upper energy range of the XUV bandwidth. ](fig4.pdf){width="35.00000%"}
The matrix element for [*one-photon*]{} ionization with a photoelectron emitted along the polarization axis is $$\begin{aligned}
M^{(d)}(k\hat z;\epsilon_i+\Omega) ={}&
\frac{(8\pi)^{3/2}}{i}
E_\Omega
\sum_{\lambda}
\left(-i\right)^{\lambda}
\sqrt{\frac{2\lambda+1}{3}}
e^{i\eta_{\lambda}(k)}
\nonumber \\ \times~&
{\langleY_{\lambda,0} |Y_{1,0}| Y_{\ell_i,0} \rangle}
{\langleR_{k,\lambda} |r| R_{n_i,\ell_i} \rangle}~
\delta_{m_i,0},
\label{Lforward}\end{aligned}$$ where the XUV frequency satisfies $\epsilon_i+\Omega = \epsilon=k^2/2$ for process (d). Note that only initial states with zero magnetic quantum number, $m_i=0$, will contribute to photoelectron emission along $\hat z$, as indicated by the Kronecker delta at the end of Eq. (\[Lforward\]) that originates from the explicit properties of the spherical harmonics along $\hat z$: $Y_{\lambda,m_i}(\hat z)=\sqrt{(2\lambda+1)/3}~\delta_{m_i,0}$.
The [*two-photon*]{} matrix element for photoelectrons along the polarization direction is $$\begin{aligned}
M(k\hat z;\epsilon_i+\Omega_{<})={}&
\frac{(8\pi)^{3/2}}{i}
E_{\Omega_<}E_\omega
\sum_{L}
\left(-i\right)^{L}
\sqrt{\frac{2L+1}{3}}
e^{i\eta_L(k)}
\nonumber \\ \times ~ &
\sum_{\lambda}
{\langleY_{L,0} |Y_{1,0}| Y_{\lambda,0} \rangle}
{\langleY_{\lambda,0} |Y_{1,0}| Y_{\ell_i,0} \rangle}
\nonumber \\ \times ~ &
T_{L,\lambda,\ell_i}(k;\epsilon_i+\Omega_{<}) ~
\delta_{m_i,0}
,
\label{Mforward}\end{aligned}$$ where $\epsilon_i+\Omega_{<}+\omega=\epsilon$ for process (a). A similar expression, $M(k\hat z,\epsilon_i+\Omega_{>})$, can be written for process (e) where $\epsilon_i+\Omega_{>}-\omega=\epsilon$, and where the complex conjugate of the laser field, $E_\omega^*$, is used for emission of a laser photon. Again, only initial states with zero magnetic quantum number contribute.
The PROOF method [@Chini2010] relies on the interference between these three quantum paths, denoted $M^{(d)}$, $M^{(a)}$ and $M^{(e)}$, as a function of the XUV-laser delay, $\tau$. The probability for emission of an electron with final momentum $k$ along the polarization axis $\hat z$ is $$\begin{aligned}
P(k \hat z) \propto |M^{(d)}+M^{(a)}+M^{(e)}|^2,
\label{streakingProb}\end{aligned}$$ where only cross-terms lead to delay dependent modulations. Provided that the laser field is [*relatively*]{} weak, we expect the interference cross-terms (d)-(a) and (d)-(e), will dominate over the cross-term (a)-(e), because the former involve the exchange of only one laser photon while the latter contributes to higher order terms in a perturbative treatment, involving the exchange of two laser photons $\omega$.
In order to illustrate the origin of the attosecond delay within this framework, we consider a final energy that is higher than the central energy of the photoelectron wave packet, $\epsilon=\epsilon_i+\Omega > \epsilon_i+\Omega_0$. In this region, the $\omega$ modulation is dominated by the (d)-(a) interference, because path (a) goes through a central part of the XUV distribution, while path (e) passes through a higher frequency range, in the upper part of the XUV bandwidth \[Fig. \[fig4\]\]. Increasing the bandwidth of the attosecond pulse or decreasing the probe photon frequency makes this distinction less pronounced. We restrict our analysis to the case of an initial $s$-state, $\ell_i=0$, and study the interference of the cross term (d)-(a). The relevant phase reads $$\begin{aligned}
\arg[M^{(a)*} M^{(d)}] \approx {} &
-\omega\tau +
\phi_\Omega-\phi_{\Omega_<}
\nonumber \\ +~&
\eta_{\lambda}(k)-\eta_{\lambda}(\kappa_{<})
-\phi_{cc}(k,\kappa_<)
-\frac{\pi}{2},
\label{LMcross term}\end{aligned}$$ where we have used the asymptotic approximation for the phase of the two-photon matrix element as in Eq. (\[phiMas1\]), thereby, introducing the continuum–continuum phase into the framework of laser-assisted photoioization by single attosecond pulses. Also, we note that the phase-shifts present in the two-photon matrix element are those of $p-$waves ($\lambda=1$). In terms of the temporal delays, the $\omega$ modulation is displaced by $$\begin{aligned}
\tau \approx {} &
\frac{\phi_\Omega-\phi_{\Omega_<}}{\omega}
+\frac{\eta_{\lambda}(k)-\eta_{\lambda}(\kappa_<)}{\omega}
-\frac{\phi_{cc}(k,\kappa_<)}{\omega} - \frac{\pi}{2\omega}
\nonumber \\ \equiv{}&
\tau_\Omega ~+~ \tau_\lambda(k) ~+~ \tau_{cc}(k;\omega) ~-~ \frac{\pi}{2\omega},
\label{streakingdisplaced}\end{aligned}$$ where we have used the definition of the continuum–continuum delays $\tau_{cc}$, Eq. (\[Taucc\]), and the following relation for the continuum–continuum phases $\phi_{cc}$: $$\begin{aligned}
\phi_{cc}(k,\kappa_<)\approx -\phi_{cc}(k,\kappa_>)
\label{ccrelation},\end{aligned}$$ which is exact in the soft-photon limit. Similarly to Eq. (\[streakingdisplaced\]), we can compute the modulation at the low energy end of the electron distribution using the cross term (d)-(e), and we find the same result but shifted by a half laser period, [*i.e.*]{} out of phase by $\pi$. This $\pi-$shift of the modulation is important to explain the onset of streaking, because it ensures that the high-energy probability of the electron spectra is maximized when the low-energy part is minimized. In the central region, $\epsilon\approx\epsilon_i+\Omega_0$, the (d)-(a) and (d)-(e) contributions will be comparable, leading to a relative cancellation of the $\omega$ modulation, leaving only the $2\omega$ modulation. In total, the electron momentum distribution is slightly shifted up or down depending on the sub-cycle delay between the laser field and the attosecond pulse: It is [*streaked*]{} as expected from the classical (or strong-field) picture. Using Eq. (\[streakingdisplaced\]), we find that this streaking modulation is displaced by a Wigner-like delay, $\tau_{\lambda}(k)$, and by the continuum–continuum delay, $\tau_{cc}(k;\omega)$. We can interpret this delay as the time it takes for the electron to be photoionized [*plus*]{} the time it takes for the measurement process to occur, [*i.e.*]{} for a lower energy electron to absorb one probe photon so that it may interfere with the direct path. We stress that this analysis was made assuming a monochromatic probe field, while it is common in streaking experiments to use few-cycle IR laser pulse, which leads to a convolution (a blurring effect) of the momentum modulation and of the attosecond time-delays.
Comparison between the two measurements
---------------------------------------
We now briefly discuss the small differences of $\tau_{cc}$ occuring between the RABBIT method and the streaking method. Eq. (\[ccrelation\]) implies that $$\begin{aligned}
\tau_{cc}(k;\omega) \equiv {} &
\frac{\phi_{cc}(k,\kappa_>)~-~\phi_{cc}(k,\kappa_<)}{2\omega}
\nonumber \\ \approx {} &
~\frac{\phi_{cc}(k,\kappa_>)}{\omega},
\label{ccphases}\end{aligned}$$ where the first line corresponds to the RABBIT method and the second line corresponds to the streaking method. These different “flavours” of $\tau_{cc}$ merge completely in the soft-photon limit, $\epsilon\gg \omega$, but differ slightly at low kinetic energies, where the phases are not exact opposites. In both methods, $\tau_{cc}$ varies with $k$ inside the bandwidth of the photoelectron wavepacket.
Furthermore, it is interesting to note that the group delays of the attosecond pulses and Wigner delays of the photoelectrons appear as discrete derivatives over one and two $\omega$ photons in the streaking and RABITT methods respectively. Here, we have assumed that all delays are slowly varying so that the discrete derivatives are equivalent. We also note that the streaking method relies on the cross-terms: (d)-(a) and (d)-(e); while the RABBIT method relies on the (a)-(e) contribution. The weaker signal in the RABBIT method is not a problem, since it is recorded between the harmonics, on “zero background”. Consequenty, the modulation frequency of the streaking/PROOF signal is $\omega$ due to the single laser photon involved in each appropriate cross term, while the corresponding RABBIT frequency is $2\omega$ due to the two laser photons involved in the latter cross term. Another important difference between the methods is related to symmetry (parity): The $\omega$ signal is somewhat restricted to the $\hat z$ direction and it is not observed if photoelectrons are collected in all directions; while the $2\omega$ signal is more general and it prevails also when electrons in all directions are collected. Interestingly, we have shown that all three attosecond characterization methods (RABITT, PROOF and streaking) provide equivalent temporal information about the XUV ionization process, even though they are built from different sets of cross-terms associated to different interfering quantum paths.
Results {#sec:results}
=======
Calculations of attosecond time-delays
--------------------------------------
In Fig. \[results5\], we plot atomic delays relevant for laser-assisted photoionization.
![\[results5\] Atomic delays of laser-assisted photoionization in hydrogen for three different angular momenta of the wave-packet wavefunction in the intermediate state ($s$, $p$ and $d$ : $\lambda=0$, 1 and 2). The delays are calculated using the regularized approximation. The data correspond to $Z=1$ and a laser probe with $\omega=1.55$ eV. ](results5.pdf){width="45.00000%"}
The Wigner-like delays, $\tau_{\lambda}$, are calculated from the finite-difference derivative of the scattering phase of hydrogen for angular momentum, $\lambda=[0,1,2]$, corresponding to $s$, $p$ and $d$ continuum wave packets. The universal continuum–continuum delay, $\tau_{cc}$, plus these Wigner-like delays yields the total atomic delay, $\tau_\theta$. Notice that the sign of $\tau_\lambda$ and $\tau_{cc}$ are opposite, so that $\tau_\theta$ is smaller than either of the contributions individually. The increase of $\tau_\lambda$ with the angular momentum can be understood as due to the repulsive, short-range, centrifugal potential. The total delay is negative in this example, which implies that the electron appears as being [*advanced*]{} compared to the probe field.
In Fig. \[results1b\], we evaluate the accuracy of $\tau_{cc}$ as derived by various degrees of the asymptotic approximation by comparison with exact calculations in hydrogen from the $1s$ state along two different angular momentum sequences: $s \rightarrow p \rightarrow s$ ($+$) and $s \rightarrow p \rightarrow d$ ($\times$).
![\[results1b\] Continuum–continuum delays calculated using the asymptotic approximation (dashed curve), Eq. (\[phicc\]), the long-range amplitude-corrected asymptotic approximation (thick curve), Eq. (\[phicctilde\]), and the regularized asymptotic approximation (dash-dot curve). The approximate delays are compared with exact delays computed from the $1s$ state in hydrogen by subtracting the intermediate Wigner delay. The data correspond to $Z=1$ and to a laser probe with $\omega=1.55$ eV. ](results1b.pdf){width="45.00000%"}
Indeed, we find that the exact $\tau_{cc}$ are almost completely independent of the final state angular momentum. The asymptotic approximation (dashed curve) predicts the correct qualitative behavior of the delay, but it slightly overestimates its magnitude. By taking into account the long-range amplitude effects, the agreement is excellent at high kinetic energies. The disagreement at low energy can be removed by avoiding the radial singularity in Eq. (\[normalizationOfr\]), namely by an [*ad hoc*]{} substitution of $r \rightarrow r+i(1-|k-\kappa|/2)$. In this way, we obtain a “regularized” continuum–continuum delay (dot–dashed curve), which is excellent at [*all*]{} energies in the range.
In Fig. \[results2\], we address the question of the universality of $\tau_{cc}$ by computing the exact two-photon phases from three different intial states in hydrogen: $1s$, $2s$ and $2p$.
![\[results2\] Continuum–continuum delays from exact calculations in hydrogen from different intial states: $1s$ (black symbols), $2s$ (red symbols) and $2p$ (blue symbols). This data demonstrates the [*universality*]{} of the continuum–continuum delay and that it is valid not only for initial states of $s-$character. The exact data is in excellent agreement with the regularized approximation. The data correspond to $Z=1$ and a laser probe with $\omega=1.55$ eV. ](results2.pdf){width="45.00000%"}
We find that all seven different angular-momentum sequences, $\ell_i\rightarrow \lambda\rightarrow L$, line up on the same universal curve, in excellent agreement with the regularized continuum–continuum delay. In this way, we have verified that not only $s-$type initial states have similar $\tau_{cc}$, but also initial $p-$states with non-zero angular momentum.
In Fig. \[results3a\], we present a contour plot that provides a rough overview of the magnitude of $\tau_{cc}$ as a function of the kinetic energy of the photoelectron and of the wavelength of the probe field.
![\[results3a\] Contour plot of continuum–continuum delays at different kinetic energy and laser probe wavelength. The delays are calculated using the regularized asymptotic approximation. ](results3a.pdf){width="45.00000%"}
In general, $\tau_{cc}$ decreases with the kinetic energy and it increases with the wavelength of the probe field. We note that a softer probe photon leads to an increased delay. A few selected $\tau_{cc}$ curves are displayed in Fig. \[results3b\] for a more quantitative comparison at some experimentally relevant wavelengths. We conclude that $\tau_{cc}$ is not extremely sensitive to the probe wavelength, except at low kinetic energies, where it takes larger negative values when the probe wavelength increases. As already mentioned, the delays converge to zero as the kinetic energy is increased, however, the convergence is rather slow and there is still $\sim -10$ as of delay remaining at a relatively high kinetic energy of $\sim100$ eV.
![\[results3b\] Continuum–continuum delays for different IR laser photon energies: 1.55 eV ($\times$ symbol), 0.62 eV ($+$ symbol) and 0.41 eV ($*$ symbol). The delays are calculated using the regularized asymptotic approximation. ](results3b.pdf){width="45.00000%"}
Finally, in Fig. \[results4\], we examine the validity of the soft-photon approximation by comparing $\tau_{cc}$ and $\tau_{cc}^{(soft)}$, calculated using Eq. (\[phicc\]) and Eq. (\[phiccsoft\]), respectively.
![\[results4\] Continuum–continuum delays calculated using the asymptotic approximation (dashed curve) using Eq. (\[phicc\]). They are compared to the corresponding soft-photon limit (curve) using Eq. (\[phiccsoft\]). The data correspond to $Z=1$ and to a laser probe with $\omega=1.55$ eV. ](results4.pdf){width="45.00000%"}
The soft-photon limit over-estimates the magnitude of the delay at low energies, but it converges rapidly towards the quantum mechanical result as the kinetic energy is increased.
Conclusions
-----------
The main result of this work is the determination of the [*phase*]{} associated with two-photon transition matrix elements for laser-assisted XUV photoionization, namely Eq. (\[phiMas1\]). These phases have broad applications in attosecond science as they enter naturally in most characterization methods, as well as in quantum control schemes and delay experiments of more general character. We have demonstrated that the phase is composed of two distinct atomic contributions: i) The one-photon scattering phase of the intermediate state and ii) a quantity, given in Eq. (\[phicc\]), that we call the continuum–continuum phase. The latter is [*universal*]{} and it describes the added phase induced by the transition from the intermediate continuum state to the final continuum state. It is independent of the short-range behavior of the atomic potential and it depends only on three quantities: the final momentum, the laser-probe frequency and the charge of the remaining ionic core. As expected from the strong-field approach, the transition phase is reduced to that of the one-photon (intermediate) scattering phase when the charge of the remaining ion is neglected, [*i.e.*]{} for a short-range potential. Other interesting findings are related to the classical–quantum correspondence: First, the phase of the classical dipole, for exchange of radiation in the continuum, Eq. (\[C\_phase\]), is closely related, but not identical, to the soft-photon limit of the quantum mechanical continuum–continuum phase, Eq. (\[phiccsoft\]). Further, as epitomized in Eq. (\[r2C\]), there exists a relationship between the initial radial position of the ejected electron (a classical concept) and the asymptotic quantum phase. We have also demonstrated how the theory of complex transition matrix elements, originally developed for RABBIT, can help to better understand attosecond streaking measurements. In our interferometric interpretation of streaking, we stressed the benefits of a weak and monochromatic laser probe field. Given such experimental conditions, quantitative analysis of more complex systems can be carried out using two-photon transition matrix elements, corrected by many-body perturbation theory.
Acknowledgements {#acknowledgements .unnumbered}
----------------
This research was supported by the Marie Curie program ATTOFEL (ITN), the European Research Council (ALMA), the Joint Research Programme ALADIN of Laserlab-Europe II, the Swedish Foundation for Strategic Research, the Swedish Research Council, the Knut and Alice Wallenberg Foundation, the French ANR-09-BLAN-0031-01 ATTO-WAVE program, COST Action CM0702 (CUSPFEL).
Parts of the computations have been performed at IDRIS, and part of the work was carried out at the NORDITA workshop: “Studying Quantum Mechanics in the Time Domain”, in Stockholm. One of us (J.M.D.) would like to thank Professor Eva Lindroth for helpful discussions.
Appendix A: Formal derivation of the approximate expression for the radial component of the 2-photon ATI transition amplitudes {#appendix-a-formal-derivation-of-the-approximate-expression-for-the-radial-component-of-the-2-photon-ati-transition-amplitudes .unnumbered}
==============================================================================================================================
The radial component of the two-photon ATI matrix elements given in Eq. (\[Tell\]) can be rewritten: $$\begin{aligned}
T_{L,\lambda,\ell_i}(k; \epsilon_i +\Omega) = \lim_{\varepsilon \to 0^+} \langle R_{k ,L} | r\: G_\lambda (\epsilon_i +\Omega+i\varepsilon)\: r |R_{n_i,\ell_i}\rangle.
\label{TGreen}\end{aligned}$$ Here the initial atomic state is represented by the radial wavefunction $R_{n_i,\ell_i} (r)$, while $ R_{k ,L} (r) $ is the wavefunction of the photoelectron with energy $k^2 /2 = \epsilon_i +\Omega \pm \omega$. $G_\lambda (r_2, r_1;\epsilon_i +\Omega)$ is the radial component, of the Green’s function given in Eq. (\[Green\]) for angular momentum $\lambda$ and positive energy argument $\kappa^2 /2=\epsilon_i +\Omega > 0$. We note the presence of the positive imaginary infinitesimal $ i\varepsilon$ in the argument of $G$: It corresponds to the change $\kappa \to \kappa + i \varepsilon/\kappa$, which ensures the presence of a converging factor $e^{-\epsilon r}$ in the integrals below, which otherwise would be divergent. For the sake of conciseness, we will omit this factor in the following.
In the limit of large values of the coordinates $r_1$ and $r_2$, the Green’s function takes the following limiting form, written for the case of a Coulomb potential, [@Zon; @Edwards]: $$\begin{aligned}
G_\lambda (r_2, r_1;\epsilon_i +\Omega) \approx -{2\over r_1 r_2 \kappa} e^{i[\kappa r_> + \Phi_{\kappa,\lambda} (r_> )]}\nonumber \\ \times \sin [ \kappa r_< + \Phi_{\kappa,\lambda} (r_< )]\end{aligned}$$ where $r_>$ (resp. $r_<$) denotes the larger (resp. smaller) of $(r_1, r_2)$ and the phases $\Phi_{\kappa,\lambda} (r )$ are defined in Eq. (\[Phase\]).
We consider now the case of two-photon ATI from the $1s$ state of a hydrogenic system with $R_{1,0}(r) = C_{1,0} e^{-Zr}$ and normalization constant $ C_{1,0} = 2 Z^{3/2}$, while the photoelectron is described by a continuum wavefunction with asymptotic form given in Eq. (\[RkL\]). In the case considered here, $\lambda =1$ while $L = 0,2$. Replacing in the general expression of the amplitude Eq. (\[TGreen\]), one has: $$\begin{aligned}
T_{L,1,0}(k; \epsilon_{1s} +\Omega) = -{2\over k}C_{1,0} N_k \int_0^\infty \!\! \!dr_2 r_2 \sin[k r_2+\Phi_{k,L}(r_2)]\nonumber \\ \int_0^\infty\!\! \!dr_1 r_1^2 e^{-Zr_1} e^{i[\kappa r_> + \Phi_{\kappa,\lambda} (r_> )]} \sin [ \kappa r_< + \Phi_{\kappa,\lambda} (r_< )]\end{aligned}$$ and splitting the integration ranges over $r_>$ and $r_<$, one gets: $$\begin{aligned}
T_{L,1,0}(k; \epsilon_{1s} +\Omega) = -{2\over k}C_{1,0} N_k \int_0^\infty \!\! \!dr_2 r_2 \sin[k r_2+\Phi_{k,L}(r_2)]\nonumber \\ \times \{ e^{i[\kappa r_2+\Phi_{\kappa,1}(r_2)]} \int_0^{r_2}\!\! \!dr_1 r_1^2 e^{-Zr_1} \sin [\kappa r_1 + \Phi_{\kappa,1} (r_1)] \nonumber \\
+ \sin [\kappa r_2 + \Phi_{\kappa,1} (r_2)] \int_{r_2}^\infty \!\! \!dr_1 r_1^2 e^{-Zr_1} e^{i[\kappa r_1 + \Phi_{\kappa,1} (r_1)]} \} \end{aligned}$$ From the two $r_1-$integrals present within the braces, the first one is by far dominant. This comes from the presence of the exponentially decaying wavefunction of the ground state, which ensures that the integration range containing the origin is dominant. Thus, one can neglect the second term and one has: $$\begin{aligned}
T_{L,1,0}(k; \epsilon_{1s} +\Omega) \approx - {2\over k}C_{1,0} N_k \nonumber \\ \int_0^\infty \!\! \!dr_2 r_2 \sin [k r_2+\Phi_{k,L}(r_2)] e^{i[\kappa r_2+\Phi_{\kappa,1}(r_2)]} \nonumber \\ \times \int_0^{r_2}\!\! \!dr_1 r_1^2 e^{-Zr_1} \sin [\kappa r_1 + \Phi_{\kappa,1} (r_1)]\end{aligned}$$ We remark that the $r_1$-integral contained in this expression is real: $\int dr_1 ... \in \mathbf{R}$ and its precise value does not affect the overall phase of the amplitude which becomes: $$\begin{aligned}
\arg [T_{L,1,0}(k; \epsilon_{1s} +\Omega)] \approx \nonumber \\ \arg \{ - \int_0^\infty \!\! \!dr r \sin [k r +\Phi_{k,L}(r)] e^{i[\kappa r + \Phi_{\kappa,1} (r)]}\}\end{aligned}$$ or, making explicit the expressions of the Coulomb phases $\Phi$, writing the sine function under its exponential form and keeping only the $r-$dependent terms in the integrand, the amplitude can be expressed in terms of the integrals $J_\pm$ defined in Eq. (\[Jint\]). Following the same line of reasoning as the one followed in Sec \[sec:asympapp\], one can neglect the contribution of the $J_+$ integral, so that: $$\begin{aligned}
\arg [T_{L,1,0}(k; \epsilon_{1s} +\Omega)] \approx \arg \{ (-)^{{L+2\over2}} (2\kappa)^{iZ/\kappa} e^{i\sigma_1 (\kappa)}\nonumber \\ (2k)^{-iZ/k} e^{-i\sigma_L (k)} \int_0^\infty \!\! \!dr r^{1 +iZ(1/\kappa - 1/k)} e^{i(\kappa -k ) r} ] \} .\end{aligned}$$ where the last integral represents a Gamma function of complex argument times algebraic factors, so that one recovers the expression of the phase of the radial amplitude given in Eq. (\[argTas1\]).
This expression, which has been established for an initial $1s$ hydrogenic state with nuclear charge $Z$, remains valid for any $s$ bound state with an exponentially decaying wavefunction. As already mentioned, for non-hydrogenic systems, the Coulomb phase-shifts have to be replaced by the relevant scattering phase-shifts .
For the sake of reference, we give the complete expression of the asymptotic form of the amplitude, including real factors: $$\begin{aligned}
T_{L,1,0}(k; \epsilon_{1s} +\Omega) \approx (-)^{{L+2 \over2}} {1\over k}C_{1,0} N_{k } e^{-{\pi Z\over2}({1\over \kappa} -{1\over k}) } \nonumber \\ {(2\kappa)^{iZ/\kappa}\over (2k)^{iZ/k} } e^{i[\sigma_1 (\kappa) -\sigma_L (k)]} {\Gamma[2 +iZ(1/\kappa - 1/k)] \over (\kappa-k)^{2 +iZ(1/\kappa - 1/k)}}\nonumber \\ \times \int_0^\infty \!\! \!dr r^2 e^{-Zr} \sin [\kappa r + \Phi_{\kappa,1} (r)] \end{aligned}$$
[00]{} M. Schultze [*et al.*]{}, Science [**328**]{}, 1658 (2010).
E. Goulielmakis [*et al.*]{}, Science [**305**]{}, 1267 (2004).
K. Klünder [*et al.*]{}, Phys. Rev. Lett. [**106**]{}, 143002 (2011).
V. Véniard, R. Taïeb and A. Maquet, Phys. Rev. Lett. [**74**]{}, 4161 (1995).
V. Véniard, R. Taïeb and A. Maquet, Phys. Rev. A [**54**]{}, 721 (1996).
H. G. Muller, Appl. Phys. B [**74**]{}, S17 (2002).
V.S. Yakovlev [*et al.*]{}, Phys. Rev. Lett. [**105**]{}, 073001 (2010).
A.S. Kheifets and I.A. Ivanov, Phys. Rev. Lett. [**105**]{}, 233002 (2010).
I.A. Ivanov, Phys. Rev. A [**83**]{}, 023421 (2011).
S. Nagele [*et al.*]{}, J. Phys. B: Atom. Molec. Opt. Phys. [**44**]{}, 081001 (2011).
C.-H. Zhang and U. Thumm, Phys. Rev. A [**84**]{}, 033401 (2011).
Misha Ivanov and Olga Smirnova, Phys. Rev. Lett. [**107**]{}, 213605 (2011).
R. Pazourek, S. Nagele, K. Doblhoff-Dier, J. Feist, C. Lemell, K. T[ö]{}kési and J. Burgd[ö]{}rfer, arXiv:1111.4172 v1 \[physics.atom-ph\] 17 Nov 2011.
E.P. Wigner, Phys. Rev. [**98**]{}, 145 (1955).
C. A. A. de Carvalho and H. M. Nussenzveig, Physics Reports [**364**]{}, 83 (2002).
O. Smirnova, M. Spanner and M.Y. Ivanov, J. Phys. B: Atom. Molec. Opt. Phys. [**39**]{}, S323 (2006).
O. Smirnova [*et al.*]{}, J. Phys. B [**40**]{}, F197 (2007).
C.-H. Zhang and U. Thumm, Phys. Rev. A [**82**]{}, 043405 (2010).
J.C. Baggesen and L.B. Madsen, Phys. Rev. Lett. [**104**]{}, 209903 (2010).
M. Chini, S. Gilbertson, S.D. Khan and Z. Chang, Opt. Express, [**18**]{}, 13006 (2010).
P.M. Paul, E.S. Toma, P. Breger, G. Mullot, F. Augé, Ph. Balcou, H.G. Muller, and P. Agostini, Science [**292**]{}, 1689 (2001).
Y. Mairesse, A. de Bohan, L.J. Frasinski, H. Merdji, L.C. Dinu, P. Monchicourt, P. Breger, M. Kovacev, R. Taïeb, B. Carré, H.G. Muller, P. Agostini, P. Salières, Science [**302**]{}, 1540 (2003).
A. Maquet and R. Ta[ï]{}eb, J. Mod. Opt. [**54**]{}, 1847 (2007).
L.D. Landau and E.M. Lifshitz, [*Quantum Mechanics: Nonrelativistic Theory*]{} 3rd edition, Pergamon Press, Oxford (1977); §136.
E.S. Toma and H.G. Muller, J. Phys. B: Atom. Molec. Opt. Phys. [**35**]{}, 3435 (2002).
D.J. Kennedy and S.T. Manson, Phys. Rev. A [**5**]{}, 227 (1972).
A. L’Huillier, L. Jönsson and G. Wendin, Phys. Rev. A 33, 3938 (1986)
Ref. [@Kennedy] uses an Hartree-Fock calculation. In case of strong intershell correlations, a better description is needed, A. Kheifets, private communication (to be published).
W. Zernik and R.W. Klopfenstein, J. Math. Phys. [**6**]{}, 262 (1965).
S. Klarsfeld and A. Maquet, Phys. Lett. [**73A**]{}, 100 (1979).
S. Klarsfeld and A. Maquet, J. Phys. B: Atom. Molec. Phys. [**12**]{}, L553 (1979).
M. Aymar and M. Crance, J. Phys. B: Atom. Molec. Phys. [**13**]{}, L287 (1980).
A. Maquet, V. Véniard and T. Marian, J. Phys. B: Atom. Molec. Opt. Phys. [**31**]{}, 3743 (1998).
M. Rotenberg, in [*Advances in Atomic and Molecular Physics*]{} edited by D.R. Bates and I. Estermann (Academic Press, New-York, 1970) Vol. 6, pp. 233.
A. Maquet, Phys. Rev. A [**15**]{}, 1088 (1977).
H. Friedrich, ’Theoretical Atomic Physics’, Springer-Verlag Berlin Heidelberg (2006).
B.A. Zon, N.L. Manakov, and L.P. Rapoport, Sov. Phys. Doklady [**14**]{} 904 (1970).
M. Edwards, X. Tang and R. Shakeshaft, Phys. Rev. A [**35**]{}, 3758 (1987).
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abstract: 'We study a model of mass-bearing coagulating planar Brownian particles. Coagulation occurs when two particles are within a distance of order ${\varepsilon}$. We assume that the initial number of particles $N$ is of order $|\log {\varepsilon}|$. Under suitable assumptions of the initial distribution of particles and the microscopic coagulation propensities, we show that the macroscopic particle densities satisfy a Smoluchowski-type equation.'
bibliography:
- 'bibcoagII.bib'
---
[**The kinetic limit of a system of coagulating planar Brownian particles** ]{}\
[Alan Hammond$^1$]{}\
Department of Statistics\
University of California\
Berkeley, California 93720\
[Fraydoun Rezakhanlou[^1]]{}\
Department of Mathematics\
University of California\
Berkeley, California 93720–3830
[Introduction]{}
A colloid consists of a large number of small particles that are suspended in an environment of far smaller and more numerous molecules. Large numbers of molecules bombard each particle, and random fluctuations among these collisions tend to give rise to an Ornstein-Uhlenbeck motion of the particle, in which its velocity is forced by a Brownian motion, with a drag force acting in the direction opposite to its velocity. On a long time scale, the colloidal particles move according to Brownian motions, because an Ornstein-Uhlenbeck process approximates such a motion over a long period of time. The particles of a colloid may also be liable to interact. In [@HR], we studied a model of a colloid in which this means of interaction took the form of a coagulation, this reaction being liable to take place between a pair of particles if they come to lie close enough to one another. The density of particles at the initial time was chosen so that the dynamics occur in a regime of mean free path, wherein a typical particle meets a bounded number of other particles in a unit of time. Speaking in rough terms, this choice of scaling causes the effects of diffusion and interaction on the macroscopic evolution of the system to be comparable. In common with much of non-equilibrium statistical mechanics, we interpret the macroscopic behaviour of the system in terms of the evolution of a small number of thermodynamic parameters, in this case, the density of particles of a given mass, as a function of macroscopic space and time. In [@HR], we proved that, when the initial number of particles is chosen to be high, this density typically evolves as the solution of the Smoluchowski system of PDE, $$\label{syspde}
\frac{\partial{f_n}}{\partial t}(x,t) = d(n)\Delta f_n (x,t) +
Q^n_1(f) (x,t) -
Q^n_2(f) (x,t) . \qquad \qquad n = 1,2, \ldots$$ The first term on the right-hand-side of (1.1) corresponds to the diffusion among particles of mass $n$, with $d(n)$ being one-half of the diffusion rate of such particles. The terms in (1.1) corresponding to the interaction of pairs of particles are given by the gain term $$\label{gainterm}
Q^n_1(f) (x,t) = \sum_{m=1}^n {\beta}(m,n-m) f_m(x,t)
f_{n-m}(x,t),$$ and the loss term $$\label{lossterm}
Q^n_2(f) = 2 f_n (x,t) \sum_{m=1}^{\infty} {\beta}(m,n) f_m (x,t).$$ Here, the collection of constants ${\beta}:\mathbb{N}^2 \to (0,\infty)$ quantify the macroscopic propensity of mass at a pair of values to combine.
The arguments of [@HR] were valid in the case where the dimension $d$ of the system was assumed to be at least three. The question of the behaviour of such a system in two dimensions is significantly different, and it is this topic that we address in this paper. We now turn to describe the model in more detail, after which, we will discuss the ways in which the two-dimensional case differs from that of higher dimensions.
We will be working with a collection of microscopic models, each model carrying an index $N \in \mathbb{N}$, this being the total number of particles present in the system at the initial time. Each of these $N$ particles is independently assigned a random integer mass and placed at the initial time at a random location whose law depends on that mass. More precisely, we will be describing the state of the system at any given moment in time by a configuration, by which we mean a map $q: I_q \to \mathbb{R}^2 \times \mathbb{N}$, whose domain $I_q$ is some finite set of a countable index set $I$. That is, if $i \in I_q$ has $q(i) = (x_i,m_i)$, then the system currently contains a particle of mass $m_i$ at $x_i \in \mathbb{R}^2$. To define the initial configuration, we choose a sequence of continuous functions $\big\{
h_n: \mathbb{R}^2 \to [0,\infty), \, n \in \mathbb{N} \big\}$ that must satisfy some conditions that we will shortly specify. We set $Z = \sum_{n=1}^{\infty}{\int_{\mathbb{R}^2}}{h_n} \in (0,\infty)$, and choose $N$ points in $\mathbb{N} \times \mathbb{R}^2$ indepedently according to a law whose density at $(x,n)$ is equal to $h_n(x)/Z$. Selecting arbitrarily a set of $N$ symbols $\{i_j: j \in \{1,\ldots,
N\} \}$ from $I$, we define the initial configuration $q_0$ by insisting that $q_0(i_j)$ is equal to the $j$-th of the randomly chosen members of $\mathbb{N} \times \mathbb{R}^2$.
Each particle moves according to an independent Brownian motion whose diffusion rate $2d(m)$ depends on its mass $m \in \mathbb{N}$. As we will explain later, we require some conditions on the choice of the function $d: \mathbb{N} \to \mathbb{R}$, although the restriction imposed by these conditions is far from prohibiting the physically reasonable choice where $d$ is decreasing. Any pair of particles that approach to within a certain range of interaction are liable to coagulate, at which time, they disappear from the system, to be replaced by a particle whose mass is equal to the sum of the colliding particles, and whose location is at some point nearby the place where the collision took place. This range of interaction is taken to be equal to a parameter $\epsilon$, whose dependence on the total particle number $N$ must be stipulated. We make the choice $N = \vert \log \epsilon \vert Z$. This will ensure that a particle randomly chosen from those initally present experiences an expected number of collisions in a given unit of time that remains bounded away from zero and $\infty$ as $N$ is taken to be high. The effects of motion and reaction determine the macroscopic evolution of the system to comparable extents in this scaling.
We now describe the mathematical details of these dynamics. Let $F: \{ \mathbb{R}^2 \times \mathbb{N} \}^I \to {\mathbb R}$ denote a smooth function, whose domain is given the product topology. The dynamics is such that the action on $F$ of the model’s infinitesimal generator $\mathbb{L}$ is given by $$(\mathbb{L} F)(q) = \mathbb{A}_0 F (q) + \mathbb{A}_C F (q),$$ where the diffusion and collision operators are given by $$\label{diff}
\mathbb{A}_0 F (q) =
\sum_{i \in I_q}{d(m_i) \Delta_{x_i} F }$$ and $$\begin{aligned}
\mathbb{A}_C F (q) & = &
\sum_{i,j \in I_q}{\epsilon^{-2} \big\vert \log \epsilon
\big\vert^{-1}
V \Big( \frac{ x_i - x_j
}{\epsilon} \Big) {\alpha}(m_i,m_j)} \label{dyna} \\
& & \qquad \qquad
\bigg[ \frac{m_i}{m_i +
m_j} F \big( S^1_{i,j} q \big) + \frac{m_j}{m_i +
m_j} F \big( S^2_{i,j} q \big) - F(q) \bigg]. \nonumber\end{aligned}$$ Note that:
- the function $V: \mathbb{R}^2 \to [0,\infty)$ is assumed to be Hölder continuous of compact support, and with $\int_{\mathbb{R}^2}{V(x) dx} = 1$.
- we denote by $S^1_{i,j}q$ that configuration formed from $q$ by removing the indices $i$ and $j$ from $I_q$, and adding a new index from $I$ to which $S^1_{i,j}q$ assigns the value $(x_i,m_i + m_j)$. The configuration $S^2_{i,j}q$ is defined in the same way, except that it assigns the value $(x_j,m_i + m_j)$ to the new index. The specifics of the collision event then are that the new particle appears in one of the locations of the two particles being removed, with the choice being made randomly with weights proportional to the mass of the two colliding particles.
We will denote by $\mathbb{P}_N$ the measure on functions from $t \in [0,\infty)$ to the configurations determined by the process at time $t$. Its expectation will be denoted $\mathbb{E}_N$.
The form of the collision term in (\[dyna\]) differs from that used in the case of higher dimensions, in that the factor of $\big\vert
\log \epsilon \big\vert^{-1}$ is absent in the latter case. To explain why we make this change, we firstly recall the reason for the form of the collision operator in the case when $d \geq 3$. Suppose that, for some such choice of the dimension, two particles $(x_i,m_i)$ and $(x_j,m_j)$ have, at some time $t_0$, just become liable to interact, in the sense that the difference $x_i - x_j$ has become of order $\epsilon$. This state of affairs is liable to persist for a time of order $\epsilon^2$, but not much longer: for $d \geq 3$ and $C$ a large constant, the Brownian displacement $x_i - x_j$ would return a distance of $\epsilon$ from the origin with only a small probability after a time of $C \epsilon^2$ after the moment $t_0$. This means that, by choosing a form of collision dynamics in which the factor of $\vert \log \epsilon \vert^{-1}$ is absent from (\[dyna\]), we ensure that the integral $$I_T = \int_{t =t_0}^{T}{{\alpha}(m_i,m_j) \epsilon^{-2}
V \Big( \frac{x_i - x_j}{\epsilon} \Big)} dt,$$ reaches its eventual value after a time of order $\epsilon^2$. Other particles are unlikely to interfere with this pair in such a short period of time, and, as such, we may neglect their influence. The probability of collision between the pair before time $T$ is equal to $1 - \exp \big\{ - I_T \big\}$. Thus, for $d \geq 3$, our choice of dynamics is such that, among all the pairs of particles that at some moment lie within $\epsilon$ of each other, the fraction that eventually coagulate is bounded in $N$ away from $0$ and $1$, with this fraction being close to $0$ or $1$ depending on whether the relevant constant ${\alpha}(m_i,m_j)$ is high or low.
Turning to the planar case, note firstly that, in order that the probability of pair collision may remain equal to $1 - \exp \big\{ - I_T \big\}$, we alter the definition of $I_T$ by introducing a factor of $\big\vert \log \epsilon \big\vert^{-1}$. The two-dimensional case differs, because a planar Brownian motion returns almost surely to any open set at indefinitely later times. As such, the difference $x_i - x_j$ will endlessly re-enter the $\epsilon$-ball centred at the origin, ensuring that $I_T \to \infty$ as $T \to
\infty$. In a system of two particles, their coagulation is inevitable. In the system that we consider, where the regime of constant mean free path has been selected, a pair of particles at $\epsilon$ distance may find that their ongoing efforts to coagulate, as measured by the increase of $I_T$, are interrupted by the arrival of a third particle, the probability of appearance of such an intruder becoming appreciable at a small time, independent of $\epsilon$, after that at which the pair in question first came close to each other. The factor of $\big\vert \log \epsilon \big\vert^{-1}$ that appears in (\[dyna\]) ensures that, during this short fixed time, the probability of coagulation between the pair is of unit order, with the constant ${\alpha}(m_i,m_j)$ determining whether this probability is high or low. To write a statement analogous to that for the higher dimensional case: among the set of pairs of particles that are at some moment at a distance of order $\epsilon$, the fraction that combine with each other, rather than with some other particles, is bounded in $N$ away from $0$ and $1$, with the value of the constant ${\alpha}$ determining whether this fraction is high or low, similarly to the earlier case.
Our main result is conveniently expressed in terms of the empirical measures on the locations of particles of a given mass. For each $n
\in \mathbb{N}$ and $t\in [0,\infty)$, we write $g_n(dx,t)$ for the measure on $\mathbb{R}^2 $ given by $$g_n(dx,t)
=|\log{\varepsilon}|^{-1}\sum_{i\in I_{q(t)}}{\delta}_{x_i(t)}(dx)1\!\!1\big(m_i(t)=n\big).$$ We also require a mild hypothesis on the diffusion coefficients $d:\mathbb{N} \to (0,\infty)$ (see the first remark after Theorem 1.1 below). Namely, we suppose that there exists a function ${\gamma}: \mathbb{N}^2 \to
(0,\infty)$ such that ${\alpha}\le {\gamma}$, with ${\gamma}$ satisfying $$\label{hypo}
n_2 {\gamma}\big( n_1, n_2 + n_3 \big) \max{ \Big\{ 1, \Big[ \frac{d (
n_2 + n_3 )}{d(n_2)}
\Big]^{3}\Big\}} \leq \big( n_2 + n_3 \big) {\gamma}(n_1,n_2).$$
The initial random configuration of $N$ particles is formed by scattering particles of numerous masses independently in $\mathbb{R}^2$ according to densities that are prescribed for each mass. These densities will be chosen as continuous functions $\{ h_n: \mathbb{R}^2 \to [0,\infty), n \in \mathbb{N} \}$, and should satisfy some fairly weak bounds. To be specific, we insist that
- $k\in L^1({\mathbb R}^2)$ and $\bar k\in L^{{\infty}}_{loc}({\mathbb{R}^2})$ where $k:=\sum_{n=1}^{{\infty}} nh_n$ and $$\bar k(x)=\iint_{|x_1-x_2-y|\le 1}\big|\log |x_1-x_2-y|\big|k(x_1)k(x_2)dx_1dx_2.$$
- For every $m$, $\sum_{n=1}^{\infty}d(n)^{2/3}{\gamma}(n,m){\hat h_n} \in
L^\infty_{loc}({\mathbb{R}^2})$ for $\hat h_n(x)=\int h_n(y)|x-y|^{-4/3}dy$.
- For every $m$, $\sum_{n=1}^{\infty}d(n)^{3/4}{\gamma}(n,m){\tilde h_n} \in
L^\infty_{loc}({\mathbb{R}^2})$ for $\tilde h_n(x)=\int h_n(y)|x-y|^{-3/2}dy$.
We then set $Z=\sum_{n=1}^{\infty}\int_{\mathbb{R}^2}{h_n} \in (0,\infty)$ and choose $N$ points in $\mathbb{N} \times \mathbb{R}^2$ independently according to a law whose density at $(x,n)$ is equal to $h_n(x)/Z$. Selecting arbitrarily a set of $N$ symbols $\{i_j: j \in \{1,\ldots,
N\} \}$ from $I$, we define the initial configuration $q_0$ by insisting that $q_0(i_j)$ is equal to the $j$-th of the randomly chosen members of $\mathbb{N} \times \mathbb{R}^2$. [**Remark**]{} It is not hard to show that our assumptions on the initial data $\{h_n\}$ are satisfied if $k$ is bounded, has a bounded support, $d(\cdot)$ is bounded and $\gamma(n,m)\le C(m)n$ for a function $C(\cdot)$. Indeed if $k$ is bounded and has a bounded support, then $\bar k L^{\infty}_{loc}$ and $\hat h_n,\tilde h_n\in
L^{\infty}_{loc}$ for every $n$. It is worth mentioning that if $k$ belongs to the negative Sobolev Space $H^{-1}=W^{-1,2}$, then $\bar k\in L^{\infty}$.
The main theorem is now stated.
\[thmo\] Let $J: \mathbb{R}^2
\times [0,\infty) \to \mathbb{R}$ be a bounded and continuous test function. Then, for each $n \in \mathbb{N}$ and $t \in (0,\infty)$, $$\label{resu}
\limsup_{N \to \infty } \mathbb{E}_N \bigg\vert \int_{\mathbb{R}^2}
J(x,n,t) \big( g_n(dx,t) - f_n (x,t) dx \big) \bigg\vert = \ 0,$$ where we recall that $\epsilon$ is related to $N$ by means of the formula $N
|\log{\varepsilon}|^{-1} = Z$, with the constant $Z \in (0,\infty)$ being given by the expression $Z = \sum_{n \in
\mathbb{N}}{\int_{\mathbb{R}^2}h_n}$. In (1.7), $\{ f_n: \mathbb{R}^2 \times [0,\infty) \to [0,\infty), n \in \mathbb{N} \}$ denotes a weak solution to the system of partial differential equations (1.1) with the initial data $f_n(\cdot,0)
=h_n(\cdot)$. The collection of constants ${\beta}: \mathbb{N}^2 \to [0,\infty)$ is given by $$\label{recp}
{{\beta}}(n,m)= \frac{2\pi \big( d(n) + d(m) \big){\alpha}(n,m)}{2\pi \big( d(n)
+ d(m) \big) + {\alpha}(n,m)}.$$
[**Remarks**]{}\
- Conditions for uniqueness among weak solutions of (1.1) are provided in Proposition 2.6 of [@wrzd]. Although these conditions are slightly stronger than those we have proved the macroscopic equations to satisfy in our case, we anticipate that this uniqueness holds. In the lack of uniqueness, the best that can be said is that if $\mathcal P_N$ denotes the probability law of $g(dx,t)$ with respect to the underlying probability distribution $\mathbb P_N$, then $\{\mathcal P_N\}$ is tight and any limit point $\mathcal P$ of the sequence $\{\mathcal P_N\}$ is concentrated on the space of measures $g_n(dx,t) = f_n(x,t)dt$ with $\{f_n: n \in \mathbb N\}$ solving (1.1) in weak sense, subject to the initial condition $f_n(\cdot,0) = h_n(\cdot)$.
- Included in the space of parameter values that satisfy (\[hypo\]) is the case where the diffusion rate $d$ is a decreasing function of the mass, and the coagulation propensities ${\alpha}$ satisfy ${\alpha}\big( n , m \big) \leq C nm$. In fact for a nonincreasing $d(\cdot)$, the condition (\[hypo\]) is equivalent to saying that ${\alpha}(n,m)\le C(n)m$ for a function $C(n)$. Also, if the microscopic coagulation rate ${\alpha}$ is identically constant, then the condition (\[hypo\]) is equivalent to saying that the function $d(n)n^{-1/4}$ is nonincreasing.
- Note that the macroscopic coagulation propensities ${\beta}$ depend only on the total integral of $V$ that is assumed to be $1$ for convenience. However when the dimension is $3$ or more the propensity ${\beta}(n,m)$ does depend on $V$ in a nontrivial way and is given as ${\alpha}(n,m)\int (1+u)Vdx$, where $u$ solves the PDE $\Delta u=\tau (1+u)V$ with $\tau={\alpha}(n,m)/(d(n)+d(m))$.
- Our technique of proof also yields a kinetic limit derivation for the model in which particles are assumed to have a range of interaction that is mass-dependent. To give an example of such a variant, suppose that each particle of mass $m$ has a radius $r(m)$, where $r(m) =\sqrt m$. We stipulate that particles of mass $m$ and $n$ are liable to react when their displacement reaches the order of $\big( r(m) + r(n) \big) \epsilon$. More precisely, we modify the definition (\[dyna\]) of the collision operator $\mathbb{A}_{C}$ by replacing the appearance of $V $ by $ \big( r(n) + r(m) \big)^{-2} V \big(
\cdot/(r(n) + r(m))\big)$, (the factor that multiplies $V$ being introduced so that, roughly speaking, the altered collision mechanism respects the spatial-temporal scaling of Brownian motion). Theorem 1.1 is still valid for this modified model with the same macroscopic coagulation propensities ${\beta}$. This is in sharp contrast with the case $d\ge 3$ for which the mass dependence affects the macroscopic coagulation propensities ${\beta}$.
In common with the proof for $d \geq 3$, a central element in deriving Theorem \[thmo\] is establishing that, at any given moment after the initial time, the presence of a particle of some given mass at some fixed point in space significantly affects the likelihood of a particle being at some other point in space only if that other point is at a short distance from the first particle. That is, on distances of short order, the presence of a particle makes it less likely to find another nearby, because the pair would have been liable to coagulate shortly beforehand. However, the distribution of particle at a given time is similar to one in which they were scattered independently, except for this short-range repulsion. The following proposition, whose form differs from that in the case $d \geq 3$ only in its scaling factor, formalises this assertion.
\[szprop\] Set $$\label{qudef}
Q = |\log{\varepsilon}|^{-2} \sum_{(i,j) \in
I_{q}}{{\alpha}(m_i,m_j) V_{\epsilon}(x_i - x_j) J(x_i,m_i,t)
\overline{J}(x_j,m_j,t)},$$ where $J,\overline{J}: \mathbb{R}^2 \times \mathbb{N}
\times [0,\infty) \to [0,\infty)$ are test functions satisfying the same conditions as those stated in Theorem \[thmo\]. We also assume that $J(x,m,t)=0$ unless $m=M_1$ and $\overline J(x,m,t)=0$ unless $m=M_2$. Let $\eta:
\mathbb{R}^2 \to [0,\infty)$ denote a smooth function of compact support for which $\int_{\mathbb{R}^2}{\eta(x)dx} = 1$. We have that $$\begin{aligned}
\label{szrec}
& & \int_0^T Q (t) dt \\
& = &
\int_0^T{} dt \int_{\mathbb{R}^2}
d\omega \, {\beta}(M_1,M_2) J(\omega,M_1,t)
\overline{J}(\omega,M_2,t) \bigg[ |\log {\varepsilon}|^{-1} \sum_{i \in I_q: m_i = M_1}{
\delta^{-2} \eta \Big( \frac{x_i - \omega}{\delta} \Big) }
\bigg] \nonumber \\
& & \qquad \quad \bigg[ |\log{\varepsilon}|^{-1} \sum_{j \in I_q; m_j = M_2}{
\delta^{-2} \eta \Big(\frac{x_j - \omega}{\delta} \Big)
} \bigg] \, + \rm{Err} \big( \epsilon, \delta
\big), \nonumber\end{aligned}$$ where the constants ${{\beta}}: \mathbb{N}^2 \to [0,\infty)$ were defined in (\[recp\]), and where the function $\rm{Err}$ satisfies $$\lim_{\delta \downarrow 0}\limsup_{\epsilon\downarrow 0} \mathbb{E}_N
\big\vert Err \big( \epsilon, \delta \big) \big\vert = 0.$$
Why is this statement a mathematical rendering of the claim discussed before it was made? The quantity $\int_0^T Q(t) dt$ can be thought of as the total propensity of particles to combine during the interval of time $[0,T]$. Proposition 1 asserts it may be approximated by a time-averaged product of empiricial approximations to the density of particles (of the appropriate mass). That is, particles are arranged independently enough near most of the collision events that the rate of these collisions is roughly proportional to that arising in a system in which particles are scattered indepedently at random according to densities given by measuring the system in question on scale $\delta$ that is much larger than the reaction range $\epsilon$. There is, however, a constant of proportion corresponding to the change from microscopic reaction propensity ${\alpha}$ appearing in the definition of $Q$ to its macroscopic counterpart ${{\beta}}$. Its presence may be explained by the negative short-range correlation between particles discussed before the statement of Proposition \[szprop\].
The analogue of Theorem \[thmo\] that appears in [@HR] for the case $d \geq 3$ is derived as a consequence of Proposition \[szprop\]. In Section 2 of [@HR], a sketch of the proof of Theorem \[thmo\] may be found. The details of the derivation of Theorem \[thmo\] from Proposition \[szprop\] do not differ in the two-dimensional case, so that we do not present these arguments again in this paper. Our task here is rather to present a detailed derivation of Proposition \[szprop\] in the case when $d=2$. Before reading further, however, the reader may wish to consult Section 2 of [@HR]. We refer the reader to [@HR] also for a discussion of previous work related to the problem. Here, we mention only Sznitman [@sznitman], in which a model of Brownian spheres that annihilate as soon as they touch is studied. The partial differential equation by which the density of particles evolves was derived for the kinetic limit, in each dimension $d \geq 2$. In this work, the macroscopic annihilation rate is exactly ${2\pi}$ when the dimension is $2$. This is compatible with our main results because if $d(\cdot)$ is identically $1/2$ and ${\alpha}\to{\infty}$, the macroscopic coagulation rate ${\beta}$ approches $2\pi$. Note that our model approximates the [*hard core*]{} model as ${\alpha}$ gets large.
[Establishing the Stosszahlansatz]{}\[sectwo\]
For any given pair $(n,m) \in \mathbb{N}^2$, ${u^{\epsilon}}=
{u^{\epsilon}}_{n,m}: \mathbb{R}^2 \to [0,\infty)$ is a function whose existence is ensured by Theorem \[th3.1\] that lies in $C^2
\big( \mathbb{R}^2 \big)$ satisfying $$\label{eq3.1}
{u^{\epsilon}}_{n,m} \big( x \big) = \frac 1{2\pi}\tau \big( n,m \big) \int
\log |x-y|\Big[
V_{\epsilon} \big(
y \big) {u^{\epsilon}}_{n,m} \big( y \big) + V^{\epsilon} \big( y \big) \Big],$$ where ${\tau}(n,m)={\alpha}(n,m)/(d(n)+d(m))$. As a consequence, $$\big(d(n)+d(m)\big)\Delta {u^{\epsilon}}_{n,m} \big( x \big) = {\alpha}\big( n,m \big) \Big[
V_{\epsilon} \big(
x \big) {u^{\epsilon}}_{n,m} \big( x \big) + V^{\epsilon} \big( x \big) \Big].$$ We are using the notations $$V^{\epsilon}(x) = \epsilon^{-2} V \Big( \frac{x}{\epsilon} \Big)$$ and $$V_{\epsilon}(x) = \epsilon^{-2}|\log{\epsilon}|^{-1} V \Big(
\frac{x}{\epsilon} \Big).$$
We present the conditions on the two test functions $J,\overline{J}:
\mathbb{R}^2 \times \mathbb{N} \times [0,{\infty})
\to {\mathbb R}$ that appear in Proposition 1. It suffices to work with functions that take non-zero values for only one value in the second argument, such functions measuring the presence of particles of a given mass. By a temporary abuse of notation, we write $$\begin{aligned}
J(x,M_1,t) & = & J(x,t) 1 \! \! 1 \big\{ m = M_1 \big\} \nonumber \\
\overline{J}(x,M_1,t) & = & \overline{J}(x,t) 1 \! \! 1 \big\{ m =
M_2 \big\}, \nonumber\end{aligned}$$ where on the right-hand-side, $J$ and $\overline{J}$ denote smooth maps from $\mathbb{R}^2 \times [0,\infty)$ to ${\mathbb R}$ of compact support. We will suppress the appearance of the $t$-variable when writing the arguments of $J$ and $\overline{J}$.
In seeking to verify Stosszahlansatz, we define $$\label{exzo}
X_z (q) = |\log\epsilon|^{-2} \sum_{i,j \in I_q}{
{u^{\epsilon}}_{M_1,M_2} (x_i- x_j + z) J (x_i,M_1,t) \overline{J}(x_j,M_2,t)}
1\! \! 1 \big\{ m_i = M_1,m_j = M_2 \big\} .$$ The relevance of the expression (2.2) for our purposes is that the term $Q$ and its variations appear as we apply the infinitesimal generaor on the expression $X_z-X_0$. We refer the reader to Section 2 of [@HR] for some heuristic justification of the special form of $X_z$.
Numerous terms arise when the operators $\mathbb{A}_0$ and $\mathbb{A}_C$ act on the expression $X_z - X_0$ (recall that the functions of configurations $X_z$, indexed by $z \in \mathbb{R}^2$, were defined in (2.2)). We now label these terms. Unless stated otherwise, we will adopt a notation whereby all the index labels appearing in sums should be taken to be distinct. This includes the case of multiple sums. For example, $\sum_{k,l \in I_q}
\sum_{i \in I_q} f(x_k,x_l,x_i)$ denotes the sum of the evaluation of the function $f$ over all arguments that are triples $(x_k,x_l,x_i)$ where $k$, $l$ and $i$ are distinct indices in $I$. Note also that, unless otherwise stated, whenever the symbol ${u^{\epsilon}}$ appears in a summand, we mean ${u^{\epsilon}}_{M_1,M_2}$.
Firstly, we label those terms arising from the action of the diffusion operator. To do so, note that, for a time-dependent functional $F$ of the configuration space, this action is given by $$\Big( \frac{\partial}{\partial t} + \mathbb{A}_0 \Big) F =
\frac{\partial}{\partial t} F + \sum_{i \in I_q}{d \big(m_i)
\Delta_{x_i} } F.$$ Thus, we label as follows: $$\left(\frac{\partial}{\partial t}+\mathbb{A}_0\right) (X_z - X_0)( q(t) ) =
H_{11} + H_{12} + H_{13} + H_{14}
+ H_2 + H_3 + H_4,$$ with $$\begin{aligned}
H_{11} & = & {\big\vert \log \epsilon \big\vert^{-2}}\sum_{i,j \in I_q}{\alpha(m_i,m_j)
\Big[ V^{\epsilon} \big( x_i - x_j + z \big) -
V^{\epsilon} \big( x_i - x_j \big) \Big]} J(x_i,m_i,t)
\overline{J}(x_j,m_j,t) \nonumber \\
H_{12} & = & - {\big\vert \log \epsilon \big\vert^{-2}}\sum_{i,j \in I_q}{\alpha(m_i,m_j)
V_{\epsilon} \big( x_i - x_j \big) {u^{\epsilon}}\big( x_i - x_j
\big) J(x_i,m_i,t) \overline{J}(x_j,m_j,t) } \nonumber \\
H_{13} & = & {\big\vert \log \epsilon \big\vert^{-2}}\sum_{i,j \in I_q}{\alpha(m_i,m_j)
V_{\epsilon} \big( x_i - x_j + z \big) {u^{\epsilon}}\big( x_i - x_j+z
\big) J(x_i,m_i,t) \overline{J}(x_j,m_j,t) }, \nonumber \\
H_{14} & = & {\big\vert \log \epsilon \big\vert^{-2}}\sum_{i,j \in I_q}{
\Big[ {u^{\epsilon}}\big( x_i - x_j + z \big) - {u^{\epsilon}}\big(
x_i - x_j \big) \Big]} \nonumber \\
& & \qquad \quad \Big[ J_t \big(x_i,m_i,t\big)
\overline{J}(x_j,m_j,t) + J \big(x_i,m_i,t\big)
\overline{J}_t(x_j,m_j,t) \Big] , \nonumber\end{aligned}$$ along with $$\begin{aligned}
H_2 & = & 2 {\big\vert \log \epsilon \big\vert^{-2}}\sum_{i,j \in I_q}{ d(m_i)
\overline{J}(x_j,m_j,t)} \nonumber \\
& & \qquad \qquad \qquad \quad \Big[ {u^{\epsilon}}_x (x_i - x_j + z ) - {u^{\epsilon}}_x
(x_i - x_j ) \Big]
\cdot J_x(x_i,m_i,t), \nonumber \\
H_3 & = & - 2 {\big\vert \log \epsilon \big\vert^{-2}}\sum_{i,j \in I_q}{ d(m_j)
J(x_i,m_i,t)} \nonumber \\
& & \qquad \qquad \qquad \quad \Big[ {u^{\epsilon}}_x (x_i - x_j + z ) -
{u^{\epsilon}}_x (x_i - x_j ) \Big]
\cdot \overline{J}_x (x_j,m_j,t) , \nonumber\end{aligned}$$ and $$\begin{aligned}
H_4 & = & {\big\vert \log \epsilon \big\vert^{-2}}\sum_{i,j \in I_q}{} \Big[
{u^{\epsilon}}(x_i - x_j + z ) - {u^{\epsilon}}(x_i - x_j ) \Big]
\nonumber\\
& & \qquad \qquad \quad \Big[ d(m_i) \Delta_x J(x_i,m_i,t)
\overline{J}(x_j,m_j,t) + d(m_j) J(x_i,m_i,t)
\Delta_x \overline{J}(x_j,m_j,t) \Big] , \nonumber\end{aligned}$$ where $f_x$ denotes the gradient of $f$, and $\cdot$ the scalar product. As for those terms arising from the action of the collision operator, $$\mathbb{A}_C (X_z - X_0)(q) = G_z (1) + G_z (2) - G_0 (1) - G_0 (2),$$ where $G_z (1)$ is set equal to $$\begin{aligned}
& & \sum_{k,l \in
I_q}{\alpha(m_k,m_l) V_{\epsilon}( x_k - x_l)} {\big\vert \log \epsilon \big\vert^{-2}}\sum_{i
\in I_q}{} \nonumber \\
& & \quad \bigg\{
\frac{m_k}{m_k + m_l} \Big[ {u^{\epsilon}}( x_k - x_i
+ z) J( x_k , m_k + m_l,t)
\overline{J}(x_i,m_i,t) \nonumber \\
& & \qquad \qquad \qquad + \, {u^{\epsilon}}( x_i - x_k
+ z) J( x_i , m_i ,t) \overline{J}(x_k,m_k + m_l,t) \Big]
\nonumber \\
& & \quad \, \, + \, \frac{m_l}{m_k + m_l} \Big[ {u^{\epsilon}}( x_l - x_i
+ z) J( x_l , m_k + m_l,t)
\overline{J}(x_i,m_i,t) \nonumber \\
& & \qquad \qquad \qquad + \, {u^{\epsilon}}( x_i - x_l
+ z) J( x_i , m_i ,t) \overline{J}(x_l,m_k + m_l,t) \Big]
\nonumber \\
& & \qquad - \ \
\Big[ {u^{\epsilon}}(x_k - x_i + z) J( x_k , m_k,t)
\overline{J}(x_i,m_i,t) \nonumber \\
& & \qquad \qquad \qquad \qquad + \, {u^{\epsilon}}(x_i - x_k + z)
J( x_i , m_i ,t) \overline{J}(x_k,m_k,t) \Big]
\nonumber \\
& & \qquad - \ \
\Big[ {u^{\epsilon}}(x_l - x_i + z) J( x_l , m_l,t)
\overline{J}(x_i,m_i,t) \nonumber \\
& & \qquad \qquad \qquad \qquad + \, {u^{\epsilon}}(x_i - x_l + z) J( x_i ,
m_i ,t) \overline{J}(x_l,m_l,t) \Big] \bigg\}
\nonumber ,\end{aligned}$$ and where $$\begin{aligned}
\label{eqnzeta}
G_z (2) & = & - {\big\vert \log \epsilon \big\vert^{-2}}\sum_{k,l \in
I_q}{ \alpha(m_k,m_l) V_{\epsilon}(x_k - x_l)} \\
& & \qquad \quad
{u^{\epsilon}}(x_k -
x_l + z ) J(x_k,m_k,t) \overline{J}(x_l,m_l,t). \nonumber\end{aligned}$$ The terms in $G_z(1)$ arise from the changes in the functional $X_z$ when a collision occurs due to the influence of the appearance and disppearance of particles on other particles that are not directly involved. Those in $G_z(2)$ are due to the absence after collision of the summand in $X_z$ indexed by the colliding particles.
Note that $$\label{igt}
H_{12} + G_0 \big( 2 \big) = 0.$$ The process $\big\{ \big( X_z - X_0 \big)(t): t \geq 0 \big\}$ satisfies $$\begin{aligned}
\big( X_z - X_0 \big) \big( T \big) & = &
\big( X_z - X_0 \big) \big( 0 \big) +
\int_{0}^{T}{ \Big( \frac{\partial}{\partial t} + \mathbb{A}_0 \Big)
\big(X_z - X_0\big) (t)dt} \label{hact} \\
& & \qquad + \,
\int_{0}^{T}{ \mathbb{A}_C (X_z - X_0) (t)dt} \, + \, M(T), \nonumber\end{aligned}$$ with $\big\{ M(t) : t \geq 0 \big\}$ being a martingale. By using the labels for the various terms that we just introduced, we find from (\[hact\]) by use of (\[igt\]) that $$\begin{aligned}
& & \Big\vert
\int_{0}^{T}{ H_{11} \big(t\big) dt} +
\int_{0}^{T}{ H_{13} \big(t\big) dt} \Big\vert \nonumber \\
& \leq & \big\vert X_z - X_0 \big\vert \big( q(T) \big) + \big\vert
X_z - X_0 \big\vert \big( q(0) \big) \nonumber \\
& & + \int_{0}^{T}{\big\vert H_{14} \big\vert (t)
dt}+\int_{0}^{T}{\big\vert H_2
\big\vert (t) dt} +
\int_{0}^{T}{\big\vert H_3 \big\vert (t) dt} +
\int_{0}^{T}{\big\vert H_4 \big\vert (t) dt} \nonumber \\
& & +
\int_{0}^{T}{ \big\vert G_z(1) - G_0(1) \big\vert (t) dt} +
\int_{0}^{T}{ \big\vert G_z(2) \big\vert (t) dt} \, + \, \big\vert
M(T) \big\vert. \label{abc}\end{aligned}$$ Since $J$ is of compact support, we have that $X_z(q(T))=0$ for $T$ sufficiently large. We aim to prove the following estimates: for each $T > 0$, $$\begin{aligned}
& & \int_{0}^{T}{ \mathbb{E}_N \vert H_{14} \vert (t) dt} \leq C
|z|^{1/2}\big|\log |z|\big|, \nonumber \\
& & \int_{0}^{T}{ \mathbb{E}_N \vert H_2 \vert (t) dt} \leq C \vert z
\vert^{1/9}\big|\log|z|\big|,
\nonumber \\
& & \int_{0}^{T}{ \mathbb{E}_N \vert H_3 \vert (t) dt} \leq C \vert z
\vert^{1/9}\big|\log|z|\big|, \nonumber \\
& & \int_{0}^{T}{ \mathbb{E}_N \vert H_4 \vert (t) dt} \leq C
|z|^{1/2}\big|\log |z|\big|, \nonumber \\
& & \int_{0}^{T}{ \mathbb{E}_N \vert G_z(1) - G_0 (1) \vert (t) dt} \leq
C |z|^{1/2}\big|\log |z|\big| | , \nonumber \\
& & \int_{0}^{T}{ \mathbb{E}_N \vert G_z (2) \vert (t) dt} \leq C
\big\vert \log \vert z
\vert \big\vert |\log{\varepsilon}|^{-1},
\nonumber\\
& &\mathbb{E}_N|X_z-X_0(0)|\le C|z|. \label{jayeye}\end{aligned}$$ Later, we apply the limit $\vert z \vert \to 0$ after sending ${\varepsilon}$ to $0$. We will also show that, for each $T \in (0,\infty)$, $$\label{martest}
\mathbb{E}_N \Big[ M \big( T \big)^2 \Big] \leq
C \big\vert \log {\epsilon}\big\vert^{-1} .$$
Lemmas bounding collision propensity {#stone}
------------------------------------
In this subsection, we discuss three lemmas that in essence serve as the backbone of the proof of the various inequalities that appear in (2.7). These lemmas allow us to reduce the proof to a calculation involving the initial configuarions for which the independence of particles and our assumptions on the initial densities can be used. In fact the proof of Lemmas 2.1 and 2.3 is very similar to the corresponding Lemmas 3.1 and 3.3 of [@HR]. For this reason, their proofs are omitted. It is Lemma 2.2 that is somewhat different from what we have in [@HR] as Lemma 3.2 and we provide a detailed proof for it. In fact this difference explains to some extent a major technical difficulty that is two dimensional and is not encountered when the dimension is 3 or more. To explain this further, let us observe that if the dimension $d$ is 3 or more and $J$ is a nonnegative function, then we can find a solution to the Poisson equation $-\Delta H=J$ that satisfies $H\ge 0$. Indeed the solution $H$ is defined by $$c_0(d)\int |x-y|^{2-d}J(y)dy,$$ where $c_0(d)=(d(d-2){\omega}(d))^{-1}$ with ${\omega}(d)$ denoting the volume of the unit ball in ${\mathbb R}^d$. This is no longer true in dimension 2 because the solution is given by $$-\frac 1{2\pi}\int \log|x-y|\ J(y)dy.$$ This causes some difficulty in treating various terms that appear in (2.7). To get around this, let us define $$\label{eq2.8}
H(x)=-\frac 1{2\pi}\int_{|x-y|\le 1} \log|x-y|\ J(y)dy.$$ We now have that $-\Delta H=J-\tilde J$ where $$\label{eq2.9}
\tilde{J}(x) = \frac 1{2\pi}\int_{\vert z \vert = 1}{J(x-z) dS(z)},$$ where $dS$ denotes the $1$-Lebesgue measure on the unit circle $S^1$. The point is that by using Lemma 2.2, we reduce bounding an expression involving $J$ to an expression involving $H$ at time $t=0$, and a similar expression involving $\tilde J$. Since the funcion $\tilde J$ is an average of $J$, we have an easier task to bound the expression involving $\tilde J$. In the case of the terms $H_2$ and $H_3$, we need to apply this process three times so that the final $\tilde J$ has a simple pointwise bound. Our three lemmas are:
\[lembc\] For any $T \in [0,\infty)$, $${\big\vert \log \epsilon \big\vert^{-1}}\mathbb{E}_N \int_0^T dt \sum_{i,j \in I_q}{\alpha(m_i,m_j)}
V_{\epsilon} \big( x_i - x_j \big) \leq Z.$$
\[lemtwop\] Let $J: \mathbb{R}^2 \to [0,\infty)$ be continuous, and let $H:
\mathbb{R}^2 \to [0,\infty)$ be given by (2.9) We also define $\tilde{J}: \mathbb{R}^2 \to \mathbb{R}$ according to (2.10). Then we have the following inequality, $$\begin{aligned}
& & \big\vert \log \epsilon \big\vert^{-2} \mathbb{E}_N \int_0^T
\sum_{i,j \in I_q}{J(x_i - x_j)
m_i m_j (d(m_i) + d(m_j))} dt \nonumber \\
& & + \big\vert \log \epsilon \big\vert^{-2} \mathbb{E}_N \int_0^T
\sum_{i,j \in I_q}{V_{{\varepsilon}}(x_i-x_j)\alpha(m_i,m_j)H(x_i - x_j)
m_i m_j } dt \nonumber \\
\ \ \ & \leq & \big\vert \log \epsilon \big\vert^{-2}
\mathbb{E}_N \sum_{i,j \in I_{q(0)}}{H(x_i - x_j)
m_i m_j} \nonumber \\
\ \ \ & & \, + \,
\big\vert \log \epsilon \big\vert^{-2} \mathbb{E}_N \int_0^T
\sum_{i,j \in I_q}{\tilde{J}(x_i - x_j)
m_i m_j (d(m_i) + d(m_j))} dt. \nonumber\end{aligned}$$
\[lemthreep\] Assume that the function ${\gamma}:\mathbb{N}^2\to (0,{\infty})$ satisfies $$\label{hypo2}
n_2 {\gamma}\big( n_1, n_2 + n_3 \big) \max{ \Big\{ 1, \Big[ \frac{d (
n_2 + n_3 )}{d(n_2)}
\Big]^{2} \Big\}} \leq \big( n_2 + n_3 \big) {\gamma}(n_1,n_2),$$ There exists a collection of constants $C: \mathbb{N}^2 \to (0,\infty)$, such that, for any smooth function $J : \mathbb{R}^{4} \to [0,\infty)$, and any given $n_1,n_3 \in \mathbb{N}$, $$\begin{aligned}
& &
\mathbb{E}_N{\int_0^T{dt \sum_{k,l,i \in I_{q(t)}}{{\gamma}(m_i,m_j)
V_{\epsilon} \big(
x_i - x_j \big) J ( x_i , x_k ) 1 \! \! 1 \{ m_i = n_1, m_k =
n_3 \} }}} \nonumber \\
& \leq & C_{n_1,n_3}\big\vert \log {\epsilon}\big\vert^{3}
\sum_{n_2 \in \mathbb{N}}{\int{A^{\epsilon}_{n_1,n_2,n_3} \big( x_1,x_2,x_3
\big)h_{n_1}(x_1)h_{n_2}(x_2)h_{n_3}(x_3) dx_1 dx_2 dx_3}},\end{aligned}$$ where, also given $\epsilon > 0$ and $n_2 \in \mathbb{N}$, the function $ A_{n_1,n_2,n_3}^{\epsilon}: \mathbb{R}^{6} \to
[0,\infty) $ is defined by $$\begin{aligned}
& & \big( d(n_1 )\Delta_{x_1} + d(n_2) \Delta_{x_2} + d(n_3)
\Delta_{x_3} \big)
A_{n_1,n_2,n_3}^{\epsilon} (x_1,x_2,x_3) \nonumber \\
& & \qquad = - {\gamma}(n_1,n_2)
V_{\epsilon} ( x_1 - x_2 ) J ( x_1 ,
x_3 ) .\end{aligned}$$
It is worth mentioning that the function $A_{n_1,n_2,n_3}^{\epsilon}(x_1,x_2,x_3)$ of Lemma 2.3 is given by $$c_0 ( 6) \int_{\mathbb{R}^2} \int_{\mathbb{R}^2}
\int_{\mathbb{R}^2}
\bigg( \frac{\vert x_1 - z \vert^2}{d(n_1)} + \frac{\vert x_2 -
y \vert^2}{d(n_2)} + \frac{\vert x_3 - y' \vert^2}{d(n_3)}
\bigg)^{-{2}} {\gamma}(n_1,n_2) J ( z , y' ) V_{\epsilon} ( z - y )
dz dy dy',$$ where $c_0(d)=(d(d-2)\omega(d))^{-1}$ with $\omega(d)$ denoting the volume of the unit ball in ${\mathbb R}^d$. Note that for Lemma 2.3 we are dealing with a solution to a Laplace type equation in ${\mathbb R}^6$ as opposed to Lemma 2.2 for which the pecularity of the Laplace equation in ${\mathbb R}^2$ played a role. This is why the proof of [@HR] in the case of Lemma 2.3 can be repeated line by line.
[**Proof of Lemma \[lemtwop\]**]{} Set $$X_q = \big\vert \log \epsilon \big\vert^{-2} \sum_{i,j \in I_q}{H(x_i
- x_j)m_i m_j}.$$ Recall the mechanism of the dynamics at collision: the location of the newly created particle is one of the two locations of the colliding particles, with weights proportional to the masses of the incident particles. We see that when $\mathbb{A}_C$ acts on $X_q$, all those terms indexed by pairs of particles one of which is not involved in the collision cancel. Thus, $$\label{locc}
\mathbb{A}_C X = - \big\vert \log \epsilon \big\vert^{-2} \sum_{i,j
\in I_q}{ V_{\epsilon} ( x_i
- x_j ) \alpha(m_i,m_j) m_i m_j H(x_i - x_j)}.$$ By $\Delta H=-J+\tilde J$, $$\begin{aligned}
\mathbb{A}_0 X & = & \big\vert \log \epsilon \big\vert^{-2}
\sum_{i,j \in I_q}{ \Delta H (x_i
- x_j) m_i m_j (d(m_i) + d(m_j))} \\
& = & - \big\vert \log \epsilon \big\vert^{-2} \sum_{i,j \in
I_q}{ J (x_i - x_j) m_i m_j (d(m_i) + d(m_j))} \nonumber \\
& & + \big\vert \log \epsilon \big\vert^{-2} \sum_{i,j \in
I_q}{ \tilde{J} (x_i - x_j) m_i m_j (d(m_i) + d(m_j))} \nonumber
\\
&=&:\mathbb{A}_0^1 X+\mathbb{A}_0^2 X\end{aligned}$$ From the non-positivity of $\mathbb{A}_0^1 X$, the non-positivity of $\mathbb{A}_C X$, apparent from (\[locc\]), and the non-negativity of $X$, follows $$\label{lokc}
- \mathbb{E}_N \int_0^T \mathbb{A}_0^1 X(t) dt
- \mathbb{E}_N \int_0^T \mathbb{A}_C X(t) dt \leq \mathbb{E}_N X(0)+
\mathbb{E}_N \int_0^T \mathbb{A}_0^2 X(t) dt .$$ $\Box$\
Bounds on functionals of $u_{n,m}$ {#sttwo}
----------------------------------
We will verify the assertions presented in (\[jayeye\]). The following lemma provides the bounds on the behaviour of the functions $\big\{ u^\epsilon_{n,m}: \mathbb{R}^2 \to [0,\infty): (n,m) \in
\mathbb{N} \big\}$ and other functions that will be used in this section. We choose the constant $R_0$ so that $V(x)=0$ whenever $|x|\ge R_0$. Recall that $k=\sum_n nh_n$.
\[phiest\] There exists a collection of constants $C:
\mathbb{N}^2 \to (0,\infty)$ for which the following bounds hold.\
- for $x \in \mathbb{R}^2$ satisfying $|x|\le 2R_0{\varepsilon}$, $\big\vert u_{n,m}^{{\epsilon}}(x) \big\vert \leq C_{n,m} |\log {\varepsilon}|$, and for all $x \in \mathbb{R}^2$, $\big\vert u_{n,m}^{{\epsilon}}(x) \big\vert \leq C_{n,m} |\log
\vert x \vert |$.
- for $x \in \mathbb{R}^2$, $\big\vert \nabla u_{n,m}^{{\epsilon}} (x) \big\vert \leq
C_{n,m} \min \Big\{\frac{1}{\vert x \vert},\frac 1{{\varepsilon}}\Big\}$.
- for $x \in \mathbb{R}^2$, $$\label{lept}
\Big\vert u^\epsilon_{n,m} \big( x + z \big) - u^\epsilon_{n,m} \big( x
\big) \Big\vert \leq C_{n,m}\vert z \vert \min\left\{
\frac{1} {\vert x \vert},\frac 1{\varepsilon}\right\}$$
- for $x \in \mathbb{R}^2$ satisfying $\vert x \vert \geq \max \big\{ 2
\vert z \vert + R_0 \epsilon , 2 R_0 \epsilon \big\}$, $$\label{lepta}
\Big\vert \nabla u^\epsilon_{n,m} \big( x + z \big) - \nabla
u^\epsilon_{n,m} \big( x
\big) \Big\vert \leq \frac{C_{n,m} \vert z \vert}{\vert x \vert^2}.$$
- let $H=H_{n,m}: \mathbb{R}^2 \times {\mathbb R}^2\to [0,\infty)$ be given by $$H (x;z) = \frac{-1}{2 \pi} \int_{\vert x - y \vert \leq
1}{\log {\vert x - y \vert}}
u_{n,m}^\epsilon \big( y+z\big) 1 \! \! 1 \big\{ \vert y
\vert \leq \rho \big\} dy .$$ Then, $$\sup_{|z|\le 1}\int{ H (x_1-x_2;z)k(x_1)k(x_2)dx_1
dx_2} \leq C_{n,m} \rho^2 |\log \rho| .$$
- Let $\hat{H} \big( =
\hat{H}_{n,m} \big): \mathbb{R}^2\times {\mathbb R}^2 \to
[0,\infty)$ be given by $$\hat{H} (x;z) = \frac{-1}{2\pi} \int_{\vert x - y \vert \leq
1} {\log {\vert x - y \vert}} \ \Big\vert \nabla
u_{n,m}^\epsilon \big( y+z\big) \Big\vert 1 \! \! 1 \big\{ \vert y
\vert \leq \rho \big\} dy.$$ Then, for every $z$ with $|z|\le 1$, $$\label{lepto}
\int{ \hat{H} (x_1-x_2;z)k(x_1)k(x_2)dx_1
dx_2} \leq C_{n,m} \big( \rho + \vert z \vert \big) .$$
- for $(x,z) \in \mathbb{R}^2\times {\mathbb R}^2$, let $L (x;z)$ be given by $$\frac{-1}{2 \pi} \int_{\vert x - y \vert \leq 1} \log
{\vert x - y \vert} \left[\big|\log|1-|y+z||\big|+1\right]
1 \! \! 1 \big\{ 1 - \rho \leq
\vert y + z \vert \leq 1 + \rho \big\} dy.$$ Then we have the bound $$\label{eq2.20}
L(x;z)\le C\rho\big(\log\rho\big)^2 .$$
- for any positive integers $n$ and $m$ and a nonnegative smooth function $\overline J$ of compact support, there exists a constant $C_{n,m}(\overline J)$ such that, for any given $z \in
\mathbb{R}^2$, the function $A_{n_1,n_2,n_3}^{\epsilon}: \mathbb{R}^6 \to [0,\infty)$ defined by $$\begin{aligned}
& & \big( d(n_1 )\Delta_{x_1} + d(n_2) \Delta_{x_2} + d(n_3)
\Delta_{x_3} \big)
A_{n_1,n_2,n_3}^{\epsilon} (x_1,x_2,x_3) \nonumber \\
& & \qquad = - u_{n,m}^{\varepsilon}(x_1 - x_3 + z) V_{{\epsilon}}
\big( x_1 - x_2 \big) 1\!\!1 \big\{ |x_1-x_3|\le \rho \big\}
\overline{J}(x_3) \nonumber\end{aligned}$$ satisfies $$\begin{aligned}
& &\sum_{n_1,n_2,n_3}\int_{\mathbb{R}^6}{ A_{n_1,n_2,n_3}^{\epsilon} \big(
x_1,x_2,x_3 \big)
h_{n_1}(x_1)h_{n_2}(x_2)h_{n_3}(x_3) d x_1
d x_2 d x_3 } \nonumber\\
& & \qquad \qquad\leq C_{n,m}(\overline J) {\big\vert \log \epsilon \big\vert^{-1}}(\rho+|z|) \log
\big( \rho + \vert z \vert \big) .
\label{leptu}\end{aligned}$$
[**Proof**]{} Throughout the proof, we write $u^{\varepsilon}$ for the function $u_{n,m}^{\varepsilon}$ and $\tau$ for the constant ${\alpha}(n,m)/(d(n)+d(m))$. The dependence of the constants on $n$ and $m$ arises from that of $\tau$, and is also omitted. The first part of the Lemma is a straightforward consequence of our results in Section 3. As a consequence of Theorems 3.1-3.2 and Lemma 3.1 we know that there exists a constant $c_1\in (0,1)$ such that for small ${\varepsilon}$ and $y$ satisfying $|y|\le 2R_0$, $$\log {\varepsilon}\le u^{\varepsilon}({\varepsilon}y)\le c_1\log{\varepsilon},$$ or equivalently, $$\label{eq2.30}
0\le u^{\varepsilon}({\varepsilon}y)|\log {\varepsilon}|^{-1}+1\le 1-c_1.$$ From (2.1), we learn that for $x$ satisfying $|x|\ge 2{\varepsilon}R_0$, $${\Lambda}_{\varepsilon}\log(|x|/2)\le u^{\varepsilon}(x)\le \log(2|x|){\Lambda}_{\varepsilon},$$ where $${\Lambda}_{\varepsilon}=\frac 1{2\pi}\tau \int \Big( V_{\epsilon} (y ) {u^{\epsilon}}_{n,m} ( y ) + V^{\epsilon}( y ) \Big) dy.$$ From this and (2.22) we learn that there are two positive constants $k_1$ and $k_2$ such that if $|x|\ge 2{\varepsilon}R_0$, then $$k_1\log(|x|/2))\le u^{\varepsilon}(x)\le k_2 \log(2|x|).$$ To prove the second part of the lemma, recall firstly that $$u^{\epsilon} \big( x \big) = \frac \tau {2\pi}\int_{\mathbb{R}^2}{
\log{\vert x - y \vert} \Big( u^{\epsilon}(y)
V_{\epsilon}(y) + V^{\epsilon}(y) \Big) dy}.$$ As a result, $$\label{wopdo}
\nabla {u^{\epsilon}}(x) = \frac \tau {2\pi}\int_{\mathbb{R}^2}{
\frac{x-y}{\big\vert x - y \big\vert^2}
\Big( {u^{\epsilon}}(y)V_{{\epsilon}}(y) + V^{{\epsilon}}(y) \Big) dy}.$$ If $\vert x \vert\ge 2R_0 {\varepsilon}$, then $\big\vert x - y \big\vert \geq
\vert x \vert / 2$, and $$\bigg\vert \frac{x - y}{\vert x - y \vert^2} \bigg\vert =
\frac{1}{\vert x - y \vert} \le \frac{2}{\vert x \vert},$$ implying that $$\big\vert \nabla {u^{\epsilon}}(x) \big\vert \leq \frac{ 2{\Lambda}_{\varepsilon}}{\vert x \vert}\le
\frac 3{|x|} ,$$ for small ${\varepsilon}$. If $\vert x \vert\le 2R_0{\varepsilon}$, then we use (2.22) to deduce $$\big\vert \nabla {u^{\epsilon}}(x) \big\vert \le c_1
\int_{|x-y|\le 3R_0{\varepsilon}}{\frac{1}{\vert x - y \vert}
V^{\varepsilon}(y)dy} \leq c_2 {\epsilon}^{-2}
\int_0^{3R_0{\epsilon}}{dr} \leq c_3{\varepsilon}^{-1}.$$ Thus, $$\big\vert \nabla {u^{\epsilon}}(x) \big\vert \leq C \min \Big\{\frac 1{\varepsilon}, \frac{1}{\vert x
\vert} \Big\},$$ as claimed in the second part of the lemma.
To prove the third part of the lemma, note that $$\begin{aligned}
& & \Big\vert u^\epsilon \big( x + z \big) - u^\epsilon \big( x
\big) \Big\vert
\nonumber \\
& \leq & \frac 1{2\pi} \int_{\mathbb{R}^2}
\bigg\vert \log \frac{\vert x - y + z \vert}{\vert x - y
\vert} \bigg\vert
\bigg( u^{\epsilon}(y) V_{\epsilon}(y) + V^{\epsilon}(y) \bigg) dy
\nonumber \\
& \leq & \frac 1{2\pi} \int_{\mathbb{R}^2}
\bigg\vert \log \frac{\vert x - y + z \vert}{\vert x - y
\vert} \bigg\vert V^{\epsilon}(y) dy,
\label{dtc}\end{aligned}$$ the latter inequality by means of (2.22). From this and the elementary inequalities $$\label{rpp}
\log \frac{\vert x - y + z \vert}{\vert x - y \vert} -1
\leq \frac{\vert x-y+z
\vert}{\vert x - y \vert} \leq \frac{ \vert z \vert}{\vert x - y
\vert},$$ we deduce that $$\Big\vert u^\epsilon \big( x + z \big) - u^\epsilon \big( x
\big) \Big\vert
\leq \frac 1{2\pi} |z| \int_{\mathbb{R}^2}
\frac 1{2\pi} \int_{\mathbb{R}^2}
\frac{ V^{\epsilon}(y)}{|x-y|} dy$$ We now use this and argue as in the proof of the second part of the lemma to deduce the third part of the lemma.
In seeking to prove the fourth part of the lemma, note that $$\frac{x + z - y}{\big\vert x + z - y \big\vert^2} - \frac{x -
y}{\big\vert x - y \big\vert^2} = \frac{\big\vert x - y \big\vert^2 \big(x + z - y \big)
- \big\vert x + z - y
\big\vert^2\big( x - y \big) }{ \big\vert x + z - y \big\vert^2 \big\vert x - y \big\vert^2 }.$$ Note that, for any $a \in \mathbb{R}^2$, $$\begin{aligned}
\Big\vert \vert a \vert^2\big( a + z \big) - \vert a + z \vert^2 a
\Big\vert & \leq & \vert a \vert \Big\vert \vert a + z \vert^2 -
\vert a \vert^2 \Big\vert + \vert z \vert \vert a \vert^2 \nonumber
\\
& \leq & c_1 \vert z \vert \vert a \vert^2, \label{cule}\end{aligned}$$ so long as $|z|\le|a|$. Note that since by our assumption $|x|\ge 2|z|+R_0{\varepsilon}$, we have that $|x-y|\ge 2|z|$. We may apply (\[cule\]) with the choice $a = x - y$ to the formula (\[wopdo\]), hereby obtaining $$\Big\vert \nabla{u^{\epsilon}}( x + z ) - \nabla{u^{\epsilon}}(x)
\Big\vert \leq C \vert z \vert \int_{\mathbb{R}^2}{
\frac{V^{{\varepsilon}}(y)}{\big\vert x + z - y \big\vert^2}} dy,$$ where we used (2.22). From the inequality $\vert x \vert \geq \max \big\{ 2 \vert z \vert +
R_0 \epsilon , 2 R_0 \epsilon \big\}$, we deduce that $\big\vert x +
z - y \big\vert \geq \vert x -y \vert/2$ and $\vert x - y \vert \geq
\vert x \vert/2$. We conclude that $$\Big\vert \nabla{u^{\epsilon}}( x + z ) - \nabla{u^{\epsilon}}(x)
\Big\vert \leq \frac{C \vert z \vert}{\vert x \vert^2} ,$$ as required.
To prove the fifth part of the lemma, note that $$|H (x;z)| \leq C \int_{|x-y|\le 1}{ |\log |x-y|| \log \vert y + z
\vert\vert
1 \! \! 1 \big\{ \vert y \vert \leq \rho \big\} dy} \, ,$$ by the first part of the lemma. Hence, $$\begin{aligned}
& & \int{|H(x_1-x_2;z)|k(x_1)k(x_2)dx_1dx_2} \nonumber \\
& \leq & C \int_{\vert y \vert \leq
\rho}{ \big|\log \vert y + z
\vert \big| \int_{|x_1-x_2-y|\le 1}\big| \log
{\vert x_1 - x_2 - y \vert} \big| k(x_1) k(x_2) dx_1
dx_2 dy}
\nonumber \\
& \leq & C \int_{\vert
y \vert \leq \rho}{\big|\log \vert y \vert\big| dy} \nonumber \\
& \leq & C \int_0^{\rho}{r |\log r| d r}
\leq C \rho^2 |\log \rho| \nonumber,\end{aligned}$$ where in the second inequality, we used our first assumption on the initial data and the fact that if $\rho+|z|\le 1$, then the expression $\int_{\vert y \vert \leq \rho}|{\big|\log \vert
y+z \vert\big| dy}|$ is maximized as a function of $z \in \mathbb{R}^2$ when $z = 0$. This establishes the third part of the lemma.
To prove the sixth part of the lemma, note that, by the first part, $$\hat{H} (x;z) \leq
\frac{1}{2\pi} \int_{\vert x - y \vert \leq 1}{\big|
\log {\vert x - y \vert} \big| \frac{1}{\vert y + z
\vert} 1 \! \! 1 \big\{ \vert y \vert \leq \rho \big\} }dy.$$ It follows that $$\begin{aligned}
& & \int{|\hat H(x_1-x_2;z)|k(x_1)k(x_2)dx_1dx_2} \nonumber \\
& \leq & c_1 \int_{\vert y \vert \leq
\rho}{ \frac{1}{\vert y + z \vert} \int_{\vert x_1 - x_2 - y \vert
\leq 1}{ \big|\log {\vert x_1 - x_2 - y \vert} \big|
k(x_1)k(x_2)dx_1dx_2}dy} \nonumber \\
& \le & c_2 \int_{\vert y \vert \leq
\rho}{\frac{1}{\vert y + z \vert} dy}
\le c_2 \int_{\vert y+z \vert \leq
\rho+|z|}{\frac{1}{\vert y + z \vert} dy}\nonumber \\
& \leq & c_3 \big( \rho + \vert z \vert \big).\end{aligned}$$ We have deduced (\[lepto\]).
As for the seventh part of the lemma, first observe that $L(x;z)=L(x+z;0)$. Hence we only need to verify (\[eq2.20\]) when $z=0$. In this case we divide the domain of integration into the sets $|x-y|\le |1-|y||$ and $|1-|y||\le |x-y|\le 1$. Hence, $$\begin{aligned}
L(x;0) & \leq & \frac{1}{2\pi} \int_{\vert x - y \vert \leq \rho}
\left[\big|\log {\vert x - y \vert} \big|+\big|\log {\vert x - y \vert} \big|^2
\right] dy \nonumber\\
& & +
\frac{1}{2\pi} \int_{\big|1-|y|\big|\le \vert x - y \vert \le 1 }
\left[\big|\log {|1-\vert y \vert}| \big|+\big|\log {|1-\vert y \vert}| \big|^2
\right]1\!\!1\big(|y|\in (1-\rho,1+\rho)\big) dy
\nonumber\\
& \leq & \int_0^{\rho}{r \left[|\log r|+|\log r|^2\right] dr} \, + \,
\int_{1-\rho}^{1+\rho}\left[\big|\log|1-r|\big|+
\big|\log|1-r|\big|^2\right]rdr\nonumber \\
& \leq & C \rho (\log \rho)^2 , \nonumber\end{aligned}$$ establishing (\[eq2.20\]).
As for the eighth part of the lemma, let us write $J(a)$ for $1\!\!1 \big\{ |a|\le \rho \big\}$ and define the quantity $I$ according to $$\begin{aligned}
I & = & c_0(6)\sum_{n_1,n_2,n_3} {\gamma}(n_1,n_2)\int_{} dx_1 dx_2 dx_3
\int_{\mathbb{R}^{6}} \Big( \frac{\vert x_1 - z' \vert^2}{d(n_1)}
+ \frac{\vert x_2 -
y \vert^2}{d(n_2)} + \frac{\vert x_3 - y' \vert^2}{d(n_3)}
\Big)^{-2} \nonumber \\
& & \qquad h_{n_1}(x_1)h_{n_2}(x_2)h_{n_3}(x_3) u^{\varepsilon}( z' - y' +
z) V_{{\epsilon}} ( z' - y ) J (z'-y')
\overline{J}(y') dz' dy dy'. \nonumber\end{aligned}$$ We write $$I=c_0(6) \int_{\mathbb{R}^{6}} u^{\varepsilon}(z' - y' +
z) V_{{\epsilon}} ( z' - y ) J (z'-y')
\overline{J}(y') G(z',y,y')
dz' dy dy',$$ where $G(z',y,y')$ is given by $$\sum_{n_1,n_2,n_3 {\gamma}(n_1,n_2)\in \mathbb{N}}
\int
\bigg( \frac{\vert x_1 - z' \vert^2}{d(n_1)} + \frac{\vert x_2 -
y \vert^2}{d(n_2)} + \frac{\vert x_3 - y' \vert^2}{d(n_3)}
\bigg)^{-2} h_{n_1}(x_1)h_{n_2}(x_2)h_{n_3}(x_3) dx_1 dx_2 dx_3.$$ Using the elementary inequality $abc\le (a^2+b^2+c^2)^{3/2}$ we deduce that $G(z',y,y')$ is at most $$\begin{aligned}
& &
\sum_{n_1,n_2,n_3 \in \mathbb{N}} {\gamma}(n_1,n_2){\left(d(n_1)d(n_2)d(n_3)\right)}^{2/3}
\int
{\vert x_1 - z' \vert}^{-4/3}\ {\vert x_2 -
y \vert}^{-4/3} {\vert x_3 - y' \vert}^{-4/3} \nonumber \\
& & \qquad \qquad \qquad h_{n_1}(x_1)h_{n_2}(x_2)h_{n_3}(x_3) dx_1
dx_2 dx_3.\nonumber\end{aligned}$$ From our assumptions on $h_n$ we deduce that $G\in L^{{\infty}}_{loc}$. Hence, $$I \leq C \int_{\mathbb{R}^{6}} u^{\varepsilon}(z' - y' +
z) V_{{\epsilon}} ( z' - y ) J (z'-y')
\overline{J}(y') dz' dy dy'.$$ Note that, for fixed $z' \in \mathbb{R}^3$, $$\int_{\mathbb{R}^2}{V \Big( \frac{z' - y}{\epsilon}\Big) dy} =
\epsilon^2.$$ Thus, $$\begin{aligned}
I & \leq & C {\big\vert \log \epsilon \big\vert^{-1}}\int_{\mathbb{R}^4} u^{\varepsilon}(z' - y' +
z) J (z'-y') \overline{J}(y') dz' dy' \nonumber \\
& \leq & C {\big\vert \log \epsilon \big\vert^{-1}}\int_{K} \int_{\mathbb{R}^2} u^{\varepsilon}(z' - y' +
z) J(z'-y')dz' dy' \nonumber \\
& \leq & C {\big\vert \log \epsilon \big\vert^{-1}}\int_{K} dy' \int_{\mathbb{R}^2}\big| \log
\vert z' - y' + z\vert\big| J(z'-y')dz' \\
&\leq& C {\big\vert \log \epsilon \big\vert^{-1}}\int_{|a|\le \rho+|z|}\big| \log \vert a \vert \big|da
\leq C {\big\vert \log \epsilon \big\vert^{-1}}(\rho+|z|)^2 \big|\log \big( \rho + \vert z \vert \big)\big| , \nonumber\end{aligned}$$ where $K \subseteq \mathbb{R}^2$ denotes a compact set containing the support $\overline{J}$, and where we made use of the first part of the lemma in the third inequality. This is the bound stated in (\[leptu\]). $\Box$
Estimating the terms {#stthree}
--------------------
### The case of $H_{14}$ and $H_4$
The estimate of $\mathbb{E}_N \int_{0}^{T}{\vert H_4 (t) \vert dt}$ is derived in an identical fashion to that of $\mathbb{E}_N \int_{0}^{T}{\vert H_{14} (t) \vert dt}$. Note that $$\begin{aligned}
\mathbb{E}_N \Big\vert \int_0^T H_{14} (t) dt \Big\vert & \leq &
C {\big\vert \log \epsilon \big\vert^{-2}}\mathbb{E}_N \int_0^T dt \sum_{i,j \in
I_q}{ \Big\vert {u^{\epsilon}}\big( x_i - x_j + z \big) - {u^{\epsilon}}\big( x_i - x_j \big) \Big\vert } \nonumber \\
& & \qquad \quad 1\!\!1 \Big\{ m_i = M_1, m_j = M_2 \Big\},
\nonumber\end{aligned}$$ where the constant $C$ depends on the $L^{\infty}$ bounds satisifed by $J,\overline{J}$ and their time derivatives. Hence $$\mathbb{E}_N \Big\vert \int_0^T H_{14} (t) dt \Big\vert
\leq K_1 + K_2,$$ where $K_1$ is given by $$\begin{aligned}
& & C {\big\vert \log \epsilon \big\vert^{-2}}\mathbb{E}_N \int_0^T{} dt\ \sum_{i,j \in I_q:
\vert x_i - x_j \vert >
\rho}{\Big\vert {u^{\epsilon}}(
x_i - x_j + z ) - {u^{\epsilon}}(
x_i - x_j ) }\Big\vert \nonumber \\
& & \qquad \qquad \qquad \qquad \qquad \qquad 1 \! \! 1 \{ m_i =
M_1 \} 1 \! \! 1 \{ m_j =
M_2 \} \nonumber\end{aligned}$$ and $K_2$ is given by $$\begin{aligned}
& & {\big\vert \log \epsilon \big\vert^{-2}}\mathbb{E}_N \int_0^T{} dt\ \sum_{i,j \in I_q:
\vert x_i - x_j \vert \leq
\rho}{\Big\vert {u^{\epsilon}}(
x_i - x_j + z ) - {u^{\epsilon}}(
x_i - x_j ) } \Big\vert \nonumber \\
& & \qquad \qquad \qquad \qquad \qquad \qquad 1 \! \! 1 \{ m_i =
M_1 \} 1 \! \! 1 \{ m_j = M_2 \} . \nonumber\end{aligned}$$ Firstly, we treat $K_1$. Note that $$K_1 \leq \frac{C \vert z \vert {\big\vert \log \epsilon \big\vert^{-2}}}{\rho}
\mathbb{E}_N \int_0^T{}
dt\ \sum_{i,j \in I_q: \vert x_i - x_j \vert > \rho}{ 1 \! \! 1 \{ m_i = M_1
\} 1 \! \! 1 \{ m_j = M_2 \} } \leq \frac{C
\vert z \vert Z^2 }{\rho} ,$$ where the first inequality follows from the third part of Lemma \[phiest\], and the final one from the initial number of particles $N$ equals $Z|\log{\varepsilon}|$.
We now treat the term $K_2$. By writing, $$\begin{aligned}
K_2 & \leq & C {\big\vert \log \epsilon \big\vert^{-2}}\mathbb{E}_N \int_0^T{} dt\ \sum_{i,j \in
I_q: \vert x_i - x_j \vert \leq
\rho}{
\bigg[ \big|{u^{\epsilon}}\big(
x_i - x_j + z \big)\big| + \big|{u^{\epsilon}}\big(
x_i - x_j \big)\big| \bigg] } \nonumber \\
& & \qquad \qquad \qquad \qquad \qquad \qquad 1 \! \! 1 \{ m_i =
M_1 \} 1 \! \! 1 \{ m_j = M_2 \} , \nonumber\end{aligned}$$ we obtain an expression on the right-hand-side which may be bounded by applying Lemma \[lemtwop\]. As a result we can write $K_2\le K_{21}+K_{22}$ where $K_{21}$ and $K_{22}$ represent the first and the second term on the right-hand-side in Lemma \[lemtwop\]. For $K_{21}$, the relevant estimate is provided by the fifth part of Lemma \[phiest\], with a bound of $C\rho^2 |\log \rho|$. To bound the term $K_{22}$, note that, with the function $J$ in Lemma \[lemtwop\] chosen to be $J(x) = u^{\epsilon}(x + z)1\!\!1(|x|\le \rho)$, we have that $$\begin{aligned}
|\tilde{J}(x) |& = &\left| \int_{\vert y \vert = 1}{u^{\epsilon}\big( x + y +
z \big) 1 \! \! 1 \big\{ \vert x + y \vert \leq \rho \big\} }S(dy)\right| \nonumber
\\
& \leq & C \int_{\vert y \vert = 1}{\big|\log \vert x + y + z \vert\big| 1 \!
\! 1 \big\{ \vert x + y \vert \leq \rho \big\}}S(dy) \nonumber \end{aligned}$$ Let us assume that $|z|\le \rho$ ans set $a=x+z$. We then have $$\begin{aligned}
|\tilde{J}(x) |
& \leq & C \int_{\vert y \vert = 1}{|\log \vert a +y \vert| 1 \!
\! 1 \big\{ \vert a + y \vert \leq 2\rho \big\}}S(dy) \\
& \leq & 2C
\int_0^{c_2\rho}{|\log (c_1 r)| dr} \leq c_3 \rho |\log \rho| ,\end{aligned}$$ where for the second inequality we used that fact that the conditions $|a+y|\le \rho, \ |y|=1$ mean that the point $y$ belongs to an arc on the unit circle of center $\sigma=a/|a|$ and length $2c_2 \rho$, and if the lenght of the arc $\sigma y$ is $r$, then $|a+y|\ge c_1 r$ for positive constants $c_1 $ and $c_2$.
In this way, we find that $\tilde J$ is uniformly bounded by $C\rho |\log\rho|$ and this in turn implies that the term $K_{22}$ is bounded above by $$C \big\vert \log {\epsilon}\big\vert^{-2} \rho |\log\rho|
\mathbb{E}_N \int_0^T dt \Big\vert \Big\{ (i,j ) \in I_q^2 : m_i =
M_1, m_j = M_2 \Big\} \Big\vert \leq C Z^2 \rho |\log\rho|,$$ the latter inequality following from $N \leq Z \big\vert \log {\epsilon}\big\vert$. We find that $$K_2 \leq C \rho |\log\rho| \, .$$ Hence, $$\mathbb{E}_N \left|\int_0^T{} H_{13}(t) \right| dt \leq K_1 + K_2
\leq \frac{C
\vert z \vert }{\rho} + C \rho |\log\rho| .$$ Setting $\rho = \vert z \vert^{1/2}$, we find that $$\mathbb{E}_N \left|\int_0^T{} H_{13}(t) \right| dt \leq C \vert z
\vert^{\frac{1}{2}} \big|\log {\vert z \vert} \big| .$$
### The cases of $H_2$ and $H_3$
The estimate of $\mathbb{E}_N \int_{0}^{T}{\vert H_3 (t) \vert dt}$ is derived in an identical fashion to that of $\mathbb{E}_N \int_{0}^{T}{\vert H_2 (t) \vert dt}$. Picking $\rho \in \mathbb{R}$ that satisfies $\rho \geq \max \big\{ 2 \vert z
\vert + R_0 \epsilon , 2 R_0 \epsilon \big\}$, we write $$\label{eq2.28}
\int_{0}^{T}{\mathbb{E}_N{\vert H_2 (t) \vert} dt} \leq R_1 + R_2,$$ where $$\begin{aligned}
R_1 & = & {\big\vert \log \epsilon \big\vert^{-2}}\mathbb{E}_N{} \int_{0}^{T}{\sum_{i,j \in
I_q: \vert x_i
- x_j \vert > \rho}{d(m_i) \left|\overline{J}(x_j,m_j,t)\right|}} \nonumber \\
& & \qquad \quad \Big\vert {u^{\epsilon}}_x ( x_i - x_j + z ) -
{u^{\epsilon}}_x ( x_i - x_j ) \Big\vert
\big\vert J_x \big( x_i,m_i,t \big) \big\vert dt
, \nonumber\end{aligned}$$ and $$\begin{aligned}
R_2 & = & {\big\vert \log \epsilon \big\vert^{-2}}\mathbb{E}_N \int_{0}^{T}{\sum_{i,j \in
I_q: \vert x_i
- x_j \vert \leq \rho}{d(m_i) \big|\overline{J}(x_j,m_j,t)\big|}} \nonumber \\
& & \qquad \quad
\Big\vert {u^{\epsilon}}_x( x_i - x_j + z ) -
{u^{\epsilon}}_x ( x_i - x_j ) \Big\vert
\big\vert J_x \big( x_i,m_i,t \big) \big\vert dt . \nonumber\end{aligned}$$ Firstly, we examine the sum $R_1$. Recalling that we consider test functions $J$ and $\overline{J}$ respectively supported on particles of mass $M_1$ and $M_2$, $$R_1 \leq C {\big\vert \log \epsilon \big\vert^{-2}}\frac{\vert
z \vert}{\rho^2} d(M_1)
\Big\vert \{ (i,j) \in I_q^2: m_i = M_1, m_j = M_2 \} \Big\vert,$$ where the lower bound on $\rho$ allowed us to apply the fourth part of Lemma \[phiest\]. Thus, $$\label{eq2.29}
R_1 \leq C \frac {\vert z \vert}{\rho^2}.$$
Secondly, we bound the sum $R_2$. Note that $$\begin{aligned}
R_2 & \leq & {\big\vert \log \epsilon \big\vert^{-2}}\vert \vert J_x \vert \vert \ \vert \vert
\overline{J} \vert \vert \ d(M_1)
\ \mathbb{E}_N \int_0^T{}dt \sum_{i,j \in I_q: \vert x_i - x_j
\vert \leq \rho} \nonumber \\
& & \qquad \Big\vert
{u^{\epsilon}}_x(x_i - x_j + z ) - {u^{\epsilon}}_x(
x_i - x_j ) \Big\vert 1 \! \! 1 \{ m_i = M_1, m_j = M_2 \} \nonumber \\
& \leq & C {\big\vert \log \epsilon \big\vert^{-2}}\ \mathbb{E}_N \int_0^T{}dt \sum_{i,j \in I_q}{m_i
m_j \big( d(m_i) + d(m_j) \big)} \nonumber \\
& & \qquad \qquad \quad \bigg[ \Big\vert
{u^{\epsilon}}_x(x_i - x_j + z ) \Big\vert + \Big\vert {u^{\epsilon}}_x(
x_i - x_j ) \Big\vert \bigg] 1 \! \! 1 \Big\{ \big\vert x_i - x_j
\big\vert \leq \rho \Big\} , \label{fhg}\end{aligned}$$ where $\|\cdot \|$ denotes the $L^{\infty}$ norm and the constant $C$ depends on the test functions $J$ and $\overline{J}$. The expression (\[fhg\]) is written in a form to which Lemma \[lemtwop\] may be applied. Doing so yields $$\begin{aligned}
R_2 & \leq & C {\big\vert \log \epsilon \big\vert^{-2}}\mathbb{E}_N \sum_{i,j
\in I_q}{m_i m_j \Big( \hat{H} (x_i - x_j;0) + \hat{H} (x_i - x_j;z)
\Big) } \nonumber \\
& & \, + \, C {\big\vert \log \epsilon \big\vert^{-2}}\ \mathbb{E}_N \int_0^T{}dt \sum_{i,j \in I_q}{m_i
m_j \big( d(m_i) + d(m_j) \big)} \nonumber\\
& & \qquad \qquad \quad \Big[
\tilde{J}\big( x_i - x_j ;z \big) +
\tilde{J}\big( x_i - x_j ;0 \big) \Big] \nonumber\\
&=& : R_{21}+R_{22}\label{eq2.31}\end{aligned}$$ where the function $ \hat{H} $ appears in the sixth part of Lemma \[phiest\], and where $\tilde{J}: \mathbb{R}^2\times {\mathbb R}^2 \to [0,\infty)$ in this case is given by $$\tilde{J}(x;z) = \frac 1{2\pi}\int_{\vert y \vert = 1}{\big\vert u_x^{{\epsilon}}(x+y+z)
\big\vert 1 \! \! 1 \big\{ \vert x + y \vert \leq \rho \big\} dS(y) }.$$
From the sixth part of Lemma \[phiest\] and our assumptions on the initial data, we deduce that $$\label{eq2.32}
R_{21}\le C(\rho + \vert z \vert).$$
It follows from the second part of Lemma \[phiest\] that the function $\tilde{J}$ satisfies the bound $$|\tilde{J}(x;z)|\le C\int_{\vert y \vert = 1}\frac 1{\big\vert x+y+z \big\vert}
1 \! \! 1 \big\{ \vert x + y \vert \leq \rho \big\} dS(y) .$$ By our assumption, we certainly have $|z|\le \rho/2$. Hence, $$|\tilde{J}(x;z)|\le C\int_{\vert y \vert = 1}\frac 1{\big\vert a+y
\big\vert }1 \! \! 1 \big\{ \vert a + y \vert \leq 2\rho \big\} dS(y) ,$$ for $a=x+z$. Note that there exists a positive constant $c_1$ such that the conditions $|a+y|\le 2\rho,\ |y|=1$ mean that $y\in {\Gamma}$, where ${\Gamma}$ is an arc of the unit circle with the center $\sigma=-a/|a|$. It is not hard to show that there exist positive constants $c_1$ and $c_2$ such that $$c_1\big(\ell+\big|1-|a|\big|\big)\le |a+y|\le c_2\big(\ell+\big|1-|a|\big|\big),$$ where $\ell$ denotes the length of the arc from $\sigma=-a/|a|$ to $y$ on the unit circle. From this we deduce $$\begin{aligned}
|\tilde{J}(x;z)|&\le &1 \! \! 1 \big\{ 1-\rho\le
\vert a \vert \leq 1+\rho \big\}\int
1\!\!1\left({c_1c_2^{-1}\big|1-|a|\big|}\le \ell\le {2c_1^{-1}\rho}\right)
\frac {c_3}\ell d\ell\\
&\le& c_3 1 \! \! 1 \big\{ 1-\rho\le \vert x+z \vert \leq 1+\rho \big\}
\left(\big|\log|1-|x+z||\big|+\log\big(c_2c_1^{-2}\big)\right) \end{aligned}$$
Using this bound on $\tilde J$, and then applying Lemma \[lemtwop\], we learn that the term $R_{22}$ is bounded above by $R_{221}+R_{222}$, where $$R_{221} = C \big\vert \log \epsilon \big\vert^{-2} \mathbb{E}_N
\sum_{i,j \in I_{q(0)}}{\Big( L \big( x_i - x_j;0 \big) + L \big(
x_i - x_j;z \big) \Big) m_i m_j},$$ and, $$R_{222}= 2C \big\vert \log \epsilon \big\vert^{-2} \mathbb{E}_N
\int_0^T \Big( \tilde{L} \big( x_i - x_j \big) + \tilde{L}
\big( x_i - x_j+z \big) \Big) m_i m_j \big( d(m_i) + d(m_j) \big) dt,$$ where the function $L: \mathbb{R}^2 \to [0,\infty)$ appears in the seventh part of Lemma \[phiest\], and where the function $\tilde{L} : \mathbb{R}^2 \to [0,\infty)$ is given by $$\tilde{L}(x) =\frac 1{2\pi} \int_{\vert y \vert = 1}{ 1 \! \! 1 \big\{ 1 - \rho
\leq\vert x + y \vert \leq 1 + \rho \big\} \left[\big|\log|1-|x+y||\big|+1
\right] d S(y) }$$ By the seventh part of Lemma \[phiest\] and our assumption on the initial total density $k=\sum_n nh_n\in L^1$ we obtain $$\label{eq2.33}
R_{221} \leq C\rho
(\log \rho )^2.$$
We decompose $R_{222}=R_{2221}+R_{2222}$ where $$R_{222r}= 2 \big\vert \log \epsilon \big\vert^{-2} \mathbb{E}_N
\int_0^T \Big( {L}_r \big( x_i - x_j \big) + {L}_r
\big( x_i - x_j+z \big) \Big) m_i m_j \big( d(m_i) + d(m_j) \big) dt,$$ for $r=1,2$, where $L_1(x)=\tilde L(x)1\!\!1(|x|\ge \rho^{1/4})$ and $L_2(x)=\tilde L(x)1\!\!1(|x|\le\rho^{1/4})$. Let us now analyse the behaviour of the function $\tilde L$. First observe that since the Lebsegue measure on the circle is rotationally invariant, we have that the function $\tilde L$ is radially symmetric. Because of this, let us assume that $x=(a,0)$ and $y=(cos\theta,\sin\theta)$. Note that $$|x+y|^2=1+a^2+2a\cos\theta.$$ As a result, the condition $|x+y|\in (1-2\rho,1+2\rho)$ for $\rho\le 1$ implies that $|a^2+2a\cos\theta|\le 8\rho$. Let us first examine the case $|a|=|x|\ge \rho^{1/4}$. In this case we have that $|a+2\cos\theta|\le 8\rho^{3/4}$. This condition is not satisfied unless $|a|\le 2$. In that case, choose $\theta_0\in [0,\pi]$ such that $a+2\cos\theta_0
=0$. Note that for small$\rho$, the set $\{\theta:|a+2\cos\theta|\le 8\rho^{3/4}\}$ is a union of two disjoint $\theta$–intervals about the points $\theta_0$ and $\theta_1=2\pi-\theta_0$. We now argue that there exists a positive constant $c_1$ such that the length of these intervals is bounded above by $c_1\rho^{1/2}$. To see this, first observe that the condition $|a|\ge \rho^{1/4}$ implies that for a positive constant $c_2$ we have that $|\theta_0|,|\theta_0-\pi|\ge 2c_2\rho^{1/4}$. This and $|a+2\cos \theta|\le 8 \rho^{3/4}$ implies that we also have $|\theta|,|\theta-\pi|\ge c_2\rho^{1/4}$ provided that $\rho$ is sufficiently small. On the other hand since $$2\cos \theta+a=-2(\sin \tau) (\theta-\theta_0),$$ for some $\tau$ between $\theta$ and $\theta_0$, we deduce that for some positive constant $c_1$, we have that $|\theta-\theta_0|\le c_1\rho^{1/2}$. Also, there exist positive constants $c_3$ and $c_4$ such that if $\theta $ is close to $\theta_r$, then $$\big|1-|x+y|\big|\ge c_3|a^2+2a\cos\theta|\ge c_4 \sqrt \rho |\theta-\theta_r|,$$ for $r=0$ or $1$. From this we learn that if $|x|\ge \rho^{1/4}$, then the term $|\tilde L(x)|$ is bounded above by $$\frac 1{2\pi}\int_{\theta_0-c_1\sqrt\rho}^{\theta_0+c_1\sqrt\rho}
\left[|\log \left(c_4\sqrt \rho |\theta-\theta_0|\right)|+1\right]d\theta
+\frac 1{2\pi}\int_{\theta_1-c_1\sqrt\rho}^{\theta_1+c_1\sqrt\rho}
\left[|\log \left(c_4\sqrt \rho |\theta-\theta_1|\right)|+1\right]d\theta.$$ As a result, $|\tilde L(x)|\le C\sqrt\rho|\log\rho|$. This in turn implies that $$R_{2221}\le C\sqrt \rho|\log\rho|$$ We now turn to $R_{2222}$. For this, observe that the support of the function $L_2$ is contained in the set of points $x$ for which $|x|\le
\rho^{1/4}$. Note that if $|a|\le \rho^{1/4}$, then we can find a positive constant $c_5$ such that $|\theta_0-\pi/2|\le c_5\rho^{1/4}$, $|\theta_1-3\pi/2|\le c_5\rho^{1/4}$, where $\theta_1=2\pi-\theta_0$. Forthermore, we can find a positive constant $c_6$ such that if $\theta\in (0,\pi)$, then $$\label{eq2.37}
\big|a^2+2a\cos\theta\big|=\big|2a(\cos\theta_0-\cos\theta)\big|
=2\left|a\sin\frac {\theta+\theta_0}2\sin\frac{\theta-\theta_0}2\right|
\ge c_6 |a(\theta-\theta_0)|.$$ The same is true if $\theta\in [\pi,2\pi]$ (use $\theta_1$ in place of $\theta_0$ in (\[eq2.37\]).) As a result, $$\big|1-|x+y|\big|\ge c_3 |a^2+2a\cos\theta|\ge c_3c_6 |a(\theta-\theta_0)|.$$ From this we learn that indeed $$L_2(x)\le C\big|\log |x|\big|1\!\!1\big\{|x|\le \rho^{1/4}\big\}.$$ To bound $R_{2222}$, let us apply Lemma 2.2 one more time to write $R_{2222}\le
R_{22221}+R_{22222}$, where $$R_{22221} = C \big\vert \log \epsilon \big\vert^{-2} \mathbb{E}_N
\sum_{i,j \in I_{q(0)}}{\Big( \Gamma \big( x_i - x_j \big) + \Gamma \big(
x_i - x_j+z \big) \Big) m_i m_j}$$ and $$R_{22222}= \big\vert \log \epsilon \big\vert^{-2} \mathbb{E}_N
\int_0^T \Big( \tilde{\Gamma} \big( x_i - x_j \big) + \tilde{\Gamma}
\big( x_i - x_j+z \big) \Big) m_i m_j \big( d(m_i) + d(m_j) \big) dt,$$ where the function $\Gamma: \mathbb{R}^2 \to [0,\infty)$ is very similar to the function $H$ that appeared in the fifth part of Lemma \[phiest\] (except that $\rho$ in the definition $H$ is replaced with $\rho^{1/4}$), and where the function $\tilde{\Gamma} : \mathbb{R}^2 \to [0,\infty)$ is given by $$\tilde{\Gamma}(x) = \int_{\vert y \vert = 1}{ 1 \! \! 1 \big\{ \vert
x + y \vert \leq \rho^{1/4} \big\} \big|\log|x+y|\big| d S(y) }.$$ As in the fifth part of Lemma 2.4 we show $$\label{eq2.35}
R_{22221}\le C\rho|\log\rho|.$$ In just the same way that we bounded $\tilde J$ in the subsection 3.3.1, we can readily show that $\tilde \Gamma(x)\le
C\sqrt\rho|\log \rho|$. This in turn implies $$\label{eq2.36}
R_{22222}\le C\rho^{1/4}|\log \rho|.$$ Putting all the pieces together we learn from (\[eq2.28\])–(\[eq2.29\]) and (\[eq2.31\])–(\[eq2.36\]) that $$\int_0^T{\mathbb{E}_N{\vert H_2(t) \vert} dt}
\leq C \left[ \frac{\vert z \vert}{\rho^2} + \rho + \vert z \vert
+{\rho}^{1/4}
\big|\log\rho \big|,\right]$$ for $\rho\le 1$. By making the choice $\rho = \vert z
\vert^{\frac{4}{9}} $, we find that $$\int_0^T{\mathbb{E}_N{\vert H_2(t) \vert} dt} \leq C \vert z
\vert^{\frac{1}{9}}\big|\log|z|\big|.$$
### The case of $G_z(1) - G_0(1)$
We now estimate the term $$\int_{0}^{T}{ \mathbb{E}_N \big\vert G_z(1) - G_0(1) \big\vert (t) dt}.$$ To ease the notation, we do not display the dependence of $J$ and $\overline J$ on the variable $t$. Note that $$\label{eq2.40}
\int_{0}^{T}{ \mathbb{E}_N \big\vert G_z(1) - G_0(1) \big\vert (t) dt}
\leq \sum_{i=1}^{8}{D_i},$$ where $$\begin{aligned}
D_1 & = & \mathbb{E}_N \int_0^T{} dt \sum_{k,l \in
I_q}{{\alpha}(m_k,m_l)V_{\epsilon}(x_k - x_l)} \big\vert
J(x_k,m_k)\big\vert \nonumber\\
& & \quad \epsilon^{2(d-2)} \sum_{i \in I_q}{
\big\vert\overline{J}(x_i,m_i)\big\vert \Big\vert
{u^{\epsilon}}(x_k - x_i + z) - {u^{\epsilon}}(x_k - x_i )
\Big\vert} , \nonumber\end{aligned}$$ each of the other seven terms on the right-hand-side of (\[eq2.40\]) differing from $D_1$ only in an inessential way. Given this. the estimates involved for each of the eight cases are in essence identical, and we examine only the case of $D_1$. We write $D_1 =
D^1 + D^2$, where we have decomposed the inner $i$-indexed sum according to the respective index sets $$\{ i \in I_q, i \not= k,l, \vert x_k - x_i \vert > \rho \} \
\textrm{and} \ \{ i \in I_q, i \not= k,l, \vert x_k - x_i \vert \leq \rho \}.$$ By the second part of Lemma \[phiest\], we have that $$D^1 \leq \frac{C \vert z \vert
{\big\vert \log \epsilon \big\vert^{-1}}}{\rho} \mathbb{E}_N \int_0^T{} dt \sum_{k,l \in
I_{q}}{\alpha(m_k,m_l) V_{\epsilon}(x_k - x_l)}, \nonumber$$ where we have also used the fact that the test functions $J$ and $\overline{J}$ are each supported on the set of particles of respective masses $M_1$ and $M_2$, and the fact that the total number of particles living at any given time is bounded above by $Z \big\vert \log \epsilon \big\vert$. From the bound on the collision that is provided by Lemma \[lembc\], it follows that $$D^1 \leq \frac{C \vert z \vert}{\rho} .$$
Note that $D^2$ is bounded above by $$\begin{aligned}
& & C \mathbb{E}_N \int_0^T{} dt \sum_{k,l \in
I_{q}}{\alpha(m_k,m_l) V_{\epsilon}(x_k - x_l) } 1 \! \! 1 \{ m_k =
M_1 \} {\big\vert \log \epsilon \big\vert^{-2}}\sum_{i \in I_q}\nonumber \\
& & \quad {} 1 \! \! 1 \{ \vert x_i - x_k
\vert \leq \rho\} 1 \! \! 1 \{ m_i = M_2 \} \Big\vert
{u^{\epsilon}}(x_k - x_i + z) - {u^{\epsilon}}(x_k - x_i ) \Big\vert \big|J(x_i,m_i)\big|
\nonumber \\
& \leq & C \mathbb{E}_N \int_0^T{} dt \sum_{k,l \in
I_{q}}{\alpha(m_k,m_l)V_{\epsilon}(x_k - x_l)} 1 \! \! 1 \{ m_k = M_1
\} {\big\vert \log \epsilon \big\vert^{-2}}\sum_{i \in I_q}\nonumber \\
& & \quad {} 1 \! \! 1 \{ \vert x_i - x_k
\vert \leq \rho \} 1 \! \! 1 \{ m_i = M_2 \} \Big[
|{u^{\epsilon}}(x_k - x_i + z)| + |{u^{\epsilon}}(x_k - x_i )| \Big]\big|J(x_i,m_i)\big| . \nonumber\end{aligned}$$ Note that the last expectation is bounded by Lemma \[lemthreep\] because, by our assumption on ${\alpha}$, we can find ${\gamma}$ such that ${\alpha}\le {\gamma}$, with ${\gamma}$ satisfying the assumption of Lemma 2.3. The upper bound provided by this Lemma in this particular application is computed in the last part of Lemma \[phiest\]. We find that $D^2 \leq C (\rho+|z|) \log \big( \rho +
\vert z \vert \big)$.
Combining these estimates yields $$D_3 \leq D^1 + D^2 \leq C \frac{\vert z \vert}{\rho} +
C (\rho+|z|) \big|\log \big( \rho + \vert z \vert \big) \big|.$$ Making the choice $\rho = \vert z \vert^{\frac{1}{2}} $ leads to the inequality $D_3 \leq \vert z \vert^{\frac{1}{2}}\big|\log \vert z \vert\big|$. Since each of the cases of $\big\{ D_i : i \in \{ 1,\ldots, 8 \} \big\}$ may be treated by a nearly verbatim proof, we deduce that $$\int_{0}^{T}{ \mathbb{E}_N \big\vert G_z(1) - G_0(1) \big\vert (t) dt}
\leq C \vert z \vert^{\frac{1}{2}} \log \vert z \vert .$$
### The case of $G_z(2)$
Recall that $$G_z(2) = - {\big\vert \log \epsilon \big\vert^{-2}}\sum_{k,l \in
I_{q}}{ \alpha(m_k,m_l) V_{\epsilon}(x_k - x_l)} {u^{\epsilon}}(x_k -
x_l + z ) J(x_k,m_k)
\overline{J}(x_l,m_l).$$ If $k,l \in I_q$ satisfy $V_\epsilon \big( x_k - x_l \big) \not= 0$, then $\vert x_k -
x_l \vert \leq R_0 \epsilon$, and so $$\vert x_k - x_l + z \vert \geq \vert z \vert - R_0 \epsilon
\geq \vert z \vert/2 ,$$ provided that $|z|\ge 2R_0{\varepsilon}$. This implies that $$\big|{u^{\epsilon}}(x_k - x_l + z )\big| \leq
C \Big\vert \log \vert x_k - x_l + z \vert \Big\vert
\leq C \Big\vert \log \vert z \vert \Big\vert ,$$ where in the first inequality, we used the first part of Lemma \[phiest\] (restated). Applying this bound, and using the fact that the test functions $J$ and $\overline{J}$ have compact support, we find that $$\int_0^T{} \mathbb{E}_N \vert G_z (2) \vert dt \leq C \Big\vert \log
\vert z \vert \Big\vert
{\big\vert \log \epsilon \big\vert^{-2}}\mathbb{E}_N
\int_0^T{} \sum_{k,l \in I_q}{ \alpha(m_k,m_l) V_{\epsilon}(x_k - x_l)} dt$$ whose right-hand-side is bounded above by $C \big\vert \log \vert z
\vert \big\vert {\big\vert \log \epsilon \big\vert^{-1}}$, according to Lemma \[lembc\]. That is, $$\int_0^T{\vert G_z(2) \vert dt} \leq C \big\vert \log \vert z
\vert \big\vert {\big\vert \log \epsilon \big\vert^{-1}}.$$
### The case of $\mathbb{E}_N \vert X_z - X_0 \vert $ {#lak}
We now turn to $\mathbb{E}_N \vert X_z - X_0 \vert $. Assume that $|z|\ge R_0{\varepsilon}$. Using the second part of Lemma 2.4, we have that $$\mathbb{E}_N \vert X_z - X_0 (0)\vert \le C|z|
\iint_{L^2} h_{M_1}(x)h_{M_2}(y)|x-y|^{-1}
dxdy,$$ where $L$ is a bounded set that contains the support of $J$ and $\overline J$. Using our second assumption on the initial data $h_n$ we obtain the bound $C|z|$ for $\mathbb{E}_N \vert X_z - X_0 \vert(0)$.
[The martingale term]{}\[stfour\] This section is devoted to proving the estimate (\[martest\]). Note that $$M_z(T) = X_z \big( q(T),T \big) - X_z \big( q(0),0 \big) -
\int_{0}^{T}{\left(\frac \partial {\partial t}+\mathbb{L} \right)X_z
\big( q(t),t) \big) dt}$$ is a martingale which satisfies $$\mathbb{E}_N \Big[ M_z(T)^2 \Big] = \sum_{i=1}^{3}{}
\mathbb{E}_N\int_{0}^{T}{A_i \big( q(t)
,t
\big) dt},$$ where $A_1(q,t)$ and $A_2(q,t)$ are respectively set equal to $$\begin{aligned}
& & 2 \big\vert \log {\epsilon}\big\vert^{-4} \sum_{i \in I_q, m_i = M_1}{d(M_1)}
\nonumber \\
& & \qquad \qquad \quad
\bigg| \nabla_{x_i}
\sum_{j \in I_q, m_j = M_2}{{u^{\epsilon}}(x_i - x_j + z)
J(x_i,M_1,t) \overline{J}(x_j,M_2,t) } \bigg|^2, \nonumber\end{aligned}$$ and $$\begin{aligned}
& & 2 \big\vert \log {\epsilon}\big\vert^{-4}
\sum_{i \in I_q, m_i = M_2}{d(M_2)}
\nonumber \\
& & \qquad \qquad \quad
\bigg| \nabla_{x_i}
\sum_{j \in I_q, m_j = M_1}{{u^{\epsilon}}(x_j - x_i + z)
J(x_j,M_1,t) \overline{J}(x_i,M_2,t) } \bigg|^2, \nonumber\end{aligned}$$ while $A_3$ is given by $$\begin{aligned}
& & \big\vert \log {\epsilon}\big\vert^{-4} \sum_{i,j \in I_q}{\alpha(m_i,m_j)
V_{{\epsilon}} ( x_i - x_j ) } \label{aatwo} \\
& & \qquad \quad \bigg\{ \sum_{k \in I_q}{} \Big[
\frac{m_i}{m_i + m_j} {u^{\epsilon}}(x_i - x_k+z ) J(x_i,m_i +
m_j) \overline{J}(x_k,m_k) \nonumber \\
& & \qquad \qquad \qquad + \
\frac{m_i}{m_i + m_j} {u^{\epsilon}}(x_k - x_i + z) J(x_k,m_k)
\overline{J}(x_i,m_i + m_j) \nonumber \\
& & \qquad \qquad \qquad + \
\frac{m_j}{m_i + m_j} {u^{\epsilon}}(x_j - x_k + z) J(x_j,m_i +
m_j) \overline{J}(x_k,m_k) \nonumber \\
& & \qquad \qquad \qquad + \
\frac{m_j}{m_i + m_j} {u^{\epsilon}}(x_k - x_j + z) J(x_k,m_k)
\overline{J}(x_j,m_i + m_j) \nonumber \\
& & \qquad \qquad \qquad - \ {u^{\epsilon}}(x_i - x_k + z)
J(x_i,m_i) \overline{J}(x_k,m_k)
- \ {u^{\epsilon}}(x_k - x_i + z)
J(x_k,m_k) \overline{J}(x_i,m_i) \nonumber \\
& & \qquad \qquad \qquad - \ {u^{\epsilon}}(x_j
- x_k + z) J(x_j,m_j) \overline{J}(x_k,m_k)
- \ {u^{\epsilon}}(x_k
- x_j + z) J(x_k,m_k) \overline{J}(x_j,m_j) \Big] \nonumber \\
& & \qquad \qquad \qquad \qquad - \ {u^{\epsilon}}(x_i
- x_j + z) J(x_i,m_i) \overline{J}(x_j,m_j)
\bigg\}^2. \nonumber\end{aligned}$$ Recall that, by our convention, we do not display the dependence of $J$ and $\overline J$ on the $t$-variable. To bound these terms, we require two variants of Lemma 2.3 :
\[tpln\] There exists a collection of constants $C: \mathbb{N}^2 \to
(0,\infty)$ such that, for any continuous functions $t,v,a_1,a_2,a_3: \mathbb{R}^2 \to
[0,\infty)$ and any $z\in {\mathbb R}^2$, $$\begin{aligned}
& & \mathbb{E}_N\int_0^T{}dt \sum_{i,j,k \in I_q(t)}{}{\gamma}(m_i,m_j)t ( x_i -
x_j+z ) v ( x_i - x_k+z )
\nonumber \\
& & \qquad \qquad \qquad a_1(x_i)
a_2(x_j) a_3(x_k) 1 \! \! 1 \big\{ m_i = n_1 ,
m_k = n_3 \big\} \nonumber \\
& & \qquad \qquad \quad \leq C_{n_1,n_3} |\log{\varepsilon}|^{3}\sum_{n_2}
\mathbb{E}_N
\sum_{i,j,k \in I_{q(0)} }A_{n_1,n_2,n_3}^{\epsilon}(x_i,x_j,x_k) , \nonumber\end{aligned}$$ where $A_{n_1,n_2,n_3}^{\epsilon}: \mathbb{R}^6 \to [0,\infty)$ is given by $$\begin{aligned}
A_{n_1,n_2,n_3}^{\epsilon}(x_1,x_2,x_3) & = &
c_0(6){\gamma}(n_1,n_2)\int_{\mathbb{R}^6} \bigg(
\frac{\vert x_1 - z' \vert^2}{d(n_1)} + \frac{\vert x_2 - y
\vert^2}{d(n_2)} + \frac{\vert x_3 - y' \vert^2}{d(n_3)}
\bigg)^{-2} \nonumber \\
& & \quad t ( z' - y+z ) v
( z' - y'+z ) a_1(z') a_2(y) a_3(y') dz' dy dy, \nonumber\end{aligned}$$ with ${\gamma}$ as in Lemma 2.3.
\[fpln\] There exists a collection of constants $C: \mathbb{N}^3 \to
[0,\infty)$ such that, for any $z\in {\mathbb R}^d$, any continuous functions $v,w: \mathbb{R}^2 \to [0,\infty)$ and another $(a_1,a_2,a_3): \mathbb{R}^6 \to [0,\infty)$, $$\begin{aligned}
& & \mathbb{E}_N\int_0^T{}dt \sum_{k,l,i,j \in I_q}{}{\gamma}(n_i,n_j)V_{{\varepsilon}}( {x_i -
x_j}) v ( x_i - x_k+z ) w
( x_i - x_l+z ) a_1( x_i)a_2(x_k)a_3(x_l) \nonumber \\
& & \qquad \qquad \qquad 1 \! \! 1 \big\{ m_i = n_1 , m_k = n_3,
m_l = n_3 \big\} \nonumber \\
& & \qquad \quad \leq C_{n_1,n_3,n_3} \big\vert \log \epsilon \big\vert^4
\sum_{n_2}\mathbb{E}_N \sum_{i,j,k,l
\in I_{q(0)}}{B_{m_i,m_j,m_k,m_l}^{\epsilon}(x_i,x_j,x_k,x_l)} \nonumber
\\
& & \qquad \qquad \qquad \qquad 1 \! \! 1 \big\{ m_i \leq n_1 , m_j
\leq n_2, m_k \leq n_3,
m_l \leq n_3 \big\} , \nonumber\end{aligned}$$ where $B_{n_1,n_2,n_3,n_3}^{\epsilon}: \mathbb{R}^8 \to [0,\infty)$ is given by $$\begin{aligned}
& & B_{n_1,n_2,n_3,n_3}^{\epsilon} \big( x_1,x_2,x_3,x_3 \big)
\nonumber \\
& = & c_0(8)\int_{\mathbb{R}^{8}} \bigg(
\frac{\vert x_1 - \hat{z} \vert^2}{d(n_1)} + \frac{\vert x_2 - z'
\vert^2}{d(n_2)} + \frac{\vert x_3 - y \vert^2}{d(n_3)} +
\frac{\vert x_3 - y' \vert^2}{d(n_3)}
\bigg)^{-3} \nonumber \\
& & \qquad {\gamma}(n_1,n_2)V_{\varepsilon}( {\hat{z} - z'}) v ( \hat{z} - y+z )
w ( z' - y'+z ) a_1( \hat{z})a_2(y)a_3( y') d \hat{z}
dz' dy dy', \nonumber\end{aligned}$$ with the function ${\gamma}:\mathbb{N}^2\to (0,{\infty})$ satisfying $$n_2 {\gamma}\big( n_1, n_2 + n_3 \big) \max{ \Big\{ 1, \Big[ \frac{d (
n_2 + n_3 )}{d(n_2)}
\Big]^3 \Big\}} \leq \big( n_2 + n_3 \big) {\gamma}(n_1,n_2).$$
The proof of Lemma 2.5 is identical to that of Lemma 2.3. The proof of Lemma 2.6 is very similar to the proof of Lemma 2.3 and is omitted.
We now bound the three terms. Of the first two, we treat only $A_1$, the other being bounded by an identical argument. By multiplying out the brackets appearing in the definition of $A_1$, we obtain that this quantity is bounded above by $$\begin{aligned}
& & C \big\vert \log \epsilon \big\vert^{-4}
\sum_{i,j,k \in I_q} \big\vert \nabla {u^{\epsilon}}\big\vert ( x_i - x_j + z ) \big\vert
\nabla {u^{\epsilon}}\big\vert ( x_i - x_k + z ) J^2(x_i,m_i) \nonumber \\
& & \qquad \qquad
\big|\overline{J}(x_j,m_j)\big| \big|\overline{J}(x_k,m_k)\big| 1
\! \! 1 \big\{
m_i = M_1 , m_j = m_k = M_2
\big\}\\
& + & C \big\vert \log \epsilon \big\vert^{-4}
\sum_{i,j,k \in I_q} \left|{u^{\epsilon}}( x_i - x_j + z )\right|\left| {u^{\epsilon}}( x_i -
x_k + z )\right| |\nabla J(x_i,m_i)|^2 \nonumber \\
& & \qquad \qquad
\big|\overline{J}(x_j,m_j)\big| \big|\overline{J}(x_k,m_k)\big| 1
\! \! 1 \big\{
m_i = M_1 , m_j = m_k = M_2
\big\} . \label{yhj}\end{aligned}$$ Let us assume that $z=0$ because this will not affect our arguments. We are required to bound the quantity appearing in the statement of Lemma \[tpln\], for each of the following cases: $$\label{twocases}
\big( t,v,a_1,a_2,a_3 \big) \in \Big\{ \big( \vert\nabla {u^{\epsilon}}\vert , \vert \nabla {u^{\epsilon}}\vert,
J^2, \big|\overline{J}\big|, \big|\overline{J}\big|\big) \ , \ \big(
{u^{\epsilon}}, {u^{\epsilon}}, \vert \nabla J \vert^2 , \big|\overline{J}\big|,
\big|\overline{J}
\big|
\big) \Big\}.$$ Recall that each of the test functions $J$, $\overline{J}$, and their gradients, is assumed to be uniformly bounded with compact support. To each of the two cases, Lemma \[tpln\] applies. For either of them, the right-hand-side of the inequality in Lemma \[tpln\] may be written as a finite sum of the expectations appearing there, with the sum being taken over triples of given masses $n_1,n_2$ and $n_3$. Such an expectation is bounded above by $$\begin{aligned}
& & C \big\vert \log \epsilon \big\vert^{-4} \sum_{n_2 \in \mathbb{N}}
\int \int_{K^3} \bigg(
\frac{\vert x_1 - z' \vert^2}{d(n_1)} + \frac{\vert x_2 - y
\vert^2}{d(n_2)} + \frac{\vert x_3 - y' \vert^2}{d(n_3)}
\bigg)^{-2} \label{yun} \\
& & \qquad \qquad t ( z' - y )
v ( z' - y' ) h_{n_1}(x_1)h_{n_2}(x_2)h_{n_3}(x_3) dz'
dy dy' d x_1 d x_2 d x_3 , \nonumber\end{aligned}$$ where $K=\{x:|x|\le \ell\} \subseteq \mathbb{R}^2$ is chosen to contain the support of $J$ and $\overline{J}$. As in Section 3.4 of [@HR], we can use our bounds in the first two parts of Lemma 2.4 and repeat the proof of the eighth part of Lemma 2.4 to obtain $$\label{eq2.44}
\mathbb{E}_N\int_0^T\left[A_1 \big( q(t) \big)+A_2 \big( q(t) \big)
\right] dt \leq C {\big\vert \log \epsilon \big\vert^{-1}}.$$ We must treat the third term, $A_3$. An application of the inequality $$\Big( a_1 + \ldots + a_n \Big)^2 \leq n \Big( a_1^2 + \ldots + a_n^2 \Big)$$ to the bound on $A_3$ provided in (\[aatwo\]) implies that $$\label{aathr}
A_3(q) \leq 9 \big\vert \log \epsilon \big\vert^{-4}
\sum_{i,j \in I_q}{\alpha (m_i,m_j)
V_{{\epsilon}} ( x_i - x_j ) \bigg[
\sum_{n=1}^{8}{\Big( \sum_{k \in I_q}Y_n \Big)^2 } + Y_9^2 \bigg]},$$ where $Y_1$ is given by $$\frac{m_i}{m_i + m_j} {u^{\epsilon}}(x_i - x_k + z ) J(x_i,m_i +
m_j) \overline{J}(x_k,m_k),$$ and where $\big\{ Y_i : i \in \{ 2, \ldots, 8 \} \big\}$ denote the other seven expressions in (\[aatwo\]) that appear in a sum over $k
\in I_q$, while $Y_9$ denotes the last term in (\[aatwo\]) that does not appear in this sum. There are nine cases to consider. The first eight are practically identical, and we treat only the fifth. Note that $$\begin{aligned}
& & \big\vert \log \epsilon \big\vert^{-4}
\sum_{i,j \in I_q}{\alpha (m_i,m_j) V_{\varepsilon}(
x_i - x_j ) \Big(
\sum_{k \in I_q}Y_5 \Big)^2 } \nonumber\\
& =& C \big\vert \log \epsilon \big\vert^{-4} \sum_{i,j \in I_q}{\alpha(m_i,m_j)
V_{\varepsilon}( x_i - x_j )} \nonumber \\
& & \qquad \Big[ \sum_{k,l \in I_q}{{u^{\epsilon}}( x_i - x_k + z )
{u^{\epsilon}}( x_i - x_l + z )
J^2(x_i,m_i) \overline{J}(x_k,m_k) \overline{J}(x_l,m_l) } \Big].
\label{zvc}\end{aligned}$$ In the sum with indices involving $k,l \in I_q$, we permit the possibility that these two may be equal, though they must be distinct from each of $i$ and $j$ (which of course must themselves be distinct by the overall convention).
Note that the expression (\[zvc\]) appears in the statement of Lemma \[fpln\], provided that the choice $$\Big( v,w ,a_1,a_2,a_3\Big) = \Big( \big|{u^{\epsilon}}\big|, \big|{u^{\epsilon}}\big|,
J^2 ,\big| \overline{J}\big|,
\big|\overline{J}\big|\Big)$$ is made. Again we set $z=0$ because this does not affect the estimates. Given that the support of each of the functions $a_1,a_2,a_3:
\mathbb{R}^{6} \to
[0,\infty)$ is bounded, we must bound $$\begin{aligned}
& & \sum_{n_2 \in \mathbb{N}}\int \int_{L^4} \Big(
\frac{\vert x_1 - \hat{z} \vert^2}{d(n_1)} + \frac{\vert x_2 - z'
\vert^2}{d(n_2)} + \frac{\vert x_3 - y \vert^2}{d(n_3)} +
\frac{\vert x_3 - y' \vert^2}{d(n_3)}
\Big)^{-3} V \Big( \frac{\hat{z} - z'}{\epsilon} \Big)\nonumber \\
& & \qquad \big|{u^{\epsilon}}( \hat{z} - y )\big|
\big| {u^{\epsilon}}( z' - y' )\big| h_{n_1}(x_1)h_{n_2}(x_2)h_{n_3}(x_3)h_{n_3}(x_3)
d \hat{z} dz' dy dy' d x_1 d x_2 d x_3 dx_3 , \nonumber\end{aligned}$$ for a compact set $L$. This expression is bounded above by $$\begin{aligned}
& & \int_{L^3} V \Big( \frac{\hat{z} - z'}{\epsilon} \Big) {u^{\epsilon}}( \hat{z} - y ) {u^{\epsilon}}( z' - y' ) d \hat{z} dz' dy dy' \nonumber \\
& & \qquad \sum_{n_2 \in \mathbb{N}}{ \int_{K^3} \bigg(
\frac{\vert x_1 - \hat{z} \vert^2}{d(n_1)} + \frac{\vert x_2 - z'
\vert^2}{d(n_2)} + \frac{\vert x_3 - y \vert^2}{d(n_3)} +
\frac{\vert x_3 - y' \vert^2}{d(n_3)}
\bigg)^{-3} } \nonumber \\
& & \qquad \qquad
h_{n_1}(x_1)h_{n_2}(x_2)h_{n_3}(x_3)h_{n_3}(x_3)dx_1 dx_2 dx_3
dx_3 , \nonumber\end{aligned}$$ which is less than $$C \int_{L^3} V \Big( \frac{\hat{z} - z'}{\epsilon} \Big) {u^{\epsilon}}( \hat{z} - y ) {u^{\epsilon}}( z' - y' ) d \hat{z} dz' dy dy'.$$ The proof of this follows the proof of the eighth part of Lemma 2.4; we use the elementary inequality $abcd\le (a^2+b^2+c^2+d^2)^2$ and the fact that the function $$\hat k(x)=\sum_nd(n)^{3/4}\int h_n(y)|x-y|^{-3/2}dy$$ is locally bounded. Noting that the bound $\vert {u^{\epsilon}}(x) \vert \leq
\big\vert \log\vert x \vert \big\vert$ implies that $$\int_{L}{u^{\varepsilon}(\hat z-y )dy}$$ is bounded above by a constant, we find that $$\int_{L^3} V \Big( \frac{\hat{z} - z'}{\epsilon} \Big)
{u^{\epsilon}}( \hat{z} - y ) {u^{\epsilon}}( z' - y' ) d\hat{z} dz' dy dy'
\leq C \int_{L^2} V \Big( \frac{\hat{z} -
z'}{\epsilon} \Big) d \hat{z} dz'.$$ This is at most $ C{\varepsilon}^2$. Applying Lemma \[fpln\], we find that the contribution to $$\mathbb{E}_N{\int_{0}^{T}{A_3 \big(q(t)\big) dt}}$$ arising from the fifth term in (\[aathr\]) is at most $$C {\epsilon}^{-2}
\big\vert^{-1} {\epsilon}^2
= C \big\vert \log {\epsilon}\big\vert^{-1}.$$
We now treat the ninth term, as they are classified in (\[aathr\]). It takes the form $$\begin{aligned}
& & \big\vert \log \epsilon \big\vert^{-4}
\sum_{i,j \in I_q}{\alpha(m_i,m_j)
V_{\varepsilon}\Big( {x_i - x_j} \Big)} \nonumber \\
& & \qquad \qquad \qquad
{u^{\epsilon}}\big(x_i - x_j + z \big)^2
J \big( x_i,m_i \big)^2 \overline{J} \big( x_j,m_j \big)^2. \nonumber\end{aligned}$$ This is bounded above by $$C \big\vert \log \epsilon \big\vert^{-3}
\sum_{i,j \in I_q}{\alpha(m_i,m_j)
V_{\varepsilon}\Big( {x_i - x_j} \Big)},$$ because $u^{\varepsilon}\le C \big\vert \log \epsilon \big\vert $ in the support of $V_{\varepsilon}$ by the first part of Lemma 2.4. The expected value of the integral on the interval of time $[0,T]$ of this last expression is bounded above by $$C \big\vert \log \epsilon \big\vert^{-3}
\mathbb{E}_N \int_0^T{dt} \sum_{i,j \in I_q}{\alpha(m_i,m_j)
V_{{\varepsilon}}(x_i - x_j)}\le C \big\vert \log \epsilon \big\vert^{-1} ,$$ where we used Lemma \[lembc\] for the last inequality. This completes the proof of (\[martest\]).
Using the estimates {#stfive}
-------------------
The inequalities (\[abc\]), (\[jayeye\]) and (\[martest\]) imply that, for large $T$, $$\lim_{\vert z \vert \to 0}\limsup_{\epsilon \downarrow
0}\mathbb{E}_N \bigg\vert\int_0^T H_{11} \big( t \big) dt +
\int_0^T H_{13} \big( t \big) dt\bigg\vert dt = 0.$$ That is, $$\begin{aligned}
& & \lim_{\vert z \vert \to 0}{\limsup_{\epsilon \downarrow
0}} {\big\vert \log \epsilon \big\vert^{-2}}\mathbb{E}_N \bigg\vert
\int_{0}^{T}{dt \sum_{i,j \in
I_{q(t)}}{\alpha(m_i,m_j) J(x_i,m_i) \overline{J}(x_j,m_j)} } \nonumber \\
& & \qquad \qquad \Big[ V^{\epsilon} (x_i - x_j + z) -
V^{\epsilon} ( x_i - x_j ) +
V_{\epsilon}(x_i - x_j + z) {u^{\epsilon}}(x_i - x_j + z)
\Big] \ \bigg\vert =\ 0. \nonumber\end{aligned}$$ (Recall that we simply write $J(x_i,m_i)$ and $\overline J(x_i,m_i)$ for $J(x_i,m_i,t)$ and $\overline J(x_i,m_i,t)$.) This implies that $$\begin{aligned}
& &
{\big\vert \log \epsilon \big\vert^{-2}}\int_{0}^{T} {\sum_{i,j \in
I_{q(t)}}{\alpha(m_i,m_j)
V^{\epsilon} ( x_i - x_j ) J(x_i,m_i)}
\overline{J}(x_j,m_j) } dt\nonumber \\
& = & {\big\vert \log \epsilon \big\vert^{-2}}\int_{0}^{T} {\sum_{i,j \in
I_{q(t)}}{\alpha(m_i,m_j)
V^{\epsilon}( x_i - x_j + z ) J(x_i,m_i)
\overline{J}(x_j,m_j)}}\nonumber \\
& & \qquad \qquad \qquad \ \Big[ 1 +
{\big\vert \log \epsilon \big\vert^{-1}}{u^{\epsilon}}(x_i - x_j
+ z) \Big] dt \ + \ Err_1(\epsilon,z) , \label{otwdfw}\end{aligned}$$ where $Err_1$ satisfies $$\label{lati}
\lim_{\vert z \vert \to 0}{\limsup_{\epsilon \downarrow
0}}{ \, \mathbb{E}_N \big\vert Err_1
\big( \epsilon,z \big) \big\vert} = 0.$$ By Theorem 3.2, the expression $1 + |\log{\varepsilon}|^{-1} u_{m_1,m_2}^{\varepsilon}\big( a
\big)$ is uniformly close to $$\left(1-\frac {{\tau}(m_1,m_2)}{2\pi+{\tau}(m_1,m_2)}\right),$$ for $a$ satisfying $V^{\varepsilon}(a)\neq 0$ and ${\tau}(m_1,m_2)={\alpha}(m_1,m_2)/(d(m_1)+
d(m_2))$. Recalling from (\[qudef\]) that $$Q = {\big\vert \log \epsilon \big\vert^{-1}}\sum_{(i,j) \in
I_{q}}{\alpha(m_i,m_j) V_{\epsilon}(x_i - x_j) J(x_i,m_i)
\overline{J}(x_j,m_j)}$$ and writing $$\label{quzdef}
\overline{Q} (z) = {\big\vert \log \epsilon \big\vert^{-1}}\sum_{i,j \in I_q}{{\beta}(m_i,m_j) V^{\varepsilon}(
{x_i - x_j + z
}) J(x_i,m_i)
\overline{J}(x_j,m_j) },$$ it follows from (\[otwdfw\]) and Theorem 3.2 that $$\label{weakstoss}
\int_0^T Q (t) dt =
\int_0^T \overline{Q} (z)(t) dt + Err_2\big(\epsilon,z \big),$$ where $Err_2$ satisfies $$\lim_{\vert z \vert \to 0}{\limsup_{\epsilon \downarrow
0}}{ \, \mathbb{E}_N \big\vert Err_2
\big( \epsilon,z \big) \big\vert} = 0.$$ From this, it is not hard to deduce that $$\begin{aligned}
\overline{Q}(z_2 - z_1) & = & |\log{\varepsilon}|^{-2}
\sum_{i,j \in I_q}{{\beta}(m_i,m_j) V^{\varepsilon}({x_i - x_j + z_2 - z_1 })}
\label{strc} \\
& & \qquad \qquad \qquad J(x_i - z_1,m_i)
\overline{J}(x_j - z_2,m_j) \, + \, Err(\epsilon,z_1,z_2), \nonumber\end{aligned}$$ where $$\mathbb{E}_N \big\vert Err(\epsilon,z_1,z_2)\big\vert \leq
C \big( \vert z_1 \vert + \vert
z_2 \vert \big).$$ (See Section 3.5 of [@HR].) By (\[weakstoss\]) and (\[strc\]), $$\begin{aligned}
& & \int_0^T Q (t) dt \nonumber \\
& = & |\log{\varepsilon}|^{-2}\int_{0}^{T}dt
\sum_{i,j \in I_{q(t)}}{\beta}(m_i,m_j) J(x_i - z_1,m_i)
\overline{J}(x_j - z_2,m_j)\nonumber \\
& & \quad \int_{\mathbb{R}^2}{\int_{\mathbb{R}^2}{V^{\varepsilon}( (x_i - z_1) - (x_j - z_2)) {\delta}^{-2}
\eta \Big( \frac{z_1}{\delta} \Big) {\delta}^{-2} \eta \Big(
\frac{z_2}{\delta} \Big) dz_1
dz_2 }} \, + \, Err_3\big(\epsilon,\delta\big) \nonumber \\
& = & |\log{\varepsilon}|^{-2} \int_{0}^{T} dt
\int_{\mathbb{R}^{2d}} d \omega_1 d \omega_2 \sum_{i,j
\in I_{q(t)}}{}V^{\varepsilon}( {\omega_1 -\omega_2}) {\beta}(m_i,m_j) J(\omega_1,m_i)
\overline{J}(\omega_2,m_j) \nonumber \\
& & \qquad \quad
\delta^{-2} \eta \Big( \frac{x_i - \omega_1}{\delta} \Big)
\delta^{-2} \eta \Big( \frac{x_j - \omega_1}{\delta} \Big)
\, + \, Err_3 \big( \epsilon, \delta \big) \nonumber \\
& = & \int_0^T{} dt \int_{\mathbb{R}^2} \int_{\mathbb{R}^2}
d\omega_1 d\omega_2 v^{\varepsilon}( {\omega_1 -
\omega_2}) {\beta}(M_1,M_2) J(\omega_1,M_1)
\overline{J}(\omega_2,M_2) \nonumber \\
& & \qquad \quad \bigg[ {\big\vert \log \epsilon \big\vert^{-1}}\sum_{i \in I_q: m_i = M_1}{
\delta^{-2} \eta \Big( \frac{x_i - \omega_1}{\delta} \Big) }
\bigg] \nonumber \\
& & \qquad \quad \bigg[ {\big\vert \log \epsilon \big\vert^{-1}}\sum_{j \in I_q; m_j = M_2}{
\delta^{-2} \eta \Big(\frac{x_j - \omega_2}{\delta} \Big)
} \bigg] \, + \, Err_3 \big( \epsilon, \delta
\big) ,\nonumber\end{aligned}$$ where $Err_3$ satisfies $$\lim_{\delta \downarrow 0}\limsup_{\epsilon_\downarrow 0} \mathbb{E}_N
\big\vert Err_3 \big( \epsilon, \delta \big) \big\vert = 0,$$ and where in the last equality, we made use of the fact that the test functions $J$ and $\overline{J}$ take non-zero values only on particles of a given mass, respectively $M_1$ and $M_2$. Thus, $$\begin{aligned}
& & \int_0^T Q (t) dt \nonumber \\
& = &
\int_0^T{} dt \int_{\mathbb{R}^2}
d\omega {\beta}(M_1,M_2) J(\omega,M_1)
\overline{J}(\omega,M_2) \bigg[ {\big\vert \log \epsilon \big\vert^{-1}}\sum_{i \in I_q: m_i = M_1}{
\delta^{-2} \eta \Big( \frac{x_i - \omega}{\delta} \Big) }
\bigg] \nonumber \\
& & \qquad \quad \bigg[ {\big\vert \log \epsilon \big\vert^{-1}}\sum_{j \in I_q; m_j = M_2}{
\delta^{-2} \eta \Big(\frac{x_j - \omega}{\delta} \Big)
} \bigg] \, + Err \big( \epsilon, \delta
\big) \nonumber\end{aligned}$$ where $Err=Err_5+err$ with the function $err=O\big({\varepsilon}{\delta}^{-5}\big)$ also satisfies $$\lim_{\delta \downarrow 0}\limsup_{\epsilon_\downarrow 0} \mathbb{E}_N
\big\vert err \big( \epsilon, \delta \big) \big\vert = 0.$$ This completes the proof of Proposition 1.
[Potential theory]{} \[sec3\]
The purpose of this section is twofold. Firstly, we show the existence of the function $u^{\varepsilon}$ that satisfies (3.1). Secondly we evaluate the limit of $u^{\varepsilon}|\log{\varepsilon}|^{-1}$ in the support of $V^{\varepsilon}$, as ${\varepsilon}\to 0$. This limit was used in the evaluation of ${\beta}$ in Section 2.5. We start with the statements of the main results of this section. Let $V: {{\mathbb R}}^d \to {{\mathbb R}}$ be a continuous function of compact support with $V \ge
0$ and ${\displaystyle}{\int_{\mathbb{R}^2} V(x)dx = 1}$. We also write $K_0$ for the topological closure of $U_0$ where $$U_0 = \{x: V(x) \ne 0\}.$$ Given a measure $\mu$, let us define $${{\mathcal G}}\mu(x)=\int \log |x-y| \mu(dy).$$ When the measure $\mu$ is absolutely continuous with respect to the Lebesgue measure with a density $g$, we simply write ${{\mathcal G}}g$ for ${{\mathcal G}}\mu$.
\[th3.1\] There exists a number ${\gamma}_0>0$ such that for every ${\gamma} \in (0, {\gamma}_0)$ and $a\in {\mathbb R}$, there exists a unique function $u \in
C^2({{\mathbb R}}^2)$ such that $u(x)= O\big(\big|\log |x|\big|\big)$ as $|x|\to{\infty}$ and $$\label{eq3.2}
u = {\gamma} {{\mathcal G}}\big((u+a )V\big).$$ Moreover $Z={\gamma} \int (u+a)Vdx\ne 0$ and $(u+a)Z^{-1}\ge 0$.
Recall that we are searching for a function $u^{\varepsilon}$ such that $$\Delta u^{\varepsilon}={\tau}(n,m)\left[V_{\varepsilon}u^{\varepsilon}+V^{\varepsilon}\right],$$ where $\tau={\tau}(n,m)={\alpha}(n,m)/(d(n)+d(m))$. For this it suffices to have $$\label{eq3.3}
u^{\varepsilon}={{\mathcal G}}\mu^{\varepsilon},$$ where $\mu^{\varepsilon}(dx)=\frac 1{2\pi} {\tau} (V_{\varepsilon}u^{\varepsilon}+V^{\varepsilon}) dx$. This can be rewritten as $$u^{\varepsilon}=\frac 1{2\pi} {\tau}^{\varepsilon}{{\mathcal G}}\left(V^{\varepsilon}\left[ u^{\varepsilon}+|\log{\varepsilon}|\right]\right),$$ where ${\tau}^{\varepsilon}={\tau} |\log{\varepsilon}|^{-1}$. Evidently we can apply Theorem 3.1 to deduce the existence of the function $u^{\varepsilon}$ for sufficiently small ${\varepsilon}$.
Our next theorem was used in the previous section for the evaluation of ${\beta}$.
\[th3.2\] For every positive $k$, $$\lim_{{\varepsilon}\to 0}\sup_{|x|\le k}\left|u^{{\varepsilon}}({\varepsilon}x)|\log {\varepsilon}|^{-1}+\frac {\tau}{2\pi+{\tau}}\right|=0.$$
[**Proof of Theorem \[th3.1\]**]{}\
[**Step 1.**]{} Let $J$ be a bounded continuous function with $J > 0$ and $${\int_{|x| \ge 1}
J(x)\left(\log |x|\right)^{2}dx <
{\infty}}.$$ Define $$\label{eq3.5}
{{\mathcal H}}= \left\{ u : \mbox{ $u$ is measurable and }\int_{\mathbb{R}^2}
u^2(x) J(x) dx <
{\infty}\right\}.$$ We then define ${{\mathcal F}}: {{\mathcal H}}\to {{\mathcal H}}$ by ${{\mathcal F}}(u)={{\mathcal G}}(uV)$. Observe that ${{\mathcal H}}$ is a Hilbert space with respect to the inner product $${\langle}u,v{\rangle}= \int_{\mathbb{R}^2} u(x)v(x) J(x)dx.$$ Let us verify that ${{\mathcal F}}$ is a bounded operator. To see this, write $$\label{eq3.6}
{\Gamma}(x) = \int_{\mathbb{R}^2} \big |\log|x-y|\big |V(y)dy.$$ When $|x|$ is sufficiently large, we have that ${\Gamma}(x)\le
\log (2|x|)$ because $0<\log|x-y|\le \log(2|x|)$ whenever $V(y)\ne 0$. Otherwise we have $${\Gamma}(x)\le c_0\int_{|x-y|\le c_1
}\big |\log|x-y|\big |dy\le c_2 ,$$ for constants $c_0,c_1$ and $c_2$. As a result $$\label{eq3.7}
{\Gamma}(x)\le c+\log^+|x|$$ for a constant $c$. Also, we may use Hölder’s inequality to assert $$\begin{aligned}
\label{eq6.10}
({{\mathcal F}}(u)(x))^2 &\le &\left[
{\Gamma}(x) \int_{\mathbb{R}^2} \big |\log|x-y|\big |V(y)|u(y)| \frac
{dy}{{\Gamma}(x)} \right]^2
\\
&\le & {\Gamma}(x)\int_{\mathbb{R}^2} \big |\log|x-y|\big |V(y)u^2(y)dy. \nonumber\end{aligned}$$ From this we deduce $$\int_{\mathbb{R}^2} ({{\mathcal F}}(u)(x))^2J(x)dx \le \int_{\mathbb{R}^2}
V(y)u^2(y) \left[
\int_{\mathbb{R}^2}
{\Gamma}(x)\big |\log|x-y|\big |J(x)dx\right]dy.$$ If $V(y) \ne 0$ then $|y| \le R_0$ for a suitable $R_0$. Define $$I(y) = \int_{\mathbb{R}^2} {\Gamma}(x)\big |\log|x-y|\big |J(x)dx,$$ Note $$\begin{aligned}
I(y) &= &\int_{|x| \le 2R_0} + \int_{|x| > 2R_0}
{\Gamma}(x)\big|\log|x-y|\big| J(x)dx \\
&\le &c_1 \int_{|x-y| \le 3R_0} \big |\log|x-y|\big |dx + c_1 \int_{|x| > 2R_0}
\big|\log|x|\big|^2 J(x)dx ,\end{aligned}$$ where for the second line we have used . From this and our assumption on $J$ we deduce that ${\displaystyle}{\sup_{|y| \le R_0} I(y) < {\infty}}$. As a result, $$\int_{\mathbb{R}^2}({{\mathcal F}}(u)(x))^2 J(x)dx \le c_1 \int_{\mathbb{R}^2}
V(y)u^2(y)dy \le c_2 \int_{\mathbb{R}^2}
u^2(y)J(y)dy$$ because $V$ is of compact support and $J>0$. This shows the boundedness of the operator ${{\mathcal F}}:{{\mathcal H}}\to {{\mathcal H}}$.\
[**Step 2.**]{} Since the operator ${{\mathcal F}}$ is bounded, the equation $$(id - {\gamma}{{\mathcal F}})(u) = g$$ has a solution, where $g(x) = -{\gamma}a{\Gamma}(x)$ with ${\Gamma}$ as in and $id$ denotes the identity transformation. Note that our assumption on ${\Gamma}$ implies that ${\Gamma}\in
{{\mathcal H}}$ because of . So far we have shown the existence of a unique solution $u \in {{\mathcal H}}$ of $u - {\tau} {{\mathcal F}}(u) = g$. From this and the Hölder continuity of $V$ we can readily show that in fact $u\in C^2$ and that $u$ is a classical solution of $$\Delta u=2\pi \gamma (u+a)V.$$ (See for example Section 4.2 of [@GT].)\
[**Step 3.**]{} In this step we verify $Z\ne 0$. Observe that $u={{\mathcal G}}\mu$ for a measure $\mu$ with a bounded support. From this we can readily deduce $$\label{eq3.9}
u(x)=\mu\left({\mathbb R}^2\right)\log|x|+O\left(|x|^{-1}\right),$$ $$\label{eq3.10}
\nabla u(x)=
\mu\left({\mathbb R}^2\right)\frac {x}{|x|^2}+O\left(|x|^{-2}\right).$$ We now choose $R > R_0$ and use ${\Delta}u =2\pi{\gamma} (u+a)V$ to write $$\int_{|x| \le R} (u+a){\Delta}u dx =2\pi {\gamma}\int_{|x| \le R} V(u+a)^2dx.$$ After an integration by parts we obtain $$- \int_{|x| \le R} |\nabla u|^2dx + \int_{|x| = R} (u+a) \nabla u
\cdot n dS =2\pi{\gamma}
\int_{|x| \le R} V(u+a)^2dx,$$ where ${\displaystyle}{n = \frac {x}{|x|}}$ is the normal vector and $dS$ is the Lebesgue measure on $|x| = R$. Now if $Z=\mu\big({\mathbb R}^2\big)=0$, then we can use – to deduce that $$\int_{|x| = R} (u+a) \nabla u \cdot n \ dS = O(R^{-1}).$$ As a result, $$- \int_{\mathbb{R}^2} |\nabla u|^2dx =2\pi\gamma \int_{\mathbb{R}^2} V(u+a)^2dx.$$ From this we deduce that ${\displaystyle}{\int_{\mathbb{R}^2} |\nabla u|^2dx = \int_{\mathbb{R}^2}
(u+a)^2Vdx = 0}$. This in turn implies that $u \equiv 0$. But this contradicts $u={{\mathcal G}}(V(u+a))$. Hence we can not have $Z=0$.\
[**Step 4.**]{} It remains to show that $(u+a)Z^{-1}\ge 0$. We only establish this when $Z>0$. The case $Z<0$ can be treated likewise. First take a smooth function $\varphi_{{\delta}}: {{\mathbb R}}
\to (-{\infty},0]$ such that $\varphi'_{{\delta}} \ge 0$ and $$\varphi_{{\delta}}(r) = \begin{cases}
0 &r > -a, \\
a + r &r < -a-{\delta}.
\end{cases}$$ We then have $$\label{eq3.11}
\int_{|x|= R}\varphi_{{\delta}}(u)\nabla u. n dS-\int_{|x|\le R}
\varphi'_{{\delta}}(u)|\nabla u|^2dx = 2\pi{\gamma} \int_{|x|\le R}
\varphi_{{\delta}}(u)(u+a)Vdx$$ by an integration by parts. (Here $n=x/|x|$.) If $Z>0$, then we can use to assert that $u(x)>0$ and $\varphi_{\delta}(u(x))=0$ whenever $|x|=R$ and R is sufficiently large. Since the left-hand side of is negative for such large $R$ and $(u+a)\varphi_{{\delta}}(u) \ge 0$ we deduce $$\int_{\mathbb{R}^2} \varphi'_{{\delta}}(u)|\nabla u|^2dx =
\int_{\mathbb{R}^2} V(u+a)\varphi_{{\delta}}(u)dx = 0.$$ We now send ${\delta}\to 0$ to deduce $$0 = \int_{\mathbb{R}^2} |\nabla u|^21\!\!1(u+a \le 0)dx
= \int_{\mathbb{R}^2} V(u+a)^21\!\!1(u+a \le 0)dx.$$ As a result, on the set $A = \{x: a + u(x) < 0\}$ we have $\nabla u =
0$. Hence $u$ is constant on each component $B$ of $A$. But this constant can only be $-a$ because on the boundary of $A$ we have $u+a=0$. This is impossible unless $A$ is empty and we deduce that $u \ge -a$ everywhere.
502em $\Box$=0
We now turn to the proof of Theorem 3.2. We first state and prove a lemma. Let us write ${\Lambda}_{\varepsilon}$ for $\mu^{\varepsilon}\big({\mathbb R}^2\big)$ where $\mu^{\varepsilon}$ was defined right after (3.3).
\[lem3.1\] We have that ${\Lambda}_{\varepsilon}>0$ for small ${\varepsilon}$. Moreover $$\limsup_{{\varepsilon}\to 0}{\Lambda}_{\varepsilon}\le 1.$$
[**Proof**]{} Let us write $\hat u^{\varepsilon}$ for $(u^{\varepsilon}+|\log {\varepsilon}|){\Lambda}_{\varepsilon}^{-1}$ and $\hat \mu^{\varepsilon}$ for ${\Lambda}_{\varepsilon}^{-1} \mu^{\varepsilon}$. By Theorem 3.1 we have that $\hat \mu^{\varepsilon}$ is a probability measure and $u^{\varepsilon}\ge 0$. Note that the support of the probability measure $\hat \mu^{\varepsilon}$ is the set ${\varepsilon}K_0$. Moreover $u^{\varepsilon}$ is harmonic off ${\varepsilon}K_0$ and $$\label{3.17}
\hat u^{\varepsilon}={{\mathcal G}}\hat \mu^{\varepsilon}+\frac {|\log{\varepsilon}|}{{\Lambda}_{\varepsilon}}.$$ By a well-known theorem in potential theory we have $${\varepsilon}Cap(K_0)=Cap({\varepsilon}K_0)\ge \exp\left(-\frac {|\log{\varepsilon}|}{{\Lambda}_{\varepsilon}}\right),$$ where $Cap$ denotes the logarithmic capacity. (See for example Theorem 9.8 of [@Po].) As a result $$\frac {\log Cap(K_0)}{|\log{\varepsilon}|}-1\ge -\frac 1{{\Lambda}_{\varepsilon}}.$$
From this, we can readily deduce the claims of the Lemma.
502em $\Box$=0
[**Proof of Theorem 3.2**]{} It suffices to show that for every positive $k$, $$\label{eq3.15}
\lim_{{\varepsilon}\to 0}\sup_{|x|\le k}\left|u^{{\varepsilon}}({\varepsilon}x)|\log {\varepsilon}|^{-1}+{\Lambda}_{\varepsilon}\right|=0,$$ and $$\label{eq3.16}
\lim_{{\varepsilon}\to 0}{\Lambda}_{\varepsilon}=\frac {\tau}{2\pi+{\tau}}.$$
Recall that by Theorem 3.1 and Lemma 3.1 the expression $ u^{\varepsilon}({\varepsilon}y)|\log{\varepsilon}|^{-1}+1$ is nonnegative Also recall that $R_0$ is defined so that the ball $B_{R_0}(0)$ contains the support of $V$. Let us write $\ell_{\varepsilon}$ for the maximum of $u^{{\varepsilon}}({\varepsilon}x)$ over the ball $B_{R_0}(0)$. We then have $$\begin{aligned}
\label{eq3.17}
u^{\varepsilon}({\varepsilon}x)&=&\frac{\tau} {2\pi}\int \log |{\varepsilon}x-{\varepsilon}y|
\left(u^{\varepsilon}({\varepsilon}y)|\log{\varepsilon}|^{-1}+1\right)V(y)dy\\
&=& {\Lambda}_{\varepsilon}\log {\varepsilon}+\frac{\tau} {2\pi}\int \log | x-y|
\left(u^{\varepsilon}({\varepsilon}y)|\log{\varepsilon}|^{-1}+1\right)\ V(y)dy\nonumber\\
&\le&{\Lambda}_{\varepsilon}\log {\varepsilon}+
\frac{\tau} {2\pi}\left(\ell_{\varepsilon}|\log{\varepsilon}|^{-1}+1\right)\int_{|x-y|>1}
\log | x-y|\ V(y)dy.\nonumber\end{aligned}$$ Hence for every $x$ with $|x|\le k$ we have $$u^{\varepsilon}({\varepsilon}x)\le {\Lambda}_{\varepsilon}\log {\varepsilon}+
c\left(\ell_{\varepsilon}|\log{\varepsilon}|^{-1}+1\right),$$ where $c$ is a constant that depends on $k$ only. By choosing $k=R_0$ we deduce that $$\left (1-c|\log {\varepsilon}|^{-1}\right)\ell_{\varepsilon}\le {\Lambda}_{\varepsilon}\log{\varepsilon}+c.$$ Hence $$\label{eq3.18}
\ell_{\varepsilon}=\max_{|x|\le R_0}u^{\varepsilon}({\varepsilon}x)\le 2{\Lambda}_{\varepsilon}\log{\varepsilon}+2c$$ for sufficiently small ${\varepsilon}$. This in turn implies $$\label{eq3.19}
\ell_{\varepsilon}|\log{\varepsilon}|^{-1}+1\le 2,$$ for small ${\varepsilon}$. Moreover, by the second equality in (\[eq3.17\]), $$u^{\varepsilon}({\varepsilon}x)|\log {\varepsilon}|^{-1}+{\Lambda}_{\varepsilon}=\frac {{\tau}}{2\pi |\log{\varepsilon}|}
\int\log|x-y|\left(u^{\varepsilon}({\varepsilon}y)|\log{\varepsilon}|^{-1}+1\right)V(y)dy=X_1+X_2,$$ where $$\begin{aligned}
X_1&=&\frac {{\tau}}{2\pi |\log{\varepsilon}|}
\int_{|x-y|\le 1}\log|x-y|\left(u^{\varepsilon}({\varepsilon}y)|\log{\varepsilon}|^{-1}+1\right)V(y)dy,\\
X_2&=&\frac {{\tau}}{2\pi |\log{\varepsilon}|}
\int_{|x-y|\ge 1}\log|x-y|\left(u^{\varepsilon}({\varepsilon}y)|\log{\varepsilon}|^{-1}+1\right)V(y)dy.\end{aligned}$$ Since the expression $ u^{\varepsilon}({\varepsilon}y)|\log{\varepsilon}|^{-1}+1$ is nonnegative and bounded above for $y$ in the ball $B_{R_0}(0)$, we deduce that both $X_1$ and $X_2$ converge to $0$ in low ${\varepsilon}$ limit. This completes the proof of (3.15).
We now turn to the proof (3.16). By the definition of ${\Lambda}_{\varepsilon}$ and (3.15), $$\begin{aligned}
{\Lambda}_{\varepsilon}&=&\frac {{\tau}}{2\pi}\int
\left(u^{\varepsilon}({\varepsilon}y)|\log{\varepsilon}|^{-1}+1\right)V(y)dy\\
&=&\frac {{\tau}}{2\pi}(1-{\Lambda}_{\varepsilon})\int V(y)dy+o(1)=\frac {{\tau}}{2\pi}(1-{\Lambda}_{\varepsilon})+o(1).\end{aligned}$$ This immediately implies (3.16).
502em $\Box$=0
[^1]: Research supported in part by NSF grant DMS0307021
|
---
abstract: 'Characteristics of the muon component in EAS are analyzed together with their fluctuations. The aim of this analysis — a comparison of experimental data with computational results obtained within frameworks of various hadron interaction models for protons and iron nuclei and an estimation of cosmic ray mass composition in the ultra-high energy region.'
author:
- 'S. P. Knurenko'
- 'A. K. Makarov'
- 'M. I. Pravdin'
- 'A. Sabourov'
title: The Relation Between Charged Particles and Muons With Threshold Energy 1 GeV in Extensive Air Showers Registered at the Yakutsk EAS Array
---
Introduction
============
The Yakutsk complex array for many years measures three main observables of extensive air showers (EAS): total charged component, muons with ${\varepsilon_{\text{thr}}}\ge 1$ GeV and Cherenkov light [@bib1]. Using these data we estimated the EAS energy with model independent quasi-calorimetric method [@bib2] and determined the depth of maximum shower development (by the measured Cherenkov light lateral distribution, using the parameter $p = \lg{Q(200) / Q(550)}$ and by the shape of the Cherenkov light pulse, $\tau_{1/2}$) [@bib3; @bib4]. The relative muon content at different core distances was measured [@bib5; @bib6] and the cosmic ray (CR) mass composition was estimated by various EAS characteristics [@bib7; @bib8; @bib9].
In this paper we analyze the muon component of EAS: mean characteristics, muon content and its fluctuations at fixed energy. The analysis is conducted within the framework of QGSJet II [@bib10] and EPOS [@bib11] hadron interaction models involving computations for primary particles of different masses using CORSIKA-6.900 code [@bib12].
Muon lateral distribution function
==================================
![Lateral distribution of charged particles for fixed energy values $10^{17}, 10^{28}, 10^{19}$ eV and zenith angle $\theta = 15^{\circ}$. Symbols refer to the Yakutsk experiment. Solid lines denote computational results with QGSJet+FLUKA [@bib13] for protons, dotted line — for iron, red line — EPOS+UrQMD [@bib14] for proton, green — carbon, blue — iron.[]{data-label="fig1"}](fig4){width="49.00000%"}
In Figure \[fig1\] examples of mean lateral distributions for muons at different energies are displayed. The muon lateral distribution function (LDF) is significantly lower then that of the charged component and can be effectively measured in individual events at ${E_{0}}\ge 10^{17}$ eV within the core distance range $100-800$ m. Thus, as a classification parameter in this energy region, a parameter ${\rho_{\mu}}(600)$ could be used — the density of muon flux at $600$ m from shower core.
Solid and dotted lines on the figure denote computational results obtained with QGSJet(UrQMD) models for proton (solid) and iron (dotted). It is seen from Figure \[fig1\] that the muon LDF from protons is steeper than that from iron nuclei and this difference is especially pronounced at large core distances. Qualitative comparison of computational results with the experiment reveals a better agreement with a heavier component of primary CR at ${E_{0}}\le 10^{18}$ eV and with lighter at ${E_{0}}\sim 10^{19}$ eV. This feature could be stressed out if one puts parameter $r^{2} \cdot \rho(r)$ on the $y$-axis of a plot instead of simple $\rho(r)$.
Muon portion and its dependence on angle, energy and the depth of maximum EAS development
=========================================================================================
We considered the dependence of ${\rho_{\mu}}/{\rho_{ch}}$ on the length of shower development after the maximum — $\Delta\lambda = {x_{0}}/ \cos{\theta} - {x_{\text{max}}}$. In highly inclined showers the muon content increases proportionally to ${x_{0}}/\cos{\theta}$ value, where ${x_{0}}= 1020$ [g/cm$^{2}$]{}for Yakutsk.
It is a known fact that the depth of maximum EAS development differs significantly, depending on the kind of primary particle and, hence, this feature could be used in the analysis of the CR mass composition: for instance, by fixing the $\Delta\lambda$ parameter and studying the fluctuations of ${\rho_{\mu}}/{\rho_{ch}}$ value. This technique is rather similar to one proposed by @bib15.
Shower parameters calculated with CORSIKA code were modified by applying distortions according to experimental errors. Parameters measured in experiment (e.g. $\cos{\theta}, {x_{\text{max}}}, {\rho_{ch}}(r), {\rho_{\mu}}(r)$) for every shower were rolled with the normal distribution with $\sigma$ parameter according to the experiment:
$$\begin{aligned}
\sigma(\theta) &= 3 \cdot \sec{\theta};\\
\sigma({x_{\text{max}}}) &= 40\text{g/cm}^{2};\\
\sigma(\rho_{r}) &= \sqrt{\rho_{r}^{2} \cdot \left(0.025 + \frac{1.2}{s_{\text{det}} \cdot \rho_{r} \cdot \cos{\theta}}\right)}
\end{aligned}$$
where $s_{\text{det}}$ is the area of the detector.
{width="49.00000%"} {width="49.00000%"}
Figure \[fig2\] shows the dependence of ${\rho_{\mu}}/{\rho_{ch}}$ on the length of cascade development after the shower maximum compared with computational results. A strong correlation is observed between the muon content and the length of track in the atmosphere after the shower maximum. It is also seen that experimental data are in good agreement with simulation results.
Mean characteristics
====================
![Muon density ${\rho_{\mu}}(600)$ measured in Yakutsk experiment as a function of primary energy compared to model calculations.[]{data-label="fig3"}](fig7){width="45.00000%"}
Figure \[fig3\] shows the energy dependence of ${\rho_{\mu}}(600)$ obtained in the Yakutsk and MIA EAS experiments [@bib16]. A good agreement is observed between the two.
Computational results obtained with EPOS and QGSJet01 from the work by @bib16 are denoted with lines, dotted line represent our simulations with QGSJet II for protons and iron nuclei. A comparison of our computations with the results obtained by @bib16 reveals that virtually there is no difference between QGSJet01 and QGSJet II. A significant discrepancy is observed between EPOS and QGSJet II and it, as we believe, is associated with different amounts of muons generated by models. For example, the ${\rho_{\mu}}(600)$ value calculated with EPOS for proton coincides with ${\rho_{\mu}}(600)$ obtained with QGSJet II for iron. Thus, a comparison of experimental data with model calculations result in controversial conclusions on CR mass composition. According to EPOS, at energies up to $2 \times 10^{17}$ eV CRs consist of iron nuclei and above that energy, up to $10^{19}$ eV — of protons. In the energy interval $10^{17} - 3 \times 10^{18}$ eV QGSJet II computations agree with the experiment quite well if primary particles are iron nuclei and above $3 \times 10^{18}$ eV the mass composition might be mixed with portion of protons and helium nuclei not less than $50-60$%. More precise estimation of CR mass composition could be derived after improvement of theoretical models and selecting one, that describes experimental EAS data better then others.
{width="49.00000%"} {width="49.00000%"}
Fluctuations of ${\rho_{\mu}}/{\rho_{ch}}$ relation on the ground level at the energy $\sim 10^{18}$ eV
=======================================================================================================
Showers initiated by different nuclei have differing altitudes of the maximum which in turn means that different numbers of muons are generated in these showers. It also means that they cover different paths in the atmosphere. By analyzing the tracks of muons that they pass in the atmosphere after the maximum of shower development we can try to estimate the composition of cosmic rays. With this aim in view, by choosing the mean zenith angle $36^{\circ}$ (which corresponds to the track length after the maximum $\Delta\lambda = 500$ [g/cm$^{2}$]{}), let us normalize the values of muon content to this level and consider their fluctuations.
Results are presented in Figure \[fig4\]. Also shown are the computational results obtained with QGSJet II and EPOS models for various nuclei. Measured values of fluctuations are presented in Table \[tab1\]. Obtained results have shown that within this method fluctuations of ${\rho_{\mu}}(600)/{\rho_{ch}}(600)$ parameters do not allow to estimate the CR mass composition. However, mean values from different nuclei differ. Besides, QGSJet II hints at a heavier composition than that of EPOS: according to first one, the composition of selected showers shifts towards heavier nuclei; according to second one, showers correspond to nuclei of intermediate group. On the whole, both models argue for a mixed composition.
However, if one takes into account gamma-photons generated in ground covering muon detectors, the mean value of ${\rho_{\mu}}(600)/{\rho_{ch}}(600)$ relation decreases and the composition shifts towards lighter nuclei (protons-helium-carbon) [@bib18].
------------------------------------------------ ---------- ------------- ---------- ---------- ---------- ---------- ---------- ----------
Yakutsk Yakutsk[^1] p C Fe p C Fe
$\left< \rho_{\mu} / \rho_{\text{ch}} \right>$ $0.3145$ $0.2768$ $0.2687$ $0.3025$ $0.3193$ $0.2893$ $0.3170$ $0.3381$
$\sigma$ $0.0747$ $0.0657$ $0.0517$ $0.0541$ $0.0536$ $0.0563$ $0.0511$ $0.0539$
------------------------------------------------ ---------- ------------- ---------- ---------- ---------- ---------- ---------- ----------
Conclusions
===========
Within the framework of QGSJet II and EPOS hadron interaction models using the CORSIKA code the values of muon portion ${\rho_{\mu}}/{\rho_{ch}}$ at core distance $r = 600$ m were obtained. A relation between the muon portion and a distance to the depth of shower maximum $\Delta\lambda$ was also obtained. A comparison of the dependency with experiment has shown that taking account of the experimental errors in the simulation data, a good agreement is observed between simulation and experiment.
A comparison of the muon portion distribution with computational results points towards a mixed cosmic ray composition near ${E_{0}}\ge 10^{18}$ eV. Large fluctuations of the muon portion prevent revealing of a single determined group of nuclei. A more detailed analysis is required, involving possible systematics of muon density measurement in the Yakutsk experiment.
The work is supported by RFBR grants 08-02-00348-a, 09-02-12028 ofi-m and FANI g.k. 02.740.11.0248, 02.518.11.7173.
[99]{}
V. P. Artamonov, B. N. Afanasiev, A. V. Glushkov. Izv. RAN, ser. fiz., 1994, t. 58, No.12, s.92-97. (in Russian)
S. P. Knurenko, A. A. Ivanov, I. E. Sleptsov et al. Pisma v ZhETF, 2006, V. 83, 11, s.563-567. (in Russian)
M. N. Dyakonov, S. P. Knurenko, V. A. Kolosov et al. Proc. 23th ICRC, Calgary, V. 3, p. 303, 1993.
S. P. Knurenko, V. A. Kolosov, I. T. Makarov et al. Int. Jour. of Modern Physics A. Vol. 20, No.29 (2005). pp. 6900-6902.
S. P. Knurenko, V. A. Kolosov, I. T. Makarov et al. Proc. 27th ICRC, Hamburg, V. 1, pp. 157-160, 2001
S. P. Knurenko, A. A. Ivanov, I. E. Sleptsov et al. Izv. RAN, ser. fiz. 2005, t. 69, No.3, s. 363-365. (in Russian)
S. P. Knurenko, V. A. Kolosov, I. T. Makarov et al. Int. Jour. of Modern Physics A. Vol. 20, No.29 (2005) pp. 6894-6896.
S. P. Knurenko, A. A. Ivanov, I. E. Sleptsov. Izv. RAN, ser. fiz., 2007, t. 71, No.4. s. 467-469. (in Russian)
S. P. Knurenko, A. A. Ivanov, I. E. Sleptsov. Pisma v ZhETF, 2007, t. 86, vyp. 10, s. 709-712. (in Russian)
S. S. Ostapchenko. Nucl. Phys. B (Proc. Suppl.) 151, 143 (2006).
K. Werner, F. M. Liu, T. Pierog. Phys. Rev. C 74 (2006) 044902.
D. Heck, J. Knapp, J.-N. Capdevielle et al. Forschungszentrum Karlsruhe Thechnical Report No.6019 (1998).
A. Fassó, A. Ferrari, J. Ranft and P. R. Salpa., CERN-2005-10 (2005), INFN/TC\_05/11, SLAC-R-773
S. A. Bass et al. Prog. Part. Nucl. Phys. 41 (1998) 225. M. Bleicher et al. J. Phys. G25 (1999) 1859.
V. B. Atrashkevich, N. N. Kalmykov, G. B. Khristiansen. Pisma v ZhETF, 1981, t. 33, vyp. 4, s. 236-239. (in Russian)
T. Abu-Zayyad et al. Astroph. J. 557, 686 (2001)
A. V. Glushkov, D. S. Gorbunov, I. T. Makarov et al. JETPl. 87, 220 (2008)
L. G. Dedenko et al. Private discussion at Moscow CR conference. (2010)
[^1]: With respect to contribution from gammas generated in the shielding of detector (Dedenko, 2010)
|
---
abstract: |
The aim of this paper is to establish an analogue of Logvinenko-Sereda’s theorem for the Fourier-Bessel transform (or Hankel transform) $\ff_\alpha$ of order $\alpha>-1/2$. Roughly speaking, if we denote by $PW_\alpha(b)$ the Paley-Wiener space of $L^2$-functions with Fourier-Bessel transform supported in $[0,b]$, then we show that the restriction map $f\to f|_\Omega$ is essentially invertible on $PW_\alpha(b)$ if and only if $\Omega$ is sufficiently dense. Moreover, we give an estimate of the norm of the inverse map.
As a side result we prove a Bernstein type inequality for the Fourier-Bessel transform.
address:
- |
S.G.: Département Mathématiques\
Faculté des Sciences de Tunis\
Université de Tunis El Manar\
Campus Universitaire\
1060 Tunis\
Tunisie\
and Université d’Orléans\
Faculté des Sciences\
MAPMO - Fédération Denis Poisson\
BP 6759\
F 45067 Orléans Cedex 2\
France
- 'P.J.: Institut de Mathématiques de Bordeaux UMR 5251, Université Bordeaux 1, cours de la Libération, F 33405 Talence cedex, France'
author:
- Saifallah Ghobber
- Philippe Jaming
title: 'The Logvinenko-Sereda Theorem for the Fourier-Bessel transform'
---
[^1]
Introduction
============
The classical uncertainty principle was established by Heisenberg bringing a fundamental problem in quantum mechanics to the point: The position and the momentum of particles cannot be both determined explicitly but only in a probabilistic sense with a certain ”uncertainty”. The mathematical equivalent is that a function and its Fourier transform cannot both be arbitrarily localized. This is a fundamental problem in time-frequency analysis. Heisenberg did not give a precise mathematical formulation of the uncertainty principle, but this was done in the late 1920s by Kennard [@ph:Ke] and Weyl (who attributes the result to Pauli) [@ph:We Appendix 1]. This leads to the classical formulation of the uncertainty principle in form of the lower bound of the product of the dispersions of a function and its Fourier transform $$\bigl\||x|f\bigr\|_{L^2(\R^d)}\,\bigl\||\xi|\ff(f)\bigr\|_{L^2(\R^d)}\geq c\bigl\|f\bigr\|^2_{L^2(\R^d)}.$$ A considerable attention has been devoted recently to discovering new mathematical formulations and new contexts for the uncertainty principle (see the surveys [@BD; @folland] and the book [@HJ] for other forms of the uncertainty principle). Our aim here is to consider uncertainty principles in which concentration is measured in sense of smallness of the support (the notion of *annihilating pairs* in the terminology of [@HJ] or *qualitative uncertainty principles* in the terminology of [@folland Section 7] which also surveys extensions of this notion to various generalizations of the Fourier transform). Further, the transform under consideration is the Fourier-Bessel transform (also known as the Hankel transform) on $\R^+$. This transform arises as [*e.g.*]{} a generalization of the Fourier transform of a radial integrable function on Euclidean $d$-space as well as from the eigenvalues expansion of a Schrödinger operator.
Let us now be more precise and describe our results. To do so, we need to introduce some notation. For $1\leq p < \infty$ and $\alpha>-1/2$, we denote by $L^p_\alpha(\R^+)$ the Banach space consisting of measurable functions $f$ on $\R^+$ equipped with the norms $${{\left\|{f}\right\|}}_{L^p_\alpha}=\left( \int_0^\infty {{\left|{f(x)}\right|}}^p\,\mbox{d}\mu_\alpha(x) \right)^{1/p},$$ where $\mbox{d}\mu_\alpha (x)=\frac{2\pi^{\alpha+1}}{\Gamma(\alpha+1)}x^{2\alpha+1}\,\mbox{d}x$. For $f\in L^1_\alpha(\R^+)$, the Fourier-Bessel (or Hankel) transform is defined by $$\ff_\alpha(f) (y)=\int_0^\infty f(x)j_\alpha(2\pi xy)\d\mu_\alpha (x),$$ where $j_\alpha$ is the Bessel function given by $$j_\alpha(x)=2^\alpha\Gamma(\alpha+1)\frac{J_\alpha (x)}{x^\alpha}
:=\Gamma(\alpha+1)\sum_{n=0}^{\infty}\frac{(-1)^n}{n!\Gamma(n+\alpha+1)}\left(\frac{x}{2}\right)^{2n}.$$ Note that $J_\alpha$ is the Bessel function of the first kind and $\Gamma$ is the gamma function. The function $j_\alpha$ is even and analytic. It is well known that the Fourier-Bessel transform extends to an isometry on $ L^2_\alpha(\R^+)$ $$\|\ff_\alpha(f) \|_{L^2_\alpha}= \|f \|_{L^2_\alpha}.$$
Uncertainty principles for the Fourier-Bessel transform have been considered in various places, [*e.g.*]{} [@Bo; @RV] for a Heisenberg type inequality, [@om1; @om2] for the “local uncertainty principle” and Pitt’s inequality or [@V] for Hardy type uncertainty principles when concentration is measured in terms of fast decay. Our main concern here is uncertainty principles of the following type:
[*A function and its Fourier-Bessel transform cannot both have small support*]{}.
In other words, we are interested in the following adaptation of a well-known notion from Fourier analysis:
\
\[num\] Let $S$, $\Sigma$ be two measurable subsets of $\R^+$. Then
- $(S,\Sigma)$ is a *weak annihilating pair* if, $\supp f \subset S$ and $\supp \ff_\alpha(f)\subset \Sigma$ implies $f=0$.
- $(S,\Sigma)$ is called a *strong annihilating pair* if there exists $C_\alpha(S,\Sigma)$ such that $$\label{int,str}
{{\left\|{f}\right\|}}^2_{L^2_\alpha}\leq C_\alpha(S,\Sigma)\Big({{\left\|{f}\right\|}}^2_{L^2_\alpha(S^c)} +{{\left\|{\ff_\alpha(f)}\right\|}}^2_{L^2_\alpha(\Sigma^c)} \Big),$$
where $A^c=\R^+\backslash A$. The constant $C_\alpha(S,\Sigma)$ will be called the *annihilation constant of $(S,\Sigma)$*.
Of course, every strong annihilating pair is also a weak one. Let us also recall that, to prove that a pair $(S,\Sigma)$ is strongly annihilating, it is enough to show that there exists a constant $D_\alpha(S,\Sigma)$ such that for every $f \in L^2_\alpha(\R^+)$ whose Fourier-Bessel transform is supported in $\Sigma$, $${{\left\|{f}\right\|}}_{L^2_\alpha}\leq D_\alpha(S,\Sigma){{\left\|{f}\right\|}}_{L^2_\alpha(S^c)}.$$ In other words, if we denote by $PW_\alpha(\Sigma)=\left\{f\in L^2_\alpha(\R^+)\,:\supp\ff_\alpha(f)\subset\Sigma\right\}$ then the pair $(S,\Sigma)$ is strongly annihilating if the restriction map $f\to f|_{S^c}$ is invertible and $D_\alpha(S,\Sigma)$ is the norm of the inverse.
There are several examples of the Uncertainty Principle of form for the Fourier transform $\ff$. One of them is the Amrein-Berthier theorem [@AB] which is a quantitative version of a result due to Benedicks [@Be]. In this theorem sets of finite measure play the role of small sets, [*i.e.*]{} if a function $f$ is supported on a set of finite measure then $\ff(f)$ cannot be concentrated on a set of finite measure unless $f$ is the zero function. An added example is the Shubin-Vakilan-Wolff theorem [@SVW Theorem 2.1], where so called $\eps$-thin sets are considered. The extension of the results of Benedicks-Amrein-Berthier and of the Shubin-Vakilan-Wolff for the Fourier-Bessel transform were shown by the authors in [@GJ]. Our first task here will be to slightly extend our version of Shubin-Vakilan-Wolff’s theorem.
Another Uncertainty Principle which is of particular interest to us is the Logvinenko-Sereda theorem [@LS], see also [@HJ page 102] and [@Ko]. This result characterizes the sets $\Omega$ such that $(\Omega^c,[0,b])$ is an annihilating pair and gives the (essentially optimal) annihilation constant. In the case of the Fourier transform, $\Omega$ is then the complement of a so called *relatively dense* subset. For the Fourier-Bessel transform, we adapt this notion as follows: [*a measurable subset $\Omega\subset \R^+$ is called *relatively dense* (for $\mu_\alpha$) if there exist $\gamma, a>0$ such that $$\label{cond}
\mu_\alpha(\Omega\cap [x-a,x+a])\ge \gamma \mu_\alpha([x-a,x+a]),$$ for all $x\ge a$.*]{}
Our main result is then the following:
[**Theorem.**]{} \
[*Let $\alpha\geq 0$ and let $a,b,\gamma>0$. Then there is a constant $C(\alpha,a,b,\gamma)$ such that, for every $f \in L^2_\alpha (\R^+)$ with $\supp \ff_\alpha (f) \subset [0,b]$ and every $(\gamma,a)$-relatively dense subset $\Omega$ of $\;\R^+$,*]{} $$\label{form}
{{\left\|{f}\right\|}}^2_{ L^2_\alpha(\Omega)} \geq
C(\alpha,a,b,\gamma) {{\left\|{f}\right\|}}^2_{ L^2_\alpha}.$$
We will show in Lemma \[lemcn\] that condition is also necessary for an inequality of the form to hold. Our proof is inspired by the proof in the Euclidean case by Kovrijkine [@Ko] who obtained an essentially sharp estimate that is polynomial in $\gamma$ (rather then a previously known exponential one). This proof allows us to obtain an estimate on $C(\alpha,a,b,\gamma)$ as well.
The remaining of the paper is organized as follows. Next section is devoted to some preliminaries on the Fourier-Bessel transform and the corresponding “translation” operator. The section is completed with a version of Bernstein’s Inequality for the Fourier-Bessel transform. In section 3, we complete our previous extension of Shubin-Vakilan-Wolff’s theorem. In the last section, we prove the Logvinenko-Sereda Theorem for the Fourier-Bessel transform.
Preliminaries
=============
In this section, we will fix some notation and prove a Bernstein type inequality for the Fourier-Bessel transform.
Generalities
------------
We will denote by $|x|$ and ${{\left\langle{x,y}\right\rangle}}$ the usual norm and scalar product on $\R^d$. The Fourier transform is defined for $f\in L^1(\R^d)$ by $$\ff(f)(\xi)=\int_{\R^d}f(x)e^{-2i\pi{{\left\langle{x,\xi}\right\rangle}}}\,\mbox{d}x.$$ Note that $\|\ff(f)\|_{L^2(\R^d)}={{\left\|{f}\right\|}}_{L^2(\R^d)}$ and the definition of the Fourier transform is extended from $f\in L^1(\R^d)\cap L^2(\R^d)$ to $L^2(\R^d)$ in the usual way. With this normalization, if $f(x)=\tilde{f}(|x|)$ is a radial function on $\R^d$, then $\ff(f)(\xi)=\ff_{d/2-1}(\tilde{f})(|\xi|)$.
If $S_d$ is a measurable set in $\R^d$, we will write $|S_d|$ for its Lebesgue measure.
For $\alpha>-1/2$, let us recall the *Poisson representation formula* $$j_\alpha(x)=\frac{\Gamma(\alpha+1)}{\Gamma\left(\alpha+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\right)}
\int_{-1}^1 (1-s^2)^{\alpha-1/2}\cos(sx) \d x.$$ Therefore, $j_\alpha$ is bounded with $|j_\alpha(x)|\leq j_\alpha(0)=1$. As a consequence, $$\label{eq:L1infty}
{{\left\|{\ff_\alpha(f)}\right\|}}_\infty\leq {{\left\|{f}\right\|}}_{L^1_\alpha}.$$ Here ${{\left\|{.}\right\|}}_\infty$ is the usual essential supremum norm and $L^\infty$ will denote the usual space of essentially bounded functions. Finally, if $\ff_\alpha(f)\in L^1_\alpha(\R^+)$, the inverse Fourier-Bessel transform, is defined for almost every $x$ by $$f (x) =\int_0^\infty \ff_\alpha(f) (\xi)j_\alpha(2\pi x\xi)\,\mbox{d}\mu_\alpha (\xi).$$
For $\lambda>0$, we introduce the dilation operator $\delta_\lambda$, defined by $$\delta_\lambda f(x)=\frac{1}{\lambda^{\alpha+1}}f\left(\frac{x}{\lambda}\right).$$ Notice that $\ff_\alpha \delta_\lambda= \delta_{\lambda^{-1}}\ff_\alpha $.
Let us now gather some facts about Bessel functions that will be used throughout the paper. First, a more refined estimate that we will need is the following: when $t\to \infty$, $$\label{Ajalpha}
j_\alpha(t)=\frac{2^{\alpha+1/2}\Gamma(\alpha+1)}{\sqrt{\pi}}
t^{-\alpha-1/2}\cos\left(t-(2\alpha+1)\frac{\pi}{4}\right)+O(t^{-\alpha-3/2}).$$ In particular, there is a constant $c_\alpha$ such that $$\label{majbessel}
|j_\alpha(t)|\le c_\alpha(1+t)^{-\alpha-1/2}.$$
Further, we will make use of a few formulas involving the functions $j_\alpha(x)$ (see [*e.g.*]{} [@watson page 132-134]): $$\label{eqderive}
\frac{\d}{\d x} j_\alpha(x)= j'_\alpha(x)= -\frac{x}{2(\alpha+1)}j_{\alpha+1}(x),$$ $$\label{eqr1}
\int_0^{s} j_\alpha(tx) t^{2\alpha+1}\d t= \frac{s^{2\alpha+2}}{2\alpha+2}j_{\alpha+1}(sx),\ \ s>0,$$ and $$\label{eqr2}
\int_0^{s} j_\alpha(t)^2\, t^{2\alpha+1}\d t = \frac{s^{2\alpha+2}}{2} \Big(j'_\alpha(s)^2 +\frac{2\alpha}{s}j'_\alpha(s)j_\alpha(s)+j_\alpha(s)^2 \Big),$$ while, for $u\neq v$ we have $$\label{eqr3}
\int_0^{s} j_\alpha(ut) j_\alpha(vt) t^{2\alpha+1}\d t
= \frac{s^{2\alpha+1}}{u^2-v^2} \Big(vj'_\alpha(vs)j_\alpha(us) -uj'_\alpha(us)j_\alpha(vs)\Big).$$
Generalized translation
-----------------------
Following Levitan [@Bm], for any function $f \in C^2(\R^+)$ we define the generalized Bessel translation operator $$T^\alpha_yf(x)=u(x,y), \hspace{0.5cm}x,\;y \in \R^+,$$ as a solution of the following Cauchy problem: $$\left(\frac{\partial^2}{\partial x^2}
+ \frac{2\alpha+1}{x} \frac{\partial}{\partial x}\right)u(x,y)
=\left(\frac{\partial^2}{\partial y^2}+ \frac{2\alpha+1}{y} \frac{\partial}{\partial y}\right)u(x,y),$$ with initial conditions $ u(x,0)=f(x)$ and $\frac{\partial}{\partial x}u(x,0)=0$, here $\frac{\partial^2}{\partial x^2}+ \frac{2\alpha+1}{x} \frac{\partial}{\partial x}$ is the differential Bessel operator. The solution of the Cauchy problem can be written out in explicit form: $$\label{translate}
T_x^\alpha f (y)=\frac{\Gamma(\alpha+1)}{\sqrt{\pi}\Gamma(\alpha+1/2)}\int_0^\pi f(\sqrt{x^2+y^2-2xy\cos \theta})(\sin \theta)^{2\alpha } \d \theta.$$ The operator $T_x^\alpha$ can be also written by the formula $$T_x^\alpha f (y)=\int_0^\infty f(t) W(x,y,t) \d\mu_\alpha(t),$$ where $W(x,y,t) \d\mu_\alpha(t)$ is a probability measure and $W(x,y,t)$ is defined by $$W(x,y,t)= \begin{cases}\dst
\frac{2^{2\alpha-2}\Gamma(\alpha+1)^2 }{ \pi^{\alpha+3/2}\Gamma\left(\alpha+\frac{1}{2}\right) }
\frac{\Delta(x,y,t)^{2\alpha-1} }{(xyt)^{2\alpha}},&\mbox{if } {{\left|{x-y}\right|}}<t< x+y;\\
0,&\mbox{otherwise};
\end{cases}$$ where $$\Delta(x,y,t)=\bigl((x+y)^2 -t^2\bigr)^{1/2}\bigl(t^2- (x-y)^2\bigr)^{1/2}$$ is the area of the triangle with side length $x,y,t$. Further, $W(x,y,t)\,\mbox{d}\mu_\alpha(t)$ is a probability measure, so that, for $p\ge1$, $|T^\alpha_xf|^p\leq T_x^\alpha|f|^p$ thus $${{\left\|{T^\alpha_x f}\right\|}}_{L^p_\alpha}\leq{{\left\|{f}\right\|}}_{L^p_\alpha}.$$ This allows to extend the definition of $T^\alpha_xf$ to functions $f\in L^p_\alpha(\R^+)$.
It is also well known that for $\lambda>0$, $$T_x^\alpha j_\alpha(\lambda\;.)(y)=j_\alpha(\lambda x)j_\alpha(\lambda y).$$ Therefore for $f\in L^p_\alpha (\R^+)$, $p=1$ or $2$, $$\ff_\alpha\bigl(T^\alpha_xf)(y)=j_\alpha(2\pi xy)\ff_\alpha(f)(y).$$ Note also that if $f$ is supported in $[0,b]$, then $T_xf$ is supported in $[0,b+x]$.
The Bessel convolution $f\ast_\alpha g$ of two functions $f$ and $g$ in $L^1_\alpha(\R^+)\cap L^\infty $ is defined by $$f\ast_\alpha g (x)= \int_0^\infty f(t)T_x^\alpha g (t)\d\mu_\alpha(t)=\int_0^\infty T_x^\alpha f (t) g(t)\d\mu_\alpha(t),\ \ x\ge0.$$
Then, if $1\le p,q,r \le\infty$ are such that $1/p+1/q-1=1/r$, $f\ast_\alpha g\in L^r_\alpha (\R^+)$ and $${{\left\|{f\ast_\alpha g}\right\|}}_{L^r_\alpha}\leq{{\left\|{f}\right\|}}_{L^p_\alpha}{{\left\|{g}\right\|}}_{L^q_\alpha}.$$ This then allows to define $ f\ast_\alpha g$ for $f\in L^p_\alpha (\R^+) $ and $g\in L^q_\alpha (\R^+)$. Moreover for $f\in L^1_\alpha (\R^+)$ and $g\in L^q_\alpha (\R^+)$, $q=1$ or $2$ we have $$\ff_\alpha(f\ast_\alpha g) = \ff_\alpha(f) \ff_\alpha(g).$$
Bernstein’s Inequality
----------------------
Let us introduce the following notation.
Let $f$ be an entire and even function, $ f(z)= \dst\sum_{n=0}^\infty a_n z^{2n}$. We define two operations on $f$: $$Df= \frac{1}{2z}\frac{\mathrm{d}f}{\mathrm{d}z}
\quad\mbox{and}\quad
\pp f(z)= \sum_{n=0}^\infty a_n z^{n}.$$ In other words $f(z)=\pp f(z^2)$ and $Df=\dst\sum_{n=0}^\infty(n+1)a_{n+1}z^{2n}$ which is again entire and even.
It is clear that $\pp D f= \partial \pp f$ and, for every $k$, $D^kf$ exists and $\pp D^k f= \partial^k \pp f$.
We will need a variant of Bernstein’s Inequality for $\ff_\alpha$ for which we have been unable to find a proper reference.
\
Let $f$ be a function in $L^1_\alpha(\R^+)$ such that $\supp \ff_\alpha(f) \subset [0, b]$. Then $f$ is an even entire function such that $$\label{bernstein}
{{\left\|{D^k f}\right\|}}_{L_{\alpha+k}^2}\leq
\sqrt{\frac{\Gamma(\alpha+1)}{\Gamma(\alpha+k+1)}}\big(\sqrt{\pi^3}\,b\big)^k{{\left\|{f}\right\|}}_{L_\alpha^2}.$$
As $\supp \ff_\alpha(f) \subset [0, b]$ then $ \ff_\alpha(f) \in L^1_\beta(\R^+)\cap L^2_\beta(\R^+)$ for every $\beta\geq\alpha$. By the inversion formula for the Fourier-Bessel transform, we have $$f(x)= \int_0^b \ff_\alpha(f)(y) j_\alpha(2\pi xy) \d\mu_\alpha(y).$$ In particular, $f$ is an even entire function. As $\dst j'_\alpha(t)=-\frac{t j_{\alpha+1}(t)}{2(\alpha+1)} $, we may differentiate the previous formula to obtain $$f'(x)= -2\pi x \int_0^b \ff_\alpha(f)(y) j_{\alpha+1}(2\pi xy) \frac{\pi y^2}{(\alpha+1)}\d\mu_\alpha(y).$$ It follows that $\dst
Df(x)= -\pi \int_0^b \ff_\alpha(f)(y) j_{\alpha+1}(2\pi xy) \d\mu_{\alpha+1}(y)
=-\pi\ff_{\alpha+1}[\ff_\alpha(f)](x)$. Repeating the previous operation, $$D^kf(x)=(-\pi)^k\int_0^b \ff_\alpha(f)(y) j_{\alpha+k}(2\pi xy) \d\mu_{\alpha+k}(y)
=(-\pi)^k\ff_{\alpha+k}\left[\ff_\alpha(f)\right](x).$$ But then $$\begin{aligned}
{{\left\|{D^k f}\right\|}}_{L^2_{\alpha+k}}&=&
\pi^k \big\|\ff_{\alpha+k}[\ff_{\alpha}(f)]\big\|_{L^2_{\alpha+k}}\\
&=&\pi^k {{\left\|{\ff_{\alpha}(f)}\right\|}}_{L^2_{\alpha+k}} \\
&=& \pi^k\left(\int_0^b|\ff_\alpha(f)(y)|^2 \frac{\Gamma(\alpha+1)}{\Gamma(\alpha+k+1)}(\pi y^2)^k
\d\mu_\alpha(y)\right)^{1/2}\\
&\le&\sqrt{\frac{\Gamma(\alpha+1)}{\Gamma(\alpha+k+1)}}\big(\sqrt{\pi^3}\,b\big)^k\left(\int_0^b|\ff_\alpha(f)(y)|^2
\d\mu_\alpha(y)\right)^{1/2}.\end{aligned}$$ Finally, from Plancherel’s theorem we deduce, $$\begin{aligned}
\dst{{\left\|{D^k f}\right\|}}_{L^2_{\alpha+k}}
&\leq&\sqrt{\frac{\Gamma(\alpha+1)}{\Gamma(\alpha+k+1)}}\big(\sqrt{\pi^3}\,b\big)^k{{\left\|{\ff_\alpha(f)}\right\|}}_{L^2_\alpha}\\
&=&\sqrt{\frac{\Gamma(\alpha+1)}{\Gamma(\alpha+k+1)}}\big(\sqrt{\pi^3}\,b\big)^k{{\left\|{f}\right\|}}_{L^2_\alpha}\end{aligned}$$ as expected.
A results on $(\eps,\alpha)$-thin sets and sets of finite measure
=================================================================
This section is motivated by our recent results on quantitative uncertainty principles stated in [@GJ]. We consider a pair of orthogonal projections on $L^2_\alpha(\R^+)$ defined by $$E_S f= \chi_S f,\hspace{1cm} F_\Sigma f= \ff_\alpha \Big[E_\Sigma \ff_\alpha (f)\Big],$$ where $S$ and $\Sigma$ are measurable subsets of $\R^+$.
The following lemma is well known (see [@GJ Lemma 4.1]):
\
\[equi\] Let $S$ and $\Sigma$ be a measurable subsets of $\R^+$. If ${{\left\|{F_\Sigma E_S}\right\|}}<1$, then $(S,\Sigma)$ is a strong annihilating pair with an annihilation constant $\Big(1-{{\left\|{F_\Sigma E_S }\right\|}}\Big)^{-2}$.
Conversely it was shown in [@HJ I.1.1.A, page 88] that if the pair $(S,\Sigma)$ is strongly annihilating then $\|F_\Sigma E_S\|<1$. We will not use this fact here.
From [@GJ] we recall the following definition:
\
Let $\eps\in(0,1)$ and $\alpha>-1/2$. A set $S \subset \R^+$ is $(\eps,\alpha)$-thin if, for $0\leq x\leq 1$, $$\mu_\alpha \bigl( S\cap [x,x+1]\bigr)\le
\eps \mu_\alpha \bigl([x,x+1]\bigr)$$ and for $x\geq 1$, $$\mu_\alpha \left( S\cap {{\left[{x,x+\frac{1}{x}}\right]}}\right)\le
\eps\mu_\alpha \left( {{\left[{x,x+\frac{1}{x}}\right]}}\right).$$
We have shown in [@GJ] that any pair of sets of finite measure as well as any pair of $(\eps,\alpha)$-thin subsets (with $\eps$ sufficiently small) are strongly annihilating. Precisely we have the following theorem:
\[th:strong\]\
Let $\alpha>-1/2$.
– Let $S_0$, $\Sigma_0$ be a pair of measurable subsets of $\R^+$ with $0<\mu_\alpha(S_0),\;\mu_\alpha(\Sigma_0)<\infty$. Then $${{\left\|{F_{\Sigma_0} E_{S_0}}\right\|}}<1.$$ – There exists $\eps_0$ such that, for every $0<\eps<\eps_0$, there exists a positive constant $C$ such that if $S_1$ and $\Sigma_1$ are $(\eps,\alpha)$-thin subsets in $\R^+$ then $${{\left\|{F_{\Sigma_1} E_{S_1}}\right\|}}\le C \eps^{1/2}.$$
\
\[r1\] – For the Fourier transform $\ff$ on $L^2(\R^d)$, the first part of Theorem \[th:strong\] was proved by Amrein-Berthier [@AB] and the second part was proved by Shubin-Vakilian-Wolff [@SVW].\
– If $\alpha=-1/2$, then $\mu_{-1/2}$ is the Lebesgue measure and $\ff_{-1/2}$ is the Fourier-cosine transform defined for any even function $f\in L^2(\R^+)$ by $$\ff_{-1/2}(f)(\xi)=\int_0^\infty f(x)\cos(2\pi x\xi)\d x.$$ In other words, $\ff_{-1/2}$ is the Fourier transform $\ff$ restricted to even function in the sense that, if $\ffi\in L^2(\R)$ is even and $f=\ffi|_{\R^+}$ —the restriction of $\ffi$ to $\R^+$— then $\ff[\ffi](\xi)=\ff_{-1/2}[f](\xi)$ for $\xi\geq 0$. It follows that Theorem \[th:strong\] is also valid for $\alpha=-1/2$.
In the definition of $\eps$-thin sets, different conditions are asked on the part of the set included in $[0,2]$ and the remaining part. This separation is somewhat arbitrary and one expects that the first condition could be imposed in any neighborhood of $0$ and the second one at infinity. A careful and painful adaptation of the proof in [@GJ] surely gives such a result. However, we now take a simpler route by first showing that an $\eps$-thin set and a compact set form a strong annihilating pair and that an estimate of the annihilation constant is available:
\[ss\] \
Let $S$ a $(\eps,\alpha)$-thin subset of $\R^+$ and let $\Sigma =[0,b]$. Then $$\|F_\Sigma E_S\|\le C \eps^{1/2}.$$
In the next section we will obtain a stronger result by characterizing all sets $S$ for which $(S,[0,b])$ are strongly annihilating.
The proof is inspired from [@SVW Lemma 4.2]. Note that $$\|F_\Sigma E_S\|=\| E_SF_\Sigma\|= \sup_{f=F_\Sigma f}\frac{\| E_Sf\|_{L^2_\alpha}}{\|f\|_{L^2_\alpha}}.$$ Now let $f\in L^2_\alpha(\R^+)$ such that $\supp \ff_\alpha (f) \subset \Sigma$ and fix a Schwartz function $\phi$ with $\ff_\alpha (\phi)=1$ on $\Sigma$. Then $f=\phi \ast_\alpha f$.
Let $\mathcal{Q}$ to be the operator from $L^2_\alpha(\R^+)$ to $L^2_\alpha(S)$ defined by $$\mathcal{Q} f (x) = E_S(\phi \ast_\alpha g)(x)=\int_0^\infty \chi_S(x)T_x^\alpha \phi(y)g(y)\d \mu_\alpha(y).$$ As $$\sup_{x\in S} \int_0^\infty |T_x^\alpha \phi(y)| \d \mu_\alpha(y)\le \|\phi\|_{L^1_\alpha}$$ and from Inequality $(5.29)$ in [@GJ], we have $$\sup_{y\in \R^+}\int_S|T_x^\alpha \phi(y)|\d \mu_\alpha(x)\le c\eps.$$ Then by Schur’s test, $$\|\mathcal{Q} \|\le \sqrt{c\|\phi\|_{L^1_\alpha}} \eps^{1/2}.$$ Therefore, as $\|F_\Sigma\|=1$, $$\| E_S F_\Sigma f\|_{L^2_\alpha} = \|\mathcal{Q} f\|_{L^2_\alpha} \le\|\mathcal{Q} \|\| F_\Sigma f\|_{L^2_\alpha}
\le \|\mathcal{Q} \|\| f\|_{L^2_\alpha}.$$ Hence $ \|F_\Sigma E_S\|\le \sqrt{c\|\phi\|_{L^1_\alpha}} \eps^{1/2}.$
We are now in position to prove the following uncertainty principle estimate.
\[th:SVWBAB\] \
Let $\alpha >-1/2$, $a,b>0$. Then there exists $\eps_0>0$ such that, if $0<\eps<\eps_0$ and if $S,\Sigma \subset \R^+$ are subsets of the form $$S=S_0\cup S_\infty,\ \
\Sigma=\Sigma_0\cup\Sigma_\infty$$ where $S_0= [0,a]$, $\Sigma_0=[0,b]$ and $S_\infty\subset[a,\infty)$, $\Sigma_\infty\subset[b,\infty)$ are $(\eps,\alpha)$-thin. Then $$\|F_\Sigma E_S\|<1.$$ In particular, $(S,\Sigma)$ is a strong annihilating pair.
We have $$F_\Sigma
E_S=F_{\Sigma_0}E_{S_0}+F_{\Sigma_\infty}E_{S_0}+F_{\Sigma_0}E_{S_\infty}+
F_{\Sigma_\infty}E_{S_\infty}.$$ Now, according to Theorem \[th:strong\], ${{\left\|{F_{\Sigma_0}E_{S_0}}\right\|}}<1$. Further, by Lemma \[ss\], $${{\left\|{F_{\Sigma_\infty}E_{S_0}}\right\|}}+{{\left\|{F_{\Sigma_0}E_{S_\infty}}\right\|}}\leq
c_1\eps^{1/2}+c_2\eps^{1/2}\to0,\ \ \mathrm{ as }\;\eps\to 0$$ and ${{\left\|{F_{\Sigma_\infty}E_{S_\infty}}\right\|}}\leq
C\eps^{1/2}$, according to Theorem \[th:strong\]. It follows that, if $\eps$ is small enough, then ${{\left\|{F_\Sigma E_S}\right\|}}<1$, so that $(S,\Sigma)$ is still a strong annihilating pair.
1. The previous corollary remains true if $S_\infty$ and $\Sigma_\infty$ are the union of finitely many $(\eps,\alpha)$-thin subsets in $\R^+$.
2. From Remark \[r1\], Corollary \[th:SVWBAB\] is still valid for $\alpha=-1/2$. Moreover if $S,\Sigma \subset \R^d$ are subsets of the form $$S=B(0,a)\cup S_\infty;\ \ \Sigma=B(0,b)\cup \Sigma_\infty,$$ with $S_\infty\subset \R^d\backslash B(0,a)$ and $\Sigma_\infty\subset \R^d\backslash B(0,b)$ are $\eps$-thin (see [@SVW page 1] for the definition of $\eps$-thin set). Then the same technique used here shows that there is an $\eps_0$ depending on $a$ and $b$ such that, if $\eps<\eps_0$, then the pair $(S,\Sigma)$ is strongly annihilating for the Fourier transform $\ff$.
A Logvinenko-Sereda type theorem
================================
A direct adaptation of the definition of relatively dense sets in the $\R^d$ setting (when applied to radial sets) leads us to the introduction of the following definition:
\
Let $\alpha \ge0 $. A measurable subset $\Omega\subset \R^+$ is called *relatively dense* if there exist $\gamma,\; a>0$ such that for all $ x\ge a$ $$\label{dense}
\mu_\alpha \Big( \Omega\cap [x-a,x+a] \Big) \geq \gamma \mu_\alpha\Big([x-a,x+a]\Big).$$ In this case $\Omega$ will be called a *$(\gamma,a)$-relatively dense* subset.
1. Let $S\subset\R^+$ be a subset with $\mu_\alpha(S)<\infty$. Then there exists $a>0$ such that $$\mu_\alpha \Big([x-a,x+a] \Big) \geq 2 \mu_\alpha(S),$$ for all $x\ge a$. Thus $\Omega=S^c$ is $\left(\frac{1}{2},a\right)$-relatively dense.
2. Let $S$ a $(\eps,\alpha)$-thin subset in $\R^+$. A simple covering argument shows that there is a constant $c$ depending only on $\alpha$ such that, for all $x\ge \frac{1}{2}$ $$\mu_\alpha \left( S\cap {{\left[{x-\frac{1}{2},x+\frac{1}{2}}\right]}}\right)\le c\eps\mu_\alpha \left({{\left[{x-\frac{1}{2},x+\frac{1}{2}}\right]}}\right).$$ Thus $$\mu_\alpha \left( S^c\cap {{\left[{x-\frac{1}{2},x+\frac{1}{2}}\right]}}\right)\ge (1-c\eps)\mu_\alpha \left({{\left[{x-\frac{1}{2},x+\frac{1}{2}}\right]}}\right).$$ Hence $\Omega=S^c$ is $\left((1-c\eps),\frac{1}{2}\right)$-relatively dense, provided $\eps$ is small enough.
For these two examples, we already know that $(S,[0,b])$ is a strong annihilating pair. We will show that for every $\Omega$ relatively dense, and every $b>0$, $(\Omega^c,[0,b])$ is a strong annihilating pair. But let us first prove that if $(\Omega^c,[0,b])$ is a strong annihilating pair, then $\Omega$ is relatively dense.
\[lemcn\] \
Let $\Omega\subset \R^+$ be a measurable subset. Suppose there exists a constant $c$ such that, for every $f\in L^2_\alpha(\R^+)$ with $\supp \ff_\alpha(f)\subset[0,b]$, $$\label{st}
c \int_0^\infty |f(x)|^2 \d \mu_\alpha(x) \le \int_\Omega |f(x)|^2 \d \mu_\alpha(x),$$ then $\Omega$ is relatively dense.
By considering $\delta_{\frac{1}{2\pi b}}f$ instead of $f$ we can assume that $\ff_\alpha(f)$ is supported in $[0,\frac{1}{2\pi}]$. Now take $$f_0= \ff_\alpha\left(\vartheta_\alpha\chi_{[0,\frac{1}{2\pi}]}\right),\\ \mathrm{with} \ \ \vartheta_\alpha= (4\pi)^{\alpha+1}\Gamma(\alpha+2),$$ then from $$\label{fzero}
f_0(x)=j_{\alpha+1}(x).$$ Let $s'_0 =0$ and denote by $0<s'_1 <s'_2 < \cdots$ the sequence of all nonnegative zeros of the function $j'_\alpha$. Note that by , $(s'_n)_{1\le n\le \infty}$ is the sequence of nonnegative zeros of the function $j_{\alpha+1}$ and the asymptotic form of $s'_n$ follows from (see [@watson page 618]): $$\label{eqasump}
s'_n=\pi \left(n+\frac{2\alpha+1}{4} +O(n^{-1})\right).$$ As a consequence, $s'_{n+1}-s'_n=\pi+O(n^{-1})$, thus there exists $n_0$, depending only on $\alpha$, such that, if $n\geq n_0$ then $s'_{n+1}-s'_n\leq 4$. In particular, if $x\geq s'_{n_0}$, there exists $n$ such that $|x-s'_n|\leq 2$
Let $f_n$ be the function defined by $$\label{eqfn}
f_n(x)= T^\alpha_{s'_n}f_0(x)=\ff_\alpha\left(\vartheta_\alpha j_\alpha(2\pi s'_n \,.)
\chi_{[0,\frac{1}{2\pi}]}\right)(x)
= j_\alpha(s'_n)\;\frac{x^2j_{\alpha+1}(x)}{x^2-s^{'2}_n},$$ with .
In particular, by , and , we have $$\label{eqs}
\begin{cases}
f_0(s'_0)=1= \vartheta_\alpha^{-1}\|f_0\|^2_{L^2_\alpha},& n=0;\\
f_n(s'_n)=(\alpha+1)j_\alpha(s'_n)^2= \vartheta_\alpha^{-1}\|f_n\|^2_{L^2_\alpha},& n\ge1;\\
f_n(s'_k)=0, & n,\,k\ge1:\;n\neq k;\\
f'_n(0)=0, &n\ge0.
\end{cases}$$ First we will prove that there is an appropriate choice of $a$ such that holds for $x=s'_n$.
But, let $a\geq s'_{n_0}$ be fixed, the precise value being given below. Let $n\ge 1$ be such that $s'_n\ge a$. To simplify notation, write $s=s'_n$. Then, from and , we have $$\int_{0}^{s-a}|f_n(t)|^2\d\mu_\alpha(t)\le \frac{2\pi^{\alpha+1}c_\alpha^2}{\Gamma(\alpha+1)} j_\alpha(s)^2\int_0^{s-a}\frac{\d t}{(t-s)^2}\le
\frac{2\pi^{\alpha+1}c_\alpha^2}{\Gamma(\alpha+1)} j_\alpha(s)^2 \int_a^\infty \frac{\d t}{t^2}\le
C_{\alpha}\frac{\|f_n\|^2}{a},$$ and $$\int_{s+a}^\infty|f_n(t)|^2\d\mu_\alpha(t) \le \frac{2\pi^{\alpha+1}c_\alpha^2}{\Gamma(\alpha+1)} j_\alpha(s)^2 \int_a^\infty \frac{\d t}{t^2}\le C_{\alpha}\frac{\|f_n\|^2}{a},$$ where $C_{\alpha}=\frac{2\pi^{\alpha+1}c_\alpha^2}{\vartheta_\alpha\Gamma(\alpha+2)}$ is a constant that depends only on $\alpha$. Now if we take $$a=\max\left(5,s'_{n_0},4\dfrac{C_{\alpha}}{c}\right),$$ so that $a$ depends only on $\alpha$ and $c$, then $$\label{eqcomp}
\int_{[s-a,s+a]^c} |f_n|^2\d\mu_\alpha\le \frac{c}{2}\|f_n\|^2.$$
But then, it follows from and that $$\label{eqst2}
\int_{\Omega\cap[s-a,s+a]}|f_n(t)|^2\d\mu_\alpha(t)\ge \frac{c}{2}\|f_n\|^2,$$ which implies that $$\label{eqst3}
\int_{\Omega\cap[s-a,s+a]}{{\left|{\frac{tj_{\alpha+1}(t)}{t-s}}\right|}}^2\d\mu_\alpha(t)\ge \frac{(\alpha+1)c}{2\vartheta_\alpha}.$$ On the other hand, by the mean value theorem, for every $t\in [s-a,s+a]$, there exists $u$ with $u\in[s,t]$ (or $[t,s]$) such that $$j_{\alpha+1}(t)=j_{\alpha+1}(s)+j'_{\alpha+1}(u)(t-s)=-\frac{uj_{\alpha+2}(u)}{2(\alpha+2)}(t-s)$$ since $j_{\alpha+1}(s)=0$ and . It follows from that $$\begin{aligned}
{{\left|{\frac{tj_{\alpha+1}(t)}{t-s}}\right|}}^2&=&\frac{|tuj_{\alpha+2}(u)|^2}{4(\alpha+2)^2}
\leq \frac{c_{\alpha+2}^2}{4(\alpha+2)^2}\frac{t^2u^2}{(1+u)^{2\alpha+5}}.$$ Further, as $u\leq\max(s,t)\leq s+a\leq 2s$ and $$1+u\geq 1+s-a\geq\begin{cases}s-a \geq s/2,&\mbox{if }s\geq 2a;\\
1\geq s/2a,&\mbox{if }s\leq 2a;\end{cases}\geq \frac{s}{2a}$$ we get $${{\left|{\frac{tj_{\alpha+1}(t)}{t-s}}\right|}}^2\leq \frac{2^{2\alpha+7}c_{\alpha+2}^2 a^{2\alpha+5}}{(\alpha+2)^2}s^{-2\alpha-1}.$$ Inserting this into , we obtain $$\label{eqfnphil}
\mu_\alpha\bigl(\Omega\cap[s-a,s+a]\bigr)\ge \frac{(\alpha+2)^2(\alpha+1)c}{2^{2\alpha+8}a^{2\alpha+5}\vartheta_\alpha c_{\alpha+2}^2}s^{2\alpha+1}.$$ On the other hand, $$\label{eqmes}
\mu_\alpha([s-a,s+a])\le\frac{4a\pi^{\alpha+1}}{\Gamma(\alpha+1)}(s+a)^{2\alpha+1}\le \frac{2^{2\alpha+3}\pi^{\alpha+1}}{\Gamma(\alpha+1)}as^{2\alpha+1}.$$
Comparing and shows that there exists a constant $\gamma>0$ depending only on $\alpha$ and $c$ such that $$\mu_\alpha(\Omega\cap[s-a,s+a])\ge \gamma \mu_\alpha([s-a,s+a]).$$ Now let $x\ge a$, then there exists $s=s'_n\ge a$ such that $|x-s|<2$. Thus $$\begin{aligned}
\mu_\alpha(\Omega\cap[x-a-2,x+a+2])&\ge& \mu_\alpha(\Omega\cap[s-a,s+a]) \ge \gamma \mu_\alpha([s-a,s+a])\\
&\ge& \gamma \mu_\alpha([x-a+2,x+a-2])\ge \gamma' \mu_\alpha([x-a-2,x+a+2]),\end{aligned}$$ with $ 0<\gamma'<\gamma$. Here we used the fact that $a\geq 5$ and that $\mu_\alpha$ is a doubling measure. This finishes the proof of the lemma.
We are now in position to prove our main theorem:
\
\[Sereda\] Let $\alpha\geq 0$ and let $a,b,\gamma>0$. Let $f \in L^2_\alpha (\R^+)$ such that $\supp \ff_\alpha (f) \subset [0,b]$. If $\Omega$ is a $(\gamma,a)$-relatively dense subset of $\;\R^+$, then $$\label{eq:sereda}
{{\left\|{f}\right\|}}^2_{ L^2_\alpha(\Omega)} \geq
\frac{2}{3}\left(\frac{\gamma}{300\times 9^\alpha}\right)^{\frac{160\sqrt{3}\pi}{\ln2}ab+\alpha\frac{\ln3}{\ln2}+1}{{\left\|{f}\right\|}}^2_{ L^2_\alpha}.$$
The proof here is inspired by O. Kovrijkine’s proof and improvement of Logvinenko-Sereda’s Theorem [@Ko].
First, we will reduce the problem by proving the following:
[**Claim.**]{} \
[*Fix $\gamma>0$. It is enough to prove that, there is a function $\psi\,:\R^+\to\R^+$ such that, if $\Omega$ is $(\gamma,1)$-relatively dense and $f\in L^2_\alpha(\R^+)$ with $\supp\ff_\alpha(f)\subset[0,ab]$, then*]{} $$\label{eq:LSabs}
{{\left\|{f}\right\|}}^2_{ L^2_\alpha(\Omega)} \geq
\psi(ab){{\left\|{f}\right\|}}^2_{ L^2_\alpha}.$$
As $\mu_\alpha([0,ab])=\dst\frac{\pi^{\alpha+1}}{\Gamma(\alpha+2)}(ab)^{2(\alpha+1)}= a^{2\alpha+2}\mu_\alpha([0,b])$, we may write $\psi(ab)=\ffi\bigl(\mu_\alpha([0,ab])\bigr)$ where $\ffi(s)=\dst\psi\left(\frac{\bigl(\Gamma(\alpha+2)s\bigr)^{1/(2\alpha+2)}}{\pi^{1/2}}\right)$.
Assume now that an inequality of the form $$\label{eq:LSabs2}
{{\left\|{f}\right\|}}^2_{ L^2_\alpha(\Omega)} \geq
\psi(ab){{\left\|{f}\right\|}}^2_{ L^2_\alpha}$$ holds, for every $f\in L^2_\alpha(\R^+)$ with $\supp\ff_\alpha(f)\subset[0,ab]$ and every $(\gamma,1)$-relatively dense subset $\Omega$ of $\R^+$.
Now let $a>0$ and let $\tilde\Omega$ be a $(\gamma,a)$-relatively dense subset of $\R^+$. Then $\Omega=\{ x/a\,:\ x\in \tilde\Omega\}$ is $(\gamma,1)$-relatively dense in $\R^+$. On the other hand, if $f\in L^2_\alpha(\R^+)$ is a function with $\supp \ff_\alpha(f)\subset[0,b]$, then as $\ff_\alpha \delta_{a^{-1}}= \delta_{a}\ff_\alpha $, we have $\supp \ff_\alpha\left(\delta_{a^{-1}} f\right)\subset [0,ab]$ and ${{\left\|{f}\right\|}}_{L^2_\alpha(\tilde\Omega)}={{\left\|{\delta_{a^{-1}} f}\right\|}}_{L^2_\alpha(\Omega)}.$ It follows then from Inequality that
$
{{\left\|{f}\right\|}}^2_{L^2_\alpha(\tilde\Omega)}={{\left\|{\delta_{a^{-1}} f}\right\|}}^2_{L^2_\alpha(\Omega)}
\geq \psi(ab){{\left\|{\delta_{a^{-1}}f}\right\|}}^2_{ L^2_\alpha}
=\ffi\bigl(a^{2\alpha+2}\mu_\alpha([0,b])\bigr){{\left\|{f}\right\|}}^2_{ L^2_\alpha}.$
We will now reformulate the problem so as to be able to apply our Bernstein type inequality.
Let $\Omega'\subset\R^+$ be a subset defined by the relation $\Omega=\{x\geq 0\,:\ x^2\in \Omega'\}$ and $\d\nu_\alpha(s) =\frac{\pi^{\alpha+1}}{\Gamma(\alpha+1)} s^\alpha \d s$. Then condition is equivalent to $$\label{dense2}
\nu_\alpha \Big( \Omega'\cap [(x-1)^2,(x+1)^2]\Big) \geq \gamma \nu_\alpha([(x-1)^2,(x+1)^2])$$ for all $x \ge 1$. Finally, let $g=\pp f$ $f(x)=g(x^2)$.
Let us first reformulate what we want to prove. A simple change of variables shows that to show it is enough to prove an inequality of the form $$\label{eq:sereda2}
\int_{\Omega'}|g(s)|^2s^\alpha\d s\geq \frac{2}{3}\left(\frac{\gamma}{300\times 9^\alpha}\right)^{\frac{160\sqrt{3}\pi}{\ln2}ab+\alpha\frac{\ln3}{\ln2}+1}
\int_0^\infty|g(s)|^2s^\alpha\d s.$$ Note that $$\label{eqm}
{{\left\|{g}\right\|}}^2_{L^2_s}=\int_0^\infty {{\left|{g (s)}\right|}}^2 s^{\alpha} \d s=\frac{\Gamma(\alpha+1)}{\pi^{\alpha+1}} {{\left\|{f}\right\|}}^2_{L^2_\alpha}.$$
We will now reformulate Bernstein’s Inequality. First, a simple computation shows that $$\begin{aligned}
{{\left\|{D^k f}\right\|}}_{L^2_{\alpha+k}}^2&=&
\frac{2\pi^{\alpha+k+1}}{\Gamma(\alpha+k+1)}\int_0^\infty {{\left|{D^k f (t)}\right|}}^2 t^{2(\alpha+k)+1}\d t\\
&=& \frac{\pi^{\alpha+k+1}}{\Gamma(\alpha+k+1)}\int_0^\infty {{\left|{\pp D^k f (s)}\right|}}^2 s^{\alpha+k}\d s\\
&=& \frac{\pi^{\alpha+k+1}}{\Gamma(\alpha+k+1)}\int_0^\infty {{\left|{ \partial^k\pp f (s)}\right|}}^2 s^{\alpha+k} \d s.\end{aligned}$$ Then by and , Bernstein’s Inequality reads $$\label{Bernstein}
\int_0^\infty {{\left|{ \partial^k g (s)}\right|}}^2 s^{\alpha+k} \d s
\leq (\pi ab)^{2k} \int_0^\infty {{\left|{g (s)}\right|}}^2 s^{\alpha} \d s.$$
\
We will say that $x\ge1$ and that the corresponding interval $I_x=[(x-1)^2,(x+1)^2]$ are *bad* if there exists $k \geq 1$ such that $$\int_{I_x} {{\left|{ \partial^{k} g(s)}\right|}}^2 s^{\alpha+k}\d s
\geq \left(2\pi ab\right)^{2k} \int_{I_x} {{\left|{ g(s)}\right|}}^2
s^{\alpha}\d s.$$
Let us now show that the bad intervals only count for a fraction of the norm of $g$: $$\begin{aligned}
\int_{\cup_{x \; is \; bad}I_x}{{\left|{ g(s)}\right|}}^2 s^{\alpha}\d s
&\leq & \sum _{k\geq 1} \frac{1}{(4\pi^2 a^2b^2)^{k}}\int_{\dst\cup_{x \; is \; bad}I_x}{{\left|{ \partial^{k} g(s)}\right|}}^2 s^{\alpha+k}\d s\\
&\leq & \sum _{k\geq 1} \frac{1}{(4\pi^2 a^2b^2)^{k}}\int_{0}^\infty{{\left|{ \partial^{k} g(s)}\right|}}^2 s^{\alpha+k}\d s\\
&\leq&\sum _{k\geq 1} \frac{1}{4^{k}} \int_0^{\infty} {{\left|{g(s)}\right|}}^2 s^{\alpha}\d s
= \frac{1}{3}\int_0^{\infty} {{\left|{g(s)}\right|}}^2 s^{\alpha}\d s,\end{aligned}$$ where we have used Bernstein’s Inequality in the last line. Therefore $$\label{sumgood}
\int_{\dst\cup_{x \; is \; good}I_x}{{\left|{ g(s)}\right|}}^2 s^{\alpha}\d s \geq\frac{2}{3}\int_{0}^{\infty} {{\left|{g(s)}\right|}}^2 s^{\alpha}\d s.$$
[**Claim.**]{}\
[*If $I_x$ is a good interval then there exists $t_x \in I_x $ with the property that for every $k\geq 0$,*]{} $$t_x^{\alpha+k}{{\left|{\partial^{k} g(t_x)}\right|}}^2
\leq\left(12 \pi^2 a^2b^2\right)^k\int_{I_x} {{\left|{ g(s)}\right|}}^2 s^{\alpha}\d s.$$
Suppose towards a contradiction that this is not true. Then for every $t \in I_x$, there exists $ k_t \geq 0$ such that $$\int_{I_x} {{\left|{ g(s)}\right|}}^2 s^{\alpha}\d s \leq \frac{1}{(12\pi^2 a^2b^2)^{k_t}}
t^{\alpha+k_t}{{\left|{ \partial^{k_t} g(t)}\right|}}^2 ,$$ therefore $$\int_{I_x} {{\left|{ g(s)}\right|}}^2 s^{\alpha}\d s \leq \sum _{k\geq 0} \frac{1}{(12 \pi^2a^2b^2)^{k}}
t^{\alpha+k}{{\left|{ \partial^{k} g(t)}\right|}}^2.$$ Integrating both sides over $I_x$, we get $$4\int_{I_x}{{\left|{ g(s)}\right|}}^2 s^{\alpha}\d s
\leq \sum _{k\geq 0} \frac{1}{(12 \pi^2a^2b^2)^k}
\int_{I_x}{{\left|{ \partial^{k} g(t)}\right|}}^2t^{\alpha+k}\d t.$$ As $x$ is good, we deduce that $$4 \int_{I_x}{{\left|{ g(s)}\right|}}^2 s^{\alpha}\d s \leq \sum _{k\geq 0} \frac{1}{3^k} \int_{I_x}{{\left|{g(t)}\right|}}^2t^{\alpha}\d t
=\frac{3}{2}\int_{I_x}{{\left|{ g(s)}\right|}}^2 s^{\alpha}\d s,$$ which gives a contradiction.
The next step in the proof is the following straightforward adaptation of a result of O. Kovrijkine [@Ko Corollary, p 3041]:
\[th:ko\] \
Let $\Phi$ be an analytic function, $I$ an interval and $J\subset I$ a set of positive measure. Let $M=\max_{D_{I}}|\Phi(z)|$ where $D_{I}=\{z\in\C, dist(z,I)<4|I|\}$ and let $m=\max_{I}|\Phi(x)|$, then $$\label{eq:ko}
\int_I|\Phi(s)|^2\d s\leq\left(\frac{300|I|}{|J|}\right)^{\frac{2\ln M/m}{\ln 2}+1}\int_J|\Phi(s)|^2\d s.$$
We will apply Theorem \[th:ko\] with $I=I_x$, $x\ge2$, a good interval, $J=\Omega'\cap I_x$ and $\Phi=g$. Let $N={{\left\|{g}\right\|}}_{L^2_\alpha(I_x)}$ and note that $$N^2=\int_{(x-1)^2}^{(x+1)^2}|g(s)|^2s^\alpha\d s\leq (x+1)^{2\alpha}\max_{I_x}|g(s)|^2$$ [*i.e.*]{} $$\label{eqmm}
1/m\leq (x+1)^{\alpha}/N.$$
Let us now estimate $M$. Since $x$ is good, we can use the claim to estimate the power series of $g$: if $t\in D_{I_x}$ then $|t-t_x|\leq 5|I_x|=20x$ thus $$\begin{aligned}
|g(t)|&\leq&\sum_{k=0}^\infty\frac{|\partial^kg(t_x)|}{k!}|t-t_x|^k\\
&\leq&\frac{1}{t_x^{\alpha/2}}\sum_{k=0}^\infty\frac{1}{k!}\left(40\sqrt{3}\pi ab\right)^k \left(\frac{x}{\sqrt{t_x}}\right)^k{{\left\|{g}\right\|}}_{L^2_s(I_x)}\\
&\leq&\frac{1}{t_x^{\alpha/2}}\sum_{k=0}^\infty\frac{1}{k!}\left(80\sqrt{3}\pi ab\right)^k N\\
&\leq&\frac{N}{t_x^{\alpha/2}}e^{80\sqrt{3}\pi a b}.\end{aligned}$$ In particular by , we have $$\frac{M}{m}\le \frac{(x+1)^{\alpha}}{t_x^{\alpha/2}}e^{80\sqrt{3}\pi ab}$$ and then $$\label{eq:bbb}
\ln \frac{M}{m}\leq 80\sqrt{3}\pi ab+\frac{\alpha}{2}\ln\frac{(x+1)^2}{t_x}\le 80\sqrt{3}\pi ab+\alpha\ln3$$ since $t_x\ge(x-1)^2$.
Now, as $$\int_{\Omega'\cap I_x}{{\left|{g(s)}\right|}}^2 s^{\alpha}\d s
\geq(x-1)^{2\alpha}\int_{\Omega'\cap I_x}{{\left|{g(s)}\right|}}^2 \d s,$$ we deduce from that $$\begin{aligned}
\int_{\Omega'\cap I_x}{{\left|{g(s)}\right|}}^2 s^{\alpha}\d s&\geq& (x-1)^{2\alpha}\left(\frac{\bigl| \Omega'\cap I_x\bigr|}{300|I_x|}\right)^{\frac{2\ln M/m}{\ln 2}+1}
\int_{I_x}{{\left|{g(s)}\right|}}^2 \d s\nonumber\\
&\geq&\left(\frac{x-1}{x+1}\right)^{2\alpha}\left(\frac{\bigl|\Omega'\cap I_x\bigr|}{300|I_x|}\right)^{\frac{2\ln M/m}{\ln 2}+1}\int_{I_x}{{\left|{g(s)}\right|}}^2 s^{\alpha}\d s\nonumber\\
&\geq&3^{-2\alpha}
\left(\frac{\bigl|\Omega'\cap I_x\bigr|}{300|I_x|}\right)^{\frac{160\sqrt{3}\pi}{\ln2} ab +\frac{\alpha\ln3}{\ln2}}\int_{I_x}{{\left|{g(s)}\right|}}^2 s^{\alpha}\d s
\label{eq:aaa}\end{aligned}$$ where we have used that $\frac{x-1}{x+1}\geq \frac{1}{3}$ for $x\geq 2$, $\frac{\bigl|\Omega'\cap I_x\bigr|}{300|I_x|}\leq 1$ and in the last inequality. Since $$\bigl|\Omega'\cap I_x\bigr|\ge \frac{\Gamma(\alpha+1)}{\pi^{\alpha+1}}(x+1)^{-2\alpha} \nu_\alpha \Big(\Omega'\cap I_x \Big)$$ we get $$\bigl|\Omega'\cap I_x\bigr|\ge\left(\frac{x-1}{x+1}\right)^{2\alpha}|I_x|\gamma\geq 3^{-2\alpha}|I_x|\gamma.$$ Integrating this into , we obtain $$\int_{\Omega'\cap I_x}{{\left|{g(s)}\right|}}^2 s^{\alpha}\d s\ge3^{-2\alpha(\frac{160\sqrt{3}\pi}{\ln2}ab+\alpha\frac{\ln3}{\ln2}+1)}
\left(\frac{\gamma}{300}\right)^{\frac{160\sqrt{3}\pi}{\ln2}ab+\alpha\frac{\ln3}{\ln2}}\times\int_{I_x}{{\left|{g(s)}\right|}}^2 s^{\alpha}\d s.$$ This leads to $$\label{eqsai}
\int_{\Omega'\cap I_x}{{\left|{g(s)}\right|}}^2 s^{\alpha}\d s\ge\left(\frac{\gamma}{300\times 9^\alpha}\right)^{\frac{160\sqrt{3}\pi}{\ln2}ab+\alpha\frac{\ln3}{\ln2}+1}
\times\int_{I_x}{{\left|{g(s)}\right|}}^2 s^{\alpha}\d s.$$
Finaly, summing over all good intervals and applying , we have $$\begin{aligned}
\int_{\Omega'}{{\left|{g(s)}\right|}}^2 s^{\alpha}\d s&\ge& \int_{ \Omega'\cap\dst\cup_{x\;is\; good} I_x }{{\left|{g(s)}\right|}}^2 s^{\alpha}\d s\\
&\geq& \left(\frac{\gamma}{300\times 9^\alpha}\right)^{\frac{160\sqrt{3}\pi}{\ln2}ab+\alpha\frac{\ln3}{\ln2}+1}
\int_{\dst\cup_{x\;is\; good} I_x }{{\left|{g(s)}\right|}}^2 s^{\alpha}\d s\\
&\geq& \frac{2}{3}\left(\frac{\gamma}{300\times 9^\alpha}\right)^{\frac{160\sqrt{3}\pi}{\ln2}ab+\alpha\frac{\ln3}{\ln2}+1}\int_0^\infty{{\left|{g(s)}\right|}}^2 s^{\alpha}\d s.\end{aligned}$$ We have thus proved and the proof of Theorem \[Sereda\] is complete.
[99]{} <span style="font-variant:small-caps;">W.O. Amrein & A.M. Berthier</span> J. Funct. Anal. [**24**]{} (1977), 258–267.
<span style="font-variant:small-caps;">M. Benedicks</span> J. Math. Anal. Appl. [**106**]{} (1985), 180–183.
<span style="font-variant:small-caps;">A. Bonami & B. Demange</span> Collect. Math. [**2**]{} (2006) Vol. Extra, 1–36.
<span style="font-variant:small-caps;">P.C. Bowie</span> SIAM J. Math. Anal. [**2**]{} (1971), 601–606.
<span style="font-variant:small-caps;">G.B. Folland & A. Sitaram</span> J. Fourier Anal. Appl. [**3**]{} (1997), 207–238.
<span style="font-variant:small-caps;">S. Ghobber & P. Jaming</span> J. Math. Anal. Appl. [**377**]{} (2011), 501–515.
<span style="font-variant:small-caps;">V. Havin & B. Jöricke</span> Springer-Verlag, Berlin, 1994.
<span style="font-variant:small-caps;">E.H. Kennard</span> Zeit. Physik [**44**]{} (1927) 326–352.
<span style="font-variant:small-caps;">O. Kovrijkine</span> Proc. Amer. Math. Soc. [**129**]{} (2001), 3037–3047.
<span style="font-variant:small-caps;">B.M. Levitan</span> Uspekhi Mat. Nauk. 6. [**2**]{} (1951), 102–143.
<span style="font-variant:small-caps;">V.N. Logvinenko & Yu.F. Sereda</span> Teor. Funktsii, Funktsional. Anal. i Prilozhen [**19**]{} (1973), 234-246.
<span style="font-variant:small-caps;">S. Omri</span> Integral Transforms Spec. Funct. [**21**]{} (2010), 703–712.
<span style="font-variant:small-caps;">S. Omri</span> Integral Transforms Spec. Funct. [**22**]{} (2011), 655–670.
<span style="font-variant:small-caps;">M. Rösler & M. Voit</span> Proc. Amer. Math. Soc. [**127(1)**]{} (1999), 183–194.
<span style="font-variant:small-caps;">C. Shubin & R. Vakilian & T. Wolff</span> GAFA, Geom. funct. anal. [**8**]{} (1998) 932–964.
<span style="font-variant:small-caps;">V.K. Tuan</span> Integral Transforms Spec. Funct. [**18**]{} (2007), 369–381.
<span style="font-variant:small-caps;">G.N. Watson</span> Cambridge Univ. Press, Cambridge, 1944.
<span style="font-variant:small-caps;">H. Weyl</span> . S. Hirzel, Leipzig. Revised english edition: , Dover 1950.
[^1]: The authors wish to thank Aline Bonami for valuable conversation and the anonymous referees for there careful reading of the manuscript that lead to an improved presentation.\
Research on this paper was started while the second author was at MAPMO, Université d’Orléans.\
This work was partially sponsored by the French-Tunisian cooperation program PHC Utique/CMCU 10G 1503.
|
---
abstract: 'Population synthesis studies constitute a powerful method to reconstruct the birth distribution of periods and magnetic fields of the pulsar population. When this method is applied to populations in different wavelengths, it can break the degeneracy in the inferred properties of initial distributions that arises from single-band studies. In this context, we extend previous works to include $X$-ray thermal emitting pulsars within the same evolutionary model as radio-pulsars. We find that the cumulative distribution of the number of X-ray pulsars can be well reproduced by several models that, simultaneously, reproduce the characteristics of the radio-pulsar distribution. However, even considering the most favourable magneto-thermal evolution models with fast field decay, log-normal distributions of the initial magnetic field over-predict the number of visible sources with periods longer than $12$ s. We then show that the problem can be solved with different distributions of magnetic field, such as a truncated log-normal distribution, or a binormal distribution with two distinct populations. We use the observational lack of isolated NSs with spin periods $P>12$ s to establish an upper limit to the fraction of magnetars born with $B > 10^{15}$ G (less than 1%). As future detections keep increasing the magnetar and high-B pulsar statistics, our approach can be used to establish a severe constraint on the maximum magnetic field at birth of NSs.'
author:
- |
M. Gullón$^1$, J.A. Pons$^1$, J. A. Miralles$^1$, D. Viganò$^2$, N. Rea$^{3,2}$, R. Perna$^4$\
$^1$ Departament de Física Aplicada, Universitat d’Alacant, Ap. Correus 99, 03080 Alacant, Spain\
$^2$ Institute of Space Sciences (CSIC-IEEC), Campus UAB, Carrer de Can Magrans s/n, 08193, Barcelona, Spain\
$^3$ Anton Pannekoek Institute for Astronomy, University of Amsterdam, Postbus 94249, 1090 GE Amsterdam, The Netherlands\
$^4$ Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY, 11794, USA
bibliography:
- 'references.bib'
date: 'Accepted 00 XXXX 00. Received 00 XXXX 00; in original form 00 XXXX 00'
title: 'Population Synthesis of Isolated Neutron Stars with magneto-rotational evolution II: from radio-pulsars to magnetars'
---
\[firstpage\]
stars: neutron – pulsars: general – stars: magnetic fields.
Introduction
============
The last couple of decades have seen tremendous advances in our understanding of the properties and evolutionary paths of neutron stars (NSs), largely driven by multi-wavelength detection. Despite this progress, however, several puzzles still remain. In particular, the apparent diversity of the observational manifestations of NSs, which has led over the years to several subclassifications (i.e. rotation-powered pulsars, dim isolated NSs, Anomalous X-ray Pulsars, Soft-Gamma Ray Repeaters, Central Compact Objects, among others) has highlighted the need of understanding possible evolutionary links among different classes, as well as the role played by the birth properties of the NSs.
Studying the properties of NSs as a whole is best done via Monte Carlo simulations aimed at reproducing the galactic population. Since the fraction of pulsars that can be detected close to their birth constitutes a negligible fraction of the total sample, population synthesis studies (PS in the following) generally use the present-day observed properties of pulsars, together with some assumptions about their time evolution, to reconstruct the birth distribution of periods and magnetic fields for the pulsar population.
PS modeling of NSs has a long history, and especially so in the radio band, where the largest sample of pulsars has been detected (e.g. @Gunn1970 [@Phinney1981; @Lyne1985; @Stollman1987; @Emmering1989; @Narayan1990; @Lorimer1993; @Hartman1997; @Cordes1998; @ACC; @Vranesevic2004; @Faucher; @Ferrario2006]) and in some cases in other wavelengths [@Gon2004; @Pierbattista]. The efforts put over the years into this area of research stem from the fact that these studies allow to constrain the birth properties of NSs, which are in turn intimately related to the physical processes occurring during the supernova (SN) explosion and in the proto-NS. As such, they bear crucial information on the physics of core-collapse SNe, in which most NSs are believed to be formed.
The energy reservoir powering Rotation Powered Pulsars (RPPs) is provided by the loss of rotational energy. RPPs are mostly seen in the radio band ($\sim 2200$ objects), and we can detect more than one hundred of them also in the $X$-ray and/or $\gamma$-ray frequency range. Part of the rotational energy loss is converted into non-thermal synchrotron and curvature radiation by acceleration of charged particles, which are either extracted from the surface or created by pair production. According to the classical scenario, such acceleration takes place in some regions, called gaps, where a cascade of pairs is supported by the same high-energy photons emitted by the particles, and is thought to produce the coherent radio emission we observe. While radio emission is likely produced above the polar cap [@sturrock71; @ruderman75], high-energy photons are thought to originate in the outer magnetosphere [@cheng86; @romani96; @muslimov03] or in the wind region [@petri12]. Of particular interest is the recent FIDO model [@Kalapo2014], according to which the gamma-ray emission is produced in regions near the equatorial current sheet. These models can fit the observed values of the radio-lags and reproduce the observed light curve phenomenology and the GeV photon cut-off energies.
The majority of PS studies in the literature have assumed that the magnetic field is fixed at its value at birth, or have adopted simple, parametrized expressions for its decay. Recently, a large theoretical effort has been devoted to model the magnetic field evolution with magnetothermal simulations which follow the coupled evolution of the temperature and the magnetic field in the NS interior [@Vigano2013]. We will use results from these simulations to model the magnetic field evolution.
[@Gullon2014] (paper I, hereafter) have revisited PS modeling of isolated radio-pulsars incorporating the $B$-field evolution from the state-of-the-art magneto-thermal evolution models, together with recent results on the evolution of the angle between the magnetic and rotational axes from magnetospheric simulations. In an era where the availability of high energy satellites has enlarged considerably our understanding of the NS population, unvealing a variety of observational manifestations of NSs, it is then timely to include these further constraints in the global PS study of the NS population. A first attempt in this direction, including part of the X-ray emitting NS population and magnetic field decay models, was performed by [@PSB_Popov], with the specific goal of uncovering links among the apparently different NS manifestations.
In line with recent theoretical efforts aimed at understanding the various observational manifestations of NSs, with a unifying evolution model [@PernaPons; @PonsPerna; @Vigano2013], our goal in this work is to combine previous studies of the radio-pulsar population (paper I) with the study of the population of $X$-ray pulsars. For young and middle-age NSs ($\lesssim 1$ Myr), the thermal emission comes from the internal heat which is gradually released or, in the case of NSs with large magnetic fields ($B\gtrsim 10^{14}$ G), by Joule dissipation of the electrical currents circulating in the NS crust which enhances the thermal luminosity. We do not include in this study non-thermal emission powered by losses of rotational energy, mainly because of the lack of detailed models for the physical mechanisms. Predictions for the non-thermal luminosities are based on phenomenological correlations with the rotational energy losses, which include more free parameters that are not linked to physical properties. On the contrary, independent, physically motivated cooling models provide the information linking the model parameters with the thermal emission of NSs. We hence focus on this population of NSs and leave the study of rotation powered, non-thermal emission ($X$-ray and $\gamma$-ray pulsars) for future works.
The article is organized as follows: In § \[sec:mod\] we summarise the main results of paper I about the radio-pulsar population and we describe the assumptions made to model the X-ray thermal radiation. In § \[sec:results\] we discuss the results of our analysis and finally, in § \[sec:conclusions\] we present our conclusions and final remarks.
------- ---------------------------- ---------------- ----------------- ----------------- ------------- ---------------- ---------- -------------------- ------------------- -- -- --
Model Decay Envelope $\mu_{B_0}$ $\sigma_{B_0}$ $\mu_{P_0}$ $\sigma_{P_0}$ $\alpha$ $n_{\rm br}$ ${\cal{D}}$
$\log{B}$ \[G\] $\log{B}$ \[G\] \[s\] \[s\] \[century$^{-1}$\]
A No decay Heavy elements $12.65$ $0.50$ $0.38$ $0.35$ $0.50$ $5.70 \pm 0.16$ $0.087 \pm 0.017$
B Slow Heavy elements $13.04$ $0.55$ $0.23$ $0.32$ $0.44$ $2.63 \pm 0.05$ $0.082 \pm 0.010$
C Fast Heavy elements $13.20$ $0.72$ $0.37$ $0.33$ $0.41$ $3.82 \pm 0.14$ $0.106 \pm 0.011$
D Slow Light elements $12.99$ $0.56$ $0.16$ $0.31$ $0.43$ $2.63 \pm 0.06$ $0.078 \pm 0.009$
E Medium Light elements $13.13$ $0.68$ $0.32$ $0.19$ $0.44$ $3.16 \pm 0.11$ $0.083 \pm 0.007$
F Slow (with toroidal field) Light elements $12.99$ $0.59$ $0.21$ $0.32$ $0.44$ $2.95 \pm 0.09$ $0.083 \pm 0.008$
------- ---------------------------- ---------------- ----------------- ----------------- ------------- ---------------- ---------- -------------------- ------------------- -- -- --
![Contour plot of the ${\cal{D}}$-value computed through a 2D KS-test in the $\mu_{B_0}$ - $\sigma_{B_0}$ plane (a 2D cut of the general 5D parameter space), for models B (top), C (middle) and D (bottom). The purple crosses indicate the minima found by the simulated annealing procedure in each case, and listed in Table \[tab:par\_rad\].[]{data-label="fig:b0"}](images/b0_B.eps "fig:"){width="7.2cm"} ![Contour plot of the ${\cal{D}}$-value computed through a 2D KS-test in the $\mu_{B_0}$ - $\sigma_{B_0}$ plane (a 2D cut of the general 5D parameter space), for models B (top), C (middle) and D (bottom). The purple crosses indicate the minima found by the simulated annealing procedure in each case, and listed in Table \[tab:par\_rad\].[]{data-label="fig:b0"}](images/b0_C.eps "fig:"){width="7.2cm"} ![Contour plot of the ${\cal{D}}$-value computed through a 2D KS-test in the $\mu_{B_0}$ - $\sigma_{B_0}$ plane (a 2D cut of the general 5D parameter space), for models B (top), C (middle) and D (bottom). The purple crosses indicate the minima found by the simulated annealing procedure in each case, and listed in Table \[tab:par\_rad\].[]{data-label="fig:b0"}](images/b0_D.eps "fig:"){width="7.2cm"}
Multi-wavelength population synthesis {#sec:mod}
=====================================
From an observational point of view, there is an important qualitative difference, regarding detectability biases, between the radio band and the high-energy bands. In the former, relatively complete surveys allow to quantitatively implement the instrumental detectability in the PS study. In $X$-rays, instead, the sample is not complete, and the sky coverage is inhomogeneous. Moreover, the detectability depends on non-trivial issues, such as the presence of a counterpart with known ephemerides in other wavelengths, or whether or not it has been associated to a transient $X$-ray outburst. This implies that the potential number of visible NSs above a certain threshold flux should be larger than the actual number of detected sources.
In $X$-rays we have to account for the effect of absorption by the interstellar medium. The hydrogen column density, $N_{H}$, for each synthetic pulsar is: $$N_{H} = \int n_{H}\,dl$$ where $n_{H}$ is the local hydrogen number density, whose spatial distribution is estimated following [@nh], and the integral is performed along the line of sight $l$. The energy-dependent absorption is calculated following the models of [@phabs], which include photoelectric cross sections of $17$ elements in the $[0.1-10]$ keV range. This allows us to calculate the detected (absorbed) flux on Earth $S_{X}^{\rm abs}$, assuming that radiation is isotropically distributed.
The radio pulsar population
---------------------------
For completeness, we begin our discussion by reviewing the main results of paper I, to which we refer for technical details. An outline of the procedure to generate a NS population is sketched in the following way:
- A large number of NSs with uniformly distributed random ages are generated in the framework of a given magneto-thermal evolution model (we explore six different models), which determines the evolution of the magnetic field. The birth location distribution, kick velocity, and evolution in the galactic gravitational potential are described in paper I.
- Gaussian distributions are assumed for both the initial spin period $P_0$ and the logarithm of the initial magnetic field, $B_0$. The central values ($\mu_{B_0}$, $\mu_{P_0}$) and widths ($\sigma_{B_0}$, $\sigma_{P_0}$) are left as free parameters. The radio-luminosity (at 1400 MHz) obeys the following equation: $$L_{1400} = L_{0} \, 10^{L_{\rm corr}} \, (P^{-3} \dot{P})^{\alpha} ~,
\label{radiolum}$$ where $P$ is the spin period, $\dot{P}$ is the period time derivative, $L_{0} = 5.69 \times 10^6$ mJy kpc$^2$ and $L_{\rm corr}$ is a random correction chosen from a zero-centered gaussian of $\sigma = 0.8$ (as in @Faucher), to take into account the large observed dispersion. The index $\alpha$ is a free parameter.
- The NSs are classified as [*radio-visible*]{} according to the same flux and beam visibility criteria of the radio survey used in paper I. The generation of pulsars stops when the number of radio-visible pulsars is equal to the number of radio pulsars in the considered observational sample. This fixes our normalisation for the total number of sources or, equivalently, the birth rate $n_{\rm br}$ for each synthetic population.
- The set of free parameters ($\mu_{B_0}$, $\sigma_{B_0}$, $\mu_{P_0}$, $\sigma_{P_0}$ and $\alpha$) is determined for each model by minimizing the associated ${\cal{D}}$-value of a generalized two-dimensional Kolmogorov-Smirnov test. The ${\cal{D}}$ statistic is calculated from the four natural quadrants that define each point in the $P-\dot{P}$ plane of the observed sample. The fraction of data from both samples (observed and simulated) in the radio band, is calculated on each quadrant, and the ${\cal{D}}$-value is defined as the maximum difference between observed and simulated fractions over all points and quadrants (for a more technical description we refer the reader to @KS2D [@numericalrecipes]).
The results for the 6 evolutionary models studied are summarized in Table \[tab:par\_rad\]. Here we considered three models with iron envelopes (A, B and C, already presented in paper I), differing in the timescale of the magnetic field decay. We have also used three new models (D, E and F) with light element envelopes. The last one also has an additional strong toroidal magnetic field. Although the envelope composition barely affects the evolution of rotational properties ($P$ and $\dot{P}$), it has a relevant impact on the thermal luminosity (see next section).
In general, we find that the differences in the initial period distribution between different models are not significant. Indeed, any broad distribution of initial periods in the range $0 < P_0 < 0.5$ s can be reconciled with the data, but narrow distributions peaked at short spin periods ($P_0 < 100$ ms) are ruled out. The ${\cal{D}}-$value is more dependent on the initial magnetic field distribution, but this is not strongly constrained because its central value and width are correlated showing an interval of possible solutions: the larger the central value, the wider the distribution. This holds true for models with and without field decay, and with different assumptions for the magnetospheric torque. This correlation is visible in Fig. \[fig:b0\], where we show contour plots of the ${\cal{D}}$-value in the $\mu_{B_0}-\sigma_{B_0}$ plane. We note this is a 2D plot of a section of a 5D parameter space. For conciseness, we only show plots for models B,C, and D (other models look very similar).
The dark blue regions, where the ${\cal{D}}$-value is low, define the range of allowed parameters. The purple crosses mark the different solutions for which explicit values for the parameters have been given in Table \[tab:par\_rad\]. The main conclusion from the analysis of only the radio pulsar population is that, independently of the underlying physical model, one can find acceptable parameter sets to fit the radio population reasonably well. Thus the radio-pulsar population alone cannot help much to discriminate among different evolutionary models. The next question is whether or not the analysis of pulsar distributions in other bands can break the degeneracy.
{width="5.6cm"} {width="5.6cm"} {width="5.6cm"}
Thermally emitting NSs
----------------------
### The observational sample
We consider all NSs for which the thermal emission is dominant, or clearly contributing to the total power in the soft X-ray band, and whose origin is attributed to residual cooling from birth. The selected sample we use has been described in [@Vigano2013] and it is periodically updated in our website[^1], includes thirteen closeby RPPs (only those with a clear thermal component), seven X-ray dim Isolated NSs (XINSs), and seventeen magnetars.
The XINSs are a small group of nearby (within about 200-300 parsecs), thermally emitting, isolated NSs. They are radio quiet but relatively bright in the X-ray band, and five of them have periods similar to magnetars. Moreover, their inferred magnetic fields are between the upper end of the radio pulsar population and the magnetar candidates. At variance with the majority of X-ray emitting pulsars, XINSs spectra are rather perfect blackbodies (kT$\sim$0.1 keV, occasionally with some broad lines), which make them the key objects for research on the NS equation of state, and the cooling properties of the NS crust and core.
The “magnetars" [@olausen14] are a small group of X-ray pulsars with spin periods between $0.3-12$s, whose properties cannot be explained within the common scenario for pulsars. Their inferred magnetic fields are $B \sim10^{14-15}$G. The decay and instabilities of such large magnetic fields are thought to power their emission [@duncan92; @thompson93]. Their powerful persistent X-ray emission (much larger than the rotational energy loss) is well modelled by a blackbody with $kT \sim 0.2-0.8$keV with an additional power-law component ($\Gamma\sim2-4$), usually interpreted as the product of Resonant Compton Scattering (RCS) of the seed thermal photons in the dense magnetosphere [@thompson02]. Hence even this apparent non-thermal component has a thermal origin. Different physical models have succeeded to reproduce the effect of the RCS to fit the magnetars X-ray spectra [@rea08; @zane09; @beloborodov13]. Given the transient nature of these objects (they show flares and sometimes long radiative X-ray outbursts), we have used in this work only X-ray luminosities derived during their quiescent state (available for the 17 selected objects).
### Magneto-thermal evolution and emission model
In NSs endowed with strong magnetic fields, the evolution of the internal and surface temperatures and the magnetic field are closely linked. The thermal luminosity is strongly dependent on the time-dependent magnetic field strength and topology, while the evolution of the magnetic field depends on the local values of the magnetic diffusivity and the electron relaxation time, which strongly depend on the temperature. [@Vigano2013] performed 2D magneto-thermal simulations of NSs, and studied the evolution of magnetic field strength and thermal luminosity as a function of time. Using this code, we produce a number of tables with the effective temperature and value of the dipolar magnetic field as a function of the age of the NS. Among the many model parameters (initial magnetic field topology, mass, radius, envelope composition, etc), we will focus on the parameters that more strongly influence the observable quantities (luminosity, $P$, $\dot{P}$) at a given age. These are the envelope composition (light vs. heavy elements), the initial magnetic field strength at the pole, $B_0$, and the impurity parameter in the inner-crust $Q_{imp}^{ic}$, a measure of the average charge distribution of the nuclei present in the crust. We refer to [@Vigano2013] for more details.
The magnetic field strength determines the rotational evolution of the NS. In addition, in highly magnetised NSs, the additional energy reservoir provided by the magnetic field dissipation may further increase the surface temperature, and extend the duration of the stage in which neutron stars are bright enough to be seen as X-ray pulsars. $Q_{imp}^{ic}$ was found to be a crucial parameter to understand the clustering of spin periods in isolated X-ray pulsars and why there is an upper limit of isolated X-ray pulsars at about 12 s (@Pons2013). The composition of the envelope is also of particular relevance. Generally speaking, low-B NSs with light element envelopes have significantly larger temperatures and luminosities during the first $\approx 10^5$ yr of their life, and lower temperature/luminosity afterwards, once they enter the photon cooling era [@Page2004; @Yakovlev2004; @Potekhin2007; @Page2011].
These three parameters characterize the evolution of the magnetic field strength and temperature as the star ages, and therefore the trajectories that NSs follow in the $P$-$\dot{P}$ diagram. Having all these tabulated quantities computed via simulations, for each star generated in the synthetic sample we calculate the present value of magnetic field, $P$, $\dot{P}$, temperature and luminosity. We assume blackbody emission, and integrate over the whole star surface to calculate the luminosity. For NSs with high magnetic fields, we also add the effect of resonant Compton scattering (see next subsection) in the spectrum. Among [*all NSs generated*]{} (not only those seen as radio-pulsars), we select those for which the absorbed X-ray flux at the Earth location is above $10^{-15}$ erg s$^{-1}$ cm$^{-2}$, as potentially visible X-ray pulsars (for present or future instruments).
### Resonant Compton Scattering
The thermal emission is usually assumed to be blackbody-like. However, it is well known that magnetar spectra show a non-thermal tail, whose origin cannot be attributed to the rotational energy of the star. The most popular interpretation is that magnetospheric plasma boosts up thermal photons coming from the surface. For these reasons, it is important to account for the possible effects that spectral distortions in the form of a hard tail would have on the detectability of NSs in X-rays.
We adopt a simplified model for the resonant Compton scattering (RCS, @lyutikov06) of seed photons (assumed to have a pure blackbody spectral energy distribution) in NS magnetospheres, and successfully applied to model spectra of several magnetars [@rea08]. Although it is a 1D, plane-parallel model, far from describing the non-trivial interaction between plasma and radiation in a realistic 3D magnetosphere, it effectively fits X-ray data. RCS models are parametrized by two parameters: the resonant optical depth along a ray $\tau_{res}$, which is related to the plasma density, and the thermal velocity of electrons in units of the speed of light, $\beta_{T}$ (see @lyutikov06 for more details).
In Fig. \[fig:rcs\] (left) we show the comparison between a blackbody of $kT= 1$ keV, and the spectra predicted by the RCS model for $\beta_T=0.3$ and different values of $\tau_{res}$. We show the spectra as they would be seen when absorbed by the interstellar medium with $N_H=3\times 10^{22}$ cm$^{-2}$, a typical value for magnetars in the galaxy. Since the interstellar absorption is important below $\sim 1$ keV, an efficient RCS could enhance the NSs detectability, because it promotes low energy photons to higher energies.
[@rea08] fit the X-ray spectra in quiescence of several magnetars. We found a correlation between $B$ and the RCS parameters $\beta_T$ and $\tau_{res}$, which can be reproduced by the following analytical fits: $$\beta_T = \left\{
\begin{array}{ll}
0.001 & \mathrm{if\ } \log{B} \le 13 \\
0.3 & \mathrm{if\ } \log{B} > 13
\end{array}
\right.$$
$$\tau_{res} = \left\{
\begin{array}{ll}
0.001 & \mathrm{if\ } \log{B} \le 13 \\
\frac{B}{10^{14} {\mathrm G}} & \mathrm{if\ } \log{B} > 13~.
\end{array}
\right.$$
We implement this analytical fit to modify the spectra of the randomly generated NSs in the population synthesis code. RCS does not change the original blackbody spectrum in NSs with low magnetic field, but it generates a hard tail in magnetars. In order to show the effect of the RCS in a typical PS realisation, in the middle panel of Fig. \[fig:rcs2\] we show the ratio of the RCS flux to the pure blackbody flux in the $[0.1,10]$ keV band, as a function of the blackbody flux. This figure clearly shows the main effect of the RCS correction: a flux enhancement by a factor of a few (up to a factor of 20) is applied to a significant number of sources, which has a visible imprint in the statistics of the number of detectable NSs. This effect is more evident for the dimmest sources: since RCS affects the high-energy tail, we can only obtain a very large enhancement for sources with a large $N_H$, and these are always dim. The right panel of Fig. \[fig:rcs\] shows the same ratio as a function of the initial field strength. It shows that only for the highest fields, in the magnetar range, a significant flux enhancement in the \[0.1-10\] keV band is produced.
![Log N - Log $S$ diagrams ([*top*]{}) and period distributions ([*bottom*]{}) for the thermally emitting X-ray pulsars for models A to F and the parameters listed in Table \[tab:par\_rad\]. In the bottom panel we show the histogram of sources with $S_{Xmin} > 10^{-14}$ erg s$^{-1}$ cm$^{-2}$. In colour lines we show the population synthesis results while the black line shows the observed sources.[]{data-label="fig:lognlogs_rad"}](images/lognlogs_rad.eps "fig:"){width="8.5cm"} ![Log N - Log $S$ diagrams ([*top*]{}) and period distributions ([*bottom*]{}) for the thermally emitting X-ray pulsars for models A to F and the parameters listed in Table \[tab:par\_rad\]. In the bottom panel we show the histogram of sources with $S_{Xmin} > 10^{-14}$ erg s$^{-1}$ cm$^{-2}$. In colour lines we show the population synthesis results while the black line shows the observed sources.[]{data-label="fig:lognlogs_rad"}](images/phist_rad.eps "fig:"){width="8.5cm"}
Results {#sec:results}
=======
We begin by analyzing the solutions that best fit the radio-pulsar distribution (Table \[tab:par\_rad\]). In Fig. \[fig:lognlogs\_rad\] we compare the accumulated number of detected X-ray sources $N(S)$ with flux $S>S^{abs}_{X}$ (hereafter $\log{N}$-$\log{S}$ diagram) for each magnetothermal evolution model. The sample of sources is not complete, since a full all-sky survey is not available for the present instruments, and many sources, in particular magnetars, have been discovered while they are in outburst. We can assume that we are actually observing all of the few brightest sources, and we gradually miss more and more sources as the threshold flux decreases. From a quick inspection of the $\log{N}$-$\log{S}$ diagrams, we can immediately conclude that none of the models of Table 1 predicts a number of thermally emitting X-ray sources compatible with the observations. However, as shown in Fig. \[fig:b0\], there is a quite large region in the parameter space with similar statistical significance for the ${\cal{D}}$-value. We can then explore whether other parameter combinations (always within the region allowed by the radio-pulsar population analysis) can improve the $\log{N}$-$\log{S}$ results. Since we obtain a systematic lack of bright sources for all models, we have to modify the initial distribution to include more NSs born with high magnetic fields, which can be done if we move up-right in the dark blue region of Fig. \[fig:b0\].
We have proceeded as follows: first, we search for a pair of values of $\mu_{B_0}$, $\sigma_{B_0}$ in the dark blue region of Fig. \[fig:b0\] than can reproduce the observed distribution of X-ray pulsars for fluxes above $\approx 10^{-12}$ erg s$^{-1}$ cm$^{-2}$ (the other 3 parameters do not affect the X-ray population). Once this is done, we fix the values of these two parameters and repeat the minimization procedure to find new values of $\mu_{P_0}$, $\sigma_{P_0}$, and $\alpha$ that best fit the radio-pulsar distribution. In Table \[tab:par\_X\] and Fig. \[fig:X\] (top) we show the results for the parameters that best reproduce the $\log{N}$-$\log{S}$ diagram for all models except model A, for which field decay was not included, and therefore the luminosity is not affected by the magnetic field. In other words, the standard cooling of a neutron star without including any additional heating by magnetic field decay cannot be reconciled with the X-ray data. For the other models, we find acceptable solutions with similar statistical significance for the radio-pulsar fits. The $\cal{D}$-value obtained with the 2D KS test shown in the last column is an average of 10 realizations, and we can see that they are compatible with the results of Table 1 within 1- or 2-$\sigma$.
![Same as Fig. \[fig:lognlogs\_rad\] for the parameters listed in Table \[tab:par\_X\].[]{data-label="fig:X"}](images/lognlogs_X.eps "fig:"){width="8.5cm"} ![Same as Fig. \[fig:lognlogs\_rad\] for the parameters listed in Table \[tab:par\_X\].[]{data-label="fig:X"}](images/phist_X.eps "fig:"){width="8.5cm"}
------- ----------------- ----------------- ------------- ---------------- ---------- -------------------- -------- ------------------- --
Model $\mu_{B_0}$ $\sigma_{B_0}$ $\mu_{P_0}$ $\sigma_{P_0}$ $\alpha$ $n_{\rm br}$ $\eta$ ${\cal{D}}$
$\log{B}$ \[G\] $\log{B}$ \[G\] \[s\] \[s\] \[century$^{-1}$\]
B $13.34$ $0.76$ $0.30$ $0.29$ $0.44$ $4.98 \pm 0.11$ $3.1$ $0.089 \pm 0.009$
C $13.36$ $0.80$ $0.29$ $0.37$ $0.44$ $4.37 \pm 0.12$ $4.0$ $0.093 \pm 0.011$
D $13.25$ $0.77$ $0.28$ $0.20$ $0.43$ $3.07 \pm 0.11$ $4.0$ $0.089 \pm 0.010$
E $13.23$ $0.72$ $0.32$ $0.19$ $0.44$ $3.58 \pm 0.10$ $2.5$ $0.083 \pm 0.008$
F $13.15$ $0.70$ $0.29$ $0.13$ $0.44$ $3.30 \pm 0.08$ $5.0$ $0.087 \pm 0.012$
------- ----------------- ----------------- ------------- ---------------- ---------- -------------------- -------- ------------------- --
Below $10^{-12}$ erg s$^{-1}$ cm$^{-2}$, the observed number of sources saturates due to the sensitivity limit of the present instruments, combined with the interstellar medium absorption, which only allows us to observe dim sources if they are nearby. Thus, we conclude that the degeneracy shown in Fig. \[fig:b0\], where a large region of the parameter space was allowed by the radio-pulsar data, is severely reduced when the thermally emitting X-ray pulsars are included in the analysis. Interestingly, for different models we obtain similar solutions for the initial magnetic field distribution, in the narrow region $\mu_{B_0}=13.15-13.35$, $\sigma_{B_0} \approx 0.7-0.8$ and $\alpha=0.43-0.44$.
However, to reproduce the $\log{N}$-$\log{S}$ diagram is a necessary, but not sufficient condition to test a model. We now turn to explore in more detail the particular form of the period distributions. We remind again that the comparison is made with an incomplete observational data set, because the number of observed sources is only a fraction of the potentially detectable number of NSs. In Fig. \[fig:X\] (bottom) we show spin period histograms comparing the observed sample (black solid line) with synthetic models.
An important result can already be seen in these distributions. On one hand, one needs a significant fraction of magnetars/high-B field NSs in the initial distribution to account for the number of X-ray sources already present in our observed sample, but on the other hand, if that fraction is too large, the model over-predicts the number of X-ray pulsars with long periods, even assuming the existence of a highly resistive layer in the inner crust of NSs [@Pons2013]. We can adjust the maximum period reached by a NS by varying the impurity parameter in the innermost part of the crust. Increasing the value of $Q_{\rm imp}^{\rm ic}$ results in the decrease of the maximum period, but still not enough to successfully reproduce the observed upper limit of 12 s and, at the same time, explain the observed number of very bright sources. The strong constraint resulting from combining the need of a few very bright sources with the lack of isolated X-ray pulsars with periods longer than 12 s turns out to be a very restrictive condition. The next relevant question is whether or not unknown observational selection effects are causing this discrepancy.
To explore how the period distribution varies with flux, let us focus on model D (the other models are qualitatively similar). In Fig. \[fig:Flux\] we compare the observed distribution to the synthetic data with a different cut-off in the absorbed flux ($10^{-14}$, $10^{-13}$, and $10^{-12}$ erg s$^{-1}$ cm$^{-2}$). An interesting trend is the apparent bimodal distribution of periods, similar to the observed distribution, although with a different scale and position. There is a partial lack of NSs with $P=1-2$ s, as seen in the observations, despite the initial period distribution being a single Gaussian. However, even with the highest flux cut-off ($10^{-12}$ erg s$^{-1}$ cm$^{-2}$) there are always too many visible X-ray pulsars with long periods. Thus, this discrepancy cannot be attributed only to selection effects.
![Same as Fig. \[fig:X\] ([*bottom*]{}) for model D considering different flux thresholds: $10^{-12}$ (blue), $10^{-13}$ (yellow) and $10^{-14}$ (red). All fluxes are in units of erg s$^{-1}$ cm$^{-2}$.[]{data-label="fig:Flux"}](images/phist_X_D.eps){width="8.5cm"}
Since it is impossible to [*a priori*]{} take into account (theoretically) unknown selection effects in $X$-rays, we propose in the following a phenomenological approach. A close inspection of Fig. \[fig:X\] shows that the observational sample appears to be complete (given all the assumptions made in paper and within statistical fluctuations) for fluxes $S_{X}^{abs} \gtrsim 3 \times 10^{-12}$ erg s$^{-1}$ cm$^{-2}$. We have also empirically found that the ratio of the number of observed sources to the number of potentially visible sources in the synthetic sample scales linearly with the flux in the range $10^{-13}-10^{-12}$ erg s$^{-1}$ cm$^{-2}$. If we assume that the probability to detect an X-ray pulsar with a given flux is simply $$\label{eq:p_obs}
p_{\rm obs} = \min \left\{ \eta S_{-11}, 1 \right\}$$ where $S_{-11}$ is the absorbed flux in units of $10^{-11}$ erg s$^{-1}$ cm$^{-2}$, we can obtain for each model the value of $\eta$ that reproduces the $\log{N} - \log{S^{abs}_{X}}$ diagram. It turns out that $\eta \sim 3-5$ for all models (eight column in Table \[tab:par\_X\]). This naive approach is useful to estimate the number of missed sources by uncontrolled selection effects.
We can now generate a random [*observed sample*]{} from the synthetically generated sample by imposing that the probability to detect the thermal emission of a pulsar with a given flux is given by the above expression. The corresponding $\log{N}$-$\log{S}$ diagrams and period distributions after such selection is applied are shown in Fig. \[fig:X\_pobs\]. Although most pulsars with high periods are now [*not observed*]{}, we still find a non-negligible number of bright enough sources with long periods. Our predictions range from 4 to 20 X-ray pulsars to be observed with $P>12$ s, depending on the model. Even considering that we are dealing with low-number statistics, this appears to be a real problem.
![Same as Fig. \[fig:X\], but applying the phenomenological detection probability formula (eq. \[eq:p\_obs\]).[]{data-label="fig:X_pobs"}](images/lognlogs_X_pobs.eps "fig:"){width="8.5cm"} ![Same as Fig. \[fig:X\], but applying the phenomenological detection probability formula (eq. \[eq:p\_obs\]).[]{data-label="fig:X_pobs"}](images/phist_X_pobs.eps "fig:"){width="8.5cm"}
------- ----------------- ----------------- ------------- ---------------- ---------- -------------------- -------- ------------------- --
Model $\mu_{B_0}$ $\sigma_{B_0}$ $\mu_{P_0}$ $\sigma_{P_0}$ $\alpha$ $n_{\rm br}$ $\eta$ ${\cal{D}}$
$\log{B}$ \[G\] $\log{B}$ \[G\] \[s\] \[s\] \[century$^{-1}$\]
D $13.75$ $1.15$ $0.44$ $0.26$ $0.44$ $6.81 \pm 0.20$ $3.1$ $0.120 \pm 0.012$
E $13.58$ $0.90$ $0.33$ $0.21$ $0.44$ $5.31 \pm 0.19$ $5.0$ $0.091 \pm 0.009$
F $13.33$ $0.83$ $0.30$ $0.19$ $0.43$ $3.24 \pm 0.09$ $5.0$ $0.110 \pm 0.008$
------- ----------------- ----------------- ------------- ---------------- ---------- -------------------- -------- ------------------- --
Beyond Gaussian distributions of initial magnetic fields. {#sec:gauss}
---------------------------------------------------------
All previous results (and most studies in the literature) rely on the assumption that the initial magnetic field distribution is log-normal. As the initial magnetic field increases, old NSs reach longer periods and therefore the narrower beam size produces a strong selection of radio-pulsars. Therefore, the low field part of the distribution is what we can practically constrain using the radio-pulsar population, while the high field part of the distribution remains unconstrained, because the distribution of high field radio-pulsars is highly degenerate with other parameters (beaming angle, radio-luminosity correlation, etc.). We have seen that the optimal parameters for radio-pulsars underpredicts the number of X-ray sources with high fluxes. We hence need to increase the width of the Gaussian to generate more bright NSs with high magnetic fields, but the problem is that the Gaussian becomes too wide, and we run into an overprediction of X-ray pulsars with periods $P>12$s, which has not been observed. The easiest, and perhaps more obvious, way out is to relax the assumption of an initial log-normal magnetic field distribution. For example, if we consider model D, we can visualize the $B_0$ distribution functions for the best fit to the radio-pulsar population (green/solid) and X-ray populations (brown/dot-dashed) in Fig. \[fig:b0hist\]. The excess of long period X-ray pulsars is originated by the high $B_0$ tail of the latter. We now discuss two possibilities, to reconcile the synthetic models with the data.
![$B_0$ distributions for model D that best fit the radio-pulsar (green/solid line) and X-ray pulsar (brown/dot-dashed) populations. In red/three-dot dashed and blue/dashed we plot the results for truncated and bimodal distributions discussed in Section \[sec:gauss\], respectively.[]{data-label="fig:b0hist"}](images/b0hist.eps){width="8.5cm"}
![Same as Fig. \[fig:X\_pobs\] assuming a log-normal distribution with cutoff at $B_{0max} = 5 \times 10^{14}$ G, and the parameters of Table \[tab:par\_X\_b0cut\]. []{data-label="fig:X_b0cut"}](images/lognlogs_X_b0cut.eps "fig:"){width="8.5cm"} ![Same as Fig. \[fig:X\_pobs\] assuming a log-normal distribution with cutoff at $B_{0max} = 5 \times 10^{14}$ G, and the parameters of Table \[tab:par\_X\_b0cut\]. []{data-label="fig:X_b0cut"}](images/phist_X_b0cut.eps "fig:"){width="8.5cm"}
### A truncated magnetic field distribution. {#sec:truncated}
If the origin of the neutron star magnetic field is attributed to magneto-hydrodynamic instabilities (e.g., any dynamo process related to rotation or convection), a nonlinear saturation is expected when the amplified magnetic field becomes dynamically relevant to suppress the instability. This may happen at typical fields of the order of $10^{15}$ G, depending on the particular mechanism [@Ober2009; @Ober2014]. Thus, we can simply introduce a cut-off value $B_{0max}$ to the log-normal distribution (red/three-dot dashed curve in Fig. \[fig:b0hist\]). In Fig. \[fig:X\_b0cut\], we can see that the $\log{N}$-$\log{S}$ diagram for the X-ray population can also be successfully reproduced by a truncated distribution with $B_{0max} = 5 \times 10^{14}$G and increased $\mu_{B_0}$ and $\sigma_{B_0}$ (Table \[tab:par\_X\_b0cut\]). The period distribution is significantly modified, and now there are no visible sources with $P>20$ s, as shown in Fig. \[fig:X\_b0cut\].
------- ------- ------- -------------------- -------- ------------------- --
Model $NS1$ $NS2$ $n_{\rm br}$ $\eta$ ${\cal{D}}$
(%) (%) \[century$^{-1}$\]
D $60$ $40$ $4.34 \pm 0.13$ $6.3$ $0.087 \pm 0.007$
E $65$ $35$ $4.80 \pm 0.10$ $2.0$ $0.082 \pm 0.006$
F $70$ $30$ $3.78 \pm 0.08$ $3.2$ $0.081 \pm 0.005$
------- ------- ------- -------------------- -------- ------------------- --
: Results for models D, E and F with a bimodal distribution. We show the relative weight of the first population ($NS1$, same parameters as in Table \[tab:par\_rad\]) and second population of highly-magnetized stars ($NS2$, flat distribution in the range $\log B_0 \in [13.5,14.5]$ ).
### A second population of magnetised NSs. {#sec:bimod}
The alternative possibility that we have explored is the existence of a second population of magnetised NSs, whose origin is different from that of standard radio-pulsars. The reason for this second population could be attributed to two different evolutionary paths of massive stars if they are isolated or in binary systems. Since about 80% of massive stars are in binaries, and assuming that half of them go through a common envelope phase or some evolutionary stage that produces a magnetized stellar core, one could expect about 40% of the total pulsar population being born with a binary massive progenitors, possibly leading to the formation of more magnetized NSs [@Spruit2008; @Langer2012; @Clark2014]. We have investigated what happens if, in addition to the population generated with the parameters of Table \[fig:lognlogs\_rad\] by fitting [*only*]{} the radio pulsar population (hereafter denoted by NS1), we add a second population (denoted by NS2), uniformly distributed in the range $\log B_0 \in [13.5,14.5]$ and normalized to account for the total number of NSs observed (blue/dashed curve in Fig. \[fig:b0hist\]). The results are shown in Fig. \[fig:bimodal\], where again we can observe that both the $\log{N}$-$\log{S}$ diagram and the period distribution are satisfactorily reproduced. We have verified that this second component does not modify the radio pulsar population, since their magnetic fields are so large that the vast majority of NS2 sources remains undetectable as radio-pulsars. The total birthrate, however, will increase accordingly, still marginally compatible with the observations. We also note that one can obtain similar results by assuming other distributions for this second population, for example a narrow log-normal distribution centered in $\approx 1-3 \times 10^{14}$G.
![Same as Fig. \[fig:X\_pobs\] with a bimodal distribution of $B_0$ described in §3.1.2.[]{data-label="fig:bimodal"}](images/lognlogs_X_bimod.eps "fig:"){width="8.5cm"} ![Same as Fig. \[fig:X\_pobs\] with a bimodal distribution of $B_0$ described in §3.1.2.[]{data-label="fig:bimodal"}](images/phist_X_bimod.eps "fig:"){width="8.5cm"}
The upper limit to the birth rate of very strongly magnetized NSs.
------------------------------------------------------------------
Before concluding our analysis on the period distribution, a simple calculation can stress once more the conflict between having a significant fraction of NSs with magnetic fields at birth above a certain value, and the lack of observed isolated NSs with periods $>12$ s. To quantify this incompatibility, we begin by obtaining the probability of observing a star, born with a given initial magnetic field $B_0$, as an X-ray pulsar with a period longer than 12 s. The results for three different values of the initial magnetic field are given in Table \[tab:prob\], for the representative model D (similar values are obtained for other models). In this calculation, we assumed a uniform distribution of ages in the range 0-560 Myrs (with constant birth rate), and we generate a large number of NSs with the same magnetic field. Then, we can obtain the probability as the fraction of synthetic [*visible*]{} sources with $P>12$ s to the total number of stars generated.
Next we address the following question: what is the maximum number of stars born with $B_0$ compatible (within statistical fluctuations) with the absence of observed neutron stars with periods longer than 12 s? This number can be estimated taking into account that if $N_{B_0}$ is the number of stars born with a given magnetic field, the probability of observing none of them with periods longer than 12 s is given by $e^{-N_{B_0} p}$, according to the Poisson distribution, and the probability that we will observe [*at least one*]{} star with $P>12$ s is $1-e^{-N_{B_0} p}$.
We can use this value to reject the null hypothesis in the following sense: for a given model, we can calculate the value of $N_{B_0}$ that corresponds to a probability of $1-e^{-N_{B_0} p}=0.01$, and we know that in 99% of the realizations with $N_{B_0}$ generated stars, we will not detect any source with $P>12$ s. Therefore, we can conclude that if a model predicts $\ge N_{B_0}$ stars, the fact that none has been observed rejects the model at the 99% confidence level (statistical fluctuations may result in a no-detection only in $\le 1\%$ of the cases).
In Fig. \[fig:ndet\] we plot the probability of no-detection of NSs with periods longer than 12 s against the birthrate, for different values of $B_0$. This can be interpreted as an upper limit to the birthrate of stars with initial magnetic field $>B_0$, for a given confidence level. For a confidence level of 99% and a magnetic field of $10^{15}$ G we obtain $N_{B_0}=112000$. This value corresponds to a birthrate of $\sim 0.02$ objects/century which means, by comparison with the total galactic NS birthrate $\sim 3$ NSs/century, that less than $0.7 \%$ can be born with magnetic fields $\gtrsim10^{15}$ G. Of course, this fraction is even smaller if we reduce our confidence level. The maximum birthrate for stars with initial magnetic field greater than $5\times 10^{14}$ G can be $3$ times larger than for $B_0=10^{15}$ G for the same confidence level. This is satisfied by both alternative models described in the two subsections above, which cannot be discriminated with current data.
$B_0$ \[G\] $p$
-------------------- ----------------------
$5 \times 10^{14}$ $1.3 \times 10^{-5}$
$8 \times 10^{14}$ $3.1 \times 10^{-5}$
$1 \times 10^{15}$ $4.1 \times 10^{-5}$
: Probability of detection of stars whose period is longer than $12$ s for samples with fixed initial magnetic field $B_0$.[]{data-label="tab:prob"}
![Probability of no-detection of NSs with $P>12$ s as a function of the birthrate, for three different initial magnetic fields: $B_0 = 1 \times 10^{15}$ G (blue squares), $8 \times 10^{14}$ G (green triangles) and $5 \times 10^{14}$ G (red diamonds). The symbols show results of simulations with 20 million NSs and the dashed lines correspond to the theoretical Poissonian value $e^{-N_{B_0} p}$.[]{data-label="fig:ndet"}](images/ndet.eps){width="8.5cm"}
$\dot{P}$ distribution.
-----------------------
To complete our discussion, we comment on our results for the distribution of $\dot{P}$. In Fig. \[fig:pdhist\_X\_pobs\] we show the histograms for the models considered, which roughly agree with the observations. However, all models and initial field distributions (log-normal, truncated log-normal, bimodal) show a similar trend: synthetic samples display an excess of sources with mean values $\sim 10^{-12}$ s s$^{-1}$ and a lack of a few sources with $\dot{P} > 10^{-11}$ s s$^{-1}$. This is due to the fact that even the strongest magnetic fields ($B_0 \gtrsim 10^{15}$ G) hardly achieve $\dot{P} \sim 10^{-10}$ s s$^{-1}$.
The observed sources with very high $\dot{P}$ are magnetars and it is likely that modelling their spin evolution with the simple magnetospheric torque formulae [@Spitkovsky; @Beskin; @Philippov] is an oversimplification. For instance, strong particle winds can be non-negligible in the early $\sim 1000$ yr of their lives [@Windbraking] when these sources undergo frequent bursts and flares. As a result of additional torques, the $\dot{P}$ can increase by an order of magnitude for the same magnetic field. This would imply that the magnetic field estimate by the classical magnetodipolar formula is a factor of a few higher than the real value. This effect is only expected to work during a short time and does not change the results of the analysis of the whole NS population, but it may contribute to the anomalous high values of $\dot{P}$ of the few youngest magnetars.
![Observed and predicted $\dot{P}$ distributions. Top: models with log-normal distributions. Middle: models with truncated distributions discussed in section \[sec:truncated\]. Bottom: models with bimodal distributions discussed in Section \[sec:bimod\]. In colour lines we show the PS results while the black line shows the observed sources.[]{data-label="fig:pdhist_X_pobs"}](images/pdhist_X_pobs.eps "fig:"){width="8.5cm"} ![Observed and predicted $\dot{P}$ distributions. Top: models with log-normal distributions. Middle: models with truncated distributions discussed in section \[sec:truncated\]. Bottom: models with bimodal distributions discussed in Section \[sec:bimod\]. In colour lines we show the PS results while the black line shows the observed sources.[]{data-label="fig:pdhist_X_pobs"}](images/pdhist_X_b0cut.eps "fig:"){width="8.5cm"} ![Observed and predicted $\dot{P}$ distributions. Top: models with log-normal distributions. Middle: models with truncated distributions discussed in section \[sec:truncated\]. Bottom: models with bimodal distributions discussed in Section \[sec:bimod\]. In colour lines we show the PS results while the black line shows the observed sources.[]{data-label="fig:pdhist_X_pobs"}](images/pdhist_X_bimod.eps "fig:"){width="8.5cm"}
Summary {#sec:conclusions}
=======
We have extended previous works on population synthesis of isolated neutron stars to study the combination of initial conditions and microphysical parameters required to explain the radio-pulsars and thermally emitting $X$-ray pulsar populations within the same evolutionary model.
Our first finding is that the $\log{N}$-$\log{S}$ diagram of X-ray pulsars can be well fitted by several models that, at the same time, reproduce the radio-pulsar distribution (because the parameter space of initial magnetic field distribution of radio-pulsars has a large region with the same statistical significance). Models with iron envelopes require a higher mean value of the initial magnetic field and a wider distribution than those with light elements.
Our second relevant result is that the obtained sample of observed sources with $S_{X}^{abs} \gtrsim 3 \times 10^{-12}$ erg s$^{-1}$ cm$^{-2}$ is nearly complete, and we estimate that, with more sensitive instruments, the thermal emission of about one hundred of NSs with fluxes $S^{abs}_{X} > 10^{-13}$ erg s$^{-1}$ cm$^{-2}$ is potentially detectable. This number can be increased up to 500 sources if the sensitivity is good enough to cover the whole sky with $ S^{abs}_{X} > 10^{-14}$ erg s$^{-1}$ cm$^{-2}$.
However, when examining in detail the period distribution all models with log-normal distributions of the initial magnetic field fail to explain why there are no X-ray pulsars with periods longer than 12 seconds. This sharp upper limit cannot be reproduced, even with models with a very high resistivity in the crust/core interface, if the initial field distribution is assumed to be log-normal. The problem is solved if other distributions are invoked, such as a truncated log-normal distribution, or a binormal distribution with two distinct populations. The most interesting feature, common to the two alternative cases considered, is that [*both*]{} cases require that magnetars cannot be born with a very high magnetic field. For a confidence level of 99%, we obtain an upper limit to the birth rate of NSs with $B_0 > 10^{15}$ G of $\lesssim 0.02$ objects/century, which represents a tiny fraction of the population (less than 1%).
The existence of an upper limit to the magnetic field at birth in the range $B_0 = 5-8 \times 10^{14}$ G, combined with fast magnetic field dissipation and observational selection effects, explains the lack of isolated pulsars with long periods. Although this has a drawback, the lack of a few pulsars with very high values of $\dot{P}>10^{-11}$, note that we have assumed that NSs spindown only by the rotating magnetospheric torque. In very young, active magnetars, losses of angular momentum carried by particles (wind) can compete with the magnetospheric torque [@Windbraking], increasing the value of $\dot{P}$ for a given magnetic field. A more detailed study of these phenomena, and other modifications of the spin-down evolution is reserved to future works.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in part by the grants AYA2013-42184-P and Prometeu/2014/69, and by the [*New Compstar*]{} COST action MP1304. MG is supported by the fellowship BES-2011-049123. DV and NR acknowledges support from grants AYA2012-39303 and SGR2014-1073. NR is additionally supported by an NWO Vidi award. We gratefully acknowledge useful discussions and suggestions from Peter Gonthier.
\[lastpage\]
[^1]: [www.neutronstarcooling.info](www.neutronstarcooling.info)
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---
abstract: 'Integrated lithium niobate (LN) photonic circuits have recently emerged as a promising candidate for advanced photonic functions such as high-speed modulation, nonlinear frequency conversion and frequency comb generation. For practical applications, optical interfaces that feature low fiber-to-chip coupling losses are essential. So far, the fiber-to-chip loss (commonly $>$ 10 dB) dominates the total insertion losses of typical LN photonic integrated circuits, where on-chip propagation losses can be as low as 0.03 - 0.1 dB/cm. Here we experimentally demonstrate a low-loss mode size converter for coupling between a standard lensed fiber and sub-micrometer LN rib waveguides. The coupler consists of two inverse tapers that convert the small optical mode of a rib waveguide into a symmetric guided mode of a LN nanowire, featuring a larger mode area matched to that of a tapered optical fiber. The measured fiber-to-chip coupling loss is lower than 1.7 dB/facet with high fabrication tolerance and repeatability. Our results open door for practical integrated LN photonic circuits efficiently interfaced with optical fibers.'
author:
- Lingyan He
- Mian Zhang
- 'Amirhassan Shams-Ansari'
- Rongrong Zhu
- Cheng Wang
- Marko Loncar
bibliography:
- 'reference.bib'
title: 'Low-loss fiber-to-chip interface for lithium niobate photonic integrated circuits'
---
Integrated lithium niobate (LN) photonics has been growing rapidly over the past decade, due to the attractive material properties including a large second-order nonlinear susceptibility, a large piezoelectric response, a wide optical transparency window, a high refractive index [@ferraro2013ferroelectric], and the recently developed thin-film LN-on-insulator (LNOI) nanofabrication technology [@RN72]. A wide range of nanophotonic devices including microring resonators [@guarino2007electro; @RN66; @wolf2017cascaded; @RN55; @krasnokutska2018ultra], photonic crystal cavities [@witmer2016design; @RN33] and microdisk resonators [@RN27; @wang2015high; @wu2018lithium] have been realized and are used to achieve advanced optical functionalities on-chip such as electro-optic (EO) modulation [@RN26; @RN65; @RN50; @weigel2018bonded; @mercante2018thin; @he2018high; @escale2018extreme; @witmer2018high], second harmonic generation [@RN55; @RN25; @RN69], supercontinuum generation [@RN74], electro-optic frequency comb [@RN20] and Kerr frequency comb generation [@RN71; @he2018self]. These on-chip functionalities could have profound impact on integrated optical solutions in optical communications [@RN25; @RN55; @wang2015high], spectroscopy and sensing [@RN56], as well as microwave photonics [@RN18; @RN75; @RN52].
However, a major challenge for practical applications of integrated LN photonics is the lack of an efficient interface between the micrometer-scale LN devices and optical fibers. For many applications including electro-optic modulators and frequency comb generation, the system performance is directly associated with the total optical power delivered through the chips. For example, integrated LN modulators developed recently feature large bandwidths, low driving voltages and low on-chip losses, but at the same time suffer from large fiber-to-fiber insertion losses in excess of 10 dB [@RN50; @RN18]. This is due to a large mismatch between the mode sizes and mode indices of optical fibers and nanophotonic LN waveguides. Such a high insertion loss prevents these high-performance modulators to reach their full potential and find practical applications. Increasing the on-chip optical mode size is therefore important not only to enable LN photonic integrated circuits with low insertion losses, but also to improve the alignment tolerance during packaging and assembly.
Here we demonstrate an on-chip lithium niobate bilayer inversely tapered mode size converter and achieve fiber-to-chip coupling losses lower than 1.7 dB/facet (fiber-to-fiber insertion loss of 3.4 dB) at telecom wavelengths through an optimized nanofabrication process. This is a substantial improvement over recent fiber-chip interfaces in thin-film LN devices, including grating couplers with insertion losses of $\sim$10 dB or higher [@RN31; @RN23], and end-fire couplers on a hybrid silicon nitride/LN platform ($\sim$ 6 dB/facet)[@RN25].
![\[fig1\]**Bilayer mode size converter.(a)** Schematic of the bilayer tapered mode size converter. Top-left inset: cross-sectional view of the bilayer inverse taper. Bottom right inset: top-down view of the bilayer inverse taper. $L_1$=120 $\mu$m and $L_2$=450 $\mu$m are the tapering lengths of the top rib waveguide and the bottom slab layer, respectively. $w_1$=1.2 $\mu$m corresponds to the width of the bottom-layer taper in the conversion section and $w_2$=340 nm is the tip width. Note that the widths here are defined as the top widths of the trapezoids. **(b-d)** Cross-sectional electrical field distribution (Ez) of the TE optical modes at different tapering regions: **(b)** tip of the bottom slab layer, **(c)** conversion area where top waveguide ridge just ends, **(d)** and bilayer tapers **(d)**. ](Fig1.pdf){width="50.00000%"}
Many integrated LN devices rely on a rib waveguide geometry, with a slab beneath the ridge. The slab geometry is desired to allow the electric field to efficiently propagate through LN with has a relatively large dielectric constant. As a result, conventional single layer inverse taper designs [@RN70] are not readily suitable here since tapering the rib portion only will push optical mode to the LN slab resulting in poor coupling efficiencies. Therefore, efficient couplers for rib waveguide structures require a bilayer taper to convert the etched rib waveguide mode to a nearly circularly symmetric output mode. In our approach, this is accomplished by adiabatically tapering the bottom slab layer in addition to the top ridge waveguide. Fig. \[fig1\](a) shows the schematic view of our LN mode size converter designed to support a fundamental transverse-electric (TE) optical mode, where in-plane electric fields ($E_z$) could interact through the highest electro-optic/nonlinear-optic tensor component of the x-cut LN thin film we use. The coupling is based on a gradual variation of the waveguide cross section to a certain point such that better size overlap and index match between modes of the input fiber and nanophotonic waveguides are achieved.
![\[fig2\]**Inverse taper mode matching.** Dependence of mode field diameter (blue curve) and effective refractive index (red curve) on the tip widths of the slab layer, respectively. Inset: the optical profile of the ridge waveguide without mode size converter.](Fig2.pdf){width="50.00000%"}
Finite-difference time domain (FDTD) simulations are used to design an adiabatic coupler [@RN76] that can match the optical mode of a lensed single mode fiber (SMF) with a mode field diameter (MFD) of $\sim$ 2 $\mu$m in air ($n_\textrm{air} $= 1), to the mode of a LN rib waveguide, with MFD $\sim$ 1.6 $\mu$m and SiO$_2 $cladding ($n_\textrm{LN}=2.2$, $n_\textrm{SiO$_2$}=1.45$). To accomplish this, we utilize gradual tapering of both rib and slab portions of the waveguide following an exponential function. The top (rib) taper evolves from a nominal LN waveguide width of 800 nm down to a 30-nm tip with a height $h_\textrm{rib}$ of 350 nm over a 120-$\mu$m length, which ensures single-mode propagation along the waveguide taper. The bottom (slab) layer taper narrows down laterally from 6 $\mu$m to a tip of 340 nm wide with a slab height $h_\textrm{slab}$ of 250 nm to match the optical mode of fiber. The optical mode profiles of the fundamental TE mode at different regions along the LN coupler are shown in Fig. \[fig1\](b-d). As expected, the mode size becomes larger and more symmetric as light is transferred to the tapered slab layer such that nearly all optical power remains in the target eigenmode. Our optimal coupler design features a fiber-to-rib coupling efficiency as high as $88\%$.
It should be noted that the MFD we obtained here is not the largest possible but is chosen to match the mode size of the lensed fiber used. It is possible to interface optical fibers with larger optical modes by tapering the waveguides to a even narrower width. We calculated the MFD of different tip widths $w_2$ changing from 50 nm to 1 $\mu$m at the telecom wavelength as shown in Fig. \[fig2\] (blue curve). The mode size at the chip facet decreases greatly as the taper becomes wider, due to the stronger light confinement (higher effective index). When the taper is wider than 0.8 $\mu$m, the mode diameter grows gradually and finally reaches a plateau. By changing the widths of tip, we can achieve 1.4-4.5 $\mu$m MFD at the edge of the chip. Furthermore, the implementation of our bilayer inverse taper successfully brings the effective refractive index of the rib waveguide mode from 1.9 down to a range between 1.44-1.62 (red curve in Fig. \[fig2\]). Therefore, the index mismatch between lithium niobate photonic integrated devices and lensed fiber is well compensated.
We fabricated the converter using a commercially available x-cut LNOI substrate (NANOLN), where a LN thin film (600 nm thick) was bonded on top of silica (2 $\mu$m thick) on a silicon substrate handle (0.5 mm thick). The top-down fabrication process involves a two-step lithography and etching process of LNOI wafer. In the first step, we defined the patterns of the intended LN photonic integrated circuits and the top taper layer using electron beam lithography (EBL) and transferred the patterns 350 nm deep into LN thin film using reactive ion etching (RIE) with argon [@RN66]. The waveguides were then cladded by depositing 1 $\mu$m of SiO$_2$ using plasma-enhanced chemical vapor deposition (PECVD). Aligned photolithography and hydrofluoric acid (HF) wet etching were then employed to remove the cladding layer at the corresponding tapering regions, while the rest of the photonic integrated chip is protected from subsequent etching. After a second layer of aligned EBL and RIE, the second taper layer was defined and re-cladded with PECVD SiO$_2$. The facets of the final devices were etched to reduce roughness and to ensure good coupling from and to the optical fibers. Fig. \[fig3\] displays a scanning electron microscope (SEM) image of the mode size converters we fabricated.
![\[fig3\]**Scanning electron microscope (SEM) images of the fabricated bilayer mode size converter.** Inset: a close-up view of the bottom-layer taper. The top-layer waveguide is covered by e-beam resist after patterning the slab, so the contour is blurred. ](Fig3.pdf){width="50.00000%"}
We use a transmission setup to measure the fiber-to-fiber insertion losses of LN chips with and without mode size converters. A tunable telecom external cavity diode laser (Santec-TSL 510) is used to excite the devices, and the transmitted light is collected and detected using an InGaAs photodetector (Newport 1811). A piezo-controlled micropositioner is employed to precisely control the motion of the tapered-lensed fiber to achieve optimal coupling with the chip. To ensure TE input polarization and maximizes the transmitted laser light, a fiber polarization controller was placed after the laser. We also fabricated microring resonators on the chip to confirm that the transmission we observe are not from slab modes. Fig. \[fig4\] displays the broadband transmission spectra of the on-chip LN mode size converter coupled to the tapered-lensed fiber, which exhibits a series of dips attributed to localized TE-polarized microring resonances (red curve). The coupling loss decreases from 11-14 dB (grey curve) to $\sim$ 3-5 dB (red curve) from 1480 nm to 1680 nm after introducing the mode size converter. Note that both waveguides, with and without mode size converter, are 4 mm long and are fabricated on the same chip to avoid possible variations between different fabrication runs. We experimentally confirm the low propagation loss of our waveguides [@RN66] using the measured quality factor ($Q$) of a nearly critically coupled microring resonator on the same chip as a reference ($Q_\textrm{loaded}\sim 1.5 \times 10^6$). The high quality factor indicates that the 4 mm waveguides have a propagation loss of $<$ 0.05 dB. Therefore, we conclude that the overall insertion loss that we measure is dominantly due to fiber-to-chip coupling. We carried out a series of measurements for devices with different slab tip widths, and found a minimum fiber-to-fiber insertion loss of 3.4 dB (\[fig4\] red curve) for a tip width of 340 nm. In order to evaluate the robustness and compatibility of our devices for high power applications we tested our devices under high optical powers up to 1 W for continuous operation of more than 1 hour, and no damage to the facet and coupling region was observed.
![\[fig4\]**Calibrated transmission spectra.** The red and grey curves correspond to the spectra for LN rib waveguides coupled to ring resonators with and without couplers, showing fiber-to-fiber insertion losses of 14 dB and 3.4 dB, respectively. Dips correspond to microring cavity resonances, featuring $Q_{loaded} = 1.5 \times 10^6$.](Fig4.pdf){width="50.00000%"}
{width="90.00000%"}
We compare the experimentally measured coupling efficiencies for different taper widths, and find them to be in good agreement with theoretical predictions (Fig.\[fig5\](a)). The difference in the experimental data and simulation may be attributed to misalignment between the two-tapers and/or non-ideal etch depths of rib waveguides. We also measured the tolerance of the optical fiber position for TE polarization. The input fiber was mounted on the motorized stage and scanned over 4 $\mu$m in the horizontal plane while the distance between the input fiber and chip facet is fixed at the confocal point of lensed fiber. The measured alignment tolerance for 1 dB excess loss is $\pm$ 1 $\mu$m in the transverse directions, as shown in Fig. \[fig5\](b). The alignment tolerance agrees with our estimate of the mode size.
In conclusion, we have experimentally demonstrated a monolithic bilayer mode size converter for efficient fiber coupling to LN nanophotonic waveguides. The mode size converter consists of two-layer taper, implemented using a two-step dry etching process, that gradually decreases the width of both rib and slab portions of LN waveguide. Using this approach, the total insertion loss could be reduced from 14 dB to 3.4 dB. The high coupling efficiency will enable optical packaging of LNOI devices, and opens the door for practical applications in LN integrated photonics. We note that our approach can be extended to coupling to standard, cleaved, optical fibers (e.g. Corning SMF28, with the MFD of 10.4 $\pm$ 0.5 $\mu$m at 1550 nm) using high numerical aperture fiber (MFD$\sim$3.2 $\mu$m) as an intermediary. This method has been successfully employed to couple light efficiently from SMF28 to silicon waveguide with an overall coupling loss of less than 1.5 dB [@preble2015chip].
This work is supported by: NSF Partnership for Innovation - Technology Translation (PFI-TT) program (Award IIP-1827720), China Scholarship Council (201706470030), City University of Hong Kong (Start-up Funds). Device fabrication is performed at the Harvard University Center for Nanoscale Systems, a member of the National Nanotechnology Coordinated Infrastructure Network, which is supported by the National Science Foundation under award number ECCS-1541959.
We also thank Mengjie Yu for assistance in high optical power measurement, Linbo Shao, Christian Reimer, and Boris Desiatov for fruitful discussions and.
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